Secondary School Mathematics for Class 10 R S Aggarwal

Table of contents :
Cover......Page 1
Title......Page 2
Preface......Page 4
Contents......Page 12
Real Numbers......Page 14
Polynomials......Page 56
Linear Equations in Two Variables......Page 84
Quadratic Equations......Page 178
Arithmetic Progression......Page 257
Coordinate Geometry......Page 310
Triangles......Page 364
Circles......Page 474
Constructions......Page 525
Trigonometric Ratios......Page 541
T-Ratios of Some Particular Angles......Page 559
Trigonometric Ratios of Complementary Angles......Page 569
Trigonometric Identities......Page 580
Heights and Distances......Page 625
Perimeter and Area of Plan Figuers......Page 667
Area of Circle, Sector and Segment......Page 691
Volume and Surface Areas of Solids......Page 755
Mean, Median, Mode of Grouped Data, Cumulative Frequency Graph and Ogive......Page 839
Probability......Page 900

Citation preview

Secondary School Mathematics FOR CLASS 10 (In accordance with the latest CBSE syllabus)

R S Aggarwal, MSc, PhD Veena Aggarwal, BA, BEd

Preface We feel proud to present the revised edition of this well-acclaimed book, which is in accordance with the current Board examination pattern and includes the valuable feedback received from our esteemed readers. Though revised, the book retains its qualities which have made it so popular among the teachers and students. It now has a large number of questions from the NCERT textbook and previous years CBSE board papers with full explanatory solutions. A large number of multiple-choice questions (MCQs) on all topics have also been included. A separate exercise for Very-Short-Answer and Short-Answer Questions has also been given at the end of each chapter, to provide extra questions for ample practice. We hope that the present edition will be of immense help to students who wish to sit for the CBSE class X board examination. We hope that we shall continue to receive invaluable feedback from teachers and students for the improvement of the book. —Authors

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Mathematics Syllabus For Class 10

Unit I: Number Systems 1. Real Numbers

(15 Periods)

Euclid’s division lemma, Fundamental Theorem of Arithmetic— statements after reviewing work done earlier and after illustrating and motivating through examples, proofs of irrationality of 2 , 3 , 5 . Decimal representation of rational numbers in terms of terminating/ nonterminating recurring decimals.

Unit II: Algebra 1. Polynomials

(7 Periods)

Zeros of a polynomial. Relationship between zeros and coefficients of quadratic polynomials. Statement and simple problems on division algorithm for polynomials with real coefficients. 2. Pair of Linear Equations in Two Variables

(15 Periods)

Pair of linear equations in two variables and graphical method of their solution, consistency/inconsistency. Algebraic conditions for number of solutions. Solution of a pair of linear equations in two variables algebraically—by substitution, by elimination and by cross multiplication method. Simple situational problems. Simple problems on equations reducible to linear equations. 3. Quadratic Equations

(15 Periods)

Standard form of a quadratic equation ax  bx  c  0, (a ! 0) . Solutions of quadratic equations (only real roots) by factorisation, by completing the square and by using quadratic formula. Relationship between discriminant and nature of roots. 2

Situational problems based on quadratic equations related to day-to-day activities to be incorporated. 4. Arithmetic Progressions

(8 Periods)

Motivation for studying Arithmetic Progression. Derivation of the nth term and sum of the first n terms of AP and their application in solving daily life problems. (v)

Unit III: Coordinate Geometry 1. Lines (In two dimensions)

(14 Periods)

Review: Concepts of coordinate geometry, graphs of linear equations. Distance between two points. Section formula (internal division). Area of a triangle.

Unit IV: Geometry 1. Triangles

(15 Periods)

Definitions, examples, counterexamples of similar triangles. 1. (Prove) If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio. 2. (Motivate) If a line divides two sides of a triangle in the same ratio, the line is parallel to the third side. 3. (Motivate) If in two triangles, the corresponding angles are equal, their corresponding sides are proportional and the triangles are similar. 4. (Motivate) If the corresponding sides of two triangles are proportional, their corresponding angles are equal and the two triangles are similar. 5. (Motivate) If one angle of a triangle is equal to one angle of another triangle and the sides including these angles are proportional, the two triangles are similar. 6. (Motivate) If a perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse, the triangles on each side of the perpendicular are similar to the whole triangle and to each other. 7. (Prove) The ratio of the areas of two similar triangles is equal to the ratio of the squares on their corresponding sides. 8. (Prove) In a right triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides. 9. (Prove) In a triangle, if the square on one side is equal to the sum of the squares on the other two sides, the angles opposite to the first side is a right angle. 2. Circles

(8 Periods)

Tangents to a circle at a point of contact. 1. (Prove) The tangent at any point of a circle is perpendicular to the radius through the point of contact. (vi)

2. (Prove) The lengths of tangents drawn from an external point to circle are equal. 3. Constructions

(8 Periods)

1. Division of a line segment in a given ratio (internally). 2. Tangent to a circle from a point outside it. 3. Construction of a triangle similar to a given triangle.

Unit V: Trigonometry 1. Introduction to Trigonometry

(10 Periods)

Trigonometric ratios of an acute angle of a right-angled triangle. Proof of their existence (well defined): motivate the ratios, whichever are defined at 0 and 90. Values (with proofs) of the trigonometric ratios of 30, 45 and 60. Relationships between the ratios. 2. Trigonometric Identities

(15 Periods)

Proof and applications of the identity sin A  cos A  1. Only simple identities to be given. Trigonometric ratios of complementary angles. 2

3. Heights and Distances

2

(8 Periods)

Simple problems on heights and distances. Problems should not involve more than two right triangles. Angles of elevation/depression should be only 30, 45 and 60.

Unit VI: Mensuration 1. Areas Related to Circles

(12 Periods)

Motivate the area of a circle; area of sectors and segments of a circle. Problems based on areas and perimeter/circumference of the abovesaid plane figures. (In calculating area of segment of a circle, problems should be restricted to central angle of 60, 90 and 120 only. Plane figures involving triangles, simple quadrilaterals and circle should be taken.) 2. Surface Areas and Volumes

(12 Periods)

(i) Surface areas and volumes of combinations of any two of the following: cubes, cuboids, spheres, hemispheres and right-circular cylinders/cones. Frustum of a cone. (ii) Problems involving converting one type of metallic solid into another and other mixed problems. (Problems with combination of not more than two different solids to be taken.) (vii)

Unit VII: Statistics and Probability 1. Statistics

(18 Periods)

Mean, median and mode of grouped data (bimodal situations to be avoided). Cumulative frequency graph. 2. Probability

(10 Periods)

Classical definition of probability. Simple problems on single events (not using set notation).



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Weightage MATHEMATICS CLASS 10 Time: 3 Hours

Max. Marks: 80

The weightage or the distribution of marks over different dimensions of the question paper shall be as follows: Weightage to Content/Subject Units S. No.

Unit

1. Number Systems 

6

Real Numbers 20

2. Algebra  

Polynomials Linear Equations in Two Variables



Quadratic Equations



Arithmetic Progression

3. Coordinate Geometry 

6

Coordinate Geometry 15

4. Geometry   

Triangles Circles Constructions 12

5. Trigonometry     

Trigonometric Ratios T-Ratios of Some Particular Angles Trigonometric Identities Trigonometric Ratios of Complementary Angles Heights and Distances

6. Mensuration   

Marks

10

Perimeter and Area of Plane Figures Area of Circle, Sector and Segment Volume and Surface Areas of Solids (ix)

11

7. Statistics and Probability 



Mean, Median, Mode of Grouped Cumulative Frequency Graph and Ogive

Data,

Probability Total

80



(x)

Contents Number Systems

1. Real Numbers

1

Algebra

2. Polynomials

43

3. Linear Equations in Two Variables

71

4. Quadratic Equations

165

5. Arithmetic Progression

244

Coordinate Geometry

6. Coordinate Geometry

297

Geometry

7. Triangles

351

8. Circles

461

9. Constructions

512

Trigonometry

10. Trigonometric Ratios

528

11. T-Ratios of Some Particular Angles

546

12. Trigonometric Ratios of Complementary Angles

556

13. Trigonometric Identities

567

14. Heights and Distances

612

Mensuration

15. Perimeter and Area of Plane Figures

654

16. Area of Circle, Sector and Segment

678

17. Volume and Surface Areas of Solids

742

Statistics and Probability

18. Mean, Median, Mode of Grouped Data, Cumulative Frequency Graph and Ogive (xi)

826

19. Probability

887

Sample Paper I

929

Sample Paper II

934 

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Real Numbers

1

In class IX we studied about real numbers, especially about irrational numbers. In this chapter, we shall continue our discussion on real numbers. We begin with two important results, namely Euclid’s division lemma and the fundamental theorem of arithmetic. LEMMA

A lemma is a proven statement used for proving another statement.

EUCLID’S DIVISION LEMMA

For any two given positive integers a and b, there exist unique whole numbers q and r such that [CBSE 2009C] a = bq + r, where 0 # r < b. Here, we call a as dividend, b as divisor, q as quotient and r as remainder. Dividend = (divisor × quotient) + remainder. Example

Suppose we divide 117 by 14. Then, we get 8 as quotient and 5 as remainder. Here dividend = 117, divisor = 14, quotient = 8 and remainder = 5.

14) 117 (8 –112 5

Clearly, 117 = (14 # 8) + 5. EXAMPLE 1

A number when divided by 73 gives 34 as quotient and 23 as remainder. Find the number.

SOLUTION

Here divisor = 73, quotient = 34 and remainder = 23. By Euclid's division lemma, we have dividend = (divisor × quotient) + remainder = (73 # 34) + 23 = (2482 + 23) = 2505. Hence, the required number is 2505.

An algorithm is a series of well-defined steps which gives a method for solving a certain type of problem. ALGORITHM

It is a technique to compute the HCF of two given positive integers, say a and b with a > b, in the following steps. EUCLID’S DIVISION ALGORITHM

1

2

Secondary School Mathematics for Class 10

Step 1.

On dividing a by b, we get the quotient q and remainder r such that a  bq  r, where 0 # r  b.

Step 2.

If r  0 then HCF (a, b)  b. If r ! 0 then apply the division lemma to b and r.

Step 3.

Continue the process till the remainder is 0. The last divisor will be the required HCF.

EXAMPLE 2

Use Euclid‘s algorithm to find the HCF of 272 and 1032.

SOLUTION

We find HCF(272, 1032) using the following steps. Since 1032 > 272, we 272) 1032 (3 divide 1032 by 272 to 816 get 3 as quotient and 216) 272 (1 216 as remainder. 216 So, by Euclid‘s division 56) 216 (3 lemma, we get 168 1032 = 272 # 3 + 216. 48) 56 (1 Step 2. Since the remainder 48 216 ! 0, we divide 8) 48 (6 272 by 216 to get 1 as 48 quotient and 56 as 0 remainder.   by Euclid‘s division lemma, we get Step 1.

272 = 216 # 1 + 56. Step 3.

Since the remainder 56 ! 0, we divide 216 by 56 to get 3 as quotient and 48 as remainder.





Step 4.

Since the remainder 48 ! 0, we divide 56 by 48 to get 1 as quotient and 8 as remainder. 

Step 5.

by Euclid‘s division lemma, we get 216 = 56 # 3 + 48.

by Euclid‘s division lemma, we get 56 = 48 # 1 + 8.

Since the remainder 8 ! 0, we divide 48 by 8 to get 6 as quotient and 0 as remainder. 

by Euclid‘s division lemma, we get 48 = 8 # 6 + 0.

Real Numbers

3

The remainder has now become 0, so our procedure stops. Hence, HCF(272, 1032) = 8. Note

8 = HCF(48, 8) = HCF(56, 48) = HCF(216, 56) = HCF(272, 216) = HCF(1032, 272).

REMARK

This method is also known as successive division method.

EXAMPLE 3

Use Euclid‘s algorithm to find HCF(196, 38220).

SOLUTION

We find HCF(196, 38220), using the following steps. Since 38220 > 196, we divide 38220 by 196 to get 195 as quotient and 0 as remainder. 

by Euclid‘s division lemma, we get 38220 = 196 # 195 + 0.

Since the remainder is 0, so our procedure stops. 

HCF(196, 38220) = 196.

196) 38220 (195 196 1862 1764 980 980 0

EXAMPLE 4

Use Euclid‘s algorithm to find HCF of 1651 and 2032. Express the HCF in the form 1651m  2032n.

SOLUTION

We find HCF(1651, 2032) using the following steps. Since 2032 > 1651, we divide 1651) 2032 (1 2032 by 1651 to get 1 as quotient 1651 and 381 as remainder. 381) 1651 (4  by Euclid‘s division lemma, 1524 we get 127) 381 (3 … (i) 2032 = 1651 # 1 + 381. 381 Step 2. Since the remainder 381 ! 0, we 0 divide 1651 by 381 to get 4 as quotient and 127 as remainder. Step 1.



by Euclid‘s division lemma, we get 1651 = 381 # 4 + 127.

Step 3.

… (ii)

Since the remainder 127 ! 0, we divide 381 by 127 to get 3 as quotient and 0 as remainder. 

by Euclid‘s division lemma, we get 381  127 # 3  0.

… (iii)

The remainder is now 0, so our procedure stops. 

HCF(1651, 2032) = 127.

4

Secondary School Mathematics for Class 10

Now, from (ii), we get 1651  381# 4  127 & & &

127  1651  381# 4 127  1651  (2032  1651#1)# 4 [from (i)] 127  1651  2032 # 4  1651# 4

127  1651# 5  2032 #( 4). Hence, m  5, n   4.

&

SOME APPLICATIONS OF EUCLID’S DIVISION LEMMA EXAMPLE 1

Show that every positive even integer is of the form 2m and that every positive odd integer is of the form (2m + 1), where m is some integer.

SOLUTION

Let n be an arbitrary positive integer. On dividing n by 2, let m be the quotient and r be the remainder. Then, by Euclid‘s division lemma, we have n = 2m + r, where 0 # r < 2. 

n = 2m or (2m + 1), for some integer m.

Case I

When n = 2m. In this case, n is clearly even.

Case II When n = 2m + 1.

In this case, n is clearly odd. Hence, every positive even integer is of the form 2m and every positive odd integer is of the form (2m + 1) for some integer m. EXAMPLE 2

Show that any positive integer is of the form 3m or (3m + 1) or (3m + 2) for some integer m.

SOLUTION

Let n be an arbitrary positive integer. On dividing n by 3, let m be the quotient and r be the remainder. Then, by Euclid‘s division lemma, we have n = 3m + r, where 0 # r < 3. 

n = 3m or (3m + 1) or (3m + 2), for some int eger m.

Thus, any positive integer is of the form 3m or (3m  1) or (3m + 2) for some integer m. EXAMPLE 3

Show that any positive odd integer is of the form (4m + 1) or (4m + 3) for some integer m.

SOLUTION

Let n be an arbitrary odd positive integer.

Real Numbers

5

On dividing n by 4, let m be the quotient and r be the remainder. So, by Euclid‘s division lemma, we have n = 4m + r, where 0 # r < 4. 

n = 4m or (4m + 1) or (4m + 2) or (4m + 3) .

Clearly, 4m and (4m + 2) are even and since n is odd, so n ! 4m and n ! (4m  2) . 

n  (4m  1) or (4m  3), for some integer m.

Hence, any positive odd integer is of the form (4m  1) or (4m  3) for some integer m. EXAMPLE 4

SOLUTION

Show that every positive odd integer is of the form (6m  1) or (6m  3) or (6m  5) for some integer m. Let n be a given positive odd integer. On dividing n by 6, let m be the quotient and r be the remainder. Then, by Euclid‘s division lemma, we have n  6m  r, where 0 # r  6  n  6m  r, where r  0, 1, 2, 3, 4, 5  n  6m or (6m  1) or (6m  2) or (6m  3) or (6m  4) or (6m + 5) . But, n = 6m, (6m + 2), (6m + 4) give even values of n. Thus, when n is odd, it is of the form (6m + 1) or (6m + 3) or (6m  5) for some integer m.

EXAMPLE 5

Using Euclid‘s division lemma, show that the square of any positive integer is either of the form 3m or (3m  1) for some integer m. [CBSE 2008]

SOLUTION

Let n be an arbitrary positive integer. On dividing n by 3, let q be the quotient and r be the remainder. Then, by Euclid‘s division lemma, we have n = 3q + r, where 0 # r < 3. 

n 2 = 9q 2 + r 2 + 6qr

Case I

… (i), where 0 # r < 3.

When r  0. Putting r = 0 in (i), we get n 2 = 9q 2 = 3 (3q 2) = 3m, where m = 3q 2 is an integer.

6

Secondary School Mathematics for Class 10 Case II When r  1.

Putting r = 1 in (i), we get n 2 = (9q 2 + 1 + 6q) = 3 (3q 2 + 2q) + 1 = 3 (3q 2 + 2q) + 1 = 3m + 1, where m = (3q 2 + 2q) is an integer. Case III When r  2.

Putting r = 2 in (i), we get n 2 = (9q 2 + 4 + 12q) = 3 (3q 2 + 4q + 1) + 1 = 3m + 1, where m = (3q 2 + 4q + 1) is an int eger. Hence, the square of any positive integer is of the form 3m or (3m  1) for some integer m. EXAMPLE 6

Using Euclid‘s division lemma, show that the cube of any positive integer is of the form 9m or (9m  1) or (9m  8) for some integer m. [CBSE 2009C]

SOLUTION

Let n be an arbitrary positive integer. On dividing n by 3, let q be the quotient and r be the remainder. So, by Euclid‘s division lemma, we have n = 3q + r, where 0 # r < 3. 

n 3 = (3q + r) 3 = 27q 3 + r 3 + 9qr (3q + r) = (27q 3 + 27q 2 r + 9qr 2) + r 3

Case I

… (i), where 0 # r  3.

When r  0. Putting r  0 in (i), we get n 3 = 27q 3 = 9 (3q 3) = 9m, where m = 3q 3 is an integer.

Case II When r  1.

Putting r  1 in (i), we get n 3 = (27q 3 + 27q 2 + 9q) + 1 = 9q (3q 2 + 3q + 1) + 1 = 9m + 1, where m = q (3q 2 + 3q + 1) is an int eger. Case III When r  2.

Putting r  2 in (i), we get n 3 = (27q 3 + 54q 2 + 36q) + 8 = 9q (3q 2 + 6q + 4) + 8 = 9m + 8, where m = q (3q 2 + 6q + 4) is an integer. Hence, the cube of any positive integer is of the form 9m or (9m + 1) or (9m + 8) for some integer m.

Real Numbers EXAMPLE 7

SOLUTION

7

Show that one and only one out of n, (n  1) and (n  2) is divisible by 3, where n is any positive integer. On dividing n by 3, let q be the quotient and r be the remainder. Then, n  3q  r, where 0 # r < 3 

n = 3q + r, where r = 0, 1 or 2



n = 3q or n = (3q + 1) or n = (3q + 2) .

Case I

If n  3q then n is clearly divisible by 3.

Case II If n  (3q  1) then (n  2)  (3q  3)  3 (q  1), which is

clearly divisible by 3. In this case, (n  2) is divisible by 3. Case III If n  (3q  2) then (n  1)  (3q  3)  3 (q  1), which is

clearly divisible by 3. In this case, (n  1) is divisible by 3. Hence, one and only one out of n, (n  1) and (n  2) is divisible by 3. EXAMPLE 8

SOLUTION

Show that one and only one out of n, n  2, n  4 is divisible by 3, where n is any positive integer. [CBSE 2008C] On dividing n by 3, let q be the quotient and r be the remainder. Then, n = 3q + r, where 0 # r < 3 

n = 3q + r, where r = 0, 1, 2



n = 3q or n = 3q + 1 or n = 3q + 2.

Case I

If n = 3q then n is divisible by 3.

Case II

If n  3q  1 then (n  2)  3q  3  3 (q  1), which is divisible by 3. So, in this case, (n  2) is divisible by 3.

Case III When n  3q  2 then (n  4)  3q  6  3 (q  2), which

is divisible by 3. So, in this case, (n  4) is divisible by 3. Hence, one and only one out of n, n  2, n  4 is divisible by 3. EXAMPLE 9 SOLUTION

If n is an odd integer then show that n 2  1 is divisible by 8. We know that every odd integer is of the form 4q  1 or 4q  3 for some integer q.

8

Secondary School Mathematics for Class 10 Case I

When n  4q  1. Then, n 2  1  (4q  1) 2  1  16q 2  8q  1  1  16q 2  8q  8 (q  2q 2), which is divisible by 8.

Case II When n  4q  3.

Then, n 2  1  (4q  3) 2  1  16q 2  24q  9  1  16q 2  24q  8  8(2q 2  3q  1), which is divisible by 8. Hence, if n is an odd integer then n 2  1 is divisible by 8. f

EXERCISE 1A

1. What do you mean by Euclid‘s division lemma? 2. A number when divided by 61 gives 27 as quotient and 32 as remainder. Find the number. 3. By what number should 1365 be divided to get 31 as quotient and 32 as remainder? 4. Using Euclid’s division algorithm, find the HCF of (i) 405 and 2520

(ii) 504 and 1188

(iii) 960 and 1575.

5. Show that every positive integer is either even or odd. 6. Show that any positive odd integer is of the form (6m + 1) or (6m + 3) or (6m + 5), where m is some integer. 7. Show that any positive odd integer is of the form (4m + 1) or (4m + 3), where m is some integer. 8. For any positive integer n, prove that n 3  n is divisible by 6. 9. Prove that if x and y are both odd positive integers then x 2  y 2 is even but not divisible by 4. 10. Use Euclid’s algorithm to find HCF of 1190 and 1445. Express the HCF in the form 1190m  1445n. ANSWERS (EXERCISE 1A)

2. 1679

3. 43

10. 85; m   6, n  5

4. (i) 45

(ii) 36

(iii) 15

Real Numbers

9

HINTS TO SOME SELECTED QUESTIONS 3. Let the required divisor be x. Then, dividend = (divisor × quotient) + remainder 

1365 = (x # 31) + 32. Find x.

5. Let n be an arbitrary positive integer. On dividing n by 2, let m be the quotient and r be the remainder. Then, n = 2m + r, where 0 # r < 2 [by Euclid’s division lemma]. 

n = 2m or n = 2m + 1 for some integer m.

So, n is either even or odd.

Every composite number can be uniquely expressed as a product of primes, except for the order in which these prime factors occurs.

FUNDAMENTAL THEOREM OF ARITHMETIC

Examples

(i) 12 = 2 # 2 # 3

(ii) 69 = 3 # 23

(iii) 105 = 3 # 5 # 7

(iv) 234 = 2 # 3 # 3 # 13 (v) 462 = 2 # 3 # 7 # 11 (vi) 651 = 3 # 7 # 31 The above factorisation can easily be verified by actual division.

SOLVED EXAMPLES EXAMPLE 1

Show that each of the following is a composite number: (i) 5 # 11 # 13 + 13

SOLUTION

(ii) 6 # 5 # 4 # 3 # 2 # 1 + 5

We have (i) 5 # 11 # 13 + 13 = 13 # (5 # 11 + 1) = (13 # 56) . Clearly, it shows that the given number has more than two factors. Hence, it is a composite number. (ii) 6 # 5 # 4 # 3 # 2 # 1 + 5 = 5 # (6 # 4 # 3 # 2 # 1 + 1) = (5 # 145) . Clearly, it shows that the given number has more than two factors. Hence, it is a composite number.

EXAMPLE 2

Show that any number of the form 4 n, nd N can never end with the digit 0.

10 SOLUTION

Secondary School Mathematics for Class 10

If 4 n ends with 0 then it must have 5 as a factor. But, 4 n  (2 2) n  2 2n shows that 2 is the only prime factor of 4 n . Also, we know from the fundamental theorem of arithmetic that the prime factorisation of each number is unique. So, 5 is not a factor of 4 n . Hence, 4 n can never end with the digit 0.

EXAMPLE 3

Show that any number of the form 6 n, where nd N can never end with the digit 0.

SOLUTION

If 6 n ends with 0 then it must have 5 as a factor. But, 6 n  (2 # 3) n  (2 n # 3 n) shows that 2 and 3 are the only prime factors of 6 n . Also, we know from the fundamental theorem of arithmetic that the prime factorisation of each number is unique. So, 5 is not a factor of 6 n . Hence, 6 n can never end with the digit 0.

EXAMPLE 4

Find the HCF and LCM of 126 and 156 using prime factorisation method.

SOLUTION

We have



126 = (2 # 3 # 3 # 7) = (2 # 3 2 # 7)

and 156 = (2 # 2 # 3 # 13) = (2 2 # 3 # 13) . 

HCF(126, 156) = product of common terms with lowest

power = (21 # 31) = (2 # 3) = 6

Real Numbers

11

and LCM(126, 156) = product of prime factors with highest power = (2 2 # 3 2 # 7 # 13) = (4 # 9 # 7 # 13)  3276. 

HCF = 6 and LCM = 3276.

EXAMPLE 5

Find the HCF and LCM of 612 and 1314 using prime factorisation method.

SOLUTION

We have



612 = (2 # 2 # 3 # 3 # 17) = (2 2 # 3 2 # 17)

and 1314 = (2 # 3 # 3 # 73) = (2 # 3 2 # 73) . 

HCF(612, 1314) = product of common terms with lowest

power = (2 # 3 2) = (2 # 9) = 18 and LCM(612, 1314) = product of prime factors with highest power = (2 2 # 3 2 # 17 # 73) = (4 # 9 # 17 # 73) = 44676. Hence, HCF = 18 and LCM = 44676. AN IMPORTANT PROPERTY

Product of two given numbers = product of their HCF and LCM.

Thus, (a # b) = HCF (a, b) # LCM (a, b) . CAUTION

The above result is true for two numbers only.

12

Secondary School Mathematics for Class 10

EXAMPLE 6

The HCF of two numbers is 23 and their LCM is 1449. If one of the numbers is 161, find the other.

SOLUTION

For two numbers a and b, we know that (a # b) = {HCF(a, b)} # {LCM(a, b)} . Here a = 161, HCF = 23 and LCM = 1449. And, we have to find b. 

(161 # b)  (23 # 1449) & b 

(23 # 1449)  207. 161

Hence, the other number is 207. EXAMPLE 7

Given that HCF(252, 594) = 18, find LCM(252, 594).

SOLUTION

We have product of two given numbers their HCF (252 # 594) = = 8316. 18 Hence, LCM(252, 594) = 8316. LCM =

EXAMPLE 8

148 Find the simplest form of 185 ·

SOLUTION

We find HCF(148, 185), which is 37. So, we divide the numerator and denominator of the given fraction by 37. 

148) 185 (1 148

148 = 148 ' 37 = 4 · 185 185 ' 37 5

4 Hence, the simplest form of the given fraction is 5 ·

37) 148 (4 148 0

EXAMPLE 9

Find the HCF and LCM of 108, 120 and 252 using prime factorisation method.

SOLUTION

By prime factorisation, we get 108 = (2 2 # 3 3) 120 = (2 3 # 3 # 5) 2

2

252 = (2 # 3 # 7)

2 108

2 120

2 252

2

54

2

60

2 126

3

27

2

30

3

63

3

9

3

15

3

21

3

5

7

HCF(108, 120, 252) = product of common terms with lowest

power = (2 2 # 3) = (4 # 3) = 12.

Real Numbers

13

LCM(108, 120, 252) = product of prime factors with highest

power = (2 3 # 3 3 # 5 # 7) = 7560.  EXAMPLE 10

SOLUTION

HCF = 12 and LCM = 7560.

Find the largest number which divides 245 and 1037, leaving remainder 5 in each case. Clearly, the required number divides (245  5) = 240 and (1037  5) = 1032 exactly. So, the required number is HCF(240, 1032). 2 240 2 120 Now, 240 = (2 4 # 3 # 5) and 1032 = (2 3 # 3 # 43) . 

HCF(240, 1032) = (2 3 # 3) = 24.

2 2 3

Hence, the required number is 24. EXAMPLE 11

SOLUTION

60 30 15 5

2 2 2 3

1032 516 258 129 43

Find the largest number which divides 129 and 545, leaving remainders 3 and 5 respectively. Clearly, the required number divides (129  3) = 126 and (545 – 5) = 540 exactly. 2 126 2 540  required number = HCF(126, 540). 3 63 2 270 Now, 126 = (2 # 3 # 3 # 7) = (2 # 3 2 # 7) and 540 = (2 # 2 # 3 # 3 # 3 # 5)

3

21 7

= (2 2 # 3 3 # 5) . 

3 135 3 45 3 15 5

HCF(126, 540) = product of common terms with lowest

power = (2 # 3 2) = (2 # 9) = 18. Hence, the required number is 18. EXAMPLE 12

SOLUTION

Find the largest number that will divide 398, 436 and 542, leaving remainders 7, 11 and 15 respectively. Clearly, the required number divides (398  7) = 391, (436  11) = 425 and (542  15) = 527 exactly. 

required number = HCF(391, 425, 527). 17 391

Now, 391 = (17 # 23), 425 = (5 2 # 17), 527 = (17 # 31) .

23

5 425 5 85 17

14

Secondary School Mathematics for Class 10



HCF(391, 425, 527) = 17.

Hence, the required number is 17. EXAMPLE 13

Two tanks contain 504 and 735 litres of milk respectively. Find the maximum capacity of a container which can measure the milk of either tank an exact number of times.

SOLUTION

Resolving 504 and 735 into prime factors, we get 504 = (2 3 # 3 2 # 7) and 735 = (5 # 3 # 7 2) .

2 504 2 252 2 126 3 63 3 21 7



HCF (504, 735) = (3 # 7) = 21.



capacity of the required container = 21 litres.

5 735 3 147 7 49 7

EXAMPLE 14

An army contingent of 612 members is to march behind an army band of 48 members in a parade. The two groups are to march in the same number of columns. What is the maximum number of columns in which they can march?

SOLUTION

Clearly, the maximum number of columns = HCF(612, 48).

2 612 2 306 3 153 and 48 = (2 4 # 3) . 3 51 2  HCF(612, 48) = (2 # 3) = (4 # 3) = 12. 17

Now, 612 = (2 2 # 3 2 # 17)



required number of columns = 12.

2 48 2 24 2 12 2 6 3

EXAMPLE 15

A sweetseller has 420 kaju burfis and 150 badam burfis. He wants to stack them in such a way that each stack has the same number, and they take up the least area of the tray. How many of these can be placed in each stack? How many stacks are formed?

SOLUTION

Maximum number of burfis in each stack = HCF(420, 150). Now, 420 = (2 2 # 3 # 5 # 7) and

150 = (2 # 3 # 5 2) .

2 420 2 210 3 105 5 35 7



HCF(420, 150)  (2 # 3 # 5)  30.



maximum number of burfis in each stack = 30.



420 150 number of stacks = a 30 + 30 k = (14 + 5) = 19.

2 150 3 75 5 25 5

Real Numbers

15

EXAMPLE 16

Ravi and Sikha drive around a circular sports field. Ravi takes 16 minutes to take one round, while Sikha completes the round in 20 minutes. If both start at the same point, at the same time and go in the same direction, after how much time will they meet at the starting point?

SOLUTION

Required number of minutes = LCM(16, 20). Now, 16 = 2 4 and 20 = (2 2 # 5) . 

LCM(16, 20)  (2 4 # 5)  (16 # 5)  80.

Hence, both will meet at the starting point after 80 minutes. EXAMPLE 17

In a school there are two sections, namely A and B, of class X. There are 30 students in section A and 28 students in section B. Find the minimum number of books required for their class library so that they can be distributed equally among students of section A or section B.

SOLUTION

Clearly, the required number of books are to be distributed equally among the students of section A or B. So, the number of these books must be a multiple of 30 as well as that of 28. Consequently, the required number is LCM(30, 28). Now, 30 = 2 # 3 # 5 and 

28 = 2 2 # 7. LCM(30, 28) = product of prime factors with highest power

= (2 2 # 3 # 5 # 7) = (4 # 3 # 5 # 7) = 420. Hence, the required number of books = 420. f

EXERCISE 1B

1. Using prime factorisation, find the HCF and LCM of: (i) 36, 84 (iv) 144, 198

(ii) 23, 31

(iii) 96, 404

(v) 396, 1080

(vi) 1152, 1664

In each case, verify that: HCF × LCM = product of given numbers.

2. Using prime factorisation, find the HCF and LCM of: (i) 8, 9, 25

(ii) 12, 15, 21

(iii) 17, 23, 29

(iv) 24, 36, 40

(v) 30, 72, 432

(vi) 21, 28, 36, 45

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Secondary School Mathematics for Class 10

3. The HCF of two numbers is 23 and their LCM is 1449. If one of the numbers is 161, find the other. 4. The HCF of two numbers is 145 and their LCM is 2175. If one of the numbers is 725, find the other. 5. The HCF of two numbers is 18 and their product is 12960. Find their LCM. 6. Is it possible to have two numbers whose HCF is 18 and LCM is 760? Give reason. HINT

HCF always divides LCM completely.

7. Find the simplest form of: 69 473 (i) 92 (ii) 645

1095 (iii) 1168

368 (iv) 496

8. Find the largest number which divides 438 and 606, leaving remainder 6 in each case. 9. Find the largest number which divides 320 and 457, leaving remainders 5 and 7 respectively. 10. Find the least number which when divided by 35, 56 and 91 leaves the same remainder 7 in each case. 11. Find the smallest number which when divided by 28 and 32 leaves remainders 8 and 12 respectively. 12. Find the smallest number which when increased by 17 is exactly divisible by both 468 and 520. 13. Find the greatest number of four digits which is exactly divisible by 15, 24 and 36. 14. Find the largest four-digits number which when divided by 4, 7 and 13 leaves a remainder of 3 in each case. 15. Find the least number which should be added to 2497 so that the sum is exactly divisible by 5, 6, 4 and 3. 16. Find the greatest number that will divide 43, 91 and 183 so as to leave the same remainder in each case. 17. Find the least number which when divided by 20, 25, 35 and 40 leaves remainders 14, 19, 29 and 34 respectively. 18. In a seminar, the number of participants in Hindi, English and mathematics are 60, 84 and 108 respectively. Find the minimum number of rooms required, if in each room, the same number of participants are to be seated and all of them being in the same subject.

Real Numbers

17

19. Three sets of English, mathematics and science books containing 336, 240 and 96 books respectively have to be stacked in such a way that all the books are stored subjectwise and the height of each stack is the same. How many stacks will be there? 20. Three pieces of timber 42 m, 49 m and 63 m long have to be divided into planks of the same length. What is the greatest possible length of each plank? How many planks are formed? 21. Find the greatest possible length which can be used to measure exactly the lengths 7 m, 3 m 85 cm and 12 m 95 cm. 22. Find the maximum number of students among whom 1001 pens and 910 pencils can be distributed in such a way that each student gets the same number of pens and the same number of pencils. 23. Find the least number of square tiles required to pave the ceiling of a room 15 m 17 cm long and 9 m 2 cm broad. 24. Three measuring rods are 64 cm, 80 cm and 96 cm in length. Find the least length of cloth that can be measured an exact number of times, using any of the rods. 25. An electronic device makes a beep after every 60 seconds. Another device makes a beep after every 62 seconds. They beeped together at 10 a.m. At what time will they beep together at the earliest? 26. The traffic lights at three different road crossings change after every 48 seconds, 72 seconds and 108 seconds respectively. If they all change simultaneously at 8 a.m. then at what time will they again change simultaneously? 27. Six bells commence tolling together and toll at intervals of 2, 4, 6, 8, 10, 12 minutes respectively. In 30 hours, how many times do they toll together? ANSWERS (EXERCISE 1B)

1. (i) HCF = 12, LCM = 252

(iii) HCF = 4, LCM = 9696

(ii) HCF = 1, LCM = 713 (iv) HCF = 18, LCM = 1584

(v) HCF = 36, LCM = 11880 (vi) HCF = 128, LCM = 14976 2. (i) HCF = 1, LCM = 1800

(ii) HCF = 3, LCM = 420

(iii) HCF = 1, LCM = 11339 (iv) HCF = 4, LCM = 360 (v) HCF = 6, LCM = 2160 3. 207

4. 435

5. 720

(vi) HCF = 1, LCM = 1260

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Secondary School Mathematics for Class 10

6. No, since HCF does not divide LCM exactly

3

7. (i) 4

11 (ii) 15

15 (iii) 16

23 (iv) 31

8. 24

9. 45

10. 3647

11. 204

12. 4663

13. 9720

14. 9831

15. 23

16. 4

17. 1394

18. 21

19. 14 24. 9.6 m

20. 7 m, 22 planks

21. 35 cm 22. 91

23. 814

25. 10 : 31 hrs

26. 8 : 7 : 12 hrs

27. 16 times

HINTS TO SOME SELECTED QUESTIONS 8. Required number = HCF(432, 600) = 24. 9. Required number = HCF(315, 450) = 45. 10. Required number = {LCM(35, 56, 91) + 7} = (3640 + 7) = 3647. 11. Here, (28  8) = 20 and (32  12) = 20. 

required number = {LCM (28, 32)  20} = (224  20) = 204.

12. Required number = {LCM (468, 520)  17} = (4680  17) = 4663. 13. Greatest number of four digits = 9999. LCM(15, 24, 36) = 360.

On dividing 9999 by 360, remainder = 279. 

required number = (9999  279) = 9720.

14. Required number = (greatest number of 4-digits divisible by 4, 7 and 13) + 3. 15. LCM of 5, 6, 4 and 3 = 60. On dividing 2497 by 60, remainder = 37. 

number to be added  (60  37)  23.

16. Required number = HCF of (91  43), (183  91) and (183  43), i.e., HCF of 48, 92 and 140. 17. Here, (20  14)  6, (25  19)  6, (35  29)  6, (40  34)  6. 

required number = (LCM of 20, 25, 35, 40) – 6.

18. Maximum number of participants in each room = HCF(60, 84, 108) = 12. 60 84 108 Minimum number of rooms required = a 12 + 12 + 12 k = 21. 19. HCF(336, 240, 96) = 48. So, we make stacks of 48 books each. 336 240 96 Number of stacks = a 48 + 48 + 48 k = (7 + 5 + 2) = 14. 20. Required length of each plank in metres = HCF(42 m, 49 m, 63 m) = 7 m. 42 49 63 Number of planks = a 7 + 7 + 7 k = (6 + 7 + 9) = 22. 21. Required length = HCF(700 cm, 385 cm, 1295 cm) = 35 cm.

Real Numbers

19

22. Maximum number of students = HCF(1001, 910) = 91. 23. Side of each square tile = HCF(1517 cm, 902 cm) = 41 cm. 1517 # 902 Required number of tiles = a 41 # 41 k = 814. 24. Required length = LCM(64 cm, 80 cm, 96 cm) = 960 cm = 9.6 m. 25. Interval of beeping together = LCM (60 seconds, 62 seconds) = 1860 seconds = 1860 min = 31 min . 60 So, they will beep together again at 10 : 31 a.m. 26. Interval of change = LCM (48 seconds, 72 seconds, 108 seconds) = 432 seconds = 7 min 12 seconds. Required time of simultaneous change = 8 : 7 : 12 hours. 27. LCM of 2, 4, 6, 8, 10, 12 = 120. After every 2 hours they toll together. 30 Required number of times = a 2 + 1 k times = 16 times.

RATIONAL NUMBERS NATURAL NUMBERS

Counting numbers 1, 2, 3, 4, …, etc., are known as natural

numbers. WHOLE NUMBERS

All counting numbers together with 0 form the collection of

whole numbers. Thus, 0, 1, 2, 3, 4, 5, …, etc., are whole numbers. All counting numbers, negatives of counting numbers and 0 form the collection of all integers.

INTEGERS

Thus, …, –4, –3, –2, –1, 0, 1, 2, 3, …, etc., are integers. p RATIONAL NUMBERS The numbers of the form q , where p and q are integers, and q ! 0 are called rational numbers. Every rational number when expressed in decimal form is expressible either in terminating or in nonterminating repeating decimal form.

RATIONAL NUMBERS IN DECIMAL FORM

AN IMPORTANT OBSERVATION

To Test Whether a Given Rational Number is a Terminating or Repeating Decimal

p Let x be a rational number whose simplest form is q , where p and q are integers and q ! 0. Then,

20

Secondary School Mathematics for Class 10

(i) x is a terminating decimal only when q is of the form (2 m # 5 n) for some non-negative integers m and n. (ii) x is a nonterminating repeating decimal, if q ! (2 m # 5 n) . AN IMPORTANT TEST

p Let q be the simplest form of a given rational number. p (i) If q = (2 m # 5 n) for some non-negative integers m and n then q is a terminating decimal.

p

(ii) If q ! (2 m # 5 n) then q is a nonterminating repeating decimal.

SOLVED EXAMPLES EXAMPLE 1

Without actual division, show that each of the following rational numbers is a terminating decimal. Express each in decimal form. (i)

SOLUTION

31 (2 2 # 5 3)

33 (ii) 50

(i) The given number is

41 (iii) 1000

17 (iv) 625

31 · (2 2 # 5 3)

Clearly, none of 2 and 5 is a factor of 31. So, the given rational is in its simplest form. Clearly, (2 2 # 5 3) is of the form (2 m # 5 n) . So, the given number is a terminating decimal. 31 62  31 # 2   62  62 Now, 2 (2 # 5 3) (2 3 # 5 3) (2 # 5) 3 (10) 3 1000  0.062. 33 (ii) The given number is 50 · Now, 50 = (2 # 5 2) and none of 2 and 5 is a factor of 33. So, the given rational number is in its simplest form. Clearly, 50 = (2 # 5 2) = (2 m # 5 n), where m = 1 and n = 2. So, the given number is a terminating decimal. 33 33 = 33 # 2 = 66 Now, 50 = (2 # 5 2) (2 2 # 5 2) (2 # 5) 2 = 66 2 = 66 = 0.66. 100 (10)

Real Numbers

21

41 (iii) The given number is 1000 · Now, 1000 = (8 # 125) = (2 3 # 5 3) . Clearly, none of 2 and 5 is a factor of 41. So, the given number is in its simplest form. Now, 1000 = (2 3 # 5 3) which is of the form (2 m # 5 n), where m = 3 and n = 3. So, the given number is a terminating decimal. 41 And, 1000 = 0.041. 17 (iv) The given number is 625 · And, 625 = 5 4 and 5 is not a factor of 17. So, the given number is in its simplest form. Now, 625 = 5 4 is of the form (2 m # 5 n), where m = 0, n = 4. So, the given number is a terminating decimal. 17 17 17 # 2 4 = 17 # 16 = 272 Now, 625 = 4 = 4 5 5 # 2 4 (5 # 2) 4 (10) 4 = 272 = 0.0272. 10000 EXAMPLE 2

SOLUTION

Without actual division, show that each of the following rational numbers is a nonterminating repeating decimal. 66 17 121 53 (i) 3 (ii) 90 (iii) 343 (iv) 180 (2 # 3 2 # 7 5) (i) Given number is

121 · (2 3 # 3 2 # 7 5)

Clearly, none of 2, 3 and 7 is a factor of 121. So, the given rational number is in its simplest form. And, (2 3 # 3 2 # 7 5) ! (2 m # 5 n) . 

121 is a nonterminating repeating decimal. (2 3 # 3 2 # 7 5)

17 (ii) Given number is 90 · And, 90 = (2 # 3 2 # 5) . Clearly, none of 2, 3 and 5 is a factor of 17.

22

Secondary School Mathematics for Class 10



17 90 is in its simplest form.

Also, 90 = (2 # 3 2 # 5) ! (2 m # 5 n) . 

17 90 is a nonterminating repeating decimal.

53 (iii) Given number is 343 · Now, 343 = 7 3 and 7 is not a factor of 53. 53  343 is in its simplest form. Also, 343  7 3 ! (2 m # 5 n) . 53 343 is a nonterminating repeating decimal. 66 (iv) Given number is 180 and HCF(66, 180) = 6. 66 = 66 ' 6 = 11 ·  180 180 ' 6 30 Now, 30 = (2 # 3 # 5) and none of 2, 3, 5 is a factor of 11. 

11 30 is in its simplest form. Also, 30 = (2 # 3 # 5) ! (2 m # 5 n) .





EXAMPLE 3

66 11 30 and hence 180 is a nonterminating repeating decimal.

The decimal expansion of the rational number after how many places of decimals?

SOLUTION

43 , will terminate 24 $ 53 [CBSE 2009]

We have 43 = 43 # 5 = 215 = 215 = 215 = 0.0215. 2 4 $ 5 3 2 4 # 5 4 (2 # 5) 4 10 4 10000 So, it will terminate after 4 places of decimals.

EXAMPLE 4

Express each of the following as a rational number in simplest form. (a) 0.6

SOLUTION

(b) 1.8

(c) 0.16

(a) Let

x = 0.6. Then, x = 0.666 …  10x = 6.666 … On subtracting (i) from (ii), we get 6 2 9x  6 & x  9  3 ·

… (i) … (ii)

Real Numbers

23

2 Hence, 0.6 = 3 · (b) Let 

x = 1.8. Then, x = 1.888 … 10x = 18.888 …

… (i) … (ii)

On subtracting (i) from (ii), we get 17 8 9x  17 & x  9  1 9 · 8 Hence, 1.8 = 1 9 · (c) Let

x = 0.16. Then, x = 0.1666 …  10x = 1.6666 … And, 100x = 16.6666 …

… (i) … (ii) … (iii)

On subtracting (ii) from (iii), we get 15 1 90x  15 & x  90  6 · 1  0.16 = 6 · EXAMPLE 5

Express 0.32 as a fraction in simplest form.

SOLUTION

Let x = 0.32. Then, x = 0.3232 …

… (i)



… (ii)

100x = 32.3232 …

On subtracting (i) from (ii), we get 32 99x  32 & x  99 · 32 Hence, 0.32 = 99 · EXAMPLE 6

Express 0.254 as a fraction in simplest form.

SOLUTION

Let x = 0.254. Then, x = 0.2545454 …

… (i)



… (ii)

10x = 2.545454 … and 1000x = 254.545454 … On subtracting (ii) from (iii), we get 252 126 42 14 990x  252 & x  990  495  165  55 · 14  0.254  55 ·

… (iii)

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Secondary School Mathematics for Class 10

Show that each of the following numbers is rational. What can you say about the prime factors of their denominators? (i) 23.123456789 (ii) 32.123456789

EXAMPLE 7

(i) Clearly, the given number 23.123456789 is a terminating decimal. So, it is rational and the prime factors of its denominator are 2 or 5 or both.

SOLUTION

(ii) Clearly, the given number 32.123456789 is a nonterminating repeating decimal. So, it is rational and the prime factors of its denominator are other than 2 or 5 also. EXAMPLE 8

Decide whether the number 0.12012001200012 … is rational or not. Give reason to support your answer.

SOLUTION

Clearly, the given number 0.12012001200012 … is a nonterminating and nonrepeating decimal. So, it is not rational.

f

EXERCISE 1C

1. Without actual division, show that each of the following rational numbers is a terminating decimal. Express each in decimal form. (i)

24 (ii) 125

23 (2 3 # 5 2)

15 (iv) 1600

171 (iii) 800

19 (vi) 3125

17 (v) 320

2. Without actual division, show that each of the following rational numbers is a nonterminating repeating decimal. (i)

11 (2 3 # 3)

77 (v) 210

(ii)

73 129 (iii) (2 2 # 3 3 # 5) (2 2 # 5 3 # 7 2)

32 (vi) 147

29 (vii) 343

9 (iv) 35 64 (viii) 455

3. Express each of the following as a fraction in simplest form. (i) 0.8

(ii) 2.4

(v) 2.24

(vi) 0.365

(iii) 0.24

(iv) 0.12

ANSWERS (EXERCISE 1C)

1. (i) 0.115

(ii) 0.192

(vi) 0.00608

(iii) 0.21375

(iv) 0.009375

(v) 0.053125

Real Numbers

8

3. (i) 9

22 (ii) 9

8 (iii) 33

11 (iv) 90

101 (v) 45

25

181 (vi) 495

HINTS TO SOME SELECTED QUESTIONS 1. (i)

23 = 23 # 5 = 115 = 115 = 0.115. (2 3 # 5 2) (2 3 # 5 3) (10) 3 1000

24 24 2 3 24 # 8 = 192 = 192 = (ii) 125 = 3 # 3 = 0.192. 5 2 (5 # 2) 3 (10) 3 1000 171 171 1 21.375 (iii) 800 = 8 # 100 = 100 = 0.21375. 15 15 1 0.9375 (iv) 1600 = 16 # 100 = 100 = 0.009375. 17 17 # 5 85 85 1 5.3125 (v) 320 = 320 # 5 = 1600 = 16 # 100 = 100 = 0.053125. 19 19 # 8 152 152 1 6.08 (vi) 3125 = 3125 # 8 = 25000 = 25 # 1000 = 1000 = 0.00608. 9 9 2. (iv) 35 = # (5 7) 29 29 (vii) 343 = 3 7

77 11 11 (v) 210 = 30 = # # (2 3 5)

32 32 (vi) 147 = (3 # 7 2)

64 64 (viii) 455 = # # (5 7 13)

IRRATIONAL NUMBERS IRRATIONAL NUMBERS The numbers which when expressed in decimal form are expressible as nonterminating and nonrepeating decimals are known as irrational numbers.

Examples

Type 1.

Note that every nonterminating and nonrepeating decimal is irrational. (i) Clearly, 0.1010010001… is a nonterminating nonrepeating decimal. So, it is irrational.

and

(ii) 0.2020020002 , 0.3030030003…, etc., are all irrational. (iii) 0.12112111211112… is irrational, 0.13113111311113… is irrational, and so on. (iv) 0.232232223… is irrational, 0.343343334… is irrational, and so on. Type 2.

If m is a positive integer which is not a perfect square then m is irrational. Thus, 2 , 3 , 5 , 6 , 7 , 8 , 10 , 11 , etc., are all irrational.

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Secondary School Mathematics for Class 10

If m is a positive integer which is not a perfect cube then is irrational.

Type 3.

Thus,

3

3

m

2 , 3 3 , 3 5 , 3 6 , etc., are all irrational.

22  is irrational, while 7 is rational.

Type 4.

SOME RESULTS ON IRRATIONALS THEOREM 1

PROOF

Let p be a prime number and a be a positive integer. If p divides a 2 then show that p divides a.

Let p be a prime number and a be a positive integer such that p divides a 2 . We know that every positive integer can be expressed as the product of primes. Let a  p1 p2 … pn, where p1, p2, …, pn are primes, not necessarily all distinct. Then, a 2  (p1 p2 … pn) (p1 p2 … pn) 

a 2  (p12 p 22 … p n2) .

Now, p divides a2 

p is a prime factor of a 2



p is one of p1, p2, …, pn



p divides a

[a prime factors of a2 are p1, p2, …, pn] [a a  p1 p2 … pn] .

Thus, (p divides a 2)  (p divides a). Using the above result, we can prove the following. THEOREM 2 PROOF

Prove that 2 is irrational.

[CBSE 2008, ’09]

a If possible, let 2 be rational and let its simplest form be · b Then, a and b are integers having no common factor other than 1, and b ! 0. a a2 Now, 2  & 2  2 b b

[on squaring both sides]

& 2b 2  a 2 & 2 divides a 2 [a 2 divides 2b 2]

… (i)

Real Numbers

27

& 2 divides a [a 2 is prime and divides b 2 & 2 divides b]. Let a  2c for some integer c. Putting a  2c in (i), we get 2b 2  4c 2 & b 2  2c 2

& 2 divides b 2 [a 2 divides 2c 2] & 2 divides b [a 2 is prime and 2 divides b 2 & 2 divides b]. Thus, 2 is a common factor of a and b. But, this contradicts the fact that a and b have no common factor other than 1. The contradiction arises by assuming that 2 is rational. Hence, 2 is irrational. THEOREM 3 PROOF

Prove that 3 is irrational.

[CBSE 2008, ’09C]

a· b Then, a and b are integers having no common factor other than 1, and b ! 0. If possible, let 3 be rational and let its simplest form be

a a2 Now, 3  & 3  2 b b & 3b 2  a 2 & 3 divides a 2 & 3 divides a

[on squaring both sides] … (i) [a 3 divides 3b 2]

[a 3 is prime and 3 divides a 2 & 3 divides a]. Let a = 3c for some integer c. Putting a = 3c in (i), we get 3b 2  9c 2 & b 2  3c 2 & 3 divides b 2 [a 3 divides 3c 2] & 3 divides b [a 3 is prime and 3 divides b 2 & 3 divides b]. Thus, 3 is a common factor of a and b. But, this contradicts the fact that a and b have no common factor other than 1. The contradiction arises by assuming that 3 is rational. Hence, 3 is irrational.

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Secondary School Mathematics for Class 10

THEOREM 4 PROOF

Prove that

If possible, let

5 is irrational.

[CBSE 2008, ’09]

a 5 be rational and let its simplest form be b ·

Then, a and b are integers having no common factor other than 1, and b ! 0. a a2 Now, 5  & 5  2 b b

& 5b 2  a 2 & 5 divides a 2 & 5 divides a

[on squaring both sides] … (i) [a 5 divides 5b ] 2

[a 5 is prime and 5 divides a 2 & 5 divides a]. Let a = 5c for some integer c. Putting a = 5c in (i), we get 5b 2  25c 2 & b 2  5c 2

& 5 divides b 2 [a 5 divides 5c2] & 5 divides b [a 5 is prime and 5 divides b 2 & 5 divides b]. Thus, 5 is a common factor of a and b. But, this contradicts the fact that a and b have no common factor other than 1. The contradiction arises by assuming that Hence, THEOREM 5 PROOF

5 is rational.

5 is irrational.

Prove that

If possible, let

11 is irrational. a 11 be rational and let its simplest form be b ·

Then, a and b are integers having no common factor other than 1, and b ! 0. a a2 Now, 11  & 11  2 b b

& 11b 2  a 2 & 11 divides a 2 & 11 divides a

[on squaring both sides] … (i) [a 11 divides 11b ] 2

[a 11 is prime and 11 divides a 2 & 11 divides a]. Let a  11c for some positive integer c.

Real Numbers

29

Putting a  11c in (i), we get 11b 2  121c 2 & b 2  11c 2 [a 11 divides 11c 2]

& 11 divides b 2 & 11 divides b

[a 11 is prime and 11 divides b 2 & 11 divides b]. Thus, 11 is a common factor of a and b. But, this contradicts the fact that a and b have no common factor other than 1. The contradiction arises by assuming that Hence, THEOREM 6 PROOF

11 is rational.

11 is irrational.

If p is a prime number then prove that

p is irrational.

Let p be a prime number and if possible, let

p be rational.

m p = n , where m and n are integers having no common factor other than 1, and n ! 0.

Let its simplest form be m m2 Then, p  n & p  2 n

[on squaring both sides]

& pn 2  m 2 & p divides m 2 & p divides m

… (i) [a p divides pn 2]

[a p is prime and p divides m 2 & p divides m]. Let m = pq for some integer q. Putting m = pq in (i), we get pn 2  p 2 q 2 & n 2  pq 2

& p divides n2 & p divides n

[a p divides pq2]

[a p is prime and p divides n 2 & p divides n]. Thus, p is a common factor of m and n. But, this contradicts the fact that m and n have no common factor other than 1. The contradiction arises by assuming that p is rational. Hence, p is irrational.

30

Secondary School Mathematics for Class 10

THEOREM 7

PROOF

If a is rational and irrational.

b is irrational then prove that (a  b ) is

Let a be rational and b be irrational. Then, we have to prove that (a  b ) is irrational. If possible, let (a  b ) be rational. Then, (a  b ) is rational and a is rational  

{(a  b )  a} is rational [a difference of rationals is rational] b is rational.

This contradicts the fact that b is irrational. The contradiction arises by assuming that (a  b ) is rational. Hence, (a  b ) is irrational. THEOREM 8

PROOF

If a is a nonzero rational and b is irrational then show that a b is irrational.

Let a be a nonzero rational and let b be irrational. Then, we have to show that a b is irrational. If possible, let a b be rational. p Then, a b  q , where p and q are nonzero integers, having no common factor other than 1. p p Now, a b  q & b  aq · But, p and aq are both rational and aq ! 0. p  aq is rational. Thus, from (i), it follows that b is rational. This contradicts the fact that b is irrational. The contradiction arises by assuming that a b is rational. Hence, a b is irrational.

SOLVED EXAMPLES EXAMPLE 1

Show that (2  3 ) is an irrational number.

SOLUTION

Let us assume, to the contrary, that (2  3 ) is rational.

... (i)

Real Numbers

31

Then, there exist co-primes a and b (b ! 0) such that a (2  3 )  b a a  2b ·  3  2 & 3  b b a  2b Since a and b are integers, so is rational. b Thus, 3 is also rational. But, this contradicts the fact that assumption is incorrect.

3 is irrational. So, our

Hence, (2 + 3 ) is irrational. EXAMPLE 2

Show that 2 3 is irrational.

SOLUTION

Let us assume, to the contrary, that 2 3 is rational. Then, there exist co-primes a and b (b ! 0) such that a a · 2 3 & 3 2b b a Since a and b are integers, so is rational. 2b Thus,

3 is also rational.

But, this contradicts the fact that assumption is incorrect.

3 is irrational. So, our

Hence, 2 3 is irrational. 1 is irrational. 2

EXAMPLE 3

Show that

SOLUTION

Let us assume, to the contrary, that

1 is rational. 2

Then, there exist co-primes a and b (b ! 0) such that 1 a & 2 b

b 2a·

b Since a and b are integers, so a is rational. Thus, 2 is also rational. But, this contradicts the fact that assumption is incorrect. Hence,

1 is irrational. 2

2 is irrational. So, our

32

Secondary School Mathematics for Class 10

EXAMPLE 4

Prove that (3  5 2 ) is irrational.

SOLUTION

Let us assume, to the contrary, that (3  5 2 ) is rational. Then, there exist co-primes a and b (b ! 0) such that a 35 2  b a  3b a  5 2  3  b b  3b a ·  2 5b a  3b Since a and b are integers, so is rational. 5b Thus, 2 is also rational. But, this contradicts the fact that assumption is incorrect.

2 is irrational. So, our

Hence, (3  5 2 ) is irrational. EXAMPLE 5

Prove that ( 2  3 ) is irrational.

SOLUTION

Let us assume that ( 2  3 ) is irrational. Then, there exist co-primes a and b such that a 2 3 b a  3  2 b 2 a  ( 3 ) 2  b  2l [squaring both sides] b   

a 2  2a 22 b2 b 2a a2 2  2 1 b b 2 2 a b · 2 2ab

3

Since a and b are integers, so

a2  b2 is rational. 2ab

Thus, 2 is also rational. But, this contradicts the fact that assumption is incorrect.

2 is irrational. So, our

Hence, ( 2  3 ) is irrational. EXAMPLE 6

Prove that p  q is irrational, where p and q are primes.

SOLUTION

Let us assume that p  q is rational.

Real Numbers

33

Then, there exist co-primes a and b such that a p q  b a  p  q b 2 a  ( p) 2  b  ql [squaring both sides] b a 2 2a  p 2 q q b b 

2a a2 q  2 qp b b

a 2 b  b 3 (q  p) b · q  (a 2  b 2 q  b 2 p)#  2a 2a a 2 b  b 3 (q  p) Since a, b, p, q are integers so is rational. 2a 

Thus, q is also rational. But, q being prime, q is irrational. Since, a contradiction arises so our assumption is incorrect. Hence, ( p  q ) is irrational.

f

EXERCISE 1D

1. Define (i) rational numbers (ii) irrational numbers (iii) real numbers. 2. Classify the following numbers as rational or irrational: (i)

22 7

(v) 5.636363 …

(ii) 3.1416

(iii) 

(iv) 3.142857

(vi) 2.040040004 … (vii) 1.535335333 …

(viii) 3.121221222 … (ix)

21

(x)

3

3

3. Prove that each of the following numbers is irrational. (i)

(iiii) (3 + 2 ) (v) (5 + 3 2 ) (vii)

(ii) (2  3 )

[CBSE 2008]

[CBSE 2009]

(iv) (2 + 5 )

[CBSE 2008C]

[CBSE 2008]

(vi) 3 7

6

3 5

(ix) ( 3 + 5 )

(viii) (2  3 5 )

[CBSE 2010]

34

Secondary School Mathematics for Class 10

4. Prove that

1 is irrational. 3

1 = 1 # 3 3

HINT

3 = 1 · 3. 3 3

5. (i) Give an example of two irrationals whose sum is rational. (ii) Give an example of two irrationals whose product is rational. HINT

(i) Take (2 +

3 ) and (2  3 ) .

(ii) Take (3 +

2 ) and (3  2 ) .

6. State whether the given statement is true or false. (i) The sum of two rationals is always rational. (ii) The product of two rationals is always rational. (iii) The sum of two irrationals is always an irrational. (iv) The product of two irrationals is always an irrational. (v) The sum of a rational and an irrational is irrational. (vi) The product of a rational and an irrational is irrational. 7. Prove that (2 3  1) is an irrational number.

[CBSE 2010]

8. Prove that (4  5 2 ) is an irrational number.

[CBSE 2010]

9. Prove that (5  2 3 ) is an irrational number.

[CBSE 2010]

10. Prove that 5 2 is irrational. 11. Prove that HINT

2 is irrational. 7

2  2 # e 7 7

7 7

o  72 $ 7 . ANSWERS (EXERCISE 1D)

2. (i) rational

(ii) rational

(iii) irrational

(iv) rational

(v) rational

(vi) irrational

(vii) irrational

(viii) irrational

(ix) irrational

(x) irrational

6. (i) True

(ii) True

(iii) False

f

(iv) False

EXERCISE 1E

Very-Short-Answer Questions 1. State Euclid‘s division lemma.

(v) True

(vi) True

Real Numbers

35

2. State fundamental theorem of arithmetic. 3. Express 360 as product of its prime factors. 4. If a and b are two prime numbers then find HCF(a, b). 5. If a and b are two prime numbers then find LCM(a, b). 6. If the product of two numbers is 1050 and their HCF is 25, find their LCM. 7. What is a composite number? 8. If a and b are relatively prime then what is their HCF? a has a terminating decimal expansion, what is b [CBSE 2008] the condition to be satisfied by b?

9. If the rational number

10. Simplify:

(2 45 + 3 20 ) · 2 5

[CBSE 2010]

11. Write the decimal expansion of

73 · (24 # 53)

[CBSE 2009]

12. Show that there is no value of n for which (2n # 5n) ends in 5. 13. Is it possible to have two numbers whose HCF is 25 and LCM is 520? 14. Give an example of two irrationals whose sum is rational. 15. Give an example of two irrationals whose product is rational. 16. If a and b are relatively prime, what is their LCM? 17. The LCM of two numbers is 1200. Show that the HCF of these numbers cannot be 500. Why? Short-Answer Questions 18. Express 0.4 as a rational number in simplest form. 19. Express 0.23 as a rational number in simplest form. 20. Explain why 0.15015001500015 … is an irrational number. 2 21. Show that 3 is irrational. 22. Write a rational number between

3 and 2.

23. Explain why 3.1416 is a rational number. ANSWERS (EXERCISE 1E)

3. (23 # 32 # 5) 4. 1

5. ab

6. 42

8. 1

9. b = (2 # 5 ), where m and n are some non-negative integers m

10. 6

n

11. 0.0365 13. No 16. ab 4 23 18. 9 19. 99 22. 1.8

17. since 500 is not a factor of 1200

36

Secondary School Mathematics for Class 10 HINTS TO SOME SELECTED QUESTIONS

7. A number having at least 3 factors is called a composite number. 11.

73 = 73 # 5 = 365 = 365 = 365 = 0.0365. (24 # 53) (24 # 54) (2 # 5) 4 (10) 4 10000

12. (2n # 5n) = (2 # 5) n = 10n, which always ends in a zero. 13. HCF always divides the LCM completely. 14. (2 + 3 ) and (2  3 ) . 15. (3 + 2 ) and (3  2 ) . 20. Given number is nonterminating and nonrepeating decimal. 22. Clearly,

3 = 1.732 … . So, we may take 1.8 as the required rational number between

3 and 2. 23. Clearly, it is a nonterminating repeating decimal.

MULTIPLE-CHOICE QUESTIONS (MCQ) Choose the correct answer in each of the following questions: 1. Which of the following is a pair of co-primes? (a) (14, 35)

(b) (18, 25)

(c) (31, 93)

(d) (32, 62)

2. If a = (2 # 3 # 5 ) and b = (2 # 3 # 5) then HCF(a, b) = ? 2

3

4

(a) 90

3

2

(b) 180

(c) 360

(d) 540

3. HCF of (2 # 3 # 5), (2 # 3 # 5 ) and (2 # 3 # 5 # 7) is 3

(a) 30

2

2

3

2

(b) 48

4

3

(c) 60

(d) 105

(c) 1120

(d) 1680

4. LCM of (23 # 3 # 5) and (24 # 5 # 7) is (a) 40

(b) 560

5. The HCF of two numbers is 27 and their LCM is 162. If one of the numbers is 54, what is the other number? (a) 36

(b) 45

(c) 9

(d) 81

6. The product of two numbers is 1600 and their HCF is 5. The LCM of the numbers is (a) 8000

(b) 1600

(c) 320

(d) 1605

7. What is the largest number that divides each one of 1152 and 1664 exactly? (a) 32

(b) 64

(c) 128

(d) 256

8. What is the largest number that divides 70 and 125, leaving remainders 5 and 8 respectively? (a) 13

(b) 9

(c) 3

(d) 585

Real Numbers

37

9. What is the largest number that divides 245 and 1029, leaving remainder 5 in each case? (a) 15

(b) 16

(c) 9

(d) 5

13 (c) 16

15 (d) 16

1095 10. The simplest form of 1168 is 17 (a) 26

25 (b) 26

11. Euclid’s division lemma states that for any positive integers a and b, there exist unique integers q and r such that a = bq + r, where r must satisfy (a) 1 < r < b

(b) 0 < r  b

(c) 0  r < b

(d) 0 < r < b

12. A number when divided by 143 leaves 31 as remainder. What will be the remainder when the same number is divided by 13? (a) 0

(b) 1

(c) 3

(d) 5

13. Which of the following is an irrational number? (a)

22 7

(b) 3.1416

(c) 3.1416

(d) 3.141141114 …

14.  is (a) an integer

(b) a rational number

(c) an irrational number

(d) none of these

15. 2.35 is (a) an integer

(b) a rational number

(c) an irrational number

(d) none of these

16. 2.13113111311113 ... is (a) an integer

(b) a rational number

(c) an irrational number

(d) none of these

17. The number 3.24636363 … is (a) an integer

(b) a rational number

(c) an irrational number

(d) none of these

18. Which of the following rational numbers is expressible as a terminating decimal? 124 (a) 165

(b)

131 30

(c)

2027 625

(d)

1625 462

38

Secondary School Mathematics for Class 10

37 19. The decimal expansion of the rational number 2 will terminate 2 #5 after (a) one decimal place (c) three decimal places

(b) two decimal places (d) four decimal places

14753 20. The decimal expansion of the number 1250 will terminate after (a) one decimal place (c) three decimal places

(b) two decimal places (d) four decimal places

21. The number 1.732 is (a) an irrational number (c) an integer

(b) a rational number (d) a whole number

22. a and b are two positive integers such that the least prime factor of a is 3 and the least prime factor of b is 5. Then, the least prime factor of (a + b) is (a) 2 23.

(c) 5

(d) 8

2 is (a) (b) (c) (d)

24.

(b) 3

a rational number an irrational number a terminating decimal a nonterminating repeating decimal

1 is 2 (a) a fraction (c) an irrational number

(b) a rational number (d) none of these

25. (2 + 2 ) is (a) an integer (c) an irrational number

(b) a rational number (d) none of these

26. What is the least number that is divisible by all the natural numbers from 1 to 10 (both inclusive)? (a) 100

(b) 1260

(c) 2520

(d) 5040

ANSWERS (MCQ)

1. (b) 2. (b) 3. (c) 4. (d) 5. (d) 6. (c) 7. (c) 10. (d) 11. (c) 12. (d) 13. (d) 14. (c) 15. (b) 16. (c) 19. (b) 20. (d) 21. (b) 22. (a) 23. (b) 24. (c) 25. (c)

8. (a) 9. (b) 17. (b) 18. (c) 26. (c)

Real Numbers

39

HINTS TO SOME SELECTED QUESTIONS 2. HCF(a, b) = product of common terms with lowest power = (22 # 32 # 5) = (4 # 9 # 5) = 180. 3. HCF = product of common terms with lowest power = (2 2 # 3 # 5) = (4 # 3 # 5) = 60. 4. LCM = product of prime factors with highest power = (2 4 # 3 # 5 # 7) = (16 # 3 # 5 # 7) = 1680. 5. Other number = 6. LCM =

HCF # LCM = 27 # 162 = 81. 54 given number

product of two numbers 1600 = = 320. 5 their HCF

7. Required number = HCF(1152, 1664) = 128. 8. Required number = HCF{(70 – 5), (125 – 8)} = HCF(65, 117) = 13. 9. Required number = HCF{(245 – 5), (1029 – 5)} = HCF(240, 1024) = 16. 10. HCF{1095, 1168} = 73. 

1095 = 1095 ' 73 = 15 1168 1168 ' 73 16 $

11. On dividing a by b, let q be the quotient and r be the remainder. Then, we have a = bq + r, where 0 # r < b. 12. Let the given number when divided by 143 give q as quotient and 31 as remainder. Then, number = 143q + 31 = {13 # 11q + 13 # 2 + 5} = 13 # (11q + 2) + 5. So, the same number when divided by 13 gives 5 as remainder. 13. 3.141141114 … is a nonterminating, nonrepeating decimal. So, it is irrational. 14.  is an irrational number. 15. 2. 35 = 2.353535 …, which is a repeating decimal. 

2. 35 is rational.

16. 2.13113111311113 … is a nonterminating, nonrepeating decimal. So, it is irrational. 17. The number 3.24636363 … is a nonterminating repeating decimal. So, it is a rational number. 18.

2027 = 2027 $ 625 (5 4 # 2 0) So, it is expressible as a terminating decimal.

19.

37 = 37 # 5 = 185 = 185 = 185 = 1.85. 2 2 # 5 2 2 # 5 2 (2 # 5) 2 (10) 2 100 So, it will terminate after 2 decimal places.

40

Secondary School Mathematics for Class 10

14753 8 14753 # 8 14753 # 8 20. 1250 # 8 = 10000 = $ (10) 4 So, it will terminate after 4 decimal places. 1732 21. 1.732 = 1000 , which is a rational number. 22. Clearly, 2 is neither a factor of a nor that of b. 

a and b are both odd.

Hence, (a + b) is even.  24.

least prime factor of (a + b) is 2.

1 = 1 # 2 2

2 1 = $ 2. 2 2

1 Here, 2 is rational and

2 is irrational.

And, the product of a rational and an irrational is irrational. 1 1 2 $ 2 and hence 2 is irrational.



25. The sum of a rational and an irrational is irrational. Here, 2 is rational and 

2 is irrational.

(2 + 2 ) is irrational.

26. Required number = LCM{1, 2, 3, 4, 5, 6, 7, 8, 9, 10} = LCM{1, 2, 3, 2 2, 5, 2 # 3, 7, 2 3, 3 2, 2 # 5} = (1 # 2 3 # 3 2 # 5 # 7) = (8 # 9 # 5 # 7) = 2520.

SUMMARY OF RESULTS 1. Euclid’s Division Lemma

Given positive integers a and b, there exist whole numbers q and r satisfying a = bq + r, where 0 # r < b. According to this, we find the HCF of two positive integer a and b with a > b in following steps.

2. Euclid’s Division Algorithm

Step 1.

Apply the division lemma to find q and r such that a = bq + r, where 0 # r < b.

Step 2. If r = 0 then HCF = b. If r ! 0, apply Euclid’s division lemma to

b and r. Step 3. Continue the process till the remainder is zero. The divisor at this

stage is HCF(a, b).

Real Numbers

41

3. The Fundamental Theorem of Arithmetic

Every composite number can be expressed as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur. 4. If p is prime and p divides a 2 then p divides a, where a is a positive integer. 5. To prove that

2 , 3 , 5 , 6 , 7 , etc., are irrationals.

6. Let x be a rational number which can be expressed as a terminating decimal.

p If we put it in the simplest form q then q = (2 m # 5 n) for some non-negative integers m and n. p

7. Let x = q

be a rational number such that q ! (2 m # 5 n) then x has a

nonterminating repeating decimal expansion. 8. A number which can be expressed as a nonterminating and nonrepeating

decimal is an irrational number. 2 , 3 , 5 , 6 , 7 , 8 , 10 , …, , e, etc., are all irrational numbers.

TEST YOURSELF MCQ 71 1. The decimal representation of 150 is (a) a terminating decimal (b) a nonterminating, repeating decimal (c) a nonterminating and nonrepeating decimal (d) none of these 2. Which of the following has a terminating decimal expansion? (a)

32 91

(b)

19 80

(c)

23 45

(d)

25 42

3. On dividing a positive integer n by 9, we get 7 as remainder. What will be the remainder if (3n  1) is divided by 9? (a) 1

(b) 2

(c) 3

(d) 4

(b) 1.42

(c) 0.141

(d) None of these

4. 0.68  0.73  ? (a) 1.41 Short-Answer Questions 5. Show that any number of the form 4 n, nd N can never end with the digit 0.

42

Secondary School Mathematics for Class 10

6. The HCF of two numbers is 27 and their LCM is 162. If one of the number is 81, find the other. 17 7. Examine whether 30 is a terminating decimal. 148 8. Find the simplest form of 185 $ 9. Which of the following numbers are irrational? (a)

(b)

2

(e) 

(f)

3

(c) 3.142857

6

22 7

(d) 2.3

(g) 0.232332333… (h) 5.2741

10. Prove that (2 + 3 ) is irrational. 11. Find the HCF and LCM of 12, 15, 18, 27. 12. Give an example of two irrationals whose sum is rational. 13. Give prime factorisation of 4620. 14. Find the HCF of 1008 and 1080 by prime factorisation method. 8 10 16 15. Find the HCF and LCM of 9 , 27 and 81 $ 16. Find the largest number which divides 546 and 764, leaving remainders 6 and 8 respectively. Long-Answer Questions 17. Prove that 3 is an irrational number. 18. Show that every positive odd integer is of the form (4q + 1) or (4q + 3) for some integer q. 19. Show that one and only one out of n, (n + 2) and (n + 4) is divisible by 3, where n is any positive integer. 20. Show that (4  3 2 ) is irrational. ANSWERS (TEST YOURSELF)

1. (b) 9.

2. (b)

3. (b)

4. (b)

2 , 3 6 , , 0.232332333…

12. (2  3 ) and (2  3 ) 14. 72

15. HCF =

6. 54

7. No

8.

4 5

11. HCF = 3, LCM = 540 13. (2 2 # 3 # 5 # 7 # 11)

80 2 , LCM  9 81 

16. 108

Polynomials

43

An expression of the form p(x) = a0 + a1 x + a2 x 2 + … + an x n, where an ! 0, is called a polynomial in x of degree n.

POLYNOMIALS

Here a0, a1, a2, …, an are real numbers and each power of x is a non-negative integer. EXAMPLES

(i) 3x  5 is a polynomial in x of degree 1. (ii) 8x 2  5x + 3 is a polynomial in x of degree 2. 4 (iii) 2y 3 + 9 y 2  5y + 3 is a polynomial in y of degree 3. (iv) 3z 4  5z 3 + 2z 2  8z + 1 is a polynomial in z of degree 4.

REMARK

6 1 Note that ( x  3), , , (x  5) (x 2  3x  1) polynomials.

LINEAR POLYNOMIAL

etc.,

are

not

A polynomial of degree 1 is called a linear polynomial.

A linear polynomial is of the form p(x) = ax + b, where a ! 0. 5 Thus, (3x  5), ( 2 x + 3), a x  8 k, etc., are all linear polynomials. QUADRATIC POLYNOMIAL

A polynomial of degree 2 is called a quadratic polynomial.

A quadratic polynomial is of the form p(x) = ax 2 + bx + c, where a ! 0. Thus, (3x 2  5x  8), (2x 2  2 2 x  6), (y 2  3y  3 ), etc., are all quadratic polynomials. A polynomial of degree 3 is called a cubic polynomial. A cubic polynomial is of the form p(x) = ax 3 + bx 2 + cx + d, where a ! 0.

CUBIC POLYNOMIAL

Thus, (2x 3  3x 2 + 8x + 1), ( 2 y 3  2y 2 + y  8), (z 3 + 2z 2  3 z + 3), etc., are all cubic polynomials. BIQUADRATIC POLYNOMIAL

A polynomial of degree 4 is called a biquadratic

polynomial. A biquadratic polynomial is of the form p(x) = ax 4 + bx 3 + cx 2 + dx + e, where a ! 0. Thus, (2x 4 + 3x 3  5x 2 + 9x + 1), (4y 4  5y 3 + 6y 2  8y + 3), etc., are all biquadratic polynomials. 43

44

Secondary School Mathematics for Class 10

VALUE OF A POLYNOMIAL AT A GIVEN POINT

If p(x) is a polynomial in x and if  is any real number then the value obtained by putting x =  in p(x) is called the value of p(x) at x = . The value of p(x) at x =  is denoted by p(). Let p(x) = 2x 2  3x + 5. Then,

EXAMPLE

p(2) = {2 # 2 2  3 # 2 + 5} = (8  6 + 5) = 7, p(1)  {2 # (1) 2  3 # (1)  5}  (2  3  5)  10. ZEROS OF A POLYNOMIAL

A real number  is called a zero of the polynomial

p(x), if p() = 0. Let p(x)  x 2  2x  3. Find (i) p(3) and (ii) p(1) . What do you conclude?

EXAMPLE

We have p(x)  x 2  2x  3.

SOLUTION

 (i) p(3)  (3 2  2 # 3  3)  (9  6  3)  0 and (ii) p(1)  {(1) 2  2 # (1)  3}  (1  2  3)  0. This shows that 3 and –1 are the zeros of the polynomial p(x). RELATION BETWEEN THE ZEROS AND COEFFICIENTS OF A QUADRATIC POLYNOMIAL

Let  and  be the zeros of a quadratic polynomial p(x)  ax 2  bx  c, where a ! 0. Then, (x  ) and (x  ) are the factors of p(x).  (ax 2  bx  c)  k(x  )(x  ), where k is a constant  k $ {x 2  (  ) x  }  kx 2  k(  ) x  k() . On comparing coefficients of like powers of x on both sides, we get k  a,  k(  )  b and k()  c  a(  )  b and a()  c b c  (  )  a and   a $ 

sum of zeros 

product of zeros 

(coefficient of x) , (coefficient of x 2) constant term · coefficient of x 2

[a k  a]

Polynomials

45

SUMMARY

I. If  and  are the zeros of p(x)  ax 2  bx  c, a ! 0 then b (i)  +  = a c (ii)   a · II. A quadratic polynomial whose zeros are  and  is given by p(x)  {x 2  (  )x  } .

SOLVED EXAMPLES EXAMPLE 1

SOLUTION

Find the zeros of the polynomial 2x 2 + 5x  12 and verify the relationship between its zeros and coefficients. Let the given polynomial be denoted by f(x). Then,



f (x) = 2x 2 + 5x  12 = 2x 2 + 8x  3x  12 = 2x(x + 4)  3(x + 4) = (x + 4)(2x  3) . f (x)  0 & (x  4)(2x  3)  0

& x  4  0 or 2x  3  0 3 & x  4 or x  2 · So, the zeros of f(x) are –4 and

3· 2

5 (coefficient of x) 3 Sum of the zeros = a 4 + 2 k = 2 = , (coefficient of x 2) 3 12  constant term · product of the zeros  (4)#  2 2 (coefficient of x 2) EXAMPLE 2

SOLUTION

Find the zeros of the polynomial 6x 2  3  7x and verify the relationship between the zeros and the coefficients. [CBSE 2008] Let the given polynomial be denoted by f(x). Then, f (x) = 6x 2  3  7x = 6x 2  7x  3 [in standard form] = 6x 2  9x + 2x  3 = 3x (2x  3) + (2x  3) = (2x  3)(3x + 1) .

46

Secondary School Mathematics for Class 10

f (x)  0 & (2x  3) (3x  1)  0 & 2x  3  0 or 3x  1  0  3 & x  2 or x  31 · 1 3 · So, the zeros of f(x) are 2 and 3 3 1 3 1 7 (coefficient of x) · Sum of zeros  &  a k0  a  k   2 3 2 3 6 (coefficient of x 2) 3 1 3  constant term 2 · Product of zeros  # a k  2 3 6 coefficient of x 

EXAMPLE 3

SOLUTION

Find the zeros of the polynomial f (x) = x 2  2 and verify the relationship between its zeros and coefficients. We have f (x) = (x 2  2) = {x 2  ( 2 ) 2} = (x + 2 )(x  2 ). 

f (x)  0 & (x  2 )(x  2 )  0 & x  2  0 or x  2  0 & x   2 or x  2 .

So, the zeros of f (x) are  2 and 2 . 0 (coefficient of x) Sum of zeros = ( 2 + 2 ) = 0 = 1 = , (coefficient of x 2) 2  constant term 2 · product of zeros  ( 2 ) # ( 2 )  1 coefficient of x EXAMPLE 4

SOLUTION

Obtain the zeros of the quadratic polynomial 3 x 2  8x + 4 3 and verify the relation between its zeros and coefficients. [CBSE 2008C] We have



f (x) = 3 x 2  8x + 4 3 = 3 x 2  6x  2x + 4 3 = 3 x (x  2 3 )  2 (x  2 3 ) = (x  2 3 ) ( 3 x  2) . f (x)  0 & (x  2 3 )( 3 x  2)  0

& (x  2 3 )  0 or ( 3 x  2)  0 & x  2 3 or x  2 ·

3 2 · So, the zeros of f(x) are 2 3 and 3 2 = 8 =  (coefficient of x) Sum of zeros = d2 3 + , n (coefficient of x 2) 3 3 product of zeros  d2 3 #

2  4 3  constant term · n coefficient of x 2 3 3

Polynomials

47

EXAMPLE 5

Find a quadratic polynomial, the sum and product of whose zeros are –5 and 6 respectively.

SOLUTION

Let  and  be the zeros of the required polynomial f(x). Then, ( + ) = 5 and  = 6. 

f (x) = x 2  ( + ) x +  = x 2  (5) x + 6 = x 2 + 5x + 6.

Hence, the required polynomial is f (x) = x 2 + 5x + 6. EXAMPLE 6

Find the quadratic polynomial, the sum of whose zeros is 2 and their product is –12. Hence, find the zeros of the polynomial.

SOLUTION

Let  and  be the zeros of the required polynomial f(x). Then, ( + ) = 2 and  = 12. 

f (x) = x 2  ( + ) x +  = x 2  2 x  12.

So, the required polynomial is f (x) = x 2  2 x  12. Now, f (x) = x 2  2 x  12 = x 2  3 2 x + 2 2 x  12 [note it] = x (x  3 2 ) + 2 2 (x  3 2 ) = (x  3 2 )(x + 2 2 ) . 

f (x)  0 & (x  3 2 )(x  2 2 )  0

& x  3 2  0 or x  2 2  0 & x  3 2 or x   2 . Hence, the required polynomial is f (x) = x 2  2 x   whose zeros are 3 2 and 2 2 . EXAMPLE 7

SOLUTION

If the product of the zeros of the polynomial (ax 2  6x  6 is 4, find the value of a. [CBSE 2009] Let  and  be the zeros of the polynomial (ax 2  6x  6) . constant term  6 · Then,   a coefficient of x 2 But,  = 4 (given). 3 6  ·   6 & a  6   a 4 & 4a 2 4 3 · Hence, a  2

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Secondary School Mathematics for Class 10

EXAMPLE 8

If one zero of the polynomial (a 2 + 9) x 2 + 13x + 6a is reciprocal of the other, find the value of a. [CBSE 2008]

SOLUTION

Let one zero of the given polynomial be . 1 Then, the other zero is · 1 product of zeros = b #  l = 1. constant term  6a · But, product of zeros  coefficient of x 2 (a 2  9) 6a   1 & a 2  9  6a (a 2  9) 

Hence, a = 3. EXAMPLE 9

& a 2  9  6a  0 & (a  3)2  0 & a  3  0 & a  3.

Find a quadratic polynomial whose zeros are 1 and –3. Verify the relation between the coefficients and zeros of the polynomial. [CBSE 2008C]

SOLUTION

Let = 1 and = 3. Sum of zeros = ( + ) = 1 + (3) = 2. Product of zeros =  = 1 # (3) = 3. So, the required polynomial is x2  ( + ) x +  = x2  (2) x + (3) = x2 + 2x  3. 2  (coefficient of x) Sum of zeros = 2 = 1 = , (coefficient of x2) 3  constant term 2 · product of zeros  3  1 coefficient of x

f

EXERCISE 2A

Find the zeros of the following quadratic polynomials and verify the relationship between the zeros and the coefficients:

1. x2 + 7x + 12 3. x2 + 3x  10

2. x2  2x  8 4. 4x2  4x  3

5. 5x2  4  8x [CBSE 2008] 7. 2x2  11x + 15

6. 2 3 x2  5x + 3 8. 4x2  4x + 1

[CBSE 2008C] [CBSE 2011]

Polynomials

9. x2  5 11. 5y2 + 10y

49

10. 8x2  4 12. 3x2  x  4

13. Find the quadratic polynomial whose zeros are 2 and –6. Verify the relation between the coefficients and the zeros of the polynomial. 1 2 · Verify the 14. Find the quadratic polynomial whose zeros are 3 and 4 relation between the coefficients and the zeros of the polynomial. 15. Find the quadratic polynomial, sum of whose zeros is 8 and their product is 12. Hence, find the zeros of the polynomial. [CBSE 2008] 16. Find the quadratic polynomial, the sum of whose zeros is 0 and their product is –1. Hence, find the zeros of the polynomial. 5 17. Find the quadratic polynomial, the sum of whose zeros is b 2 l and their product is 1. Hence, find the zeros of the polynomial.

18. Find the quadratic polynomial, the sum of whose roots is 2 and their 1 product is · 3 2 19. If x = 3 and x = 3 are the roots of the quadratic equation ax2 + 7x + b = 0 [CBSE 2011] then find the values of a and b. 20. If (x + a) is a factor of the polynomial 2x2 + 2ax + 5x + 10, find the value of a. [CBSE 2009] 2 3 2 21. One zero of the polynomial 3x  16x  15x  18 is · Find the other 3 zeros of the polynomial. ANSWERS (EXERCISE 2A)

1. –4, –3

3

6. 2 , 11. 0, –2

1 3

2. 4, –2

5

3. –5, 2

1 1

7. 3, 2

8. 2 , 2

4

12. 3 , 1

3 1

4. 2 , 2 9.

5,  5

2

5. 2, 5 10.

1 1 , 2 2

13. x2 + 4x  12 14. 12x2  5x  2

15. x2  8x + 12, {6, 2}

16. (x2  1), {1, 1}

17. (2x2  5x + 2), &2, 2 0

18. 3x2  3 2 x + 1

19. a = 3, b = 6

20. a = 2

1

HINTS TO SOME SELECTED QUESTIONS 5. 5x2  8x  4 = 5x2  10x + 2x  4 = 5x (x  2) + 2 (x  2) = (x  2) (5x + 2) .

21. –3, –3

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Secondary School Mathematics for Class 10

6. 2 3 x2  5x + 3 = 2 3 x2  3x  2x + 3 = 3 x (2x  3 )  (2x  3 ) = (2x  3 ) ( 3 x  1) . 8. 4x2  4x + 1 = (2x  1) 2 . 9. (x2  5) = (x  5 )(x + 5 ) . 10. 8x2  4 = 4 (2x2  1) = 4 ( 2 x  1)( 2 x + 1) . 11. 5y2 + 10y = 5y (y + 2) . 12. 3x2  x  4 = 3x2  4x + 3x  4 = x (3x  4) + (3x  4) = (3x  4)(x + 1) . 7 2 · 19. Sum of the roots  a  3k  3 3 2 Product of roots = 3 # (3) = 2. 7 7 b  'a  and a  21 & a  3 and b  6. 3 20. Let f (x) = 2x2 + 2ax + 5x + 10. Since (x + a) is a factor of f(x), we have f ( a) = 0. 

2(a) 2 + 2a (a) + 5 (a) + 10 = 0



2a 2  2a 2  5a  10  0 & 5a  10 & a  2.

2 21. ax  k is a factor of the given polynomial and therefore, (3x  2) is also its factor. 3 On dividing the given polynomial by (3x  2), we get (x 2  6x  9) as quotient, i.e., (x  3) 2  0 & x  3.

RELATION BETWEEN THE ZEROS AND COEFFICIENTS OF A CUBIC POLYNOMIAL

Let ,  and  be the zeros of a cubic polynomial p(x) = ax3 + bx2 + cx + d, where a ! 0. Then, (x  ), (x  ) and (x  ) are the factors of p(x) . 

(ax3 + bx2 + cx + d) = k(x  )(x  )(x  ) for some constant k = k{x3  ( +  + )x2 + ( +  + )x  ()} = kx3  k( +  + ) x2 + k( +  + )x  k().

Comparing coefficients of like powers of x on both sides, we get k = a, k( +  + ) = b, k( +  + ) = c, k() = d  

a( +  + ) = b, a( +  + ) = c,  a() = d [a k  a] b d c (    )  a , (    )  a ,   a ·

Polynomials

51

SUMMARY

I. If  and  are the zeros of p(x) = ax 3 + bx 2 + cx + d then b (i) ( +  + ) = a

c (ii) ( +  + ) = a

d (iii)   a ·

II. A cubic polynomial whose zeros are  and  is given by p(x)  {x 3  (    )x 2  (    )x  } .

SOLVED EXAMPLES EXAMPLE 1

1 Verify that 3, –1 and 3 are the zeros of the cubic polynomial p(x) = 3x3  5x2  11x  3 and verify the relation between its zeros and coefficients.

SOLUTION

The given polynomial is p(x) = 3x3  5x2  11x  3. 

p(3) = {3 # 33  5 # 32  11 # 3  3} = (81  45  33  3) = 0; p(1) = {3 # (1) 3  5 # (1) 2  11 # (1)  3} = (3  5 + 11  3) = 0;

1 1 3 1 2 1 and p b 3 l = (3 # b 3 l  5 # b 3 l  11 # b 3 l  32   = '3 # b 1 l  5 # 1 + 11  31 = b 1  5 + 11  3l 27 9 3 9 9 3 (1  5 + 33  27) = = 0. 9 1  3, –1 and 3 are the zeros of p(x) . 1 Let  = 3,  = 1 and  = 3 $ Then, 2 1 5 (coefficient of x ) ( +  + ) = b3  1  3 l = 3 = 3 ; (coefficient of x ) (coefficient of x) 11 1 ( +  + ) = b3 + 3  1l = 3 = ; (coefficient of x3) 1 3 (constant term) ·   &3 # (1) # a k0  1   3 3 (coefficient of x 3) EXAMPLE 2

Find a cubic polynomial with the sum of its zeros, sum of the products of its zeros taken two at a time and the product of its zeros as 2, –7 and –14 respectively.

52 SOLUTION

Secondary School Mathematics for Class 10

Let  be the zeros of the required polynomial. Then,  +  +  = 2,  +  +  = 7 and  = 14. So, the required polynomial is p(x) = x3  ( +  + )x2 + ( +  + )x    x 3  2x 2  7x  (14)  x 3  2x 2  7x  14.

EXAMPLE 3

SOLUTION

If the zeros of the polynomial x3  3x2 + x + 1 are (a  b), a, (a + b), find a and b. Given polynomial is f (x) = x3  3x2 + x + 1. Let  = (a  b),  = a and  = (a + b) . Then,       3 & (a  b)  a  (a  b)  3 & 3a  3 & a  1.       1 & a (a  b)  a (a  b)  (a  b)(a  b)  1

& 3a 2  b 2  1 & (3 # 1 2)  b 2  1 & b2  2 & b  ! 2 .  EXAMPLE 4 SOLUTION

a = 1 and b = ! 2 .

1 Find a cubic polynomial whose zeros are 3, 2 and 1. 1 Let  = 3,  = 2 and  = 1. Then, 1 5 ( +  + ) = b3 + 2  1l = 2 , 4 3 1 ( +  + ) = b 2  2  3l = 2 = 2 3 1 and  = &3 # 2 # (1)0 = 2 $ Hence, the required polynomial is 5 3 x3  ( +  + )x2 + ( +  + )x   = x3  2 x2  2x + 2 $ Thus, 2x3  5x2  4x + 3 is the desired polynomial.

DIVISION ALGORITHM FOR POLYNOMIALS

If f(x) and g(x) are any two polynomials with g(x) ! 0 then we can find polynomials q(x) and r(x) such that f (x) = q(x) # g(x) + r(x), where r(x) = 0 or {degree of r(x)} < {degree of g(x)} . We may write it as Dividend = (Quotient × Divisor) + Remainder.

Polynomials

53

SOME MORE EXAMPLES EXAMPLE 5 SOLUTION

Divide 3  x + 2x2 by (2  x) and verify the division algorithm. First we write the terms of dividend and divisor in decreasing order of their degrees and then perform the division as shown below.  x + 2) 2x2  x + 3 ( 2x  3 2x2  4x 3x + 3 3x  6 9 Clearly, degree (9) = 0 < degree (x + 2). 

quotient = (2x  3) and remainder = 9



(quotient × divisor) + remainder = (2x  3)(x + 2) + 9 = 2x2  4x + 3x  6 + 9 = 2x2  x + 3 = dividend.

Thus, (quotient × divisor) + remainder = dividend. Hence, the division algorithm is verified. EXAMPLE 6

SOLUTION

Divide 5x3  13x2 + 21x  14 by (3  2x + x2) and verify the division algorithm. First we write the given polynomials in standard form in decreasing order of degrees and then perform the division as shown below. x2  2x + 3) 5x3  13x2 + 21x  14 (5x  3 5x3  10x2 + 15x 3x2 + 6x  14 3x2 + 6x  9 –5 Clearly, degree (5) = 0 < degree (x2  2x + 3) . 

quotient = (5x  3) and remainder = –5



(quotient × divisor) + remainder = (5x  3)(x2  2x + 3)  5

54

Secondary School Mathematics for Class 10

= 5x3  10x2 + 15x  3x2 + 6x  9  5 = 5x3  13x2 + 21x  14 = dividend. Thus, (quotient × divisor) + remainder = dividend. Hence, the division algorithm is verified. EXAMPLE 7

SOLUTION

What real number should be subtracted from the polynomial (3x3 + 10x2  14x + 9) so that (3x  2) divides it exactly? [CBSE 2009C] On dividing (3x3 + 10x2  14x + 9) by (3x  2), we get 3x  2) 3x3 + 10x2  14x + 9 (x2 + 4x  2 3x3  2x2 12x2  14x + 9 12x2  8x 6x + 9 6x + 4 5 Required number to be subtracted = 5.

EXAMPLE 8

If the polynomial (x4 + 2x3 + 8x2 + 12x + 18) is divided by another polynomial (x2 + 5), the remainder comes out to be (px + q) . Find the values of p and q. [CBSE 2009]

SOLUTION

Let f (x) = (x4 + 2x3 + 8x2 + 12x + 18) and g(x) = (x2 + 5). On dividing f (x) by g(x), we get x2 + 5) x4 + 2x3 + 8x2 + 12x + 18 (x2 + 2x + 3 + 5x2 x4 2x3 + 3x2 + 12x + 18 + 10x 2x3 2 3x + 2x + 18 + 15 3x2 2x + 3 Now, px + q = 2x + 3 & p = 2 and q = 3.

EXAMPLE 9

On dividing (x3  3x2 + x + 2) by a polynomial g(x), the quotient and remainder are (x  2) and ( 2x + 4) respectively. Find g(x) . [CBSE 2009C]

Polynomials SOLUTION

55

Let f (x) = (x3  3x2 + x + 2), q(x) = (x  2) and r(x) = ( 2x + 4). Then, f (x) = g(x) $ q(x) + r(x) {f (x)  r(x)} ·  g(x)  q(x) Now, {f (x)  r(x)} = (x3  3x2 + x + 2)  (2x + 4)

… (i)

= (x3  3x2 + 3x  2). (x3  3x2 + 3x  2) [using (i)] $ (x  2) On dividing (x3  3x2 + 3x  2) by (x  2), we get g(x). 

g(x) =

x  2) x3  3x2 + 3x  2 (x2  x + 1 x3  2x2 x2 + 3x  2 x2 + 2x x2 x2 × 

g(x) = (x2  x + 1) .

AN IMPORTANT NOTE

If  is a zero of the polynomial f(x) then (x  ) is a divisor of f(x). EXAMPLE 10

SOLUTION

It being given that 1 is a zero of the polynomial (7x  x3  6), find its other zeros. Let f (x) = x3 + 7x  6. Since 1 is a zero of f (x), so (x  1) is a factor of f (x). On dividing f (x) by (x  1), we get + 7x  6 ( x2  x + 6 x  1) x3 3 2 x + x  x2 + 7x  6  x2 + x 6x  6 6x  6 × 

f (x)  (x 3  7x  6)  (x  1)(x 2  x  6)  (x  1)(x 2  x  6)  (x  1)(x 2  3x  2x  6)  (1  x) [x(x  3)  2(x  3)]  (1  x)(x  3)(x  2) .

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Secondary School Mathematics for Class 10



f (x)  0 & (1  x)(x  3)(x  2)  0

& (1  x)  0 or (x  3)  0 or (x  2)  0 & x  1 or x  3 or x  2. Thus, the other zeros are –3 and 2. EXAMPLE 11

SOLUTION

Obtain all zeros of the polynomial (2x3  4x  x2 + 2), if two of its zeros are 2 and  2 . [CBSE 2008C] The given polynomial is f (x) = 2x3  x2  4x + 2. Since 2 and  2 are the zeros of f (x), it follows that each one of (x  2 ) and (x + 2 ) is a factor of f (x). Consequently, (x  2 )(x + 2 ) = (x2  2) is a factor of f (x). On dividing f (x) = 2x3  x2  4x + 2 by (x2  2), we get x2  2) 2x3  x2  4x + 2 (2x  1  4x 2x3  x2  x2 

×

+2 +2

f (x)  0 & (x  2)(2x  1)  0 2

& (x  2 ) (x  2 )(2x  1)  0 & (x  2 )  0 or (x  2 )  0 or (2x  1)  0 & x  2 or x   2 or x  12 · Hence, all zeros of f (x) are 2 ,  2 and EXAMPLE 12

SOLUTION

1· 2

If two zeros of the polynomial (x4  6x3  26x2 + 138x  35) are (2 + 3 ) and (2  3 ), find other zeros. Let f (x) = x4  6x3  26x2 + 138x  35. Let  = (2 + 3 ) and  = (2  3 ). Then, ( + ) = 4 and  = (4  3) = 1. So, the quadratic polynomial whose roots are  and  is given by x2  ( + ) x +  = (x2  4x + 1). 

(x2  4x + 1) is a factor of f(x).

On dividing f (x) by (x2  4x + 1), we get

Polynomials

57

x2  4x + 1) x4  6x3  26x2 + 138x  35 (x2  2x  35 x4  4x3 + x2  2x3  27x2 + 138x  35  2x3 + 8x2  2x  35x2 + 140x  35  35x2 + 140x  35 × 

f (x) = (x2  4x + 1) (x2  2x  35).



the other two zeros of f(x) are given by (x2  2x  35) = 0.

Now, x 2  2x  35  0 & x 2  7x  5x  35  0 & x(x  7)  5(x  7)  0 & (x  7)(x  5)  0 & x  7  0 or x  5  0 & x  7 or x  5. Hence, the other two zeros of f(x) are 7 and –5. EXAMPLE 13

SOLUTION

Obtain all zeros of (3x4  15x3 + 13x2 + 25x  30), if two of its zeros 5 5· are [CBSE 2009C] and  3 3 The given polynomial is f (x) = (3x4  15x3 + 13x2 + 25x  30). 5  5 are the zeros of f (x), it follows that each 3 and 3 5 5 one of cx  3 m and cx + 3 m is a factor of f (x).

Since



fx 

5 pfx + 3

(3x2  5) 5 5 is a factor of f (x). p = bx2  3 l = 3 3

Consequently, (3x2  5) is a factor of f (x). On dividing f(x) by (3x2  5), we get 3x2  5) 3x4  15x3 + 13x2 + 25x  30 (x2  5x + 6  5x2 3x4  15x3 + 18x2 + 25x  30  15x3 + 25x 18x2 18x2

×

 30  30

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Secondary School Mathematics for Class 10



f (x) = 3x4  15x3 + 13x2 + 25x  30 = (3x2  5) (x2  5x + 6) = ( 3 x + 5 )( 3 x  5 )(x  2)(x  3).



f (x)  0 & ( 3 x  5 )  0 or ( 3 x  5 )  0 or (x  2) = 0 or (x  3) = 0

5 or x  3 5 Hence, all zeros of f(x) are 3 , 

&x

f

5 or x  2 or x  3. 3 5 3 , 2 and 3.

EXERCISE 2B

1. Verify that 3, –2, 1 are the zeros of the cubic polynomial p(x) = x3  2x2  5x + 6 and verify the relation between its zeros and coefficients. 1 2. Verify that 5, 2 and 3 are the zeros of the cubic polynomial p(x) = 3x3  10x2  27x + 10 and verify the relation between its zeros and coefficients. 3. Find a cubic polynomial whose zeros are 2, –3 and 4. 1 4. Find a cubic polynomial whose zeros are 2 , 1 and 3. 5. Find a cubic polynomial with the sum, sum of the product of its zeros taken two at a time, and the product of its zeros as 5, –2 and –24 respectively. Find the quotient and the remainder when

6. f (x) = x3  3x2 + 5x  3 is divided by g(x) = x2  2. 7. f (x) = x4  3x2 + 4x + 5 is divided by g(x) = x2 + 1  x. 8. f (x) = x4  5x + 6 is divided by g(x) = 2  x2. 9. By actual division, show that x2  3 is a factor of 2x4 + 3x3  2x2  9x  12. 10. On dividing 3x3 + x2 + 2x + 5 by a polynomial g(x), the quotient and remainder are (3x  5) and (9x + 10) respectively. Find g(x). HINT

g(x) =

(3x3 + x2 + 2x + 5)  (9x + 10) · (3x  5)

11. Verify division algorithm for the polynomials f (x) = 8 + 20x + x2  6x3 and g(x) = 2 + 5x  3x2 . 12. It is given that –1 is one of the zeros of the polynomial x3 + 2x2  11x  12. Find all the zeros of the given polynomial.

Polynomials

59

13. If 1 and –2 are two zeros of the polynomial (x3  4x2  7x + 10), find its third zero. 14. If 3 and –3 are two zeros of the polynomial (x4 + x3  11x2  9x + 18), find all the zeros of the given polynomial. 15. If 2 and –2 are two zeros of the polynomial (x4 + x3  34x2  4x + 120), find all the zeros of the given polynomial. [CBSE 2008] 16. Find all the zeros of (x4 + x3  23x2  3x + 60), if it is given that two of its zeros are 3 and  3 . [CBSE 2009C] 17. Find all the zeros of (2x4  3x3  5x2 + 9x  3), it being given that two of its zeros are 3 and  3 . 18. Obtain all other zeros of (x4 + 4x3  2x2  20x  15) if two of its zeros are [CBSE 2009C] 5 and  5 . 19. Find all the zeros of the polynomial (2x4  11x3 + 7x2 + 13x  7), it being given that two of its zeros are (3 + 2 ) and (3  2 ). ANSWERS (EXERCISE 2B)

3. x3  3x2  10x + 24

4. 2x3 + 3x2  8x + 3

6. q(x) = (x  3), r(x) = (7x  9) 8. q(x) = x2  2, r(x) = 5x + 10 12. – 4, –1, 3 13. 5 16.

3 ,  3 , 4, 5

14. 1, 2, 3, 3 17.

1 19. (3 + 2 ), (3  2 ), 2 , 1 f

5. x3  5x2  2x + 24

7. q(x) = x2 + x  3, r(x) = 8 10. g(x) = x2 + 2x + 1 15. 2, 2, 6, 5

1 3 ,  3 , 1, 2

18.  1, 3

EXERCISE 2C

Very-Short-Answer Questions 1. If one zero of the polynomial x2  4x + 1 is (2 + 3 ), write the other zero. [CBSE 2010]

2. Find the zeros of the polynomial x2 + x  p(p + 1). 3. Find the zeros of the polynomial x2  3x  m(m + 3).

[CBSE 2011] [CBSE 2011]

4. If  are the zeros of a polynomial such that  +  = 6 and  = 4 then write the polynomial. [CBSE 2010] 2 5. If one zero of the quadratic polynomial kx + 3x + k is 2 then find the value of k. 6. If 3 is a zero of the polynomial 2x2 + x + k, find the value of k. [CBSE 2010]

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Secondary School Mathematics for Class 10

7. If – 4 is a zero of the polynomial x2  x  (2k + 2) then find the value of k. [CBSE 2009]

8. If 1 is a zero of the polynomial ax  3(a  1) x  1 then find the value of a. 9. If –2 is a zero of the polynomial 3x2 + 4x + 2k then find the value of k. 2

[CBSE 2010]

10. Write the zeros of the polynomial x  x  6. 2

[CBSE 2008]

11. If the sum of the zeros of the quadratic polynomial kx  3x + 5 is 1, write the value of k. 12. If the product of the zeros of the quadratic polynomial x2  4x + k is 3 then write the value of k. 13. If (x + a) is a factor of (2x2 + 2ax + 5x + 10), find the value of a. [CBSE 2010] 2

14. If (a  b), a and (a + b) are zeros of the polynomial 2x3  6x2 + 5x  7, write the value of a. 15. If x3 + x2  ax + b is divisible by (x2  x), write the values of a and b. 16. If  and  are the zeros of the polynomial 2x2 + 7x + 5, write the value of [CBSE 2010]  +  + . 17. State division algorithm for polynomials. 18. The sum of the zeros and the product of zeros of a quadratic polynomial 1 are 2 and –3 respectively. Write the polynomial. Short-Answer Questions 19. Write the zeros of the quadratic polynomial f (x) = 6x2  3. 20. Write the zeros of the quadratic polynomial f (x) = 4 3 x2 + 5x  2 3 . 21. If  and  are the zeros of the polynomial f (x) = x2  5x + k such that    = 1, find the value of k. 22. If  and  are the zeros of the polynomial f (x) = 6x2 + x  2, find the   value of d   n ·  23. If  and  are the zeros of the polynomial f (x) = 5x2  7x + 1, find the 1 1 value of c   m ·  24. If  and  are the zeros of the polynomial f (x) = x2 + x  2, find the value 1 1 of c   m · 

Polynomials

61

25. If the zeros of the polynomial f (x) = x3  3x2 + x + 1 are (a  b), a and (a + b), find a and b. ANSWERS (EXERCISE 2C)

1. (2  3 )

6

5. k = 5 10. 3 and –2

2. (p + 1) and p

3. (m + 3) and m

4. x2  6x + 4

6. k = 21

7. k = 9

9. k = 2

11. k = 3 16. –1

15. a = 2 and b = 0 20. x =

12. k = 3

8. a = 1 13. a = 2

1

18. x2 + 2 x  3 19. x =

25

3 2 or x = 4 21. k = 6 3

22. 12

23. 7

25. a = 1 and b = ! 2 HINTS TO SOME SELECTED QUESTIONS 1. Let the other zero be .  (coefficient of x) (4) = = 4. Then, sum of zeros = 1 (coefficient of x2)  2+

2. x

(2 

3 )    4 &   (2 

x  p (p + 1) = x

2+

3).

(p + 1) x  px  p (p + 1)

= x {x + (p + 1)}  p {x + (p + 1)} = {x + (p + 1)}{x  p} . So, its zeros are  (p + 1) and p. 3. x2  3x  m(m + 3) = x2  (m + 3) x + mx  m(m + 3) = x {x  (m + 3)} + m {x  (m + 3)} = {x  (m + 3)}{x + m} . So, its zeros are (m + 3) and m. 4. Required polynomial is x2  ( + ) x +  = x2  6x + 4. 5. Since 2 is a zero of kx2 + 3x + k, we have 4k + 6 + k = 0. 6 ·  5k  6 & k  5 6. Since 3 is a zero of 2x2 + x + k, we have 18 + 3 + k = 0 & k = 21. 7. Since –4 is a zero of x2  x  (2k + 2), we have 16 + 4  2k  2 = 0. 

2k = 18 & k = 9.

8. Since 1 is a zero of ax2  3 (a  1) x  1, we have a  3 (a  1)  1 = 0.  a  3a  3  1  0 & 2a  2 & a  1. 9. Since –2 is a zero of 3x2 + 4x + 2k, we have 12  8 + 2k = 0. 

2k =  4 & k = 2.

14. a = 1

1 1 or x = 2 2 3 24. 2

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Secondary School Mathematics for Class 10

10. x2  x  6 = x2  3x + 2x  6 = x (x  3) + 2 (x  3) = (x  3) (x + 2) . So, its zeros are 3 and –2. 11. Sum of the zeros  

(coefficient of x) (coefficient of x 2)

 3· k

3  1 & k  3. k

constant term  k · So, k = 3. coefficient of x 2 1 13. Let f (x) = 2x2 + 2ax + 5x + 10. Then, f ( a) = 0. 12. Product of the zero 



2a 2  2a 2  5a  10  0 & 5a  10 & a  2. (coefficient of x2) ( 6) = = 3. 2 (coefficient of x3) (a  b)  a  (a  b)  3 & 3a  3 & a  1.

14. ( +  + ) = 

15. On dividing x3 + x2  ax + b by x2  x, we get x2  x ) x3 + x2  ax + b (x + 2 x3  x2 2x2  ax 2x2  2x (2  a) x + b 

(2  a  0 and b  0) & a  2 and b  0. 7 5 and   · 2 2 (7 + 5) 2 7 5 = = 1.  +  +  = b 2 + 2 l = 2 2

16. Clearly,     

1 18.  +  = 2 and  = 3. 1 So, the polynomial is x2  ( + ) x +  = x2 + 2 x  3. 19. 6x2  3 = 3 (2x2  1) = 3 ( 2 x  1)( 2 x + 1) . 1 1 · So, its zeros are and 2 2 20. f (x) = 4 3 x2 + 5x  2 3 = 4 3 x2 + 8x  3x  2 3 = 4x ( 3 x + 2)  3 ( 3 x + 2) = ( 3 x + 2)(4x  3 ) . 2 3 · , 3 4 21. ( + ) = 5 and  = k. Also,    = 1 (given). So, its zeros are



(  ) 2  (  ) 2  4 & 5 2  1 2  4



4  (25  1)  24 &   6.

Hence, k = 6.

Polynomials

63

1 2 1 22.  +  = 6 and  = 6 = 3 · 

1 +2 (2 + 2) ( + ) 2  2 b 36 3 l   d + n = = = = b 25 # 3 l = 25 · 36 1 12      b 1l 3

7 1 23.  +  = 5 and  = 5 $ ( + ) 1 1 7 5  c  +  m =  = b 5 # 1 l = 7. 24.  +  = 1 and  = 2. ( + ) 2  (  ) 2 = 4 

(  ) 2 = ( + ) 2  4 = (1) 2  4 # (2) = 9



   = 3.



1  1 m = (  ) = 3 = 3 · c 2 2  

25.  +  +  = 3,  +  +  = 1 and  = 1 

(a  b)  a  (a  b)  3 & 3a  3 & a  1

and (a  b)# a #(a  b)  1 & a (a 2  b 2)  1

& 1  b 2  1 & b 2  2 & b  ! 2 .

MULTIPLE-CHOICE QUESTIONS (MCQ) Choose the correct answer in each of the following questions: 1. Which of the following is a polynomial? (a) x2  5x + 4 x + 3 1 (c) x + x

(b) x3/2  x + x1/2 + 1 (d)

2 x2  3 3 x + 6

2. Which of the following is not a polynomial? 3 x2  2 3 x + 5 3 1  (c) 2 x3 + 6x2  x 8 2

(b) 9x2  4x + 2 3 (d) x + x

(a)

3. The zeros of the polynomial x2  2x  3 are (a) –3, 1

(b) –3, –1

(c) 3, –1

(d) 3, 1

4. The zeros of the polynomial x  2 x  12 are 2

(a)

2,  2

(b) 3 2 , 2 2

(c) 3 2 , 2 2

(d) 3 2 , 2 2

5. The zeros of the polynomial 4x2 + 5 2 x  3 are (a) 3 2 , 2

2 (b) 3 2 , 2

(c)

3 2 2 2 , 4

(d) none of these

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Secondary School Mathematics for Class 10

1 6. The zeros of the polynomial x2 + 6 x  2 are 4 3 3 4 (a) –3, 4 (b) 2 , 3 (c) 3 , 2 (d) none of these 11x 2 7. The zeros of the polynomial 7x2  3  3 are 2 1 2 1 2 1 (b) 7 , 3 (c) 3 , 7 (d) none of these (a) 3 , 7 8. The sum and the product of the zeros of a quadratic polynomial are 3 and –10 respectively. The quadratic polynomial is (a) x2  3x + 10 (b) x2 + 3x  10 (c) x2  3x  10 (d) x2 + 3x + 10 9. A quadratic polynomial whose zeros are 5 and –3, is (a) x2 + 2x  15 (b) x2  2x + 15 (c) x2  2x  15 (d) none of these 1 3 10. A quadratic polynomial whose zeros are 5 and 2 , is (a) 10x2 + x + 3

(b) 10x2 + x  3

(c) 10x2  x + 3 (d) 10x2  x  3

11. The zeros of the quadratic polynomial x2 + 88x + 125 are (a) both positive

(b) both negative

(c) one positive and one negative (d) both equal 12. If  and  are the zeros of x2 + 5x + 8 then the value of ( + ) is (a) 5

(b) –5

(c) 8

(d) –8

13. If  and  are the zeros of 2x + 5x  9 then the value of  is 5 9 5 9 (a) 2 (b) 2 (c) 2 (d) 2 2

14. If one zero of the quadratic polynomial kx2 + 3x + k is 2 then the value of k is 6 5 5 6 (b) 6 (c) 5 (d) 5 (a) 6 15. If one zero of the quadratic polynomial (k  1) x2 + kx + 1 is – 4 then the value of k is 5 4 4 5 (b) 4 (c) 3 (d) 3 (a) 4 16. If –2 and 3 are the zeros of the quadratic polynomial x2 + (a + 1) x + b then (a) a = 2, b = 6 (b) a = 2, b = 6 (c) a = 2, b = 6 (d) a = 2, b = 6 17. If one zero of 3x2 + 8x + k be the reciprocal of the other then k = ? 1 1 (a) 3 (b) –3 (c) 3 (d) 3

Polynomials

65

18. If the sum of the zeros of the quadratic polynomial kx2 + 2x + 3k is equal to the product of its zeros then k = ? 2 1 2 1 (a) 3 (b) 3 (c) 3 (d) 3 1 1 19. If  are the zeros of the polynomial x2 + 6x + 2 then c  +  m = ? (a) 3 (b) –3 (c) 12 (d) –12 3  20. If    are the zeros of the polynomial x 6x2  x + 30 then ( +  + ) = ? (a) –1 (b) 1 (c) –5 (d) 30 3 2 21. If  are the zeros of the polynomial 2x + x  13x + 6 then  = ? 13 1 (c) 2 (d) 2 22. If    be the zeros of the polynomial p(x) such that ( +  + ) = 3, ( +  + ) = 10 and  = 24 then p(x) = ? (a) –3

(b) 3

(a) x3 + 3x2  10x + 24 (c) x3  3x2  10x + 24

(b) x3 + 3x2 + 10x  24 (d) None of these 23. If two of the zeros of the cubic polynomial ax3 + bx2 + cx + d are 0 then the third zero is b d b c (a) a (b) a (c) a (d) a 24. If one of the zeros of the cubic polynomial ax3 + bx2 + cx + d is 0 then the product of the other two zeros is c b c (a) a (b) a (c) 0 (d) a 25. If one of the zeros of the cubic polynomial x3 + ax2 + bx + c is –1 then the product of the other two zeros is (a) a  b  1 (b) b  a  1 (c) 1  a + b (d) 1 + a  b 26. If   be the zeros of the polynomial 2x2 + 5x + k such that 21 2 + 2 +  = 4 then k = ? (a) 3 (b) –3 (c) –2 (d) 2 27. On dividing a polynomial p(x) by a nonzero polynomial q(x), let g(x) be the quotient and r(x) be the remainder then p(x) = q(x) $ g(x) + r(x), where (a) r(x) = 0 always (b) deg r(x) < deg g(x) always (c) either r(x) = 0 or deg r(x) < deg g(x) (d) r(x) = g(x)

66

Secondary School Mathematics for Class 10

28. Which of the following is a true statement? (a) x2 + 5x  3 is a linear polynomial. (b) x2 + 4x  1 is a binomial. (c) x + 1 is a monomial. (d) 5x3 is a monomial. ANSWERS (MCQ)

1. 10. 19. 28.

4. (b) 5. (c) 6. (b) 7. (a) 8. (c) 9. (c) (d) 2. (d) 3. (c) (d) 11. (b) 12. (b) 13. (c) 14. (d) 15. (b) 16. (c) 17. (a) 18. (d) (b) 20. (a) 21. (a) 22. (c) 23. (a) 24. (b) 25. (c) 26. (d) 27. (c) (d) HINTS TO SOME SELECTED QUESTIONS

1. Clearly, 2 x2  3 3 x + 6 is a polynomial. 3 2. Clearly, x + x is not a polynomial. 3. x2  2x  3 = x2  3x + x  3 = x (x  3) + (x  3) = (x  3)(x + 1) . (x  3)(x  1)  0 & x  3 or x  1.



4. x2  2 x  12 = x2  3 2 x + 2 2 x 12 = x (x  3 2 ) + 2 2 (x  3 2 ) = (x  3 2 )(x + 2 2 ) . x = 3 2 or x = 2 2 .

 2+

5. 4x

5 2 x  3 = 4x2 + 6 2 x  2 x  3 = 2 2 x ( 2 x + 3)  ( 2 x + 3) = ( 2 x + 3)(2 2 x  1) .



x

2 3 2 2 2 3 1 ·   # # or x  2 4 2 2 2 2 2

6x 2  x  12 · 1 6. x 2  x  2  6 6 6x2 + x  12 = 6x2 + 9x  8x  12 = 3x (2x + 3)  4 (2x + 3) = (2x + 3)(3x  4) 

3 4 the zeros are 2 and · 3

11x 2 21x2  11x  2 · 7. 7x2  3  3 = 3 Now, 21x2  11x  2 = 21x2  14x + 3x  2



= 7x (3x  2) + (3x  2) = (3x  2)(7x + 1) . 2 1 · the zeros are , 3 7

Polynomials 8. The required polynomial is x2  ( + ) x +  = x2  3x  10. 9.  +  = 5 + (3) = 2,  = 5 # (3) = 15. Required polynomial is x2  2x  15. 3 1 3 1 3 1 · 10.     a  k  ,   # a k  5 2 5 2 10 10 1 3 Required polynomial is x2  10 x  10 , i.e., 10x2  x  3. 11. Let  and  be the zeros of the given polynomial. Then,  +  = 88 and  = 125. This happens when  and  are both negative. b 12. For ax2 + bx + c, we have     a · For x2 + 5x + 8, we have  +  =  5. c 13. For ax2 + bx + c, we have   a · 9 · For 2x2 + 5x  9, we have   2 14. x = 2 satisfies kx2 + 3x + k = 0. 

4k  6  k  0 & 5k  6 & k 

6 · 5

15. x =  4 satisfies (k  1) x2 + kx + 1 = 0. 

16 (k  1)  4k  1  0 & 12k  15 & k 

5· 4

16.  +  = 3 + (2) = 1 and  = 3 # (2) = 6. 

(a  1)  1 & a  1  1 & a  2.

Also, b = 6. k 1 & k  3. 3  2 3k 2    3 & k  2· 18.      & & 3 k k k  = = + 19.   6 and  2. ( + ) 6 1 1  c  +  m =  = 2 = 3. 17.   1 &

For ax3 + bx2 + cx + d, we have b d c (    )  a , (    )  a and   a · c 1 20. ( +  + ) = a = 1 = 1.  d 6 21.  = a = 2 = 3. Note

22. p(x) = x3  ( +  + ) x2 + ( +  + ) x   = x3  3x2  10x + 24.

67

68

Secondary School Mathematics for Class 10

23. Let , 0, 0 be the zeros of ax3 + bx2 + cx + d. Then, b b b sum of zeros  a &   0  0  a &   a · b Hence, the third zero is a · 24. Let , , 0 be the zeros of ax3 + bx2 + cx + d. Then, sum of the products of zeros, taken two at a time is given by c c (   # 0   # 0)  a &   a · c  the product of the other two zeros is a · 25. Since –1 is a zero of x3 + ax2 + bx + c, we have (1) 3  a #(1) 2  b #(1)  c  0 & a  b  c  1  0 & c  1  a  b. Also, product of all zeros is given by  #(1)  c &   c &   1  a  b. 5 k 26.     and   · 2 2 21 21  2   2    & (  ) 2    4 4 

5 k 21 &a 2 k  2  4 2

25 21 k & 2  a  k  1 & k  2. 4 4

SUMMARY OF RESULTS 1.

I. Polynomial: An expression of the form p(x) = a0 + a1 x + a2 x2 + … + an xn, where an ! 0, is called a polynomial in x of degree n. II. A polynomial is said to be linear, quadratic, cubic and biquadratic according as its degree is 1, 2, 3 and 4 respectively. General form:

(i) linear equation: ax + b. (ii) quadratic equation: ax2 + bx + c. (iii) cubic equation: ax3 + bx2 + cx + d. (iv) biquadratic equation: ax4 + bx3 + cx2 + dx + e. III. Value of a polynomial at a point: The value of a polynomial p(x) at x =  is obtained by putting x =  and it is denoted by p(). IV. Zeros of a polynomial: A real number  is called a zero of p(x), if p() = 0.

Polynomials

2.

69

I. If  are the zeros of p(x)  ax 2  bx  c, a ! 0 then b (i)  +  = a ,

c

(ii)  = a ·

II. If  and  are the zeros of a quadratic polynomial p(x) then p(x) = {x2  ( + )x + } .

3.

I. If  are the zeros of p(x)  ax 3  bx 2  cx  d, a ! 0 then d b c (i)  +  +  = a , (ii)  +  +  = a , (iii)  = a · II. If  and  be the zeros of a polynomial p(x) then p(x) = x3  ( +  + ) x2 + ( +  + ) x  .

TEST YOURSELF MCQ 1. Zeros of p (x) = x2  2x  3 are (a) 1, –3

(b) 3, –1

(c) –3, –1 (d) 1, 3 3 2 2. If  are the zeros of the polynomial x  6x  x + 30 then the value of ( +  + ) is (a) –1

(b) 1

(c) –5

(d) 30

3. If  are the zeros of kx  2x + 3k such that  +  =  then k = ? 2 1 2 1 (a) 3 (b) 3 (c) 3 (d) 3 4. It is given that the difference between the zeros of 4x2  8kx + 9 is 4 and k > 0. Then, k = ? 3 5 7 1 (a) 2 (b) 2 (c) 2 (d) 2 Short-Answer Questions 5. Find the zeros of the polynomial x2 + 2x  195. 2

6. If one zero of the polynomial (a2 + 9) x2 + 13x + 6a is the reciprocal of the other, find the value of a. 7. Find a quadratic polynomial whose zeros are 2 and –5. 8. If the zeros of the polynomial x3  3x2 + x + 1 are (a  b), a and (a + b), find the values of a and b. 9. Verify that 2 is a zero of the polynomial x3 + 4x2  3x  18. 10. Find the quadratic polynomial, the sum of whose zeros is –5 and their product is 6.

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Secondary School Mathematics for Class 10

11. Find a cubic polynomial whose zeros are 3, 5 and –2. 12. Using remainder theorem, find the remainder 3 2   = + + p(x) x 3x 5x 4 is divided by (x 2) . 13. Show that (x + 2) is a factor of f (x) = x3 + 4x2 + x  6.

when

14. If  are the zeros of the polynomial p(x) = 6x3 + 3x2  5x + 1, find the 1 1 1 value of c     m ·  15. If   are the zeros of the polynomial f (x) = x2  5x + k such that    = 1, find the value of k. 16. Show that the polynomial f (x) = x4 + 4x2 + 6 has no zero. Long-Answer Questions 17. If one zero of the polynomial p(x) = x3  6x2 + 11x  6 is 3, find the other two zeros. 18. If two zeros of the polynomial p(x) = 2x4  3x3  3x2 + 6x  2 are 2 and  2 , find its other two zeros. 19. Find the quotient when p(x) = 3x4 + 5x3  7x2 + 2x + 2 is divided by (x2 + 3x + 1) . 20. Use remainder theorem to find the value of k, it being given that when x3 + 2x2 + kx + 3 is divided by (x  3) then the remainder is 21. ANSWERS (TEST YOURSELF)

1. (a)

2. (a) + 7. x 3x  10 2

12. 14

3. (c)

4. (c) = 8. a 1, b = ! 2

14. 5

15. k = 6

5. –15, 13

6. a = 3

10. x + 5x + 6

11. x3  6x2  x + 30

17. 1, 2

18. 1, 2

2

19. 3x2  4x + 2 20. k = 9



1

Linear Equations in Two Variables

71

LINEAR EQUATIONS IN TWO VARIABLES

An equation of the form ax + by + c = 0, where a, b, c are real numbers (a ! 0, b ! 0), is called a linear equation in two variables x and y. Examples

Each of the equations (i) 3x  4y + 2 = 0

(ii) 2x + 5y = 9

(iii) 0.4x + 0.3y = 2.7

(iv)

2x 3y = 0

is a linear equation in x and y. SOLUTION OF A LINEAR EQUATION

We say that x =  and y =  is a solution of ax + by + c = 0 if a + b + c = 0. EXAMPLE

Show that x = 3 and y  2 is a solution of 5x  3y  9.

SOLUTION

Substituting x = 3 and y = 2 in the given equation, we get LHS = 5 # 3  3 # 2 = (15  6) = 9 = RHS.



x = 3 and y = 2 is a solution of 5x  3y = 9.

SIMULTANEOUS LINEAR EQUATIONS IN TWO VARIABLES

Two linear equations in two unknowns x and y are said to form a system of simultaneous linear equations if each of them is satisfied by the same pair of values of x and y. Example

Consider the system of linear equations x + y = 10, x  y = 2. By substitution, you will find that each of these equations is satisfied by the values x = 6 and y = 4. Hence, the given equations form a system of simultaneous linear equations in x and y.

SOLUTION OF A GIVEN SYSTEM OF TWO SIMULTANEOUS EQUATIONS

A pair of values of x and y satisfying each of the equations in a given system of two simultaneous equations in x and y is called a solution of the system. 71

72 EXAMPLE 1

Secondary School Mathematics for Class 10

Show that x = 5, y = 2 is a solution of the system of linear equations 2x + 3y = 16, x  2y = 1.

SOLUTION

The given equations are 2x + 3y = 16

… (i)

x  2y = 1.

… (ii)

Putting x = 5 and y = 2 in (i), we get LHS = (2 # 5 + 3 # 2) = 16 = RHS.

Putting x = 5 and y = 2 in (ii), we get LHS = (5  2 # 2) = 1 = RHS.

Thus, x = 5 and y = 2 satisfy both (i) and (ii). Hence, x = 5, y = 2 is a solution of the given system of equations. EXAMPLE 2

Show that x = 3, y = 2 is not a solution of the system of linear equations 3x  2y = 5, 2x + y = 7.

SOLUTION

The given equations are 3x  2y = 5

… (i)

2x + y = 7.

… (ii)

Putting x = 3 and y = 2 in (i), we get LHS = (3 # 3  2 # 2) = 5 = RHS.

Putting x = 3 and y = 2 in (ii), we get LHS = (2 # 3 + 2) = 8 ! RHS.

Thus, the values x = 3, y = 2 do not satisfy (ii). Hence, x = 3, y = 2 is not a solution of the given system of equations. CONSISTENT AND INCONSISTENT SYSTEMS OF LINEAR EQUATIONS CONSISTENT SYSTEM OF LINEAR EQUATIONS

A system of two linear equations in two unknowns is said to be consistent if it has at least one solution. INCONSISTENT SYSTEM OF LINEAR EQUATIONS

A system of two linear equations in two unknowns is said to be inconsistent if it has no solution at all.

Linear Equations in Two Variables Example

73

Consider the system of linear equations x + y = 3, 2x + 2y = 7. Clearly, we cannot find values of x and y which may satisfy both the given equations simultaneously. Hence, the given system is inconsistent.

SOLVING SIMULTANEOUS LINEAR EQUATIONS (GRAPHICAL METHOD) METHOD

Let the given system of linear equations be a1 x + b1 y + c1 = 0 a2 x + b2 y + c2 = 0.

… (i) … (ii)

On the same graph paper, we draw the graph of each one of the given linear equations. Each such graph is always a straight line. Let the lines L1 and L2 represent the graphs of (i) and (ii) respectively. Now, the following cases arise. Case I

When the lines L1 and L2 intersect at a point Let the graph lines L1 and L2 intersect at a point P (, ), as shown in the adjoining figure. Then, x = , y =  is the unique solution of the given system of equations.

Case II

When the lines L1 and L2 are coincident When the two graph lines L1 and L2 coincide, the given system of equations has infinitely many solutions.

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Secondary School Mathematics for Class 10

Case III When the lines L1 and L2 are parallel

In this case, there is no common solution of the given system of equations. Thus, in this case, the given system is inconsistent.

SUMMARY

A system of two linear equations in x and y has (i) a unique solution if the graph lines intersect at a point (ii) infinitely many solutions if the two graph lines coincide (iii) no solution if the two graph lines are parallel

SOLVED EXAMPLES GRAPHS OF EQUATIONS HAVING UNIQUE SOLUTIONS Note

Such graph lines intersect at a point.

EXAMPLE 1

Solve graphically the system of linear equations x + 2y = 3, 4x + 3y = 2.

SOLUTION

On a graph paper, draw a horizontal line X lOX and a vertical line YOYl as the x-axis and the y-axis respectively. Graph of x + 2y = 3

x  2y  3 & 2y  (3  x) (3  x) & y 2 ·

… (i)

Putting x = 3 in (i), we get y = 3. Putting x = 1 in (i), we get y = 2. Putting x = 1 in (i), we get

y = 1.

Thus, we have the following table for the equation x + 2y = 3. x

–3

–1

1

y

3

2

1

Now, plot the points A(3, 3), B (1, 2) and C(1, 1) on the graph paper.

Linear Equations in Two Variables

75

Join AB and BC to get the graph line ABC. Extend it on both ways. Thus, the line ABC is the graph of x + 2y = 3. Graph of 4x + 3y = 2

4x  3y  2 & 3y  (2  4x) (2  4x) & y 3 · Putting x = 4 in (ii), we get y = 6.

… (ii)

Putting x = 1 in (ii), we get y = 2. y = 2.

Putting x = 2 in (ii), we get

Thus, we have the following table for the equation 4x + 3y = 2. x

–4

–1

2

y

6

2

–2

Now, on the same graph paper as above, plot the points P(4, 6) and Q (2, 2). The point B(1, 2) has already been plotted. Join PB and BQ to get the line PBQ. Extend it on both ways. Thus, the line PBQ is the graph of 4x + 3y = 2.

The two graph lines intersect at the point B(1, 2) .  x = 1 and y = 2 is the solution of the given system of equations. EXAMPLE 2

Solve graphically the system of linear equations 4x  5y + 16 = 0 and 2x + y  6 = 0.

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Secondary School Mathematics for Class 10

Determine the vertices of the triangle formed by these lines and the x-axis. [CBSE 2006] SOLUTION

On a graph paper, draw a horizontal line X l OX and a vertical line YOYl as the x-axis and the y-axis respectively. Graph of 4x  5y + 16 = 0

4x  5y  16  0 & 5y  4x  16 (4x  16) · & y 5

… (i)

Putting x = 4 in (i), we get y = 0. Putting x = 1 in (i), we get

y = 4.

Putting x = 6 in (i), we get

y = 8.

Thus, we have the following table for 4x  5y + 16 = 0. x

–4

1

6

y

0

4

8

Now, plot the points A(4, 0), B(1, 4) and C(6, 8) on the graph paper. Join AB and BC to get the graph line ABC. Extend it on both ways. Thus, the line ABC is the graph of 4x  5y + 16 = 0. Graph of 2x + y  6 = 0

2x  y  6  0 & y  (6  2x) .

… (ii)

Putting x = 0 in (ii), we get y = 6. Putting x = 2 in (ii), we get y = 2. Putting x = 3 in (ii), we get y = 0. Thus, we have the following table for 2x + y  6 = 0. x

0

2

3

y

6

2

0

On the same graph paper as above, plot the points P(0, 6), Q(2, 2) and R(3, 0) . Join PQ and QR to get the graph line PQR. Extend it on both ways. Thus, the line PQR is the graph of 2x + y  6 = 0.

Linear Equations in Two Variables

77

The two graph lines ABC and PQR intersect at the point B(1, 4) .  x = 1 and y = 4 is the solution of the given system of equations. These lines form 3 BAR with the x-axis, whose vertices are B(1, 4), A(4, 0) and R(3, 0) . EXAMPLE 3

Solve the following system of linear equations graphically: 4x  5y  20 = 0 and 3x + 5y  15 = 0. Determine the vertices of the triangle formed by the lines representing the above equations and the y-axis. [CBSE 2004]

SOLUTION

On a graph paper, draw a horizontal line X lOX and a vertical line YOYl as the x-axis and the y-axis respectively. Graph of 4x  5y  20 = 0

4x  5y  20  0 & 5y  (4x  20) (4x  20) · & y 5 Putting x = 0 in (i), we get y = 4.

… (i)

Putting x = 2 in (i), we get y = 2.4. Putting x = 5 in (i), we get y = 0. Thus, we have the following table for 4x  5y  20 = 0. x

0

2

5

y

–4

–2.4

0

78

Secondary School Mathematics for Class 10

Now, plot the points A(0, 4), B(2, 2.4) and C (5, 0) on the graph paper. Join AB and BC to get the graph line ABC. Extend it on both ways. Thus, the line ABC is the graph of 4x  5y  20 = 0. Graph of 3x + 5y  15 = 0

3x  5y  15  0 & 5y  (15  3x) (15  3x) · 5 Putting x = 5 in (ii), we get y = 6.

& y

Putting x = 0 in (ii), we get

y = 3.

Putting x = 5 in (ii), we get

y = 0.

… (ii)

Thus, we have the following table for 3x + 5y  15 = 0. x

–5

0

5

y

6

3

0

On the same graph paper as above, plot the points P(5, 6) and Q(0, 3) . The third point C(5, 0) has already been plotted. Join PQ and QC to get the graph line PQC. Extend it on both ways.

Linear Equations in Two Variables

79

Thus, the line PQC is the graph of 3x + 5y  15 = 0. The two graph lines intersect at the point C(5, 0) . 

x = 5, y = 0 is the solution of the given system of equations.

Clearly, the given equations are represented by the graph lines ABC and PQC respectively. The vertices of 3 AQC formed by these lines and the y-axis are A(0, 4), Q(0, 3) and C(5, 0) . EXAMPLE 4

Solve the following system of equations graphically: 3x + 2y  11 = 0 and 2x  3y + 10 = 0.

[CBSE 2006C]

Shade the region bounded by these lines and the x-axis. SOLUTION

On a graph paper, draw a horizontal line X l OX and a vertical line YOYl as the x-axis and the y-axis respectively. Graph of 3x + 2y  11 = 0

3x  2y  11  0 & 2y  (11  3x) (11  3x) · & y 2 Putting x = 1 in (i), we get y = 7. Putting x = 1 in (i), we get

y = 4.

Putting x = 3 in (i), we get

y = 1.

… (i)

Thus, we have the following table for 3x + 2y  11 = 0. x

–1

1

3

y

7

4

1

Now, plot the points A(1, 7), B(1, 4) and C(3, 1) on the graph paper. Join AB and BC to obtain the graph line ABC. Extend it on both ways. Thus, the line ABC is the graph of the equation 3x + 2y  11 = 0. Graph of 2x  3y + 10 = 0

2x  3y  10  0 & 3y  2x  10 & y  Putting x = 2 in (ii), we get y = 2. Putting x = 1 in (ii), we get

y = 4.

Putting x = 4 in (ii), we get y = 6.

(2x  10) · 3

… (ii)

80

Secondary School Mathematics for Class 10

Thus, we have the following table for 2x  3y + 10 = 0. x

–2

1

4

y

2

4

6

On the same graph paper as above, plot the points P(2, 2) and Q(4, 6) . The third point B(1, 4) has already been plotted. Now, join PB and BQ to obtain the graph line PBQ. Extend it on both ways. Thus, the line PBQ is the graph of the equation 2x  3y + 10 = 0.

The two graph lines intersect at the point B(1, 4) . 

x = 1, y = 4 is the solution of the given system of equations.

These graph lines intersect the x-axis at R and S. The region bounded by these lines and the x-axis has been shaded. The shaded region is the 3 BRS with B(1, 4), R(–5, 0) and 11 Sa 3 , 0 k $ EXAMPLE 5

Solve the following system of linear equations graphically: 3x + y  11 = 0, x  y  1 = 0. Shade the region bounded by these lines and the y-axis. Find the coordinates of the points where the graph lines cut the y-axis. [CBSE 2002C]

Linear Equations in Two Variables SOLUTION

81

On a graph paper, draw a horizontal line X l OX and a vertical line YOYl as the x-axis and the y-axis respectively. Graph of 3x + y  11 = 0

3x  y  11  0 & y  (11  3x) .

… (i)

Putting x = 2 in (i), we get y = 5. Putting x = 3 in (i), we get y = 2. Putting x = 5 in (i), we get y = 4. Thus, we have the following table for equation (i). x

2

3

5

y

5

2

–4

On the graph paper, plot the points A(2, 5), B(3, 2) and C(5, 4). Join AB and BC to get the graph line ABC. Thus, the line ABC is the graph of the equation 3x + y  11 = 0. Graph of x  y  1 = 0

x  y  1  0 & y  (x  1) .

… (ii)

Putting x = 3 in (ii), we get y = 4. y = 1.

Putting x = 0 in (ii), we get

Putting x = 3 in (ii), we get y = 2. Thus, we have the following table for equation (ii). x

–3

0

3

–y

–4

–1

2

On the same graph paper as above, plot the points P(3, 4) and Q(0, 1). The third point B(3, 2) is already plotted. Join PQ and QB to get the graph line PQB. Thus, line PQB is the graph of the equation x  y  1 = 0. The two graph lines intersect at the point B(3, 2). 

x = 3, y = 2 is the solution of the given system of equations.

The region bounded by these lines and the y-axis has been shaded.

82

Secondary School Mathematics for Class 10

On extending the graph lines on both sides, we find that these graph lines intersect the y-axis at the points Q(0, 1) and R(0, 11) . GRAPH OF EQUATIONS HAVING INFINITELY MANY SOLUTIONS Note

Such graph lines just coincide.

EXAMPLE 6

Show graphically that the system of equations 3x  y = 2, 9x  3y = 6 has an infinite number of solutions.

SOLUTION

On a graph paper, draw a horizontal line X l OX and a vertical line YOYl as the x-axis and the y-axis respectively. Graph of 3x  y = 2

3x  y  2 & y  (3x  2) . Putting x = 1 in (i), we get y = 5.

… (i)

Linear Equations in Two Variables

83

Putting x = 0 in (i), we get y = 2. Putting x = 2 in (i), we get y = 4. Thus, we have the following table for 3x  y = 2. x

–1

0

2

y

–5

–2

4

Now, plot the points A(1, 5), B (0, 2) and C(2, 4) on the graph paper. Join AB and BC to get the graph line ABC. Extend the graph line ABC on both sides. Thus, the line ABC is the graph of the equation 3x  y = 2. Graph of 9x  3y = 6

9x  3y  6 & 3y  (9x  6) (9x  6) · 3 Putting x = 2 in (ii), we get y = 8. Putting x = 1 in (ii), we get y = 1. Putting x = 2 in (ii), we get y = 4.

& y

Thus, we have the following table for 9x  3y = 6. x

–2

1

2

y

–8

1

4

Now, plot the points P(2, 8) and Q(1, 1) on the same graph paper. The point C(2, 4) has already been plotted. Join PQ and QC to obtain the line PQC. Thus, the line PQC is the graph of 9x  3y = 6. Thus, we find that the two graph lines coincide. Hence, the given system of equations has an infinite number of solutions.

… (ii)

84

Secondary School Mathematics for Class 10

GRAPH OF EQUATIONS HAVING NO SOLUTION Note

Such graph lines are parallel.

EXAMPLE 7

Show graphically that the system of linear equations 2x  3y = 5, 6y  4x = 3 is inconsistent, i.e., has no solution.

SOLUTION

On a graph paper, draw a horizontal line X l OX and a vertical line YOYl as the x-axis and the y-axis respectively. Graph of 2x  3y = 5

2x  3y  5 & 3y  (2x  5) (2x  5) & y 3 · Putting x = 2 in (i), we get y = 3. Putting x = 1 in (i), we get

… (i)

y = 1.

Putting x = 4 in (i), we get y = 1. Thus, we have the following table for the equation 2x  3y = 5. x

–2

1

4

y

–3

–1

1

Now, plot the points A(2, 3), B(1, 1) and C(4, 1) on the graph paper. Join AB and BC to get the graph line ABC. Extend it on both ways. Thus, the line ABC is the graph of the equation 2x  3y = 5. Graph of 6y  4x = 3

6y  4x  3 & 6y  (3  4x) (3  4x) & y 6 ·

… (ii)

3 Putting x = 3 in (ii), we get y = 2 $ 1 Putting x = 0 in (ii), we get y = 2 $ 5 Putting x = 3 in (ii), we get y = 2 $ Thus, we have the following table for the equation 6y  4x = 3.

Linear Equations in Two Variables

x

–3

0

3

y

3 2

1 2

5 2

85

3 1 5 Now, plot the points P a 3, 2 k, Q a0, 2 k and R a 3, 2 k on the same graph paper as above. Join PQ and QR to get the graph line PQR. Extend it on both ways. Then, the line PQR is the graph of the equation 6y  4x = 3.

It is clear from the graph that the two graph lines are parallel and do not intersect even when produced. Hence, the given system of equations has no solution. EXAMPLE 8

Show graphically that the system of linear equations x  y = 8, 3x  3y = 16 is inconsistent, i.e., has no solution.

SOLUTION

On a graph paper, draw a horizontal line X l OX and a vertical line YOYl as the x-axis and y-axis respectively. Graph of x  y = 8

The first equation is x  y = 8. Now, x  y  8 & y  x  8. Putting x = 3 in (i), we get y = 5. Putting x = 4 in (i), we get y = 4. Putting x = 5 in (i), we get y = 3.

… (i)

86

Secondary School Mathematics for Class 10

Thus, we have the following table for the equation x  y = 8. x

3

4

5

y

–5

–4

–3

Now, we plot the points A (3, 5), B (4, 4) and C(5, 3) on the graph paper. Join AB and BC to get the graph line ABC. Graph of 3x  3y = 16

The second equation is 3x  3y = 16. (3x  16) · … (ii) 3 10 1 Putting x = 2 in (ii), we get y = 3 =  3 3 = 3.3. 7 1 Putting x = 3 in (ii), we get y = 3 = 2 3 = 2.3. 16 1 Putting x = 0 in (ii), we get y = 3 = 5 3 = 5.3. Thus, we have the following table for the equation 3x  3y = 16.

3x  3y  16 & 3y  3x  16 & y 

x

3

2

0

y

–2.3

–3.3

–5.3

Now, we plot D(3, 2.3), E(2, 3.3) and F(0, 5.3) on the same graph paper as above. Join DE and EF to get the graph line DEF.

Linear Equations in Two Variables

87

It is clear from the graph that the lines ABC and DEF are parallel and do not meet when produced. Hence, the given system of equations has no solution and therefore, it is inconsistent.

f

EXERCISE 3A

Solve each of the following systems of equations graphically:

1. 2x + 3y = 2, x  2y = 8.

[CBSE 2007]

2. 3x + 2y = 4, 2x  3y = 7.

3. 2x + 3y = 8, x  2y + 3 = 0.

[CBSE 2005]

4. 2x  5y + 4 = 0, 2x + y  8 = 0.

[CBSE 2005]

5. 3x + 2y = 12, 5x  2y = 4.

[CBSE 2006]

6. 3x + y + 1 = 0, 2x  3y + 8 = 0.

[CBSE 2007C]

8. 2x  3y + 13 = 0, 3x  2y + 12 = 0.

7. 2x + 3y + 5 = 0, 3x  2y  12 = 0. 9. 2x + 3y  4 = 0, 3x  y + 5 = 0.

[CBSE 2004C]

[CBSE 2006C]

10. x + 2y + 2 = 0, 3x + 2y  2 = 0.

Solve each of the following given systems of equations graphically and find the vertices and area of the triangle formed by these lines and the x-axis:

11. x  y + 3 = 0, 2x + 3y  4 = 0. 12. 2x  3y + 4 = 0, x + 2y  5 = 0. 13. 4x  3y + 4 = 0, 4x + 3y  20 = 0. 14. x  y + 1 = 0, 3x + 2y  12 = 0.

[CBSE 2005] [CBSE 2008] [CBSE 2002]

15. x  2y + 2 = 0, 2x + y  6 = 0. Solve each of the following given systems of equations graphically and find the vertices and area of the triangle formed by these lines and the y-axis:

16. 2x  3y + 6 = 0, 2x + 3y  18 = 0. 17. 4x  y  4 = 0, 3x + 2y  14 = 0. 18. x  y  5 = 0, 3x + 5y  15 = 0. 19. 2x  5y + 4 = 0, 2x + y  8 = 0. 20. 5x  y  7 = 0, x  y + 1 = 0. 21. 2x  3y = 12, x + 3y = 6.

[CBSE 2004] [CBSE 2006C] [CBSE 2009C] [CBSE 2005]

[CBSE 2008]

88

Secondary School Mathematics for Class 10

Show graphically that each of the following given systems of equations has infinitely many solutions:

22. 2x + 3y = 6, 4x + 6y = 12. 23. 3x  y = 5, 6x  2y = 10.

[CBSE 2010]

24. 2x + y = 6, 6x + 3y = 18. 25. x  2y = 5, 3x  6y = 15. Show graphically that each of the following given systems of equations is inconsistent, i.e., has no solution:

26. x  2y = 6, 3x  6y = 0. 27. 2x + 3y = 4, 4x + 6y = 12. 28. 2x + y = 6, 6x + 3y = 20. 29. Draw the graphs of the following equations on the same graph paper: 2x + y = 2, 2x + y = 6. Find the coordinates of the vertices of the trapezium formed by these lines. Also, find the area of the trapezium so formed. [HOTS] HINT

The line 2x + y = 2 cuts the x-axis at A(1, 0) and the y-axis at B(0, 2). The line 2x + y = 6 cuts the x-axis at C(3, 0) and the y-axis at D(0, 6). Area of trap. ABDC = ar(3 OCD)  ar(3 OAB) = ` 1 # 3 # 6 j  ` 1 # 1 # 2 j = 8 sq units. 2 2 ANSWERS (EXERCISE 3A)

1. x = 4, y = 2

2. x = 2, y = 1

3. x = 1, y = 2

4. x = 3, y = 2

5. x = 2, y = 3 9. x = 1, y = 2

6. x = 1, y = 2 10. x = 2, y = 2

7. x = 2, y = 3

8. x = 2, y = 3

11. (x = 1, y = 2); A(1, 2), B(3, 0), C(2, 0); ar(3 ABC) = 5 sq units 12. (x = 1, y = 2); A(1, 2), B(2, 0), C(5, 0); ar(3 ABC) = 7 sq units 13. (x = 2, y = 4); A(2, 4), B(1, 0), C(5, 0); ar(3 ABC) = 12 sq units 14. (x = 2, y = 3); A(2, 3), B(1, 0), C(4, 0); ar(3 ABC) = 7.5 sq units 15. (x = 2, y = 2); A(2, 2), B(2, 0), C(3, 0); ar(3 ABC) = 5 sq units 16. (x = 3, y = 4); A(3, 4), B(0, 2), C(0, 6); ar(3 ABC) = 6 sq units 17. (x = 2, y = 4); A(2, 4), B(0, 4), C(0, 7); ar(3 ABC) = 11 sq units 18. (x = 5, y = 0); A(5, 0), B(0, 5), C(0, 3); ar(3 ABC) = 20 sq units 19. (x = 3, y = 2); A(3, 2), B(0, 0.8), C(0, 8); ar(3 ABC) = 10.8 sq units 20. (x = 2, y = 3); A(2, 3), B(0, 7), C(0, 1); ar(3 ABC) = 8 sq units

Linear Equations in Two Variables

89

21. (x = 6, y = 0), A(6, 0), B(0, 4), C(0, 2), ar(3 ABC) = 18 sq units. 29. 8 sq units

SOLVING SIMULTANEOUS LINEAR EQUATIONS (BY ALGEBRAIC METHODS) (I) SUBSTITUTION METHOD METHOD

Suppose we are given two linear equations in x and y. For solving these equations by the substitution method, we proceed according to the following steps.

Step 1.

Express y in terms of x in one of the given equations.

Step 2.

Substitute this value of y in terms of x in the other equation. This gives a linear equation in x.

Step 3.

Solve the linear equation in x obtained in Step 2.

Step 4.

Substitute this value of x in the relation taken in Step 1 to obtain a linear equation in y.

Step 5.

Solve the above linear equation in y to get the value of y.

REMARK

We may interchange the role of x and y in the above method.

SOLVED EXAMPLES EXAMPLE 1

Solve for x and y, using substitution method: 2x + y = 7, 4x  3y + 1 = 0.

SOLUTION

The given system of equations is 2x + y = 7

… (i)

4x  3y = 1.

… (ii)

From (i), we get y = (7  2x) . Substituting y = (7  2x) in (ii), we get 4x  3 (7  2x) = 1 

4x  21 + 6x = 1

 10x = 20 

x = 2.

90

Secondary School Mathematics for Class 10

Substituting x = 2 in (i), we get 2 # 2  y  7 & y  7  4  3. Hence, the solution is x = 2, y = 3. EXAMPLE 2

Solve for x and y, using substitution method: 3x  5y =  x + y = 13 2, 3 2 2 3 6 $

SOLUTION

The given system of equations may be written as 9x  10y + 12 = 0

… (i)

2x + 3y  13 = 0.

… (ii)

13  2x From (ii), we get y = $ 3  13 2x Substituting y = in (i), we get 3 10(13  2x) + 12 = 0 9x  3  27x  10(13  2x) + 36 = 0 

27x  130 + 20x + 36 = 0

94 47x  94  0 & 47x  94 & x  47  2. Substituting x = 2 in (i), we get 

30 9 # 2  10y  12  0 & 10y  30 & y  10  3. Hence, x = 2 and y = 3 is the required solution. (II) ELIMINATION METHOD METHOD

In this method, we eliminate one of the unknown variables and proceed using the following steps.

Step 1.

Multiply the given equations by suitable numbers so as to make the coefficients of one of the unknown variables numerically equal.

Step 2.

If the numerically equal coefficients are opposite in sign then add the new equations. Otherwise, subtract them.

Step 3.

The resulting equation is linear in one unknown variable. Solve it to get the value of one of the unknown quantities.

Step 4.

Substitute this value in any of the given equations.

Step 5.

Solve it to get the value of the other unknown variable.

Linear Equations in Two Variables EXAMPLE 3

91

Solve for x and y using elimination method: 10x  3y  75, 6x  5y  11.

SOLUTION

The given equations are 10x  3y  75

… (i)

6x  5y  11.

… (ii)

Multiplying (i) by 5 and (ii) by 3, we get 50x  15y  375

… (iii)

18x  15y  33.

… (iv)

Adding (iii) and (iv), we get 408 68x  408 & x  68 & x  6. Putting x  6 in (i), we get (10 # 6)  3y  75 & 60  3y  75

EXAMPLE 4



3y  75  60 & 3y  15 & y  5.



x  6, y  5.

Solve for x and y: 11x  15y  23  0, 7x  2y  20  0.

SOLUTION

The given equations are 11x  15y   23

… (i)

7x  2y  20.

… (ii)

Multiplying (i) by 2 and (ii) by 15 and adding the results, we get 22x  105x  46  300  127x  254 254  x  127  2. Putting x  2 in (i), we get 22  15y  23  15y  23  22 45  15y  45 & y  15  3. Hence, x  2 and y  3.

92 EXAMPLE 5

Secondary School Mathematics for Class 10

Solve for x and y:

[CBSE 2000, ‘04C, ‘05]

0.4x  1.5y  6.5, 0.3x  0.2y  0.9. SOLUTION

Multiplying each of the equations by 10, we get 4x  15y  65 3x  2y  9.

… (i) … (ii)

Multiplying (i) by 2 and (ii) by 15 and adding, we get 8x  45x  130  135  53x  265 265  x  53  5. Putting x  5 in (ii), we get 15  2y  9 & 2y  9  15 & 2y  6 & y  3. Hence, x  5 and y  3. EXAMPLE 6

SOLUTION

Solve for x and y: 4   6   5. x 3y 8, x 4y 1 Putting x  u, the given equations become: 4u  3y  8

[CBSE 2010]

… (i)

6u  4y  5.

… (ii)

Multiplying (i) by 4 and (ii) by 3 and adding the results, we get 16u  18u  32  15 34u  17 17 1  u  34  2 · 1 Putting u  2 in (i), we get b 4 # 1 l  3y  8 & 2  3y  8 2 & 3y  8  2 & 3y  6 & y  2. 

… (iii)

1 1 1 Now, u  2 & x  2 & x  2. Hence, x  2 and y  2. EXAMPLE 7

Solve for x and y: 23  5  4  2 (x ! 0 and y ! 0) . x y 13, x y

[CBSE 2008C]

Linear Equations in Two Variables SOLUTION

93

1 1 Putting x  u and y  v, the given equations become 2u  3v  13

… (i)

5u  4v  2.

… (ii)

Multiplying (i) by 4 and (ii) by 3 and adding the results, we get 8u  15u  52  6 23u  46 46  u  23  2. 

Putting u  2 in (i), we get (2 # 2)  3v  13 & 3v  13  4  9 & v  3. 1 1 Now, u  2 & x  2 & 2x  1 & x  2 · 1 1 And, v  3 & y  3 & 3y  1 & y  3 · 1 1 Hence, x  2 and y  3 · EXAMPLE 8

SOLUTION

Solve for x and y: 1  1  1  1  1, x 2y 8 (x !0, y !0) . 2x y

[CBSE 2004C]

1 1 Putting x  u and y  v, the given equations become u   1 & u  2v  2 2 v v u  2  8 & 2u  v  16.

… (i) … (ii)

Multiplying (ii) by 2 and adding the result with (i), we get u  4u  2  32 5u  30 30  u  5  6. Putting u  6 in (i), we get 

6  2v  2 & 2v  8 & v  4. 1 1 Now, u  6 & x  6 & 6x  1 & x  6 · 1 1 And, v  4 & y  4 & 4y  1 & y  4 · 1 1 Hence, x  6 and y  4 ·

94 EXAMPLE 9

SOLUTION

Secondary School Mathematics for Class 10

Solve for x and y: 1  1  1  1  7x 6y 3, 2x 3y 5 (x ! 0, y ! 0). 1 1 Putting x  u and y  v, the given equations become uv    7 6 3 & 6u 7v 126 uv    2 3 5 & 3u 2v 30.

… (i) … (ii)

Multiplying (i) by 2 and (ii) by 7 and adding the results, we get 12u  21u  252  210 33u  462 462  u  33  14. Putting u  14 in (i), we get 

(6 # 14)  7v  126 42  7v  126  84  42 & v  7  6. 1 1 Now, u  14 & x  14 & 14x  1 & x  14 · 1 1 And, v  6 & y  6 & 6y  1 & y  6 · 1 1 Hence, x  14 and y  6 · EXAMPLE 10

SOLUTION

Solve for x and y: 3a  2b   a  3b   x y 5 0, x y 2 0 (x ! 0, y ! 0) .

[CBSE 2005C]

1 1 Putting x  u and y  v, the given equations become 3au  2bv  5

… (i)

au  3bv  2.

… (ii)

Multiplying (ii) by 3 and subtracting (i) from the result, we get 9bv  2bv  6  5 11  1 ·  11bv  11 & v  11b b 1 Putting v  in (ii), we get b

1 au  3  2 & au  1 & u  a ·

Linear Equations in Two Variables

95

1 1 1 Now, u  a & x  a & x  a & x  a. 1 1 1 And, v  & y  & y  b. b b   Hence, x a and y  b. EXAMPLE 11

Solve for x and y: 6x  3y  7xy, 3x  9y  11xy (x ! 0, y ! 0).

SOLUTION

On dividing each of the given equations by xy, we get 63  y x 7 39  y x 11.

… (i) … (ii)

1 1 Putting x  u and y  v in (i) and (ii), we get 6v  3u  7

… (iii)

3v  9u  11.

… (iv)

Multiplying (iii) by 3 and subtracting (iv) from the result, we get 18v  3v  21  11  15v  10 10 2  v  15  3 · 2 Putting v  3 in (iii), we get b 6 # 2 l  3u  7 & 4  3u  7 & 3u  3 & u  1. 3 1 Now, u  1 & x  1 & x  1. 2 1 2 3 And, v  3 & y  3 & 2y  3 & y  2 · 3 Hence, x  1 and y  2 · EXAMPLE 12

Solve the following system of equations for x and y: 5  1  6 3 2,     1. x1 y2 x 1 y 2

SOLUTION

Putting

[CBSE 2008C, ’09]

1  1 y and   v, the given equations become x1 y 2

5u  v  2

… (i)

6u  3v  1.

… (ii)

96

Secondary School Mathematics for Class 10

Multiplying (i) by 3 and adding the result with (ii), we get 15u  6u  6  1  21u  7 7 1  u  21  3 · 1 Putting u  3 in (i), we get b5 # 1 l  v  2 & 5  v  2 & v  2  5 & v  1 · 3 3 3 3 1 1 1 Now, u  3 &   3 & x  1  3 & x  4 x 1 [by cross multiplication]. 1 1 1 Similarly, v  3 &   3 & y  2  3 & y  5 y 2 [by cross multiplication]. Hence, x  4 and y  5. EXAMPLE 13

Solve for x and y: 15  22  40 55 5,     13, x ! y and x !  y. xy xy x y x y [CBSE 2008C]

SOLUTION

1 1 Putting   u and   v, the given equations become x y x y … (i) 15u  22v  5 … (ii) 40u  55v  13. Multiplying (ii) by 2 and (i) by 5 and subtracting the results, we get 80u  75u  26  25 5u  1 1  u 5· 1 Putting u  5 in (i), we get b15 # 1 l  22v  5 5  3  22v  5 2 1  22v  2 & v  22  11 · 1 1 1 Now, u  5 &   5 & x  y  5. x y 1 1 1 And, v  11 &   11 & x  y  11. x y 

On adding (iii) and (iv), we get 2x  16 & x  8.

… (iii) … (iv)

Linear Equations in Two Variables

97

On subtracting (iii) from (iv), we get 2y  6 & y  3. Hence, x  8 and y  3. EXAMPLE 14

Solve for x and y: 1 12 7 4    1,  2, 2 (2x  3y) 7 (3x  2y) 2 (2x  3y) (3x  2y) where (2x  3y) ! 0 and (3x  2y) ! 0.

SOLUTION

Putting

[CBSE 2004C]

1 1  v, the given equations  u and 3x  2y (2x  3y)

become u  12v  1   2 7 2 & 7u 24v 7 and 7u  4v  2.

… (i) … (ii)

Subtracting (ii) from (i), we get 5 1 20v  5 & v  20  4 · 1 Putting v  4 in (i), we get 1 7u  b 24 # 4 l  7 1  7u  6  7 & 7u  1 & u  7 · 1 1  1 & 2x  3y  7 Now, u  7 & (2x  3y) 7 1 1  1 & 3x  2y  4. and v  4 & (3x  2y) 4

… (iii) … (iv)

Multiplying (iii) by 2 and (iv) by 3 and adding the results, we get 4x  9x  14  12  13x  26 & x  2. Putting x  2 in (iii), we get (2 # 2)  3y  7 & 3y  (7  4)  3 & y  1. Hence, x  2 and y  1. EXAMPLE 15

Solve for x and y: 2  3  4  9  2, 1 (x ! 0, y ! 0) . x y x y

SOLUTION

Putting

1  1  u and v, the given equations become x y

98

Secondary School Mathematics for Class 10

2u  3v  2

… (i)

4u  9v  1.

… (ii)

Multiplying (i) by 3 and adding the result with (ii), we get 6u  4u  6  1  10u  5 5 1  u  10 & u  2 · 1 Putting u  2 in (i), we get a 2 # 12 k  3v  2 1  1  3v  2 & 3 v  1 & v  3 · 1 1 1 Now, u  2 & & x  2 & x  4. x 2 1 1 1 And, v  3 & & y  3 & y  9. y 3 Hence, x  4 and x  9. EXAMPLE 16

Solve the following system of linear equations: (a  b) x  (a  b) y  a 2  2ab  b 2, (a  b) (x  y)  a 2  b 2 .

SOLUTION

[CBSE 2004, ‘07C, ‘08]

The given equations may be written as (a  b) x  (a  b) y  a 2  2ab  b 2

… (i)

(a  b) x  (a  b) y  a  b .

… (ii)

2

2

Subtracting (i) from (ii), we get {(a  b)  (a  b)} x  2ab  2b 2 2b (a  b)  2bx  2b (a  b) & x  2b  x  (a  b) . Putting x  (a  b) in (ii), we get (a  b) 2  (a  b) y  a 2  b 2  (a  b) y  (a 2  b 2)  (a  b) 2  (a  b) y  (a 2  b 2)  (a 2  b 2  2ab) 2ab  (a  b) y  2ab & y   · (a b) 2ab  x  (a  b) and y   is the required solution. (a b)

Linear Equations in Two Variables EXAMPLE 17

Solve for x and y: ax  by  a  b  0, bx  ay  a  b  0.

SOLUTION

99

[CBSE 2000, ‘04C, ‘05]

The given equations may be written as ax  by  a  b

… (i)

bx  ay  a  b.

… (ii)

Multiplying (i) by a and (ii) by b and adding the results, we get (a 2  b 2) x  (a 2  b 2) & x  1. Putting x  1 in (i), we get (a # 1)  by  a  b 

a  by  a  b

b  1.  by  b & y  b Hence, x  1 and y  1. EXAMPLE 18

Solve the following system of linear equations: 2(ax  by)  (a  4b)  0, 2(bx  ay)  (b  4a)  0.

SOLUTION

[CBSE 2004]

The given equations may be written as 2ax  2by  a  4b

… (i)

2bx  2ay  4a  b.

… (ii)

Multiplying (i) by a and (ii) by b and adding, we get (2a 2  2b 2) x  (a 2  b 2) 1 2 (a 2  b 2) x  (a 2  b 2) & x  2 · 1 Putting x  2 in (i), we get 1 2a # a 2 k  2by   a  4b   a  2by   a  4b 

4b 2by  4b & y  2b  2. 1 Hence, x  2 and y  2. 

EXAMPLE 19

Solve for x and y: ax  by     a a b, ax by 2ab. b

[CBSE 2000, ‘04C, ‘05, ‘07C, ‘09]

100

Secondary School Mathematics for Class 10

The given equations may be written as

SOLUTION

a 2 x  b 2 y  a 2 b  ab 2 ax  by  2ab.

… (i) … (ii)

Multiplying (ii) by b and subtracting the result from (i), we get (a 2  ab) x  a 2 b  ab 2  (a 2  ab) x  b (a 2  ab) & x  b. Putting x  b in (ii), we get ab ab  by  2ab & by  ab & y  b  a. Hence, x  b and y  a. EXAMPLE 20

Solve for x and y: 2 x  3 y  0, 5 x  2 y  0.

[CBSE 2000, ‘04C, '05]

The given system of equations is

SOLUTION

2x 3y  0

… (i)

5 x  2 y  0.

… (ii)

Multiplying (i) by 2 and (ii) by 3 and adding, we get (2  15 ) x  0 & x  0. Putting x  0 in (ii), we get 2 y  0 & y  0. Hence, x  0 and y  0. A SPECIAL CASE

METHOD Step 1.

When the coefficients of x and y in one equation are interchanged in the other. Add the given equations and from it obtain an equation of the form x  y  a.

Subtract the given equations and from it obtain an equation of the form x  y  b. Now, x  y  a and x  y  b can be solved easily.

Step 2.

EXAMPLE 21

Solve for x and y: 37x  43y  123, 43x  37y  117.

SOLUTION

[CBSE 2008]

The given equations are 37x  43y  123

… (i)

43x  37y  117.

… (ii)

Linear Equations in Two Variables

101

Clearly, the coefficients of x and y in one equation are interchanged in the other. Adding (i) and (ii), we get (37  43) x  (43  37) y  (123  117)  80x  80y  240  80(x  y)  240 

240 x  y  80 & x  y  3.

… (iii)

Subtracting (i) from (ii), we get 6x  6y  6  6(x  y)  6 & x  y  1

… (iv)

Adding (iii) and (iv), we get 2x  2 & x  1. Subtracting (iv) from (iii), we get 2y  4 & y  2. Hence, x  1 and y  2. EXAMPLE 22

Solve for x and y: 152x  378y  74, 378x  152y  604.

SOLUTION

The given equations are 152x  378y  74

… (i)

 378x  152y  604.

… (ii)

Clearly, the coefficients of x and y in one equation are interchanged in the other. Adding (i) and (ii), we get {152  (378)} x  {(378)  152} y  (74  604)  (226) x  (226) y  678  (226)(x  y)  678 678  (x  y)   & x  y  3. 226

… (iii)

Subtracting (ii) from (i), we get (152  378) x  (378  152) y  (74  604) 

530x  530y  530



530(x  y)  530 & x  y  1.

Adding (iii) and (iv), we get 2x  4 & x  2.

… (iv)

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Secondary School Mathematics for Class 10

Subtracting (iv) from (iii), we get 2y  2 & y  1. Hence, x  2 and y  1. SOME MORE EXAMPLES EXAMPLE 23

SOLUTION

Solve for x and y: x1  y1  x1  y1  8, 3 9. 2 3 2

[CBSE 2007]

The given equations may be written as 3(x  1)  2 (y  1)  48 & 3x  2y  47

… (i)

and 2(x  1)  3(y  1)  54 & 2x  3y  53.

… (ii)

Multiplying (i) by 2 and (ii) by 3 and subtracting, we get (4  9) y  94  159 65  5y  65 & y   & y  13. 5 Putting y  13 in (i), we get 3x  (2 # 13)  47 & 3x  (47  26)  21 21   x 7. 3 Hence, x  7 and y  13. EXAMPLE 24

Solve for x and y: 2  3  9 4  9  21 x y xy , x y xy (x ! 0, y ! 0).

SOLUTION

Multiplying each equation throughout by xy, we get 2y  3x  9

… (i)

4y  9x  21.

… (ii)

Multiplying (i) by 3 and subtracting (ii) from the result, we get (6  4) y  (27  21) & 2y  6 & y  3. Putting y  3 in (i), we get (2 # 3)  3x  9 & 6  3x  9 & 3x  3 & x  1. Hence, x  1 and y  3. EXAMPLE 25

Solve for x and y: xy xy  6,  6 (x ! 0, y ! 0 and x ! y) . xy 5 yx

Linear Equations in Two Variables SOLUTION

The given equations may be written as xy 5  &11  5 xy y x 6 6 yx 1 1  & 1 1 xy x y 6 6

103

… (i) … (ii)

Adding (i) and (ii), we get 2  51 6   x a 6 6 k 6 1 & x 2. Subtracting (ii) from (i), we get 2  51  4  2  y a 6 6 k 6 3 & y 3. Hence, x  2 and y  3. EXAMPLE 26

Solve for x and y: 7x  2y 8x  7y  5,  15 (x ! 0, y ! 0) . xy xy

SOLUTION

We have 7x  2y 5& 7 2 5 xy y x 8x  7y  15 & 8  7  15. xy y x

… (i) … (ii)

Multiplying (i) by 7 and (ii) by 2 and adding the results, we get  16 m  (35  30) c 49 y y 

65    y 65 & 65y 65 & y 1.

Putting y  1 in (ii), we get 87  7      1 x 15 & x (15 8) 7 & 7x 7 & x 1. Hence, x  1 and y  1. f

EXERCISE 3B

Solve for x and y:

1. x  y  3, 4x  3y  26.

2. x  y  3, xy  3 2 6.

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Secondary School Mathematics for Class 10

3. 2x  3y  0, 3x  4y  5.

4. 2x  3y  13, 7x  2y  20.

5. 3x  5y  19  0,  7x  3y  1  0.

6. 2x  y  3  0, 3x  7y  10  0.

x y 7. 2  9  6, xy  7 3 5. 9. 4x  3y  8,

x y 8. 3  4  11, 5x  y   7. 6 3 3y 10. 2x  4  3,

29 6x  y  3 · 8 11. 2x  5y  3 , 5 3x  2y  6 ·

12. 2x  3y  1  0, 7  4x  y. 3

13. 0.4x  0.3y  1.7, 0.7x  0.2y  0.8.

14. 0.3x  0.5y  0.5, 0.5x  0.7y  0.74

15. 7 (y  3)  2 (x  2)  14, 4(y  2)  3(x  3)  2.

16. 6x  5y  7x  3y  1  2(x  6y  1)

17.

5x  2y  7.

x  y  8 x  2y  14 3x  y  12   2 3 11 HINT

a  b  c & a  b and b  c.

5 18. x  6y  13, 3  x 4y 7 (x ! 0) . 3 20. 2x  y  9, 7 3x  y  2 (y ! 0) . 9 4 22. x  y  8, 13  7  x y 101 (x ! 0, y ! 0) . 1 1 24. 2x  3y  2, 1  1  13 3x 2y 6 (x ! 0, y ! 0) .

6 19. x  y  6, 8 [CBSE 2007] 3x  y  5 (y ! 0) . 3 1 21. x  y  9  0, 23  x y 5 (x ! 0, y ! 0) . 5 3 23. x  y  1, 3  2  2x 3y 5 (x ! 0, y ! 0) .

26. x  y  5xy,

27.

3x  2y  13xy (x ! 0, y ! 0) .

25. 4x  6y  3xy, 8x  9y  5xy (x ! 0, y ! 0) . 5  2  1, xy xy 15  7  10. xy xy

Linear Equations in Two Variables

105

28.

3  2  2, xy xy 9  4  1. xy xy

29.

5  2 1 , x1 y1 2 10  2  5 , x ! 1 and y ! 1. x1 y1 2

30.

44  30  10, xy xy

31.

10  2  4, xy xy

55  40  13. xy xy 32. 71x  37y  253, 37x  71y  287.

15  9   2. xy xy

[CBSE 2007C]

34. 23x  29y  98, 29x  23y  110.

33. 217x  131y  913, 131x  217y  827. 35.

2x  5y  6, xy 4x  5y   3. xy  1 5   3, 2 2(x  2y) 3(3x  2y) 5 3   61 · 4(x  2y) 5(3x  2y) 60

36.

1 1   3, (3x  y) (3x  y) 4  1 1   1· 8 2(3x  y) 2(3x  y)

37.

38.

2 3   17 , 5 (3x  2y) (3x  2y) 5 1   2. (3x  2y) (3x  2y)

39. 3(2x  y)  7xy, 3(x  3y)  11xy (x ! 0 and y ! 0) . x y 41. a   2, b  ax by  a 2  b 2 . x y 43. a   0, b ax  by  (a 2  b 2) .

40. x  y  a  b, ax  by  a 2  b 2 . 42. px  qy  p  q, qx  py  p  q. 44. 6(ax  by)  3a  2b, 6(bx  ay)  3b  2a. bx ay    46. a  , b a b 0 bx  ay  2ab  0. [CBSE 2006] 48. x  y  a  b, ax  by  a 2  b 2 . y x y x 50. a   a  b, 2  2  2. b a b

45. ax  by  a 2  b 2, x  y  2a . bx ay  2  2 47. a  , b a b x  y  2ab. 49. a 2 x  b 2 y  c 2,

[CBSE 2005]

[CBSE 2006C]

[CBSE 2010]

b2 x  a2 y  d2 .

ANSWERS (EXERCISE 3B)

1. x  5, y  2

2. x  9, y  6

3. x  15, y  10

4. x  2, y  3

106

Secondary School Mathematics for Class 10

5. x  2, y  5

6. x  1, y  1 7. x  14, y  9  3 2 1 1 9. x  2 , y  3 10. x  3, y  4 11. x  2 , y  3 13. x  2, y  3 14. x  0.5, y  0.7 15. x  5, y  1

12. x  4, y  3

1

20. x  3, y  1

17. x  2, y  6

1

19. x  3, y  2

22. x  4 , y  7

1

1

23. x  2 , y  3

24. x  2 , y  3

25. x  3, y  4

26. x  2 , y  3

1

1

27. x  3, y  2

28. x  2 , y  2

29. x  4, y  5

30. x  8, y  3

33. x  3, y  2

34. x  3, y  1

31. x  8 , y  8 35. x  1, y  2

37. x  2 , y  4

38. x  1, y  1

39. x  1, y  2

40. x  a, y  b

41. x  a, y  b

42. x  1, y  1

43. x  a, y  b

44. x  2 , y  3 47. x  ab, y  ab

5

1

21

49. x 

9

3

45. x  a  b, y  a  b 48. x  a, y  b

1

16. x  3, y  2

21. x  2 , y  3

1

1

18. x  5 , y  2

8. x  6, y  36

46. x  a, y  b

1

1

5

1

32. x  2, y  3 36. x  1, y  1

1

1

(a c  b d ) (a 2 d 2  b 2 c 2) 50. x  a 2, y  b 2 ,y 4 4 (a  b ) (a 4  b 4) 2 2

2

2

HINTS TO SOME SELECTED QUESTIONS 9. The second equation is 18x  3y  29. 11. Given equations are 6x  15y  8, 18x  12y  5. 7  4x 12. 3  y & 7  4x  3y & 4x  3y  7. 13. Multiply each of the given equations throughout by 10. 14. Multiply each of the given equations throughout by 10. 15. The given equations are 2x  7y  3, 3x  4y  19. 16. 6x  5y  7x  3y  1 & x  2y  1. 7x  3y  1  2x  12y  2 & 5x  9y  3. 17.

x  y  8 x  2y  14  & 3 (x  y  8)  2 (x  2y  14) 2 3 & x  y   4. x  2y  14 3x  y  12  & 11 (x  2y  14)  3 (3x  y  12) 3 11 & 2x  19y  118. Now, solve (i) and (ii).

18. Multiply (i) by 3 and (ii) by 5 and subtract. 19. Multiply (i) by 4 and (ii) by 3 and add.

… (i)

… (ii)

Linear Equations in Two Variables

107

1 1 23. Put x  u and y  v to get 3u 2v 5u  3v  1 and 2  3  5 or 9u  4v  30. Now, solve 5u  3v  1 and 9u  4v  30. 1 1 24. Putting x  u and y  v, we get uv    2 3 2 & 3u 2v 12

… (i)

u  v  13   3 2 6 & 2u 3v 13. Add (i) and (ii). Subtract (ii) from (i).

… (ii)

4 6 8 9 25. Divide each equation throughout by xy to get y  x  3, y  x  5. 27. Put

1  1 u and   v. xy x y

29. Put

1  1 u and   v. x1 y 1

32. Add (i) and (ii) to get 108 (x  y)  540 & x  y  5. Subtract (ii) from (i) to get 34 (x  y)  34 & x  y  1. 2 5 4 5 35. y  x  6, y  x  3. Now, add. 36. Put

1 1  u and  v. (3x  y) (3x  y)

37. Put

1 1  u and  v. (x  2y) (3x  2y)

6 3 39. 6x  3y  7xy & y  x  7.

… (i)

3 9 3x  9y  11xy & y  x  11.

… (ii)

40. Multiply (i) by b and add (ii) to the result. 41. bx  ay  2ab

and ax  by  a 2  b 2.

… (ii)

and ax  by  (a 2  b 2).

… (ii)

and 6bx  6ay  3b  2a.

… (ii)

… (i)

and b 2 x  aby  2ab 2  0.

… (ii)

… (i)

and x  y  2ab.

… (ii)

… (i)

Multiply (i) by b and (ii) by a and add. 42. Multiply (i) by p and (ii) by q and add. 43. bx  ay  0

… (i)

Multiply (i) by b and (ii) by a and add. 44. 6ax  6by  3a  2b

… (i)

Multiply (i) by a and (ii) by b and add. 45. Multiply (ii) by b and add the result with (i). 46. b 2 x  a 2 y  a 2 b  ab 2  0 Now, subtract (ii) from (i). 47. b 2 x  a 2 y  a 3 b  ab 3

Multiply (ii) by b 2 and subtract the result from (i).

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Secondary School Mathematics for Class 10

48. Multiply (i) by b and add the result with (ii). 49. Multiply (i) by b 2 and (ii) by a 2 and subtract the results. 50. bx  ay  a 2 b  ab 2

… (i)

and b 2 x  a 2 y  2a 2 b 2.

… (ii)

METHOD OF CROSS MULTIPLICATION The system of two linear equations

THEOREM

a1 x  b1 y  c1  0, a2 x  b2 y  c2  0, a b where a1 ! 1 , has a unique solution, given by b2 2 (b c  b c ) (c a  c a ) x  1 2 2 1 , y  1 2 2 1 · (a1 b2 a2 b1) (a1 b2 a2 b1) PROOF

The given equations are a1 x  b1 y  c1  0 a2 x  b2 y  c2  0.

… (i) … (ii)

Multiplying (i) by b2, (ii) by b1 and subtracting, we get (a1 b2  a2 b1) x  (b1 c2  b2 c1) 

x

(b1 c2  b2 c1) (a1 b2  a2 b1)

a b [a a1 ! b1 & (a1 b2  a2 b1) ! 0] . 2 2

Multiplying (ii) by a1, (i) by a2 and subtracting, we get (a1 b2  a2 b1) y  (c1 a2  c2 a1) 

y

(c1 a2  c2 a1) (a1 b2  a2 b1)

[a (a1 b2  a2 b1) ! 0] .

Hence, a unique solution exists, which is given by x

(b1 c2  b2 c1) (c a  c a ) , y  1 2 2 1 · (a1 b2  a2 b1) (a1 b2 a2 b1)

This is customarily written as y x 1 ·   (b1 c2  b2 c1) (c1 a2  c2 a1) (a1 b2  a2 b1) REMARK

The following diagram helps in remembering the above solution.

Linear Equations in Two Variables

109

Numbers with downward arrows are multiplied first; and from this product, the product of numbers with upward arrows is subtracted.

Rule

SOLVED EXAMPLES EXAMPLE 1

SOLUTION

Solve the system of equations 2x  3y  17, 3x  2y  6 by the method of cross multiplication. The given equations may be written as 2x  3y  17  0

… (i)

3x  2y  6  0.

… (ii)

By cross multiplication, we have



y x  {3 # ( 6)  (2) # (17)} {(17) # 3  ( 6) # 2} 

  

1 {2 # (2)  3 # 3}

y x 1   (18  34) (51  12) (4  9) x  y  1 52 39 13 52 39 x    4, y    3. 13 13

Hence, x  4 and y  3 is the required solution. 4x  7y  28  0, 5y  7x  9  0.

EXAMPLE 2

Solve

SOLUTION

The given equations are 4x  7y  28  0

… (i)

7x  5y  9  0.

… (ii)

By cross multiplication, we have



y x 1   {(7)# 9  5# 28} {28#(7)  9# 4} {4 # 5  (7)#(7)}

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Secondary School Mathematics for Class 10

y x 1   (63  140) (196  36) (20  49) x  y  1   203 232 29 203  232  x  a  k  7 and y  a  k  8. 29 29 Hence, x  7 and y  8 is the required solution. 

EXAMPLE 3 SOLUTION

Solve

23  5  4  2, where x ! 0 and y ! 0. x y 13, x y

1 1 Taking x  u and y  v, the given equations become 2u  3v  13  0 5u  4v  2  0.

… (i) … (ii)

By cross multiplication, we have



u v 1   [3# 2  (4)#(13)] [(13)# 5  2# 2] [2#(4)  5# 3]

u  v  1 46 69 23 46 69  u  a  k  2, v  a  k  3 23 23 1  1   x 2, y 3 

1 1 x  2, y  3 · 1 1 Hence, x  2 and y  3 is the required solution. 

ax  by  c, bx  ay  1  c.

EXAMPLE 4

Solve

SOLUTION

The given equations may be written as ax  by  c  0

… (i)

bx  ay  (1  c)  0.

… (ii)

By cross multiplication, we have

Linear Equations in Two Variables

111

y 1 x   b(1  c)  ac cb  a (1  c) (a 2  b 2) c(a  b)  b c(a  b)  a ·  x and y  (a 2  b 2) (a 2  b 2) c(a  b)  b c(a  b)  a · Hence, x  and y  2 2 (a  b ) (a 2  b 2) 

f

EXERCISE 3C

Solve each of the following systems of equations by using the method of cross multiplication:

1. x  2y  1  0, 2x  3y  12  0.

2. 3x  2y  3  0, 4x  3y  47  0.

3. 6x  5y  16  0, 7x  13y  10  0. 5. 2x  5y  1, 2x  3y  3.

4. 3x  2y  25  0, 2x  y  10  0. 6. 2x  y  35, 3x  4y  65. x y 8. 6  15  4, x  y  19 · 3 12 4

7. 7x  2y  3, 22x  3y  16.

1 1 9. x  y  7, 23  x y 17 (x ! 0, y ! 0). 5  2   10. 1 0, (x  y) (x  y) 15  7   10 0 (x ! y, x !  y) . (x  y) (x  y) 11.

ax  by   a a b, b  ax by  2ab. [CBSE 2007C]

12. 2ax  3by  (a  2b), 3ax  2by  (2a  b) .

a b ab 2 a 2 b 13. x  y  0, x  y  (a 2  b 2), where x ! 0 and y ! 0. ANSWERS (EXERCISE 3C)

1. x  3, y  2 5. x  3, y  1

2. x  5, y  9

6. x  15, y  5 1 1 9. x  4 , y  3 10. x  3, y  2 (4a  b) (4b  a)  12. x  5a , y 5b

3. x  6, y  4

4. x  5, y  20

7. x  1, y  2

8. x  18, y  15

11. x  b, y  a 13. x  a, y  b

112

Secondary School Mathematics for Class 10 HINTS TO SOME SELECTED QUESTIONS

8. The given equations are 5x  2y  120  0 and 4x  y  57  0. 1 1 9. Putting x  u and y  v, we get u  v  7  0 and 2u  3v  17  0. Now, solve for u and v. 10. Putting

1  1 u and   v, we get (x  y) (x y)

5u  2v  1  0 and 15u  7v  10  0. Now, solve for u and v. 11. The given equations are a 2 x  b 2 y  (a 2 b  ab 2)  0 and ax  by  2ab  0. 12. 2ax  3by  (a  2b)  0 and 3ax  2by  (2a  b)  0. 1 1 13. Put x  u and y  v. Then, au  bv  0, ab 2 u  a 2 bv  (a 2  b 2)  0.

CONDITIONS FOR SOLVABILITY OF LINEAR EQUATIONS CONSISTENT AND INCONSISTENT SYSTEMS OF LINEAR EQUATIONS

A system of equations a1 x  b1 y  c1  0, a2 x  b2 y  c2  0 is said to be consistent if it has at least one solution. On the other hand, the above system is said to be inconsistent if it has no solution at all. CONDITIONS FOR SOLVABILITY OF LINEAR EQUATIONS

The system of a pair of linear equations a1 x  b1 y  c1  0, a2 x  b2 y  c2  0 (i) has a unique solution, if

a1 b1 a2 ! b2

and the solution is x  d

b1 c2  b2 c1 c a c a n, y  c 1 2  2 1 m; a1 b2  a2 b1 a1 b2 a2 b1

(ii) has an infinite number of solutions, if

a1 b1 c1   a2 b2 c2 ;

(iii) has no solution (i.e., inconsistent), if

a1 b1 c1 ·  a2 b2 ! c2

Linear Equations in Two Variables

113

SOLVED EXAMPLES EXAMPLE 1

Show that the system of equations 2x  5y  17, 5x  3y  14 has a unique solution. Find the solution.

SOLUTION

The given system of equations is 2x  5y  17  0, 5x  3y  14  0. These equations are of the form a1 x  b1 y  c1  0, a2 x  b2 y  c2  0, where a1  2, b1  5, c1  17 and a2  5, b2  3, c2  14. a 17 17 2 b 5 c  a1  5 , 1  3 , c1    14 · b2 14 2 2 a b Thus, a1 ! 1 · b2 2 Hence, the given system of equations has a unique solution. By cross multiplication, we have

y x 1   (70  51) (85  28) (6  25) y x 1     19 57 19 19 57  x    1 and y    3. 19 19   Hence, x 1 and y 3 is the required solution. 

EXAMPLE 2

Find the values of k for which the system of equations x  2y  3, 3x  ky  1 has a unique solution.

SOLUTION

The given system of equations is x  2y  3  0, 3x  ky  1  0. These equations are of the form a1 x  b1 y  c1  0 and a2 x  b2 y  c2  0, where a1  1, b1  2, c1  3 and a2  3, b2  k, c2  1. a b For a unique solution, we must have a1 ! 1 · b 2 2

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Secondary School Mathematics for Class 10

2 1  3 ! k & k ! 6. Hence, the given system of equations will have a unique solution for all real values of k, other than –6. 

EXAMPLE 3

Show that the system of equations 4x  6y  7, 12x  18y  21 has infinitely many solutions.

SOLUTION

The given system of equations is 4x  6y  7  0, 12x  18y  21  0. These equations are of the form a1 x  b1 y  c1  0 and a2 x  b2 y  c2  0, where a1  4, b1  6, c1  7 and a2  12, b2  18, c2  21. a c 7 4 1 b 6 1 1  a1  12  3 , 1  18  3 and c1  a  k  3 · b2 21 2 2 a b c Thus, a1  1  c1 · b2 2 2 Hence, the given system of equations has infinitely many solutions.

EXAMPLE 4

Find the value of k for which the following pair of linear equations has infinitely many solutions: 2x  3y  7, (k  1) x  (1  2k) y  (5k  4) .

SOLUTION

[CBSE 2008C]

The given equations are 2x  3y  7  0, (k  1) x  (1  2k) y  (4  5k)  0. These equations are of the form a1 x  b1 y  c1  0 and a2 x  b2 y  c2  0, where a1  2, b1  3, c1  7 and

a2  (k  1), b2  (1  2k), c2  (4  5k) .

Let the given system of equations have infinitely many solutions. a b c Then, a1  1  c1 b 2 2 2 2  3  7  (k  1) (1  2k) (4  5k)

Linear Equations in Two Variables

 

115

2  3 7  (k  1) (2k  1) (5k  4) 2  3 3 7  and (k  1) (2k  1) (2k  1) (5k  4)



4k  2  3k  3 and 15k  12  14k  7



k  5 and k  5.

Hence, k  5. EXAMPLE 5

Find the value of k for which the given system of equations has infinitely many solutions: kx  3y  k  3, 12x  ky  k.

SOLUTION

The given system of equations is kx  3y  (3  k)  0, 12x  ky  k  0. These equations are of the form a1 x  b1 y  c1  0 and a2 x  b2 y  c2  0, where a1  k, b1  3, c1  (3  k) and a2  12, b2  k, c2  k. Let the given system of equations have infinitely many solutions. a b c Then, a1  1  c1 b2 2 2 k  3  (3  k)  12 k k k  3  k3  12 k k k  3 3  k3  12 k and k k 2 2     k 36 and k 6k 0  (k  6 or k  6) and k(k  6)  0  (k  6 or k   6) and (k  0 or k  6) 

k  6.

Hence, the given system of equations has infinitely many solutions when k  6. EXAMPLE 6

Find the value of k for which the given system of equations has infinitely many solutions: x  (k  1) y  5, (k  1) x  9y  (1  8k)  0.

[CBSE 2003]

116 SOLUTION

Secondary School Mathematics for Class 10

The given system of equations is x  (k  1) y  5  0

… (i)

(k  1) x  9y  (1  8k)  0.

… (ii)

These equations are of the form a1 x  b1 y  c1  0 and a2 x  b2 y  c2  0, where a1  1, b1  (k  1), c1  5 and 

a2  (k  1), b2  9, c2  (1  8k) . c1 a1 b1 (k  1) 5 5  1    · a2 (k  1) , b2 9 and c2 (1  8k) (8k  1)

Let the given system of equations have infinitely many solutions. a b c Then, a1  1  c1 b2 2 2  1  (k 1)  5  9 (k  1) (8k  1) (k  1) 1  (k  1) 5   9 and 9 (k  1) (8k  1)  (k  1) 2  9 and (k  1) (8k  1)  45  (k  1  3 or k  1  3) and 8k 2  7k  46  0  (k  2 or k  4) and (k  2) (8k  23)  0 23  (k  2 or k   4) and ak  2 or k  8 k  k  2. Hence, the given system of equations will have infinitely many solutions when k  2. EXAMPLE 7

Find the values of a and b for which the following pair of linear equations have an infinite number of solutions: 2x  3y  7, (a  b) x  (a  b) y  3a  b  2.

SOLUTION

[CBSE 2008C]

The given equations are 2x  3y  7  0,

… (i)

(a  b) x  (a  b) y  (2  3a  b)  0.

… (ii)

These equations are of the form a1 x  b1 y  c1  0 and a2 x  b2 y  c2  0, where a1  2, b1  3, c1  7 and

a2  (a  b), b2  (a  b), c2  (2  3a  b) .

Linear Equations in Two Variables

117

a b c 7 2 3 7 ·  a1   , 1   and c1     (a b) b2 (a b) (2 3a b) (3a  b  2) 2 2 Let the given system of equations have infinitely many solutions. a b c Then, a1  1  c1 b2 2 2 2  3  7  (a  b) (a  b) (3a  b  2) 2  3 3 7  and   (a  b) (a  b) (a b) (3a  b  2)  2a  2b  3a  3b and 9a  3b  6  7a  7b  a  5b  0 and 2a  4b  6  a  5b  0 and a  2b  3. On solving a  5b  0 and a  2b  3, we get a  5 and b  1. Hence, the required values are a  5 and b  1. EXAMPLE 8

Find the values of m and n for which the following system of linear equations has infinitely many solutions: 3x  4y  12, (m  n) x  2 (m  n) y  (5m  1) .

SOLUTION

The given system of equations is 3x  4y  12  0, (m  n) x  2(m  n) y  (1  5m)  0. These equations are of the form a1 x  b1 y  c1  0 and a2 x  b2 y  c2  0, where a1  3, b1  4, c1  12 and a2  (m  n), b2  2(m  n), c2  (1  5m) . 

a1 b1 c1 3 2 12 ·    a2 (m  n) , b2 (m  n) and c2 (5m  1)

Let the given system of equations have infinitely many solutions. a b c Then, a1  1  c1 b2 2 2 3 2 12    (m  n) (m  n) (5m  1) 3 2 2 12    and (m  n) (m  n) (m  n) (5m  1) 

3m  3n  2m  2n and 10m  2  12m  12n

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Secondary School Mathematics for Class 10

 m  5n  0

… (i)

and m  6n  1  0.

… (ii)

On solving (i) and (ii), we get m  5 and n  1. Hence, the required values are m  5 and n  1. EXAMPLE 9

Show that the system of equations 3x  5y  7, 6x  10y  3 has no solution.

SOLUTION

The given system of equations is 3x  5y  7  0 and 6x  10y  3  0. These equations are of the form a1 x  b1 y  c1  0 and a2 x  b2 y  c2  0, where a1  3, b1  5, c1  7 and a2  6, b2  10, c2  3. a c 5 7 7 3 1 b 1  a1  6  2 , 1    2 and c1    3 · b2 10 3 2 2 a1 b1 c1 Clearly, a  ! c · b2 2 2 Hence, the given system of equations has no solution.

EXAMPLE 10

Find the value of k for which the system of equations x  2y  5, 3x  ky  15  0 has no solution.

SOLUTION

The given system of equations is x  2y  5  0

… (i)

3x  ky  15  0.

… (ii)

These equations are of the form a1 x  b1 y  c1  0 and a2 x  b2 y  c2  0, where a1  1, b1  2, c1  5 and a2  3, b2  k, c2  15. a c 5 1 b 2 1  a1  3 , 1  and c1    3 · b2 k 15 2 2 Let the given system of equations have no solution. a b c Then, a1  1 ! c1 b2 2 2   

1  2 1 3 k!3 1  2 2 1 3 k and k ! 3 k  6 and k ! 6, which is impossible.

Linear Equations in Two Variables

119

Hence, there is no value of k for which the given system of equations has no solution. EXAMPLE 11

Find the values of k for which the pair of linear equations kx  3y  k  2 and 12x  ky  k has no solution.

SOLUTION

[CBSE 2009]

The given equations are kx  3y  (2  k)  0 and 12x  ky  k  0. These equations are of the form a1 x  b1 y  c1  0 and a2 x  b2 y  c2  0, where a1  k, b1  3, c1  (2  k) and a2  12, b2  k, c2  k. a c (2  k) (k  2) k b 3 ·   a1  12 ; 1  and c1   b k k k 2 2 2 Let the given system of equations have no solution. a b c Then, a1  1 ! c1 b2 2 2 (k  2) k  3  12 k ! k (k  2) k 3 3  12  and ! k k k  k 2  36 and k 2  2k ! 3k  k 2  36 and k 2  5k ! 0  (k  6 or k  6) and k (k  5) ! 0. Case 1. When k  6. In this case, k(k  5)  6(6  5)  6 # 1  6 ! 0. When k  6. In this case, k(k  5)  (6)(6  5)  ( 6)#(11)  66 ! 0. Thus, in each case, the given system has no solution. Hence, k  6 or k  6. Case 2.

EXAMPLE 12

Find the value of k for which the pair of linear equations (3k  1) x  3y  2  0, (k 2  1) x  (k  2) y  5  0 has no solution.

SOLUTION

The given linear equations are of the form a1 x  b1 y  c1  0 and a2 x  b2 y  c2  0, where a1  (3k  1), b1  3, c1  2 and

a2  (k 2  1), b2  (k  2), c2  5.

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Secondary School Mathematics for Class 10



a1 (3k  1) b1 c1 2 2  3   ·  a2 (k 2  1) , b2 (k  2) and c2 5 5

Let the given system of e quations have no solution. a b c Then, a1  1 ! c1 b2 2 2 (3k  1) 3 2   ! 5 (k 2  1) (k  2) (3k  1) 3 2  3  … (i) and !5 · (k  2) (k 2  1) (k  2) (3k  1)  3 & (3k  1)(k  2)  3(k 2  1) Now, 2 (k  1) (k  2) & 3k 2  6k  k  2  3k 2  3

… (ii)

& 5k  5 & k  1. 3 2 ! 5· (k  2) Hence, the given system of equations has no solution when k  1. When k  1, then clearly,

EXAMPLE 13

Find the value of k for which the following pair of linear equations has no solution: 3x  y  1, (2k  1) x  (k  1) y  2k  1.

SOLUTION

The given linear equations are 3x  y  1  0 (2k  1) x  (k  1) y  (2k  1)  0. These equations are of the form a1 x  b1 y  c1  0 and a2 x  b2 y  c2  0, where a1  3, b1  1, c1   1 and a2  (2k  1), b2  (k  1), c2  (2k  1) . 

a1 b1 c1 1 3 1 ·   1   a2 (2k  1) , b2 (k  1) , c2 (2k  1) (2k  1)

Let the given system of equations have no solution. a b c Then, a1  1 ! c1 b2 2 2   

3 1  1 ! (2k  1) (k  1) (2k  1) 3 1  1 and 1 ! (2k  1) (k  1) (k  1) (2k  1) 1 1 3k  3  2k  1 and  ! (k 1) (2k  1)

… (i) … (ii)

Linear Equations in Two Variables

121

1 1 · ! (k  1) (2k  1) 1 1 Clearly, when k  2, then  ! (k 1) (2k  1) 

k  2 and

;as

1 1 E· ! (2  1) (4  1)

Hence, the given system of equations will have no solution, when k  2. EXAMPLE 14

Find the values of k for which the system of equations kx  y  2, 6x  2y  3 has (i) a unique solution, (ii) no solution. (iii) Is there a value of k for which the given system has infinitely many solutions?

SOLUTION

The given system of equations is kx  y  2  0, 6x  2y  3  0. This is of the form a1 x  b1 y  c1  0 and a2 x  b2 y  c2  0, where a1  k, b1  1, c1  2 and a2  6, b2  2, c2  3. a b (i) For a unique solution, we must have a1 ! 1 · b2 2 1 k k 1  6 !  & 6 ! 2 & k ! 3. 2 Hence, the given system of equations will have a unique solution when k ! 3. a b c (ii) For no solution, we must have a1  1 ! c1 · b2 2 2 2 k 1  6 ! 2 3 k1 2  6 2!3 k 1 k 2  6  2 and 6 ! 3 & k  3 and k ! 4. Clearly, k  3 also satisfies the condition k ! 4. Hence, the given system of equations will have no solution when k  3. (iii) For infinitely many solutions, we must have a1 b1 c1   a2 b2 c2 ,

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k 1 1 1 1 i.e., 6  2  3 , which is never possible, as 2 ! 3 · Hence, there is no real value of k for which the given system of equations has infinitely many solutions. HOMOGENEOUS SYSTEM OF EQUATIONS

The system of equations a1 x  b1 y  0 and a2 x  b2 y  0 has a b (i) the zero solution, x  0 and y  0 when a1 ! 1 · b2 2 a b (ii) an infinite number of nonzero solutions when a1  1 · b2 2 Note

The homogeneous system of equations is always consistent.

EXAMPLE 15

Find the value of k for which the system of equations 3x  5y  0, kx  10y  0 has a nonzero solution.

SOLUTION

The given equations are of the form a1 x  b1 y  0 and a2 x  b2 y  0, where a1  3, b1  5 and a2  k, b2  10. a b 3 5 1 Then, a1  and 1  10  2 · k b 2 2 Let the given system of equations have a nonzero solution. a b 3 1 Then, a1  1 &  2 & k  6. b k 2 2 Hence, the required value of k is 6.

f

EXERCISE 3D

Show that each of the following systems of equations has a unique solution and solve it:

1. 3x  5y  12, 5x  3y  4. x y 3. 3  2  3, x  2y  2.

2. 2x  3y  17, 4x  y  13.

Find the value of k for which each of the following systems of equations has a unique solution:

4. 2x  3y  5  0, kx  6y  8  0. 6. 5x  7y  5  0, 2x  ky  1  0.

5. x  ky  2, 3x  2y  5  0. 7. 4x  ky  8  0, x  y  1  0.

Linear Equations in Two Variables

8. 4x  5y  k, 2x  3y  12.

123

9. kx  3y  (k  3), 12x  ky  k.

10. Show that the system of equations 2x  3y  5, 6x  9y  15 has an infinite number of solutions. 11. Show that the system of equations 15 6x  5y  11, 9x  2 y  21 has no solution. 12. For what value of k does the system of equations kx  2y  5, 3x  4y  10 have (i) a unique solution, (ii) no solution? 13. For what value of k does the system of equations x  2y  5, 3x  ky  15  0 have (i) a unique solution, (ii) no solution? 14. For what value of k does the system of equations x  2y  3, 5x  ky  7  0 have (i) a unique solution, (ii) no solution? Also, show that there is no value of k for which the given system of equations has infinitely many solutions. Find the value of k for which each of the following systems of linear equations has an infinite number of solutions:

15. 2x  3y  7, (k  1) x  (k  2) y  3k. 16. 2x  (k  2) y  k, 6x  (2k  1) y  (2k  5) . 17. kx  3y  (2k  1), 2(k  1) x  9y  (7k  1) . 18. 5x  2y  2k, 2(k  1) x  ky  (3k  4) . 19. (k  1) x  y  5, (k  1) x  (1  k) y  (3k  1) .

[CBSE 2010]

[CBSE 2000C]

[CBSE 2000C]

[CBSE 2003C]

[CBSE 2003]

20. (k  3) x  3y  k, kx  ky  12. Find the values of a and b for which each of the following systems of linear equations has an infinite number of solutions:

21. (a  1) x  3y  2, 6x  (1  2b) y  6.

[CBSE 2002C]

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22. (2a  1) x  3y  5, 3x  (b  1) y  2. 23. 2x  3y  7, (a  b) x  (a  b  3) y  4a  b. 24. 2x  3y  7, (a  b  1) x  (a  2b  2) y  4(a  b)  1. 25. 2x  3y  7, (a  b) x  (2a  b) y  21. 26. 2x  3y  7, 2ax  (a  b) y  28.

[CBSE 2001C]

[CBSE 2002]

[CBSE 2003]

[CBSE 2001]

[CBSE 2001]

Find the value of k for which each of the following systems of equations has no solution:

27. 8x  5y  9, kx  10y  15. 28. kx  3y  3, 12x  ky  6. 29. 3x  y  5  0, 6x  2y  k  0 (k ! 0). 30. kx  3y  k  3, 12x  ky  k.

[CBSE 2008] [CBSE 2009]

31. Find the value of k for which the system of equations 5x  3y  0, 2x  ky  0 has a nonzero solution. ANSWERS (EXERCISE 3D)

1. x  1, y  3

2. x  4, y  3

2

3. x  6, y  2

4. k ! 4

14

5. k ! 3 6. k ! 5 7. k ! 4 8. k is any real number 9. k is any real number other than 6 and –6

3

3

12. (i) k ! 2 (ii) k  2 15. k = 7

13. (i) k ! 6 (ii) k = 6

14. (i) k ! 10 (ii) k = 10

16. k = 5

17. k = 2 18. k = 4 19. k = 3 20. k = 6 17 11 21. a  3, b  4 22. a  4 , b  5 23. a  5, b  1 24. a  3, b  2 25. a  5, b  1 26. a  4, b  8 27. k  16 28. k  6 6 29. k ! 10 30. k  6 31. k  5 HINTS TO SOME SELECTED QUESTIONS k 3 9. 12 ! & k 2 ! 36 & k ! 6 and k ! 6. k So, for a unique solution, we may take any real value of k, other than 6 and –6.

Linear Equations in Two Variables

125

1 3 k 2 k 12. (i) 3 !  & 3 ! 2 & k ! 2 · 4 3 k 2 5 k 1 k 1 (ii) 3   ! 10 & 3  2 and 3 ! 2 & k  2 · 4 1 2 13. (i) 3 ! & k ! 6. k 5 1 12 1 2 (ii) 3 ! 15 & 3  ! 3 & k  6. k k 1 2 14. (i) 5 ! & k ! 10. k 3 12 (ii) 5 ! 7 & k  10. k 1 2 3 1 3 Clearly, 5   7 is never true, as 5  7 is false. k k  3  3  k k3  3 3 k 20. & and  12 · k k 12 k k k 

k 2  6k  0 and k 2  36 & k (k  6)  0 and k 2  36



(k  0 or k  6) and (k  6 or k  6)

 k  6. a1 3  2  1 21. 6   1 2b 6 3 a1  1 3 1  6 3 and 1  2b 3  3a  3  6 and 1  2b  9 3a  9 and 2b  8 & a  3 and b  4. 3 k 3 k 3 3 1 28. 12 !  & 12  and ! 2 k k k 6 

& k 2  36 and k ! 6 & k  6.  3 k k 3 30. 12  ! & k 2  36 and k (k  6) ! 0 & k  6. k k

WORD PROBLEMS SOLVED EXAMPLES PROBLEMS ON MONEY MATTERS EXAMPLE 1

8 chairs and 5 tables for a classroom cost ` 10500, while 5 chairs and 3 tables cost ` 6450. Find the cost of each chair and that of each table.

SOLUTION

Let the cost of each chair be ` x and that of each table be ` y. Then, 8x  5y  10500

… (i)

and 5x  3y  6450.

… (ii)

On multiplying (ii) by 5 and (i) by 3 and subtracting the results, we get

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Secondary School Mathematics for Class 10

25x  24x  32250  31500 & x  750. Putting x  750 in (i), we get 8 # 750  5y  10500 & 6000  5y  10500 

5y  (10500  6000)  4500



y  900.



cost of each chair = ` 750 and cost of each table = ` 900.

EXAMPLE 2

The coach of a cricket team buys 7 bats and 6 balls for ` 13200. Later, he buys 3 bats and 5 balls for ` 5900. Find the cost of each bat and each ball.

SOLUTION

Let the cost of each bat be ` x and the cost of each ball be ` y. Then, 7x  6y  13200.

… (i)

And, 3x  5y  5900.

… (ii)

On multiplying (i) by 5, (ii) by 6 and subtracting the results, we get 35x  18x  66000  35400 & 17x  30600 30600   x 1800. 17 Putting x  1800 in (ii), we get 

5400  5y  5900 & 5y  5900  5400 5y  500



y  100.



cost of each bat = ` 1800 and cost of each ball = ` 100.

EXAMPLE 3

37 pens and 53 pencils together cost ` 955, while 53 pens and 37 pencils together cost ` 1115. Find the cost of a pen and that of a pencil.

SOLUTION

Let the cost of each pen be ` x and that of a pencil be ` y. Then, 37x  53y  955.

… (i)

And, 53x  37y  1115.

… (ii)

Adding (i) and (ii), we get 90x  90y  2070 & 90(x  y)  2070  

2070 90 x  y  23.

xy 

… (iii)

Linear Equations in Two Variables

127

On subtracting (i) from (ii), we get 16x  16y  160 & 16 (x  y)  160  (x  y)  10.

… (iv) 33    Adding (iii) and (iv), we get 2x 33 & x 2 16.50. 13 Subtracting (iv) from (iii), we get 2y  13 & y  2  6.50.  cost of each pen = ` 16.50 and cost of each pencil = ` 6.50. EXAMPLE 4

Taxi charges in a city consist of fixed charges and the remaining depending upon the distance travelled in kilometres. If a person travels 60 km, he pays ` 960, and for travelling 80 km, he pays ` 1260. Find the fixed charges and the rate per kilometre.

SOLUTION

Let the fixed charges be ` x and the other charges be ` y per km. Then, x  60y  960.

… (i)

And, x  80y  1260.

… (ii)

On subtracting (i) from (ii), we get 300 20y  300 & y  20 & y  15. Putting y  15 in (i), we get x  (60 # 15)  960 & x  960  900 & x  60. 

fixed charges = ` 60 and the rate per km = ` 15 per km.

EXAMPLE 5

A part of monthly hostel charges in a school is fixed and the remaining depends on the number of days one has taken food in the mess. When a student A takes food for 22 days, he has to pay ` 4250 as hostel charges, whereas a student B, who takes food for 28 days, pays ` 5150 as hostel charges. Find the fixed charges and the cost of food per day.

SOLUTION

Let the fixed charges be ` x per month and the cost of meals per day be ` y. Then, we have x  22y  4250

… (i)

and x  28y  5150.

… (ii)

On subtracting (i) from (ii), we get 6y  900 & y  150. Putting y  150 in (i), we get x  (22 # 150)  4250

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Secondary School Mathematics for Class 10

  

x  3300  4250 x  4250  3300  950. x  950 and y  150.

Hence, the fixed charges are ` 950 per month and the cost of food is ` 150 per day. EXAMPLE 6

The monthly incomes of A and B are in the ratio 8 : 7 and their expenditures are in the ratio 19 : 16. If each saves ` 5000 per month, find the monthly income of each.

SOLUTION

Let the monthly incomes of A and B be ` 8x and ` 7x respectively, and let their expenditures be ` 19y and ` 16y respectively. Then, A’s monthly savings = ` (8x  19y) . And, B’s monthly savings = ` (7x  16y) . But, the monthly saving of each is ` 5000. 8x  19y  5000

… (i)

and 7x  16y  5000.

… (ii)



Multiplying (ii) by 19, (i) by 16 and subtracting the results, we get (19 # 7  16 # 8) x  (19 # 5000  16 # 5000)  (133  128) x  5000 #(19  16)  5x  15000 & x  3000. 

A’s monthly income = ` (8x) = ` (8 # 3000)  ` 24000.

And, B’s monthly income = ` (7x) = ` (7 # 3000)  ` 21000. EXAMPLE 7

On selling a TV at 5% gain and a fridge at 10% gain, a shopkeeper gains ` 3250. But, if he sells the TV at 10% gain and the fridge at 5% loss, he gains ` 1500. Find the actual cost price of TV and that of the fridge.

SOLUTION

Let the cost price of the TV set be ` x and that of the fridge be ` y. Then, total CP of TV and fridge = ` (x  y) . 105x  110y m  c 21x  11y m · SP in first case = ` c ` 20 100 100 10 

y 21x 11y x gain in this case = ` 'c 20  10 m  (x  y)1  ` a 20  10 k ·

Linear Equations in Two Variables

129

y x So, 20  10  3250 & x  2y  65000. … (i) 110x  95y m  c11x  19y m · SP in second case = ` c ` 10 20 100 100 19 y y 11x x  gain in this case = ` 'c 10  20 m  (x  y)1  ` a10  20 k · y x So, 10  20  1500 & 2x  y  30000. … (ii) Multiplying (ii) by 2 and adding the result with (i), we get 5x  60000  65000 & 5x  125000 & x  25000. Putting x  25000 in (i), we get 25000  2y  65000 & 2y  65000  25000  40000 

y  20000.



x  25000 and y  20000.

Hence, the CP of the TV set is ` 25000 and that of the fridge is ` 20000. EXAMPLE 8

A man invested an amount at 12% per annum simple interest and another amount at 10% per annum simple interest. He received an annual interest of ` 2600. But, if he had interchanged the amounts invested, he would have received ` 140 less. What amounts did he invest at the different rates?

SOLUTION

Let the amount invested at 12% be ` x and that invested at 10% be ` y. Then, total annual interest x # 12 # 1  y # 10 # 1 m  c 6x  5y m · =`c ` 50 100 100 6x  5y  2600 & 6x  5y  130000.  … (i) 50 Again, the amount invested at 12% is ` y and that invested at 10% is ` x. Total annual interest at the new rates y # 12 # 1 x # 10 # 1 6y  5x m· m `c  =`c 50 100 100 But, interest received at the new rates = ` (2600  140) = ` 2460. 

6y  5x  2460 & 5x  6y  123000. 50

… (ii)

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Secondary School Mathematics for Class 10

Adding (i) and (ii), we get 11x  11y  253000 & 11 (x  y)  253000 

x  y  23000.

… (iii)

Subtracting (ii) from (i), we get x  y  7000.

… (iv)

Adding (iii) and (iv), we get 2x  30000 & x  15000. Putting x  15000 in (i), we get 15000  y  23000 & y  23000  15000  8000. 

x  15000 and y  8000.

Hence, the amount at 12% is ` 15000 and that at 10% is ` 8000. EXAMPLE 9

Each one of A and B has some money. If A gives ` 30 to B then B will have twice the money left with A. But, if B gives ` 10 to A then A will have thrice as much as is left with B. How much money does each have?

SOLUTION

Suppose A and B have ` x and ` y respectively. When A gives ` 30 to B.

Case I

Then, money with A = ` (x  30) money with B = ` (y  30) .

and 

(y  30)  2(x  30) & 2x  y  90.

Case II

… (i)

When B gives ` 10 to A.

Then, money with A = ` (x  10) and 

money with B = ` (y  10) . (x  10)  3(y  10) & x  3y  40.

… (ii)

On multiplying (ii) by 2 and subtracting the result from (i), we get 170 5y  170 & y  5  34. Putting y  34 in (i), we get



124 2x  34  90 & 2x  (90  34)  124 & x  2  62. x  62 and y  34.

Hence, A has ` 62 and B has ` 34.

Linear Equations in Two Variables

131

PROBLEMS ON NUMBERS EXAMPLE 10

The students of a class are made to stand in rows. If 4 students are extra in each row, there would be 2 rows less. If 4 students are less in each row, there would be 4 rows more. Find the number of students in the class.

SOLUTION

Let the number of rows be x and the number of students in each row be y. Then, the total number of students = xy. When there are 4 more students in each row.

Case I

Then, number of students in each row  (y  4). And, number of rows  (x  2). Total number of students  (x  2) (y  4). 

(x  2) (y  4)  xy & 4x  2y  8

& 2x  y  4. Case II

… (i)

When 4 students are removed from each row.

Then, number of students in each row  (y  4). And, number of rows  (x  4). Total number of students  (x  4)(y  4). 

(x  4)(y  4)  xy & 4y  4x  16

& 4(y  x)  16 & (y  x)  4.

… (ii)

Adding (i) and (ii), we get x  8. Putting x  8 in (ii), we get y  8  4 & y  12. Thus, x  8 and y  12. This shows that there are 8 rows and there are 12 students in each row. Hence, the number of students in the class  xy  8 #12  96. EXAMPLE 11

The sum of two numbers is 1000 and the difference between their squares is 256000. Find the numbers.

SOLUTION

Let the larger number be x and the smaller number be y. Then, x  y  1000

… (i)

and x  y  256000.

… (ii)

2

2

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Secondary School Mathematics for Class 10

On dividing (ii) by (i), we get x 2  y 2 256000    1000 & x y 256. xy

… (iii)

Adding (i) and (iii), we get 2x  1256 & x  628. Substituting x = 628 in (i), we get y  372. Hence, the required numbers are 628 and 372. EXAMPLE 12

If three times the larger of two numbers is divided by the smaller one, we get 4 as the quotient and 3 as the remainder. Also, if seven times the smaller number is divided by the larger one, we get 5 as the quotient and 1 as the remainder. Find the numbers.

SOLUTION

Let the larger number be x and the smaller one be y. We know that dividend = (divisor × quotient) + remainder. Using the above result and the given conditions, we have 3x  4y  3 & 3x  4y  3 and 7y  5x  1 & 5x  7y  1.

… (i) … (ii)

Multiplying (i) by 5, (ii) by 3 and subtracting, we get y  18. Putting y  18 in (i), we get 3x  (4 # 18)  3 & 3x  72  3 

3x  75 & x  25.

Hence, the required numbers are 25 and 18. EXAMPLE 13

SOLUTION

8 The sum of two numbers is 8 and the sum of their reciprocals is 15 · Find the numbers. [CBSE 2009] Let the required numbers be x and y. Then, x  y  8. xy 8 1 1 8 And, x  y  15 & xy  15 8  8 & xy 15  xy 15 . & 

(x  y)  (x  y) 2  4xy  8 2  4 # 15  64  60  4  !2.

… (i)

[using (i)]

Linear Equations in Two Variables

133

Thus, we have xy  8 xy  2

… (i) … (ii)

3

or

)

xy  8

… (iii)

x  y  2

… (iv)

On solving (i) and (ii), we get x  5 and y  3. On solving (iii) and (iv), we get x  3 and y  5. Hence, the required numbers are 5 and 3. EXAMPLE 14

The difference of two numbers is 4 and the difference of their 4 reciprocals is 21 · Find the numbers. [CBSE 2008]

SOLUTION

Let the larger number be x and the smaller number be y. Then, x  y  4.

… (i) ;a x  y & 1y  1xE

1 1 4 And, y  x  21   

xy  4 xy 21 4  4  xy 21 & xy 21

[using (i)].

(x  y)  (x  y) 2  4xy

 4 2  4 # 21  16  84  100  !10. Thus, we have xy  4

… (i)

x  y  10

… (ii)

3

or

)

xy  4

… (iii)

x  y  10

… (iv)

On solving (i) and (ii), we get x  7 and y  3. On solving (iii) and (iv), we get x  3 and y  7. Hence, the required numbers are (7 and 3) or (–3 and –7). EXAMPLE 15

The sum of the digits of a two-digit number is 12. The number obtained by interchanging its digits exceeds the given number by 18. Find the number. [CBSE 2006]

SOLUTION

Let the ten‘s digit of the required number be x and the unit‘s digit be y. Then, … (i) x  y  12. Required number  (10x  y). Number obtained on reversing the digits  (10y  x).

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Secondary School Mathematics for Class 10



(10y  x)  (10x  y)  18 & 9y  9x  18

& y  x  2.

… (ii)

On adding (i) and (ii), we get 2y  14 & y  7. Putting y  7 in (i), we get x  7  12 & x  12  7  5. 

x  5 and y  7.

Hence, the required number is 57. EXAMPLE 16

The sum of a two-digit number and the number obtained by reversing the order of its digits is 99. If the digits differ by 3, find the number. [CBSE 2002]

SOLUTION

Let the ten‘s and unit‘s digits of the required number be x and y respectively. Then, the number  (10x  y). The number obtained on reversing the digits  (10y  x) . 

(10y  x)  (10x  y)  99 & 11(x  y)  99 & x  y  9.

Also, (x  y)  !3. Thus, we have xy  9

… (i)

xy  3

… (ii)

3

or

)

xy  9

… (iii)

x  y  3

… (iv)

From (i) and (ii), we get x  6, y  3. From (iii) and (iv), we get x  3, y  6. Hence, the required number is 63 or 36. EXAMPLE 17

Seven times a two-digit number is equal to four times the number obtained by reversing the order of its digits. If the difference between the digits is 3, find the number.

SOLUTION

Let the ten‘s and unit‘s digits of the required number be x and y respectively. Then, the number  (10x  y). The number obtained by reversing the digits  (10y  x). 

7(10x  y)  4(10y  x) & 33(2x  y)  0



2x  y  0 & y  2x

… (i)

Linear Equations in Two Variables

135

Thus, unit‘s digit = 2 times the ten‘s digit. 

(unit‘s digit) > (ten‘s digit) and so y  x.

 y  x  3.

… (ii)

Using (i) in (ii), we get (2x  x)  3 & x  3. On substituting x  3 in (i), we get y  2 # 3  6. Hence, the required number is 36. EXAMPLE 18

A two-digit number is such that the product of its digits is 14. If 45 is added to the number, the digits interchange their places. Find the number. [CBSE 2005]

SOLUTION

Let the ten‘s and unit‘s digits of the required number be x and y respectively. Then, xy  14. Required number  (10x  y). Number obtained on reversing its digits  (10y  x). 

(10x  y)  45  (10y  x)



9(y  x)  45 & y  x  5.

… (i)

Now, (y  x)  (y  x)  4xy 2

2

 (y  x)  (y  x) 2  4xy  25  4 # 14  81 

yx  9

… (ii)

[£ digits are never negative].

On adding (i) and (ii), we get 2y  14 & y  7. Putting y  7 in (ii), we get 7  x  9 & x  9  7  2. 

x  2 and y  7.

Hence, the required number is 27. EXAMPLE 19

A two-digit number is four times the sum of its digits and twice the product of its digits. Find the number. [CBSE 2005]

SOLUTION

Let the ten‘s digit of the required number be x and its unit‘s digit be y. Then, 10x  y  4(x  y) & 6x  3y  0 & 2x  y  0.

… (i)

Also, 10x  y  2xy.

… (ii)

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Secondary School Mathematics for Class 10

Putting y  2x from (i) in (ii), we get 10x  2x  4x 2 & 4x 2  12x  0

& 4x(x  3)  0 & x  3  0 & x  3 [£ ten‘s digit, x ! 0] . Putting x  3 in (i), we get y  6. Thus, ten‘s digit = 3 and unit‘s digit = 6. Hence, the required number is 36. EXAMPLE 20

SOLUTION

1 A fraction becomes 3 , if 2 is added to both of its numerator and denominator. If 3 is added to both of its numerator and denominator 2 then it becomes 5 · Find the fraction. [CBSE 2009C] x· Let the required fraction be y Then, x2  1 & 3x  6  y  2 y2 3 … (i) & 3x  y  4. Also,

x3  2 & 5x  15  2y  6 y3 5 & 5x  2y  9.

… (ii)

Multiplying (i) by 2 and subtracting (ii) from the result, we get 6x  5x   8  9 & x  1 . Putting x  1 in (i), we get 3  y   4 & y  7. Thus, x  1 and y  7. 1 Hence, the required fraction is 7 · EXAMPLE 21

SOLUTION

The sum of numerator and denominator of a fraction is 3 less than twice the denominator. If each of the numerator and denominator is 1 decreased by 1, the fraction becomes 2 · Find the fraction. [CBSE 2010] x Let the required fraction be y · Then, … (i) (x  y)  2y  3 & x  y  3. And,

x1  1 & 2x  2  y  1 y1 2

& 2x  y  1.

… (ii)

Linear Equations in Two Variables

137

On subtracting (i) from (ii), we get x  4. Putting x  4 in (i), we get y  x  3  4  3  7. 

x  4 and y  7.

4 Hence, the required fraction is 7 · EXAMPLE 22

SOLUTION

In a given fraction, if the numerator is multiplied by 2 and the 6 denominator is reduced by 5, we get a 5 k · But if the numerator of the given fraction is increased by 8 and the denominator is doubled, we 2 get a 5 k · Find the fraction. x Let the required fraction be y · Then, 2x  6 & 10x  6(y  5) y5 5 & 10x  6y  30

& 5x  3y  15

… (i)

x8 2 and 2y  5 & 5(x  8)  4y & 5x  4y  40.

… (ii)

On subtracting (ii) from (i), we get y  25. Putting y  25 in (i), we get 5x  (3 # 25)  15 & 5x  75  15 & 5x  60 & x  12.  x  12 and y  25. 12 Hence, the required fraction is 25 · PROBLEMS ON AGES EXAMPLE 23

Five years ago, A was thrice as old as B and ten years later A shall be twice as old as B. What are the present ages of A and B? [CBSE 2002C]

SOLUTION

Let the present ages of B and A be x years and y years respectively. Then, B’s age 5 years ago  (x  5) years and A’s age 5 years ago  (y  5) years. 

(y  5)  3(x  5) & 3x  y  10.

… (i)

B’s age 10 years hence  (x  10) years. A’s age 10 years hence  (y  10) years. 

(y  10)  2(x  10) & 2x  y  10.

… (ii)

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Secondary School Mathematics for Class 10

On subtracting (ii) from (i), we get x  20. Putting x  20 in (i), we get (3 # 20)  y  10 & y  60  10  50. 

x  20 and y  50.

Hence, B’s present age = 20 years and

A’s present age = 50 years.

EXAMPLE 24

A man‘s age is three times the sum of the ages of his two sons. After 5 years, his age will be twice the sum of the ages of his two sons. Find the age of the man. [CBSE 2003]

SOLUTION

Let the present age of the man be x years and the sum of the present ages of his two sons be y years. Then, x  3y & x  3y  0.

… (i)

Man‘s age after 5 years  (x  5) years. Sum of the ages of his two sons after 5 years  (y  5  5) years  (y  10) years. 

(x  5)  2(y  10) & x  2y  15.

… (ii)

On subtracting (i) from (ii), we get y  15. Putting y  15 in (i), we get x  45. 

x  45 and y  15.

Hence, the present age of the man is 45 years. PROBLEMS ON TIME AND DISTANCE EXAMPLE 25

A man travels 370 km, partly by train and partly by car. If he covers 250 km by train and the rest by car, it takes him 4 hours. But, if he travels 130 km by train and the rest by car, he takes 18 minutes longer. Find the speed of the train and that of the car.

SOLUTION

Let the speed of the train be x km/hr and that of the car be y km/hr. Case I

Distance covered by train = 250 km.

Distance covered by car  (370  250) km  120 km. 250 Time taken to cover 250 km by train  x hours. 120 Time taken to cover 120 km by car  y hours.

Linear Equations in Two Variables

Total time taken = 4 hours. 250  120  125 60  4 & x  y  2. x y Case II

139

… (i)

Distance covered by train = 130 km.

Distance covered by car  (370  130) km  240 km. 130 Time taken to cover 130 km by train  x hours. 240 Time taken to cover 240 km by car  y hours. 18 3 43 Total time taken  4 60 hours  4 10 hours  10 hours. 130  240  43 1300  2400   … (ii) 43. x y y 10 & x 1 1 Putting x  u and y  v, equations (i) and (ii) become 125u  60v  2

… (iii)

and 1300u  2400v  43. … (iv)

On multiplying (iii) by 40 and subtracting (iv) from the result, we get 5000u  1300u  80  43 & 3700u  37 37 1 1 1  u  3700  100 & x  100 & x  100. 1 Putting u  100 in (iv), we get 1  a1300 # 100 k 2400v  43 & 2400v  43  13  30 30 1 1 1  v  2400  80 & y  80 & y  80. 

x  100 and y  80.

Hence, the speed of the train is 100 km/hr and that of the car is 80 km/hr. EXAMPLE 26

Places A and B are 100 km apart on a highway. One car starts from A and another from B at the same time. If the cars travel in the same direction at different speeds, they meet in 5 hours. If they travel towards each other, they meet in 1 hour. What are the speeds of the two cars? [CBSE 2009]

SOLUTION

Let the speeds of the cars from A and B be x km/hr and y km/hr respectively. Case I

When the two cars move in the same direction:

In this case, let them meet at M after 5 hours.

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Secondary School Mathematics for Class 10

Then, AM  5x km. And, BM  5y km. 

AM  BM  AB & 5x  5y  100



5(x  y)  100 & x  y  20.

Case II

… (i)

When the two cars move in the opposite directions:

Let one car move from A to B and let the other move from B to A. Let them meet at N after 1 hour. Then, AN  x km and BN  y km. 

AN  BN  AB & x  y  100.

… (ii)

Adding (i) and (ii), we get 2x  120 & x  60. Putting x  60 in (ii), we get 60  y  100 & y  40. 

speed of the car from A  60 km/hr,

and speed of the car from B  40 km/hr. EXAMPLE 27

A train covered a certain distance at a uniform speed. If the train had been 6 kmph faster, it would have taken 4 hours less than the scheduled time. And, if the train were slower by 6 kmph, it would have taken 6 hours more than the scheduled time. Find the length of the journey.

SOLUTION

Let the original speed of the train be x kmph and let the time taken to complete the journey be y hours. Length of whole journey  (xy) km. Case I

When speed  (x  6) kmph and time taken  (y  4) hours.

Length of total journey  (x  6)(y  4) km. 

xy  (x  6)(y  4) & xy  xy  4x  6y  24



4x  6y  24 & 2x  3y  12.

Case II

… (i)

When speed  (x  6) kmph and time taken  (y  6) hours.

Length of total journey  (x  6)(y  6) km.

Linear Equations in Two Variables



xy  (x  6)(y  6) & xy  xy  6x  6y  36



6x  6y  36 & x  y  6.

141

… (ii)

On multiplying (ii) by 3 and subtracting (i) from the result, we get 3x  2x  18  12 & x  30. On putting x  30 in (ii), we get y  30  6  24. 

x  30 and y  24.

So, length of journey = (xy) km  (30 # 24) km = 720 km. PROBLEMS ON BOATS AND STREAM EXAMPLE 28

A man can row downstream 20 km in 2 hours, and upstream 4 km in 2 hours. Find his speed of rowing in still water. Also, find the speed of the current.

SOLUTION

Let the speed of the man in still water be x km/hr and let the speed of the current be y km/hr. Speed downstream  (x  y) km/hr. Speed upstream  (x  y) km/hr. But, speed downstream  And, speed upstream  

x  y  10

distance  20 time 2 km/hr = 10 km/hr.

distance  4 time 2 km/hr = 2 km/hr.

… (i)

and x  y  2

… (ii).

Adding (i) and (ii), we get 2x  12 & x  6. Putting x  6 in (i), we get 6  y  10 & y  4. Thus, x  6 and y  4. Hence, the speed of the man in still water is 6 km/hr and the speed of the current is 4 km/hr. EXAMPLE 29

A boat goes 16 km upstream and 24 km downstream in 6 hours. Also, it covers 12 km upstream and 36 km downstream in the same time. Find the speed of the boat in still water and that of the stream.

SOLUTION

Let the speed of the boat in still water be x km/hr and the speed of the stream be y km/hr. Then,

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Secondary School Mathematics for Class 10

speed upstream  (x  y) km/hr and speed downstream  (x  y) km/hr. Time taken to cover 16 km upstream 

16 hours. (x  y)

Time taken to cover 24 km downstream 

24 hours. (x  y)

Total time taken = 6 hours. 

16  24  6. xy xy

… (i)

Again, time taken to cover 12 km upstream  Time taken to cover 36 km downstream 

12 hours. (x  y)

36 hours. (x  y)

Total time taken = 6 hours. 

12  36  6. xy xy

Putting

… (ii)

1  1 u and   v in (i) and (ii), we get (x  y) (x y)

16u  24v  6 & 8u  12v  3,

… (iii)

12u  36v  6 & 2u  6v  1.

… (iv)

On multiplying (iv) by 4 and subtracting (iii) from the result, we get 1 12v  1 & v  12 1  1  & x  y  12. x  y 12

… (v)

On multiplying (iv) by 2 and subtracting the result from (iii), we get



1 4u  1 & u  4 1 1 & x  y  4. xy 4

On adding (v) and (vi), we get 2x  16 & x  8. On subtracting (vi) from (v), we get 2y  8 & y  4. 

speed of the boat in still water = 8 km/hr,

and speed of the stream = 4 km/hr.

… (vi)

Linear Equations in Two Variables

143

PROBLEMS ON TIME AND WORK EXAMPLE 30

8 men and 12 boys can finish a piece of work in 5 days, while 6 men and 8 boys can finish it in 7 days. Find the time taken by 1 man alone and that by 1 boy alone to finish the work.

SOLUTION

Suppose 1 man alone can finish the work in x days and 1 boy alone can finish it in y days. 1 Then, 1 man’s 1 day’s work  x · 1 And, 1 boy’s 1 day’s work  y · 8 men and 12 boys can finish the work in 5 days 1 (8 men’s 1 day’s work) + (12 boys’ 1 day’s work)  5 8 12 1  x y 5 1 1 1  8u  12v  5 … (i), where x  u and y  v. 

Again, 6 men and 8 boys can finish the work in 7 days 1 (6 men’s 1 day’s work) + (8 boys’ 1 day’s work)  7 6 8 1  xy7 1  6u  8v  7 · … (ii) On multiplying (i) by 3, (ii) by 4 and subtracting the results, we get 3 4 1 1 1 1 4v  a 5  7 k  35 & v  140 & y  140 & y  140. 1 On putting v  140 in (ii), we get 1 8 12 12 1 1 6u  a7  140 k  140 & u  a140 # 6 k  70 1 1  x  70 & x  70.  one man alone can finish the work in 70 days,



and one boy alone can finish the work in 140 days. PROBLEMS ON AREA EXAMPLE 31

If the length of a rectangle is reduced by 5 units and its breadth is increased by 2 units then the area of the rectangle is reduced by 80 sq units. However, if we increase its length by 10 units and decrease

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Secondary School Mathematics for Class 10

the breadth by 5 units, its area is increased by 50 sq units. Find the length and breadth of the rectangle. SOLUTION

Let the length and breadth of the rectangle be x units and y units respectively. Then, area of the rectangle = xy sq units. Case I

When the length is reduced by 5 units and the breadth is increased by 2 units.

Then, new length  (x  5) units and new breadth  (y  2) units. 

new area  (x  5)(y  2) sq units.



xy  (x  5)(y  2)  80 & 5y  2x  70.

Case II

… (i)

When the length is increased by 10 units and the breadth is decreased by 5 units.

Then, new length  (x  10) units and new breadth  (y  5) units.  new area  (x  10)(y  5) sq units. 

(x  10)(y  5)  xy  50

 10y  5x  100 & 2y  x  20.

… (ii)

On multiplying (ii) by 2 and subtracting the result from (i), we get y  30. Putting y  30 in (ii), we get (2 # 30)  x  20 & 60  x  20 & x  (60  20)  40. 

x  40 and y  30.

Hence, length = 40 units and breadth = 30 units. PROBLEMS ON GEOMETRY EXAMPLE 32

In a 3ABC, +C  3+B  2(+A  +B). Find the angles.

SOLUTION

Let +A  xc and +B  yc. Then, +C  3+B  (3y)c. Now, +C  2(+A  +B) 

3y  2(x  y) & 2x  y  0.

We know that the sum of the angles of a triangle is 180c.

… (i)

Linear Equations in Two Variables

145

 +A  +B  +C  180c & x  y  3y  180

& x  4y  180.

… (ii)

On multiplying (i) by 4 and adding the result with (ii), we get 8x  x  180 & 9x  180 & x  20. Putting x  20 in (i), we get y  (2 # 20)  40. Thus, x  20 and y  40.  +A  20c, +B  40c and +C  (3 # 40c)  120c. EXAMPLE 33

Find the four angles of a cyclic quadrilateral ABCD in which +A  (2x  1)c, +B  (y  5)c, +C  (2y  15)c and +D  (4x  7)c.

SOLUTION

We know that the sum of the opposite angles of a cyclic quadrilateral is 180c.  +A  +C  180c and +B  +D  180c. Now, +A  +C  180c & (2x  1)  (2y  15)  180

& 2(x  y)  166 & x  y  83. And, +B  +D  180c & (y  5)  (4x  7)  180 & 4x  y  182.

… (i) … (ii)

On subtracting (i) from (ii), we get 3x  182  83  99 & x  33. Putting x  33 in (i), we get 33  y  83 & y  (83  33) & y  50. Thus, x  33 and y  50.  +A  {(2 # 33)  1}c  65c, +B  (50  5)c  55c. +C  {(2 # 50)  15}c  115c and +D  {(4 # 33)  7}c  125c. Hence, +A  65c, +B  55c, +C  115c and +D  125c.

f

EXERCISE 3E

1. 5 chairs and 4 tables together cost ` 5600, while 4 chairs and 3 tables together cost ` 4340. Find the cost of a chair and that of a table. 2. 23 spoons and 17 forks together cost ` 1770, while 17 spoons and 23 forks together cost ` 1830. Find the cost of a spoon and that of a fork.

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Secondary School Mathematics for Class 10

3. A lady has only 25-paisa and 50-paisa coins in her purse. If she has 50 coins in all totalling ` 19.50, how many coins of each kind does she have? 4. The sum of two numbers is 137 and their difference is 43. Find the numbers. 5. Find two numbers such that the sum of twice the first and thrice the second is 92, and four times the first exceeds seven times the second by 2. 6. Find two numbers such that the sum of thrice the first and the second is 142, and four times the first exceeds the second by 138. 7. If 45 is subtracted from twice the greater of two numbers, it results in the other number. If 21 is subtracted from twice the smaller number, it results in the greater number. Find the numbers. 8. If three times the larger of two numbers is divided by the smaller, we get 4 as the quotient and 8 as the remainder. If five times the smaller is divided by the larger, we get 3 as the quotient and 5 as the remainder. Find the numbers. 9. If 2 is added to each of two given numbers, their ratio becomes 1 : 2. However, if 4 is subtracted from each of the given numbers, the ratio becomes 5 : 11. Find the numbers. 10. The difference between two numbers is 14 and the difference between their squares is 448. Find the numbers. 11. The sum of the digits of a two-digit number is 12. The number obtained by interchanging its digits exceeds the given number by 18. Find the [CBSE 2006] number. 12. A number consisting of two digits is seven times the sum of its digits. When 27 is subtracted from the number, the digits are reversed. Find the number. 13. The sum of the digits of a two-digit number is 15. The number obtained by interchanging the digits exceeds the given number by 9. Find the number. [CBSE 2004] 14. A two-digit number is 3 more than 4 times the sum of its digits. If 18 is added to the number, the digits are reversed. Find the number. 15. A number consists of two digits. When it is divided by the sum of its digits, the quotient is 6 with no remainder. When the number is diminished by 9, the digits are reversed. Find the number. [CBSE 1999C]

Linear Equations in Two Variables

147

16. A two-digit number is such that the product of its digits is 35. If 18 is added to the number, the digits interchange their places. Find the [CBSE 2006] number. 17. A two-digit number is such that the product of its digits is 18. When 63 is subtracted from the number, the digits interchange their places. Find [CBSE 2006C] the number. 18. The sum of a two-digit number and the number obtained by reversing the order of its digits is 121, and the two digits differ by 3. Find the number. 19. The sum of the numerator and denominator of a fraction is 8. If 3 is added to both of the numerator and the denominator, the fraction 3 becomes 4 · Find the fraction. [CBSE 2003] 1 20. If 2 is added to the numerator of a fraction, it reduces to a 2 k and if 1 is 1 subtracted from the denominator, it reduces to a 3 k · Find the fraction. 21. The denominator of a fraction is greater than its numerator by 11. If 8 is 3 added to both its numerator and denominator, it becomes 4 · Find the fraction. 1 22. Find a fraction which becomes a 2 k when 1 is subtracted from the numerator and 2 is added to the denominator, and the fraction becomes a13 k when 7 is subtracted from the numerator and 2 is subtracted from the denominator. 23. The sum of the numerator and denominator of a fraction is 4 more than twice the numerator. If the numerator and denominator are increased by 3, they are in the ratio 2 : 3. Determine the fraction. [CBSE 2010] 1 24. The sum of two numbers is 16 and the sum of their reciprocals is 3 · Find the numbers. [CBSE 2005] 25. There are two classrooms A and B. If 10 students are sent from A to B, the number of students in each room becomes the same. If 20 students are sent from B to A, the number of students in A becomes double the number of students in B. Find the number of students in each room. 26. Taxi charges in a city consist of fixed charges and the remaining depending upon the distance travelled in kilometres. If a man travels

148

Secondary School Mathematics for Class 10

80 km, he pays ` 1330, and travelling 90 km, he pays ` 1490. Find the fixed charges and rate per km. 27. A part of monthly hostel charges in a college hostel are fixed and the remaining depends on the number of days one has taken food in the mess. When a student A takes food for 25 days, he has to pay ` 4500, whereas a student B who takes food for 30 days, has to pay ` 5200. Find the fixed charges per month and the cost of the food per day. 28. A man invested an amount at 10% per annum and another amount at 8% per annum simple interest. Thus, he received ` 1350 as annual interest. Had he interchanged the amounts invested, he would have received ` 45 less as interest. What amounts did he invest at different rates? 29. The monthly incomes of A and B are in the ratio 5 : 4 and their monthly expenditures are in the ratio 7 : 5. If each saves ` 9000 per month, find the monthly income of each. 30. A man sold a chair and a table together for ` 1520, thereby making a profit of 25% on chair and 10% on table. By selling them together for ` 1535, he would have made a profit of 10% on the chair and 25% on the table. Find the cost price of each. 31. Points A and B are 70 km apart on a highway. A car starts from A and another car starts from B simultaneously. If they travel in the same direction, they meet in 7 hours. But, if they travel towards each other, they meet in 1 hour. Find the speed of each car. [CBSE 2007C] 32. A train covered a certain distance at a uniform speed. If the train had been 5 kmph faster, it would have taken 3 hours less than the scheduled time. And, if the train were slower by 4 kmph, it would have taken 3 hours more than the scheduled time. Find the length of the journey. 33. Abdul travelled 300 km by train and 200 km by taxi taking 5 hours 30 minutes. But, if he travels 260 km by train and 240 km by taxi, he takes 6 minutes longer. Find the speed of the train and that of the taxi. [CBSE 2006C]

34. Places A and B are 160 km apart on a highway. One car starts from A and another from B at the same time. If they travel in the same direction, they meet in 8 hours. But, if they travel towards each other, they meet in 2 hours. Find the speed of each car. [CBSE 2009C] 35. A sailor goes 8 km downstream in 40 minutes and returns in 1 hour. Find the speed of the sailor in still water and the speed of the current.

Linear Equations in Two Variables

149

36. A boat goes 12 km upstream and 40 km downstream in 8 hours. It can go 16 km upstream and 32 km downstream in the same time. Find the speed of the boat in still water and the speed of the stream. 37. 2 men and 5 boys can finish a piece of work in 4 days, while 3 men and 6 boys can finish it in 3 days. Find the time taken by one man alone to finish the work and that taken by one boy alone to finish the work. 38. The length of a room exceeds its breadth by 3 metres. If the length is increased by 3 metres and the breadth is decreased by 2 metres, the area remains the same. Find the length and the breadth of the room. 39. The area of a rectangle gets reduced by 8 m 2, when its length is reduced by 5 m and its breadth is increased by 3 m. If we increase the length by 3 m and breadth by 2 m, the area is increased by 74 m 2 . Find the length and the breadth of the rectangle. 40. The area of a rectangle gets reduced by 67 square metres, when its length is increased by 3 m and breadth is decreased by 4 m. If the length is reduced by 1 m and breadth is increased by 4 m, the area is increased by 89 square metres. Find the dimensions of the rectangle. 41. A railway half ticket costs half the full fare and the reservation charge is the same on half ticket as on full ticket. One reserved first class ticket from Mumbai to Delhi costs ` 4150 while one full and one half reserved first class tickets cost ` 6255. What is the basic first class full fare and what is the reservation charge? [HOTS] 42. Five years hence, a man‘s age will be three times the age of his son. Five years ago, the man was seven times as old as his son. Find their present ages. 43. Two years ago, a man was five times as old as his son. Two years later, his age will be 8 more than three times the age of his son. Find their present ages. [CBSE 2008] 44. If twice the son‘s age in years is added to the father‘s age, the sum is 70. But, if twice the father‘s age is added to the son‘s age, the sum is 95. Find the ages of father and son. 45. The present age of a woman is 3 years more than three times the age of her daughter. Three years hence, the woman’s age will be 10 years more than twice the age of her daughter. Find their present ages. 46. On selling a tea set at 5% loss and a lemon set at 15% gain, a crockery seller gains ` 7. If he sells the tea set at 5% gain and the lemon set at 10% gain, he gains ` 13. Find the actual price of each of the tea set and the lemon set. [HOTS]

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Secondary School Mathematics for Class 10

47. A lending library has a fixed charge for the first three days and an additional charge for each day thereafter. Mona paid ` 27 for a book kept for 7 days, while Tanvy paid ` 21 for the book she kept for 5 days. Find the fixed charge and the charge for each extra day. 48. A chemist has one solution containing 50% acid and a second one containing 25% acid. How much of each should be used to make 10 litres of a 40% acid solution? [HOTS] 49. A jeweller has bars of 18-carat gold and 12-carat gold. How much of each must be melted together to obtain a bar of 16-carat gold, weighing 120 g? (Given: Pure gold is 24-carat). [HOTS] 50. 90% and 97% pure acid solutions are mixed to obtain 21 litres of 95% pure acid solution. Find the quantity of each type of acids to be mixed to form the mixture. [HOTS] 51. The larger of the two supplementary angles exceeds the smaller by 18. Find them. 52. In a 3ABC, +A  xc, +B  (3x  2)c, +C  yc and +C  +B  9c. Find the three angles. 53. In a cyclic quadrilateral ABCD, it is given that +A  (2x  4)c, +B  (y  3)c, +C  (2y  10)c and +D  (4x  5)c. Find the four angles. ANSWERS (EXERCISE 3E)

1. ` 560, ` 700

2. ` 40, ` 50

3. 22, 28

4. 90, 47

6. 40, 22

7. 37, 29

8. 20, 13

9. 34, 70

11. 57

12. 63 13. 78

3 3 18. 47 or 74 19. 5 20. 10 25. 100 in A and 80 in B

5. 25, 14 10. 23, 9

14. 35

15. 54 16. 57 17. 92 25 5 15 21. 36 22. 26 23. 9 24. 12 and 4 26. fixed charges = ` 50, rate per km = ` 16

27. fixed charges = ` 1000, cost of food per day = ` 140 28. ` 8500 at 10% p.a. and ` 6250 at 8% p.a.

29. ` 30000, ` 24000

30. ` 600, ` 700 31. 40 km/hr, 30 km/hr

32. 1080 km

33. 100 kmph, 80 kmph 34. 50 kmph, 30 kmph

35. 10 kmph, 2 kmph

36. 6 kmph, 2 kmph

37. 18 days, 36 days

38. length = 15 m, breadth = 12 m

39. length = 19 m, breadth = 10 m

40. length = 28 m, breadth = 19 m 41. first class full fare = ` 4090, reservation charges = ` 60 42. 40 years, 10 years

43. 42 years, 10 years

44. 40 years, 15 years

Linear Equations in Two Variables

151

45. 33 years, 10 years

46. tea set = ` 100, lemon set = ` 80

47. ` 15, ` 3 per day

48. 50% solution = 6 litres, 25% solution = 4 litres

49. 80 g, 40 g

50. 6 litres, 15 lires

51. 99, 81

52. +A  25c, +B  73c, +C  82c 53. +A  70c, +B  53c, +C  110c, +D  127c HINTS TO SOME SELECTED QUESTIONS 3. Let the number of 25-p and 50-p coins be x and y respectively. x y 1950 Then, x  y  50 and 4  2  100 · 8. Let the larger number be x and the smaller number be y. Then, 3x  4y  8 and 5y  3x  5. 10. x  y  14 and x 2  y 2  448. (x 2  y 2) 448    xy  14 32. (x  y) 15. Let the ten‘s digit be x and the unit‘s digit be y. Then, 10x  y  6 and 10x  y  9  10y  x. xy 16. Let the ten‘s digit be x and the unit‘s digit be y. Then, xy  35 and 10x  y  18  10y  x & y  x  2. 

(y  x) 2  (y  x) 2  4xy & (y  x)  (y  x) 2  4xy  144  12.

Now, solve y  x  2 and y  x  12. 18. Let the ten‘s digit be x and the unit‘s digit be y. Then, (10x  y)  (10y  x)  121 & 11 (x  y)  121 & x  y  11. (x  y  11, x  y  3) or (x  y  11, y  x  3) . x 21. Let the required fraction be y · Then, x  11  y & x  y  11. x8 3 And,   4 & 4x  32  3y  24 & 4x  3y  8. y 8 x 23. Let the required fraction be y · Then, x  y  4  2x & y  x  4. x3  2 & 3x  9  2y  6 & 2y  3x  3. y3 3 yx 1 1 1 1 24. x  y  16 and x  y  3 & xy  3 & xy  16 # 3  48. (x  y) 2  (x  y) 2  4xy  (16) 2  (4 # 48)  256  192  64 

x  y  8 or x  y  8.



(x  y  16, x  y  8) or (x  y  16, x  y  8) .

25. Let the number of students in room A and room B be x and y respectively. Then, x  10  y  10 & x  y  20.

… (i)

152

Secondary School Mathematics for Class 10 And, (x  20)  2 (y  20) & x  2y  60.

… (ii)

Now, solve (i) and (ii) to get x and y. 26. Let the fixed charges be ` x and rate per km be ` y. Then, x  80y  1330 and x  90y  1490. 27. Let the fixed charges be ` x and the charges for food be ` y per day. Then, x  25y  4500 and x  30y  5200. 28. Suppose he invested ` x at 10% p.a. and ` y at 8% p.a. (x # 10 # 1) (y # 8 # 1)   1350 & 5x  4y  67500. 100 100 (x # 8 # 1) (y # 10 # 1)   1350  45 & 4x  5y  65250. 100 100 Add (i) and (ii). Then, subtract (ii) from (i).

… (i) … (ii)

29. 5x  7y  9000 and 4x  5y  9000. Find x and y. 30. Let the CP of a chair be ` x and that of a table be ` y. Then, 125x  110y    100 100 1520 & 25x 22y 30400. 110x  125y    100 100 1535 & 22x 25y 30700.  On adding (i) and (ii), we get 47 (x y)  61100 & x  y  1300.

… (i) … (ii)

On subtracting (i) from (ii), we get 3 (y  x)  300 & y  x  100. 

x  600 and y  700.

31. Let the speed of the car from A be x kmph and that of the car from B be y kmph. Then, 7x  7y  70 & x  y  10.

… (i)

And, x  y  70.

… (ii)

32. Let the original speed of the train be x kmph and let the time taken to complete the journey be y hours. Then, length of the journey  xy km. Case I



Speed  (x  5) kmph and time taken  (y  3) hours.

xy  (x  5) (y  3) & 5y  3x  15.

Case II



… (i)

Speed  (x  4) kmph and time taken  (y  3) hours.

xy  (x  4) (y  3) & 3x  4y  12.

… (ii)

From (i) and (ii), we get x  40 and y  27. 

length of the journey  xy  40 # 27 km = 1080 km.

33. Let the speed of the train be x kmph and that of the taxi be y kmph. 300 200 11 1 1 Then, x  y  2 & 600u  400v  11 … (i), where x  u and y  v. 260 240 28 And, x  y  5 & 325u  300v  7. … (ii) 35. Let the speed of the sailor in still water be x kmph and the speed of the current be y kmph.

Linear Equations in Two Variables

153

Then, speed downstream  (x  y) km/hr. And, speed upstream  (x  y) km/hr. 8  40  2 & x  y  12 x  y 60 3 8 and   1 & x  y  8. x y 

… (i) … (ii)

Solve (i) and (ii). 1 1 37. Let 1 man‘s 1 day‘s work be x and 1 boy‘s 1 day‘s work be y · 2 5 1 1 Then, x  y  4 & 2u  5v  4 3 6 1 1 and x  y  3 & 3u  6v  3 · 38. Let length  x metres and breadth  y metres. Then, x  y  3 and (x  3)(y  2)  xy. 

x  y  3 and 3y  2x  6.

39. Let length = x metres and breadth = y metres. Then, xy  (x  5) (y  3)  8 & 3x  5y  7. And, (x  3) (y  2)  xy  74 & 2x  3y  68. 40. Let length = x metres and breadth = y metres. Then, xy  (x  3)(y  4)  67 & 4x  3y  55 and (x  1)(y  4)  xy  89 & 4x  y  93. 41. Let the basic first class full fare be ` x and the reservation charge be ` y. Then, … (i) x  y  4150. 1 And, (x  y)  a 2 x  yk  6255 & 3x  4y  12510. … (ii) 42. Let the present ages of the man and his son be x years and y years respectively. Then, x  5  3 (y  5) & x  3y  10 and x  5  7 (y  5) & x  7y  30.

… (i) … (ii)

43. Let the present ages of the man and his son be x years and y years respectively. Then, (x  2)  5 (y  2) & x  5y  8

… (i)

and (x  2)  3 (y  2)  8 & x  3y  12.

… (ii)

44. Let father‘s age be x years and son‘s age be y years. Then, x  2y  70 and 2x  y  95. 45. Let woman‘s present age be x years and daughter‘s present age be y years. Then, x  3y  3 & x  3y  3 and, x  3  2 (y  3)  10 & x  2y  13. 46. Let the cost price of the tea set be ` x and that of the lemon set be ` y. Then, 115y (19x  23y) c 95x  m  (x  y)  7 &  (x  y)  7 20 100 100 & 3y  x  140.

… (i)

154

Secondary School Mathematics for Class 10 (21x  22y) 105x 110y  (x  y)  13 And, c 100  100 m  (x  y)  13 & 20 & x  2y  260.

… (ii)

Now, solve (i) and (ii). 47. Let the fixed charge be ` x and charge for each extra day be ` y. Then, x  (7  3) y  27 & x  4y  27

… (i)

and, x  (5  3) y  21 & x  2y  21.

… (ii)

48. Let x litres of 50% solution be mixed with y litres of 25% solution. Then, x  y  10 50x 25y 40 # 10 and, 100  100  100 & 2x  y  16.

… (i) … (ii)

49. Let x g of 18-carat gold be mixed with y g of 12-carat gold to get 120 g of 16-carat gold. Then, x  y  120. 18 Gold % in 18-carat gold  a 24 # 100k %  75% 12 Gold % in 12-carat gold  a 24 # 100k %  50% 16 200 Gold % in 16-carat gold  a 24 # 100k %  3 % 200  75% of x  50% of y  3 % of 120 75x 50y 200 # 120  100  100  3 # 100 & 3x  2y  320. Now, solve (i) and (ii).

… (i)

… (ii)

50. Let x litres of 90% pure solution be mixed with y litres of 97% pure solution to get 21 litres of 95% pure solution. Then, x  y  21 90x 97y 95 # 21 and 100  100  100 & 90x  97y  1995. Now, solve (i) and (ii).

… (i) … (ii)

51. x  y  18 and x  y  180. Find x and y. 52. +A  +B  +C  180c & x  (3x  2)  y  180 & 4x  y  182. +C  +B  9c & y  3x  2  9 & y  3x  7.

… (i) … (ii)

From (i) and (ii), we get 7x  175 & x  25. Putting x  25 in (i), we get y  82. 

+A  25c, +B  (3 # 25  2)c  73c, +C  82c.

53. We have, +A  +C  180c and +B  +D  180c. (2x  4)  (2y  10)  180 & x  y  83

… (i)

and (y  3)  (4x  5)  180 & 4x  y  182.

… (ii)



From (i) and (ii), we get 

3x  99 & x  33. And, 33  y  83 & y  50.

Linear Equations in Two Variables

f

155

EXERCISE 3F

Very-Short and Short-Answer Questions 1. Write the number of solutions of the following pair of linear equations: [CBSE 2009] x  2y  8  0, 2x  4y  16. 2. Find the value of k for which the following pair of linear equations have infinitely many solutions: [CBSE 2010] 2x  3y  7, (k  1) x  (k  2) y  3k. 3. For what value of k does the following pair of linear equations have infinitely many solutions? 10x  5y  (k  5)  0 and 20x  10y  k  0. 4. For what value of k will the following pair of linear equations have no solution? [CBSE 2010] 2x  3y  9, 6x  (k  2) y  (3k  2) . 5. Write the number of solutions of the following pair of linear equations: x  3y  4  0 and 2x  6y  7  0. 6. Write the value of k for which the system of equations 3x  ky  0, 2x  y  0 has a unique solution. 7. The difference between two numbers is 5 and the difference between their squares is 65. Find the numbers. 8. The cost of 5 pens and 8 pencils is ` 120, while the cost of 8 pens and 5 pencils is ` 153. Find the cost of 1 pen and that of 1 pencil. 9. The sum of two numbers is 80. The larger number exceeds four times the smaller one by 5. Find the numbers. 10. A number consists of two digits whose sum is 10. If 18 is subtracted from the number, its digits are reversed. Find the number. 11. A man purchased 47 stamps of 20 p and 25 p for ` 10. Find the number of each type of stamps. 12. A man has some hens and cows. If the number of heads be 48 and number of feet be 140, how many cows are there? 2 3 9 4 9 21 13. If x  y  xy and x  y  xy , find the values of x and y. x x y 5 14. If 4  3  12 and 2  y  1 then find the value of (x  y) . 15. If 12x  17y  53 and 17x  12y  63 then find the value of (x  y) .

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Secondary School Mathematics for Class 10

16. Find the value of k for which the system 3x  5y  0, kx  10y  0 has a nonzero solution. 17. Find k for which the system kx  y  2 and 6x  2y  3 has a unique solution. 18. Find k for which the system 2x  3y  5  0, 4x  ky  10  0 has an infinite number of solutions. 19. Show that the system 2x  3y  1  0, 4x  6y  4  0 has no solution. 20. Find k for which the system x  2y  3 and 5x  ky  7  0 is inconsistent. 21. Solve:

3  2  9 4 2 and     1. xy xy x y x y ANSWERS (EXERCISE 3F)

1. infinitely many 2. k  7

3. k  10

4. k  11

3 6. k ! 2

8. ` 16, ` 5

9. 65 and 15

7. 9 and 4

11. 35 and 12

12. 22

16. k = 6 17. k ! 3 18. k = 6

3

2

20. k  5

15. 4

5

1

21. x  2 , y  2

1. Given equations are x  2y  8  0 and 2x  4y  16  0.

a1 b1 c1   · a2 b2 c2

So, there are infinitely many solutions. 2. 2x  3y  7  0 and (k  1) x  (k  2) y  3k  0. a b c For infinitely many solutions, we have a1  1  c1 · b2 2 2 2  3 3  7  and (k  1) (k  2) (k  2) 3k 2k  4  3k  3 and 9k  7k  14 & k  7. a b c 3. For infinitely many solutions, we have a1  1  c1 · b2 2 2 10  5  k  5 k5  1     20 10 & 2 & 2k 10 k & k 10. k k a b c 4. For no solution, we have a1  1 ! c1 · b2 2 2 

2 3 2 9 3 1 9 1 6 k  2 and 6 ! 3k  2 & k  2 3 and 3k  2 ! 3 29  k  2  9 and 3k  2 ! 27 & k  11 and k ! 3 · Hence, k  11. 

10. 64

13. x = 1, y = 3 14. 2

HINTS TO SOME SELECTED QUESTIONS



5. 0

Linear Equations in Two Variables

157

a b c 5. a1  1 ! c1 · b2 2 2 Hence, the given system has no solution. a b 6. For unique solution, we have a1 ! 1 · b2 2 3 3 k  2 ! &k! 2 · 1 7. Let the required numbers be x and y. Then, x 2  y 2 65     5 13 & x y 13. xy Solving x  y  5 and x  y  13, we get x  9 and y  4. x  y  5 and x 2  y 2  65 &

8. Let the cost of 1 pen be ` x and that of 1 pencil be ` y. Then, 5x  8y  120

… (i) and 8x  5y  153.

… (ii)

Adding (i) and (ii), we get 13 (x  y)  273 & x  y  21.

… (iii)

Subtracting (i) from (ii), we get 3 (x  y)  33 & x  y  11.

… (iv)

From (iii) and (iv), we get

x  16 and y  5.

9. Let the required numbers be x and y such that x  y. Then, x  y  80

… (i) and x  4y  5.

… (ii)

Solve (i) and (ii). 10. Let the ten‘s digit be x and unit‘s digit be y. Then, x  y  10

… (i) and (10x  y)  18  (10y  x) & x  y  2. … (ii)

Solve (i) and (ii). 11. Let x stamps of 20 p and y stamps of 25 p be purchased. Then, x  y  47

… (i) and 20x  25y  1000 & 4x  5y  200.

… (ii)

Solve (i) and (ii). 12. Let there be x cows and y hens. Then, x  y  48

… (i) and 4x  2y  140 & 2x  y  70.

… (ii)

Subtracting (i) from (ii), we get x  22. 13. The given equations are 2y  3x  9

… (i) and 4y  9x  21.

… (ii)

… (i) and x  2y  2.

… (ii)

Solve (i) and (ii). 14. Given equations are 3x  4y  5

Solve (i) and (ii) for x and y and find (x  y) . 15. Adding the given equations, we get 116 29 (x  y)  116 & x  y  29  4. 16. The given system has a nonzero solution when

3 5 & k  6. k 10

1 k 17. For a unique solution, we have 6 !  & k ! 3. 2

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Secondary School Mathematics for Class 10

18. For an infinite number of solutions, we have 2  3  5  4 k 10 & k 6. a b c 19. Here a1  1 ! c1 · Hence, the given system has no solution. b2 2 2 a b c 20. The system will be inconsistent when a1  1 ! c1 · b2 2 2 3 1 2 2  5  ! 7 & k  5· k 1  1 21. Put  u and   v to get 3u  2v  2 and 9u  4v  1. x y x y 1 1 This gives u  3 and v  2 · So, x  y  3 and x  y  2. Solve for x and y.

MULTIPLE-CHOICE QUESTIONS (MCQ) Choose the correct answer in each of the following questions: 1. If 2x  3y  12 and 3x  2y  5 then (a) x  2, y  3 (b) x  2, y  3

(c) x  3, y  2 (d) x  3, y  2

2 1 then xy 5 (b) x  5, y  3 (c) x  6, y  4 (d) (a) x  4, y  2 y y 2 2x 1 x If 3  2  6  0 and 2  3  3 then (b) x  2, y  3 (c) x  2, y  3 (d) (a) x  2, y  3 12 31  If x y 4 and y x 11 then 1 (b) x  2, y  3 (c) x  2 , y  3 (d) (a) x  2, y  3 2x  y  2 3x  y  1 3x  2y  1   If then 5 3 6 (b) x  1, y  1 (c) x  1, y  2 (d) (a) x  1, y  1 3  2  9 4 If  2 and     1 then x y xy x y x y 1 3 5 1 3 1 (a) x  2 , y  2 (b) x  2 , y  2 (c) x  2 , y  2 (d) If 4x  6y  3xy and 8x  9y  5xy then (a) x  2, y  3 (b) x  1, y  2 (c) x  3, y  4 (d)

2. If x  y  2 and

3. 4.

5. 6.

7.

x  7, y  5

x  2, y  3 1 1 x 2,y 3

x  2, y  1

1 5 x  2, y  2 x  1, y  1

8. If 29x  37y  103 and 37x  29y  95 then (a) x  1, y  2 (b) x  2, y  1 (c) x  3, y  2 (d) x  2, y  3 9. If 2 x  y  2 x  y  8 then the value of y is 3 1 (b) 2 (c) 0 (a) 2

(d) none of these

Linear Equations in Two Variables

2 3 1 1 10. If x  y  6 and x  2y  2 then 2 2 (a) x  1, y  3 (b) x  3 , y  1

159

3 3 (c) x  1, y  2 (d) x  2 , y  1

11. The system kx  y  2 and 6x  2y  3 has a unique solution only when (a) k  0 (b) k ! 0 (c) k  3 (d) k ! 3 12. The system x  2y  3 and 3x  ky  1 has a unique solution only when (a) k  6 (b) k ! 6 (c) k  0 (d) k ! 0 13. The system x  2y  3 and 5x  ky  7  0 has no solution, when 7 (a) k  10 (b) k ! 10 (c) k  3 (d) k  21

14. If the lines given by 3x  2ky  2 and 2x  5y  1  0 are parallel then the value of k is 5 2 3 15 (a) 4 (b) 5 (c) 2 (d) 4 15. For what value of k do the equations kx  2y  3 and 3x  y  5 represent two lines intersecting at a unique point? (a) k  3 (b) k  3 (c) k  6

(d) all real values except –6 16. The pair of equations x  2y  5  0 and 3x  6y  1  0 has (a) a unique solution

(b) exactly two solutions

(c) infinitely many solutions (d) no solution 17. The pair of equations 2x  3y  5 and 4x  6y  15 has (a) a unique solution

(b) exactly two solutions

(c) infinitely many solutions

(d) no solution

18. If a pair of linear equations is consistent then their graph lines will be (a) parallel

(b) always coincident

(c) always intersecting

(d) intersecting or coincident

19. If a pair of linear equations is inconsistent then their graph lines will be (a) parallel

(b) always coincident

(c) always intersecting (d) intersecting or coincident 20. In a 3ABC, +C  3+B  2 (+A  +B), then +B  ? (a) 20

(b) 40

(c) 60

(d) 80

21. In a cyclic quadrilateral ABCD, it is being given that +A  (x  y  10)c, +B  (y  20)c, +C  (x  y  30)c and +D  (x  y)c. Then, +B  ? (a) 70

(b) 80

(c) 100

(d) 110

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Secondary School Mathematics for Class 10

22. The sum of the digits of a two-digit number is 15. The number obtained by interchanging the digits exceeds the given number by 9. The number is (a) 96

(b) 69

(c) 87

(d) 78

23. In a given fraction, if 1 is subtracted from the numerator and 2 is added 1 to the denominator, it becomes 2 $ If 7 is subtracted from the numerator 1 and 2 is subtracted from the denominator, it becomes 3 $ The fraction is 16 15 16 13 (b) 26 (c) 27 (d) 21 (a) 24 24. 5 years hence, the age of a man shall be 3 times the age of his son while 5 years earlier the age of the man was 7 times the age of his son. The present age of the man is (a) 45 years

(b) 50 years

(c) 47 years

(d) 40 years

25. The graphs of the equations 6x  2y  9  0 and 3x  y  12  0 are two lines which are (a) coincident (b) parallel (c) intersecting exactly at one point (d) perpendicular to each other 26. The graphs of the equations 2x  3y  2  0 and x  2y  8  0 are two lines which are (a) (b) (c) (d)

coincident parallel intersecting exactly at one point perpendicular to each other

24 27. The graphs of the equations 5x  15y  8 and 3x  9y  5 are two lines which are (a) (b) (c) (d)

coincident parallel intersecting exactly at one point perpendicular to each other ANSWERS (MCQ)

1. (c) 2. (c) 3. (a) 4. (d) 5. (a) 6. (b) 7. (c) 8. (a) 9. (c) 10. (b) 11. (d) 12. (b) 13. (a) 14. (d) 15. (d) 16. (d) 17. (d) 18. (d) 19. (a) 20. (b) 21. (b) 22. (d) 23. (b) 24. (d) 25. (b) 26. (c) 27. (a)

Linear Equations in Two Variables

161

HINTS TO SOME SELECTED QUESTIONS 3. 4x  3y  1, 3x  4y  18. Solve. 1 1 4. Putting x  u and y  v, we get u  2v  4 and u  3v  11. 5.

2x  y  2 3x  y  1  & 6x  3y  6  15x  5y  5 & 9x  8y  1. 5 3 3x  y  1 3x  2y  1  & 18x  6y  6  9x  6y  3 & 9x  12y  3. 3 6 Solve (i) and (ii) to get x  1 and y  1.

… (i) … (ii)

1  1 u and   v to get 3u  2v  2 and 9u  4v  1. xy x y 1 1 Solve for u and v to get u  3 and v  2 ·  x  y  3 and x  y  2.

6. Put

1 1 7. Divide throughout by xy and put x  u and y  v to get … (i) and 8v  9u  5. 4v  6u  3 1 1 This gives u  3 and v  4 · Hence, x  3 and y  4. 8. Adding (i) and (ii), we get 66 (x  y)  198 & x  y  3.

… (ii)

Subtracting (ii) from (i), we get 8 (y  x)  8 & y  x  1. 3 3 9. 2 x  y  2 x  y  2 3/2 & x  y  2 and x  y  2 · So, y  0. 1 1 1 10. Put x  u and y  v. Then, 2u  3v  6 … (i) and u  2 v  2 & 2u  v  4. … (ii) a b 11. For a unique solution, we have a1 ! 1 · b2 2 1 k  6 !  & k ! 3. 2 a b 2 1 12. For a unique solution, we have a1 ! 1 · So, 3 ! & k ! 6. k b2 2 a b c 13. For no solution, we have a1  1 ! c1 · b2 2 2 3  14 · 12   5 ! 7 & k 10 and k ! 3 Hence, k  10. k a b c 14. For parallel lines, we have a1  1 ! c1 · b2 2 2 2 3  2k 15 15  2 5 ! 1 & k  4 and k ! 5 & k  4 · a b 15. For a unique intersecting point, we have a1 ! 1 · b2 2 2 k  3 ! 1 & k ! 6. Hence, correct answer is (d). c a 1 b 1 5 1 2 16. Here, a1    3 , 1    3 and c1  1 · b2 3 6 2 2 

a1 b1 c1 ·  a2 b2 ! c2 So, the given system has no solution.

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Secondary School Mathematics for Class 10

c a 5 1 2 1 b 3 1 17. Here, a1  4  2 , 1  6  2 and c1    3 · b2 15 2 2 

a1 b1 c1 ·  a2 b2 ! c2 So, the given system has no solution.

18. If a pair of linear equations is consistent then their graph lines will be intersecting or coincident. 20. Let C  3B  2(A  B)  xc. xc xc Then, C  xc, B  ` 3 j and (A  B)  ` 2 j · x (A  B)  C  180c & 2  x  180 & 3x  360 & x  120. 120 c  +B  a k 40c. 3 21. +A  +C  180c and +B  +D  180c. This gives x  y  100 and x  2y  160. 22. Let the ten‘s digit be x and the unit‘s digit be y. Then, x  y  15

… (i)

and (10y  x)  (10x  y)  9 & y  x  1.

x 23. Let the required fraction be y · Then, x1  1 & 2x  2  y  2 & 2x  y  4 y2 2 x7  1 & 3x  21  y  2 & 3x  y  19. y2 3

… (ii)

… (i) … (ii)

24. Let the present ages of the man and his son be x years and y years respectively. Then, (x  5)  3(y  5) & x  3y  10

… (i)

(x  5)  7(y  5) & x  7y  30.

… (ii)

a c 2 2 6 2 b 9 3 25. a1  3  1 , 1    1 and c1  12  4 · b2 1 2 2 a1 b1 c1 ·  a2 b2 ! c2 So, the system is inconsistent and hence the lines are parallel. 

a b a 2 b 3 26. a1  1 , 1   · So, a1 ! 1 · b2 b2 2 2 2 Thus, the system has a unique solution and therefore the lines intersect exactly at one point. 27. The equations are 5x  15y  8  0 and 15x  45y  24  0. 

a1 b c1    5  1 1  15  1  8 1 a2 15 3 , b2 45 3 and c2 24 3



a1 b1 c1   a2 b2 c2 and therefore the system has infinitely many solutions.

Hence, the graph lines are coincident.

Linear Equations in Two Variables

163

TEST YOURSELF MCQ 1. The graphic representation of the equations x  2y  3 and 2x  4y  7  0 gives a pair of (a) parallel lines

(b) intersecting lines

(c) coincident lines

(d) none of these

2. If 2x  3y  7 and (a  b)x  (a  b  3)y  4a  b have an infinite number of solutions then (a) a  5, b  1 (b) a  5, b  1 (c) a  5, b  1

(d) a  5, b  1 3. The pair of equations 2x  y  5, 3x  2y  8 has (a) a unique solution

(b) two solutions

(c) no solution (d) infinitely many solutions 4. If x   y and y  0, which of the following is wrong? (a) x 2 y  0

(b) x  y  0

(c) xy  0

1 1 (d) x  y  0

Short-Answer Questions 1 1 5. Show that the system of equations x  2y  2  0 and 2 x  2 y  1  0 has a unique solution. 6. For what values of k is the system of equations kx  3y  k  2, 12x  ky  k inconsistent? 3x 5y 7 7. Show that the equations 9x  10y  21, 2  3  2 have infinitely many solutions. 8. Solve the system of equations x  2y  0, 3x  4y  20. 9. Show that the paths represented by the equations x  3y  2 and 2x  6y  5 are parallel. 10. The difference between two numbers is 26 and one number is three times the other. Find the numbers. 11. Solve: 23x  29y  98, 29x  23y  110. 12. Solve: 6x  3y  7xy and 3x  9y  11xy. 13. Find the value of k for which the system of equations 3x  y  1 and kx  2y  5 has (i) a unique solution, (ii) no solution. 14. In a 3ABC, +C  3+B  2(+A  +B) . Find the measure of each one of +A, +B and +C.

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Secondary School Mathematics for Class 10

15. 5 pencils and 7 pens together cost ` 195 while 7 pencils and 5 pens together cost ` 153. Find the cost of each one of the pencil and the pen. 16. Solve the following system of equations graphically: 2x  3y  1, 4x  3y  1  0. Long-Answer Questions 17. Find the angles of a cyclic quadrilateral ABCD in which +A  (4x  20)c, +B  (3x  5)c, +C  (4y)c and +D  (7y  5)c . 18. Solve for x and y:

35  14  14 35 19,     37. xy xy x y x y

19. If 1 is added to both the numerator and the denominator of a fraction, 4 it becomes 5 · If, however, 5 is subtracted from both the numerator and 1 the denominator, the fraction becomes 2 · Find the fraction. 20. Solve:

ax  by     a a b, ax by 2ab. b ANSWERS (TEST YOURSELF)

1. (a)

2. (d)

3. (a)

4. (d)

8. x  4, y  2

6. k  !6

3

11. x  3, y  1 12. x  1, y  2 13. (i) k ! 6 (ii) k  6 14. +A  20c, +B  40c, +C  120c 15. ` 4 per pencil, ` 25 per pen 10. 39, 13

16. x  1, y  1

17. +A  120c, +B  70c, +C  60c, +D  110c

18. x  4, y  3

19. 9

7

20. x  b, y  a



Quadratic Equations

165

Quadratic Equations

4

HISTORY

Babylonians were the first to solve the quadratic equations of the form x 2  px + q = 0.

Brahmagupta (AD 598–665), an Indian mathematician gave an explicit formula to solve a quadratic equation of the form ax 2  bx = c. An Arab mathematician, Al-Khwarizmi in AD 800 also studied quadratic equations of various types. An ancient Indian mathematician Shridharacharya derived the well-known Quadratic Formula for solving a quadratic equation ax 2 + bx + c = 0 by the method of completing the square. This is being used as the standard formula for solving such an equation. According to this formula, the roots of ax 2 + bx + c = 0 are given by b  D b + D = 2 and  = = 2a , where D (b 4ac) is called the 2a discriminant of this equation.

A quadratic equation in the variable x is an equation of the form ax + bx + c = 0, where a, b, c are real numbers and a ! 0.

QUADRATIC EQUATION 2

A real number  is called a root of the quadratic    equation ax bx c 0, a ! 0 if a 2 + b + c = 0.

ROOTS OF A QUADRATIC EQUATION 2

NOTE 1.

If  is a root of ax 2 + bx + c = 0 then we say that (i) x =  satisfies the equation ax 2 + bx + c = 0 or (ii) x =  is a solution of the equation ax 2 + bx + c = 0.

NOTE 2.

The roots of a quadratic equation ax 2 + bx + c = 0 are called the zeros of the polynomial ax 2 + bx + c.

SOLVING A QUADRATIC EQUATION

Solving a quadratic equation means finding

its roots. 165

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Secondary School Mathematics for Class 10

SOLVED EXAMPLES EXAMPLE 1

Which of the following are quadratic equations? (i) x 2  5x  3  0 (iii) 3x 2  2 x  8  0 1 (v) x  x  x 2

SOLUTION

(ii) 2x 2  3 2 x  6  0 (iv) 2x 2  3  0 1 1 (vi) x 2  2  4 4 x

(i) Clearly, (x 2  5x  3) is a quadratic polynomial. 

x 2  5x  3  0 is a quadratic equation.

(ii) Clearly, (2x 2  3 2 x  6) is a quadratic polynomial. 

2x 2  3 2 x  6  0 is a quadratic equation.

(iii) 3x 2  2 x  8 is not of the form ax 2  bx  c  0. 

3x 2  2 x  8  0 is not a quadratic equation.

(iv) 2x 2  3  0 is of the form ax 2  bx  c  0, where a  2, b = 0 and c  3. 2x 2  3  0 is a quadratic equation. 1 (v) x  x  x 2 & x 2  1  x 3 & x 3  x 2  1  0. 

And, (x 3  x 2  1) being a polynomial of degree 3, it is not quadratic. 1 Hence, x  x  x 2 is not a quadratic equation. 1  17 & 4x 4  4  17x 2 & 4x 4  17x 2  4  0. 4 x2 Clearly, 4x 4  17x 2  4 is a polynomial of degree 4.

(vi) x 2 

 EXAMPLE 2

x2 

1  17 is not a quadratic equation. 4 x2

Check whether the following are quadratic equations: (i) (2x  1) (x  3)  (x  4) (x  2) (ii) (x  2) 3  2x (x 2  1) (iii) (x  1) 3  x 3  x  6

SOLUTION

(iv) x (x  3)  6  (x  2) (x  2)

We have (i) (2x  1) (x  3)  (x  4) (x  2)  2x 2  7x  3  x 2  2x  8 & x 2  9x  11  0. This is of the form ax 2  bx  c  0, where a  1, b  9 and c  11. Hence, the given equation is a quadratic equation.

Quadratic Equations

167

(ii) (x + 2) 3 = 2x (x 2  1)





x 3 + 8 + 6x (x + 2) = 2x 3  2x



x 3 + 6x 2 + 12x + 8 = 2x 3  2x x 3  6x 2  14x  8 = 0.



This is not of the form ax 2 + bx + c = 0. Hence, the given equation is not a quadratic equation. (iii) (x + 1) 3 = x 3 + x + 6 

x 3 + 1 + 3x (x + 1) = x 3 + x + 6



3x 2 + 2x  5 = 0.

This is of the form ax 2 + bx + c = 0, where a = 3, b = 2 and c = 5. Hence, the given equation is a quadratic equation. (iv) x (x + 3) + 6 = (x + 2) (x  2)  

x 2 + 3x + 6 = x 2  4 3x + 10 = 0.

This is not of the form ax 2 + bx + c = 0. Hence, the given equation is not a quadratic equation. EXAMPLE 3

For the quadratic equation 2x 2  5x  3 = 0, show that (i) x = 3 is its solution. (iii) x = 4 is not its solution.

SOLUTION

1 (ii) x = 2 is its solution.

The given equation is 2x 2  5x  3 = 0. (i) On substituting x = 3 in the given equation, we get LHS = 2 # 3 2  5 # 3  3 = (18  15  3) = 0 = RHS.

x = 3 is a solution of 2x 2  5x  3 = 0. 1 (ii) On substituting x = 2 in the given equation, we get 1 2  l  5 #b 1 l 3 LHS = 2 # b 2 2 



= &2 # 1 + 5 # 1  3 0 2 4 15 ' 31  0  RHS. 2 2 1 x = 2 is a solution of 2x 2  5x  3 = 0.

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Secondary School Mathematics for Class 10

(iii) On substituting x = 4 in the given equation, we get = 0. LHS = 2 # 4 2  5 # 4  3 = (32  20  3) = 9 Y Thus, LHS ≠ RHS.  x = 4 is not a solution of 2x 2  5x  3 = 0. EXAMPLE 4

SOLUTION

Show that 2 and  2 2 x 2 + 2 x  4 = 0.

are the roots of the equation

The given equation is x 2 + 2 x  4 = 0. Putting x = 2 in the given equation, we get LHS = ( 2 ) 2 + ( 2 # 2 )  4 = (2 + 2  4) = 0 = RHS.



2 is a root of the given equation.

Putting x = 2 2 in the given equation, we get LHS = (2 2 ) 2 + 2 # (2 2 )  4 = (8  4  4) = 0 = RHS.



 2 2 is a root of the given equation.

Hence,

2 and  2 2 are the roots of the given equation.

SOLVING A QUADRATIC EQUATION BY FACTORISATION

Let the given quadratic equation be ax 2 + bx + c = 0, where a ! 0. Let (ax 2 + bx + c) be expressible as the product of two linear expressions, say (px + q) and (rx + s), where p, q, r, s are real numbers such that p ! 0 and r ! 0. Then, ax 2  bx  c  0 & (px  q) (rx  s)  0 & (px  q)  0 or (rx  s)  0 q & x   p or x   rs · (x + 2)(3x  5) = 0.

EXAMPLE 5

Solve:

SOLUTION

We have (x  2)(3x  5)  0 & x  2  0 or 3x  5  0 5 & x  2 or x  3 · Hence, the roots of the given equation are –2 and 35 · 5x 2  8x = 0.

EXAMPLE 6

Solve:

SOLUTION

We have 5x 2  8x  0 & x (5x  8)  0 & x  0 or 5x  8  0 8 & x  0 or x  5 ·

Quadratic Equations

169

Hence, 0 and 85 are the roots of the given equation. REMARK

In Class IX, we have learnt how to factorise a quadratic polynomial by splitting the middle term. We shall use it for finding the roots of a quadratic equation.

EXAMPLE 7

Solve: 6x 2  x  2 = 0 by the factorisation method.

SOLUTION

We write,  x = 4x + 3x as (4x) # 3x = 12x 2 = 6x 2 # (2) .   

6x 2  x  2 = 0 6x 2  4x + 3x  2 = 0  2x (3x  2) + (3x  2) = 0 (3x  2) (2x + 1) = 0  3x  2 = 0 or 2x + 1 = 0

1 2 x = 3 or x = 2 · 1 2 Hence, 3 and 2 are the roots of the given equation. 

EXAMPLE 8

Solve: 8x   22x  21 = 0 by the factorisation method.

SOLUTION

We write,  22x = 28x + 6x as 8x 2 # (21) = 168x 2 = (28x) # 6x . 

8x 2  22x  21 = 0



8x 2  28x + 6x  21 = 0  4x (2x  7) + 3 (2x  7) = 0



(2x  7) (4x + 3) = 0  2x  7 = 0 or 4x + 3 = 0

3 7 x = 2 or x = 4 · 3 Hence, 72 and 4 are the roots of the given equation. 

EXAMPLE 9 SOLUTION

Solve:

6x 2 + 40 = 31x.

The given equation may be written as 6x 2  31x + 40 = 0. We write,  31x = 16x  15x as 6x 2 # 40 = 240x 2 = (16x) # (15x) . 

6x 2  31x + 40 = 0 6x 2  16x  15x + 40 = 0  2x (3x  8)  5 (3x  8) = 0



(3x  8) (2x  5) = 0  3x  8 = 0 or 2x  5 = 0



8 5 x = 3 or x = 2 · Hence, 83 and 25 are the roots of the given equation. 

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Secondary School Mathematics for Class 10

4 3 x 2 + 5x  2 3 = 0.

EXAMPLE 10

Solve:

SOLUTION

Here, 4 3 # (2 3 ) = 24, 8 # (3) = 24 and 8 + (3) = 5.

[CBSE 2013]



4 3 x 2 + 5x  2 3 = 0  4 3 x 2 + 8x  3x  2 3 = 0



4x ( 3 x + 2)  3 ( 3 x + 2) = 0



( 3 x + 2) (4x  3 ) = 0 



x=



x =e

3 x + 2 = 0 or 4x  3 = 0

3 2 or x = 4 3 2 # 3

2 3 3 3 o= or x = 4 · 3 3

4x 2  12x + 9 = 0.

EXAMPLE 11

Solve:

SOLUTION

Here, 4 # 9 = 36, (6) # (6) = 36, and (6) + (6) = 12. 

4x 2  12x + 9 = 0



4x 2  6x  6x + 9 = 0  2x (2x  3)  3 (2x  3) = 0

(2x  3) (2x  3) = 0  (2x  3) 2 = 0 3  2x  3  0 & x  · 2 3 = Hence, x 2 is the repeated root of the given equation. 

EXAMPLE 12

SOLUTION

Solve the following equation by using factorisation method: [CBSE 2012, ‘15] 4x 2  4ax + (a 2  b 2) = 0. We may write,  4a = {2 (a + b)} + {2 (a  b)} . Also, {2 (a + b)} # {2 (a  b)} = 4 (a 2  b 2) . 

4x 2  4ax + (a 2  b 2) = 0



4x 2  2 (a + b) x  2 (a  b) x + (a 2  b 2) = 0



2x {2x  (a + b)}  (a  b) {2x  (a + b)} = 0

 {2x  (a  b)}#{2x  (a  b)}  0 2x  (a  b)  0 or 2x  (a  b)  0 (a  b) (a  b) ·  x or x  2 2 (a  b) (a  b) Hence, and are the roots of the given equation. 2 2 

EXAMPLE 13

SOLUTION

Solve the following equation by using factorisation method: [CBSE 2015] 9x 2  6b 2 x  (a 4  b 4)  0. We may write,  6b 2 = 3 (a 2  b 2)  3 (a 2 + b 2) .

Quadratic Equations

171

Also, {3 (a 2  b 2)}#{3 (a 2  b 2)}   9 (a 4  b 4) 

9x 2  6b 2 x  (a 4  b 4) = 0



9x 2 + 3 (a 2  b 2) x  3 (a 2 + b 2) x  (a 4  b 4) = 0



3x {3x + (a 2  b 2)}  (a 2 + b 2) {3x + (a 2  b 2)} = 0



{3x + (a 2  b 2)} # {3x  (a 2 + b 2)} = 0



3x + (a 2  b 2) = 0 or 3x  (a 2 + b 2) = 0

(b 2  a 2) (a 2  b 2) · or x  3 3 (b 2  a 2) (a 2 + b 2) Hence, and are the required roots of the 3 3 given equation. 

x

x + 3 = 3x  7 · x + 2 2x  3

EXAMPLE 14

Solve:

SOLUTION

By cross multiplication, we get (x + 3) (2x  3) = (x + 2) (3x  7) 

2x 2 + 3x  9 = 3x 2  x  14  x 2  4x  5 = 0



x 2  5x + x  5 = 0  x (x  5) + (x  5) = 0



(x  5) (x + 1) = 0  x  5 = 0 or x + 1 = 0



x = 5 or x = 1.

Hence, 5 and –1 are the roots of the given equation. EXAMPLE 15 SOLUTION

14   5 1 , x ! 3, 1. x3 x1 The given equation may be written as 14  5  1 x3 x1 14 (x  1)  5 (x  3) 1  (x  3) (x  1) (9x  1)  1 & x 2  4x  3  9 x  1  2 x  4x  3  x 2  5x  4  0 & x 2  4x  x  4  0  x (x  4)  (x  4)  0 & (x  4)(x  1)  0 Solve:

 

x  4  0 or x  1  0 x  4 or x  1.

Hence, 4 and 1 are the roots of the given equation.

[CBSE 2014]

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Secondary School Mathematics for Class 10

1  1  11 , x ! 4, 7. (x  4) (x  7) 30

EXAMPLE 16

Solve:

SOLUTION

We have 1  1  11 (x  4) (x  7) 30 (x  7)  (x  4) 11 11   11   2 30 (x  4) (x  7) (x  3x  28) 30 1  1 [on dividing both sides by 11]  2 (x  3x  28) 30  (x 2  3x  28)  30

[CBSE 2008, ‘10]

[by cross multiplication]

 x  3x  2  0 & x  2x  x  2  0  x (x  2)  (x  2)  0 & (x  2)(x  1)  0  x  2  0 or x  1  0  x  2 or x  1. Hence, 2 and 1 are the roots of the given equation. 2

2

1  1  1  1 , [x ! 0, x ! (a  b)] . (a  b  x) a b x

EXAMPLE 17

Solve:

SOLUTION

We have 1 111 (a  b  x) a b x x  (a  b  x) b  a 1 1 11    (a  b  x) x a b x (a  b  x) ab  

[CBSE 2013]

 (a + b) (a + b) = ab x (a + b + x) 1 = 1 [on dividing both sides by (a + b)] x (a + b + x) ab



x (a + b + x) = ab [by cross multiplication]



x 2 + ax + bx + ab = 0  x (x + a) + b (x + a) = 0



(x + a) (x + b) = 0  x + a = 0 or x + b = 0



x = a or x = b.

Hence, –a and –b are the roots of the given equation. x2 + x4 = 1 Y 3 3 , x = 3, 5 . x3 x5

EXAMPLE 18

Solve:

SOLUTION

We have x  2  x  4  10 x3 x5 3 (x  2)(x  5)  (x  4)(x  3) 10   3 (x  3)(x  5)

[CBSE 2014]

Quadratic Equations



(x 2  7x  10)  (x 2  7x  12) 10  3 (x 2  8x  15)



(2x   14x  22) 10  3 (x   8x  15)



173

3 (2x   14x  22)  10 (x   8x  15) [by cross multiplication]

 6x 2  42x  66  10x 2  80x  150  4x 2  38x  84  0 & 2x 2  19x  42  0  2x 2  12x  7x  42  0 & 2x (x  6)  7 (x  6)  0  (x  6)(2x  7)  0 & x  6  0 or 2x  7  0 7 x  6 or x  · 2 7 Hence, 6 and 2 are the roots of the given equation. 2x  1  x  3  m 5, x ! 3, 1 · m 3c x3 2x  1 2

EXAMPLE 19

Solve: 2 c

SOLUTION

On putting

[CBSE 2014]

2x  1 = y, the given equation becomes x+3

3 2y  y  5 & 2y 2  3  5y & 2y 2  5y  3  0 & 2y 2  6y  y  3  0 & 2y (y  3)  (y  3)  0 & (y  3) (2y  1)  0 & y  3  0 or 2y  1  0 1

& y  3 or y  2 ·

2x  1  3 x3 & 2x  1  3 (x  3) [by cross multiplication] & 2x  1  3x  9 & x  10.

Case I

y3&

Case II

y

 1  & 2xx 31  21 2 & 2 (2x  1)  (x  3)

& 5x  1 1

&x 5 · 1 Hence, –10 and 5 are the roots of the given equation.

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Secondary School Mathematics for Class 10

3 2   23 , x ! 0, 1, 2. (x  1) 2 (x  2) 5x

EXAMPLE 20

Solve:

SOLUTION

We have 3 2   23 (x  1) 2 (x  2) 5x 4 (x  2)  3 (x  1) 23 7x  5   23   5x 2 (x  1) (x  2) 2 (x 2  x  2) 5x  5x (7x  5)  46 (x 2  x  2) [by cross multiplication]

[CBSE 2015]

 35x 2  25x  46x 2  46x  92  11x 2  21x  92  0  11x 2  44x  23x  92  0  11x (x  4)  23 (x  4)  0  (x  4)(11x  23)  0  x  4  0 or 11x  23  0  23 ·  x  4 or x  11 23 Hence, 4 and are the roots of the given equation. 11  1  1  11 , x ! 3, 1 , 9 · 2 7 (x  3) (2x  1) (7x  9)

EXAMPLE 21

Solve:

SOLUTION

We have 1  1  11 (x  3) (2x  1) (7x  9) (2x  1)  (x  3) (3x  2)  11  2  11  (x  3)(2x  1) (7x  9) 2x  5x  3 (7x  9)

[CBSE 2009C[

 (3x  2)(7x  9)  11(2x 2  5x  3) [by cross multiplication]  21x 2  41x  18  22x 2  55x  33  x 2  14x  51  0 & x 2  17x  3x  51  0  x (x  17)  3 (x  17)  0 & (x  17)(x  3)  0  x  17  0 or x  3  0  x  17 or x  3. Hence, –17 and 3 are the roots of the given equation. EXAMPLE 22

Solve: 5(x + 1)  5(2  x)  5 3  1.

SOLUTION

We have



5 ( x  1)  5 ( 2  x)  5 3  1 25 5 x · 5  5 2 · 5 x  126  5 x · 5  x  126 5 25 x 5y  y  126, where 5  y



5y 2  126y  25  0  5y 2  125y  y  25  0



Quadratic Equations

175



5y (y  25)  (y  25)  0  (y  25)(5y  1)  0



y  25  0 or 5y  1  0 1 y  25 or y  5 5 x  25  5 2 or 5 x  5 1 x  2 or x  1.

  

Hence, 2 and –1 are the roots of the given equation. f

EXERCISE 4A

1. Which of the following are quadratic equations in x? (i) x 2  x + 3 = 0 (iii)

2 x 2 + 7x + 5 2 = 0

(v) x 2  3x  x + 4 = 0 2 (vii) x + x = x 2 (ix) (x + 2) 3 = x 3  8

5 (ii) 2x 2 + 2 x  3 = 0 1 1 (iv) 3 x 2 + 5 x  2 = 0 6 (vi) x  x = 3 1 (viii) x 2  2 = 5 x (x) (2x + 3) (3x + 2) = 6 (x  1) (x  2)

1 2 1 (xi) b x + x l = 2 b x + x l + 3 2. Which of the following are the roots of 3x 2 + 2x  1 = 0 ? 1 (ii) 3

(i) –1

1 (iii)  2

3. (i) Find the value of k for which x  1 is a root of the equation x 2  kx  3  0. Also, find the other root. 3 (ii) Find the values of a and b for which x  and x   2 are the roots 4 of the equation ax 2  bx  6  0. 4. Show that x   ad 2 b

bc ad

is a solution of the quadratic equation

ax  2c  2  l x bc 0. b d

[CBSE 2017]

Solve each of the following quadratic equations:

5. (2x  3)(3x  1)  0 7. 3x 2  243  0 9. x 2  6x  5  0 11. x 2  12x  35  0

6. 4x 2  5x  0 8. 2x 2  x  6  0 10. 9x 2  3x  2  0 12. x 2  18x  77

176

Secondary School Mathematics for Class 10

13. 6x 2  11x  3  0 15. 3x 2  2x  1  0

14. 6x 2  x  12  0 16. 4x 2  9x  100

17. 15x 2  28  x 19. 48x 2  13x  1  0

18. 4  11x  3x 2 20. x 2  2 2 x  6  0 22.

3 x 2  11x  6 3  0

23. 3 7 x 2  4x  7  0

24.

7 x 2  6x  13 7  0

25. 4 6 x 2  13x  2 6  0

26. 3x 2  2 6 x  2  0

[CBSE 2010, ’12]

[CBSE 2011]

28. x  3 5 x  10  0

[CBSE 2011]

[CBSE 2015]

30. x  3 3 x  30  0

[CBSE 2015]

[CBSE 2013]

32. 5x  13x  8  0

21.

27.

3 x 2  10x  8 3  0

3x 2 2x2 3  0 2

29. x  ( 3  1) x  3  0 2

31.

2 x  7x  5 2  0 2

[CBSE 2017]

33. x  (1  2 ) x  2  0

2 2

2

2

35. 100x 2  20x  1  0 1 37. 10x  x  3 39. 2x 2  ax  a 2  0

[CBSE 2015]

40. 4x  4bx  (a  b )  0 2

2

[CBSE 2013C]

34. 9x  6x  1  0 1 36. 2x 2  x   0 8 2 5  38. 2 x 2 0 x

2

2

[CBSE 2015, ’17]

41. 4x  4a x  (a  b )  0

[CBSE 2015]

42. x  5x  (a  a  6)  0

[CBSE 2015]

43. x  2ax  (4b  a )  0

[CBSE 2015]

44. x  (2b  1) x  (b  b  20)  0

[CBSE 2015]

45. x  6x  (a  2a  8)  0

[CBSE 2015]

46. abx  (b  ac) x  bc  0

[CBSE 2014]

47. x  4ax  b  4a  0

[CBSE 2012]

2

2

4

2

4

2

2

2

2

2

2

2

2

2

2

2

2

2

48. 4x  2 (a  b ) x  a b  0 2

2

2

2 2

49. 12abx 2  (9a 2  8b 2) x  6ab  0

[CBSE 2006]

50. a b x  b x  a x  1  0 2 2

2

2

2

[CBSE 2005]

51. 9x  9 (a  b) x  (2a  5ab  2b )  0 2

2

16 15 52. x  1   , x ! 0, 1 x 1 3 5 4 53. x  3  , x ! 0, 2x  3 2 3 1 2 1 54.    , x ! 1, 3 x 1 2 3x  1

2

[CBSE 2009] [CBSE 2014] [CBSE 2014] [CBSE 2014]

Quadratic Equations

1  1 6 , x ! 1, 5 x1 x5 7 3 1  1  1, (ii) 1 x! , 5 2x  3 x  5 9 2 1 1 1 1    2a  b  2x 2a b 2x x3 1x  1 4 , x ! 2, 0 x x2 4 3x  4  7  5 4 , x! 7 3 3x  4 2 x x1 1 (i)   x  4 , x ! 0, 1 x 1 4   x 1  2x 1  1 (ii) 2, x !  , 1 2 2x  1 x  1 x  x1  4 2 , x ! 0, 1 x x1 15 x4  x6  1 3 , x ! 5, 7 x5 x7 3 x1  x3  1 3 , x ! 2, 4 x2 x4 3 1  2 6 , x ! 0, 1, 2 x2 x1 x 5 1 2 (i)      , x ! 1, 2, 4 x 1 x 2 x 4 3 5 1 1 (ii)      , x ! 1,  ,  4 5 x 1 5x 1 x 4 3x  1  2x  3  1 3 3 c  m 2 c  m 5, x ! , 2x 3 3x 1 3 2 3 1 7x  1  5x  3  3 c  m 4 c  m 11, x ! , 5x 3 7x 1 5 7  3 4x  3  2x  1  c m 10 c m 3, x ! 1 , 4x  3 2x  1 2 4

177

55. (i)

56. 57. 58. 59.

60. 61. 62. 63. 64.

65. 66. 67.

x 2 x 68. c  m  5 c  m  6  0, x ! 1 x 1 x 1 a  b  69. 2, x ! b, a (x  b) (x  a) 70.

a b   (a  b), x ! 1 , 1 a b (ax  1) (bx  1)

71. 3(x + 2)  3 x  10 72. 4(x + 1)  4(1  x)  10 73. 2 2x  3 · 2(x + 2)  32  0

[CBSE 2017] [CBSE 2013]

[CBSE 2010]

[CBSE 2017] [CBSE 2014] [CBSE 2014] [CBSE 2017] [CBSE 2013] [CBSE 2013C] [CBSE 2017] [CBSE 2014]

[CBSE 2014]

178

Secondary School Mathematics for Class 10 ANSWERS (EXERCISE 4A)

1 2. 1 and

1. (i), (ii), (iii), (iv), (vi), (ix) 3. (i) k  4, other root = 3 6. x  0 or x 

5 4

1 2 or x  3 3 3 1 13. x  or x  2 3 25 16. x  or x  4 4 1 1 19. x  or x  3 16 10. x 

12. x = 11 or x = 7

1 3 1 18. x  4 or x  3 15. x  1 or x 

2 3

13 or x   7 7  2 27. x  6 or x  3 24. x 

22. x  3 3 or x  25. x 

2 2 3

or x 

36. 39. 41. 43.

11. x = –7 or x = –5

3 or x  2 4 17. x  or x  3

4 3 7 5

14. x 

20. x  2 or x  3 2

2 3

 3 4 2

28. x  5 or x  2 5

30. x  2 3 or x  5 3 31. x   2 or x  33.

1 3 or x  2 3 3 8. x  2 or x  2 5. x 

(ii) a  4, b  5

7. x = 9 or x = –9

9. x = –5 or x = –1

21. x   4 3 or x 

3

5

23. x  26. x 

 7 1 or x  3 7 2 3

or x 

2 3

29. x  3 or x  1 32. x  1 or x 

8 5

2 1 1 1 1 34. x  , x  35. x  , x  x  1 or x  2 3 3 10 10 1 1 1 1 1 37. x  or x  38. x  2 or x  x ,x 2 5 2 4 4   ( a b ) ab a 40. x  or x  x  a or x  2 2 2 2 2 2 2 a b a b 42. x  (a  3) or x  (a  2) x or x  2 2 44. x = (b  5) or x = (b + 4) x = (a  2b) or x = (a + 2b) b

c

45. x  (a  4) or x  (a  2)

46. x  a or x  b

47. x  (2a  b) or x  (2a  b)

48. x 

2b 3a or x  3a 4b (a  2b) (2a  b) 51. x  or x  3 3 49. x 

a2 b2 or x  2 2 1 1 50. x  2 or x  2 a b 52. x  4 or x  4

53. x  2 or x  1

Quadratic Equations

54. x  3 or x  1

55. (i) x  6 or x  2

179

(ii) x 

37 or x  6 20

b 2 1 57. x  4 or x  58. x  6 or x  2 9 2 2 5 1 3 4 59. (i) x  or x  (ii) x   2 60. x  or x  3 3 2 2 5 4 1 61. x  8 or x  5 62. x  5 or x  63. x  3 or x  2 3 2 3 11 64. (i) x  2 or x  (ii) x   65. x  7 or x  0 or x  1 2 17 3 4 1 66. x = 1 or x = 0 67. x  68. x  or x  or x  2 3 8 2 (a  b) (a  b) 2 69. x  (a  b) or x  70. x  or x  2 ab (a  b) 1 1 71. x  2 or x  0 72. x  or x  73. x  2 or x  3 2 2 56. x  a or x 

HINTS TO SOME SELECTED QUESTIONS 16. 4x

2

9x  100  0 & 4x 2  25x  16x  100  0.

17. 15x 2  x  28  0 & 15x 2  21x  20x  28  0. 18. 3x 2  11x  4  0 & 3x 2  12x  x  4  0. 19. 48x 2  13x  1  0 & 48x 2  16x  3x  1  0. 20. x 2  2 2 x  6  0 & x 2  3 2 x  2 x  6  0 & x (x  3 2 )  2 (x  3 2 )  0. 21.

3 x 2  10x  8 3  0 & 3 x 2  12x  2x  8 3  0 & 3 x (x  4 3 )  2 (x  4 3 )  0.

22.

3 x 2  11x  6 3  0 & 3 x 2  9x  2x  6 3  0 & 3 x (x  3 3 )  2 (x  3 3 )  0.

23. 3 7 x 2  4x  7  0 & 3 7 x 2  7x  3x  7  0 & 7 x (3x  7 )  (3x  7 )  0. 24.

7 x 2  6x  13 7  0 & 7 x 2  13x  7x  13 7  0 & x ( 7 x  13)  7 ( 7 x  13)  0.

25. 4 6 x 2  13x  2 6  0 & 4 6 x 2  16x  3x  2 6  0 & 4 2 x ( 3 x  2 2 )  3 ( 3 x  2 2 )  0. 26. 3x 2  2 6 x  2  0 & 3x 2  6 x  6 x  2  0 & 3 x ( 3 x  2 )  2 ( 3 x  2 )  0. 27.

3 x2  2 2 x  2 3  0 & 3 x2  3 2 x  2 x  2 3  0 & 3 x (x  6 )  2 (x  6 )  0.

28. x 2  3 5 x  10  0 & x 2  2 5 x  5 x  10  0 & x (x  2 5 )  5 (x  2 5 )  0. 29. x 2  ( 3  1) x  3  0 & x 2  3 x  x  3  0 & x (x  3 )  (x  3 )  0. 30. x 2  3 3 x  30  0 & x 2  5 3 x  2 3 x  30  0 & x (x  5 3 )  2 3 (x  5 3 )  0. 31.

2 x 2  7x  5 2  0 & 2 x 2  5x  2x  5 2  0 & x ( 2 x  5)  2 ( 2 x  5)  0.

32. 5x 2  13x  8  0 & 5x 2  5x  8x  8  0.

180

Secondary School Mathematics for Class 10

34. 9x 2  6x  1  0 & (3x  1) 2  0 & 3x  1  0. 35. 100x 2  20x  1  0 & (10x  1) 2  0 & 10x  1  0. 36. 16x 2  8x  1  0 & (4x  1) 2  0 & 4x  1  0. 37. 10x 2  3x  1  0 & 10x 2  5x  2x  1  0 & 5x (2x  1)  (2x  1)  0. 38. 2x 2  5x  2  0 & 2x 2  4x  x  2  0 & 2x (x  2)  (x  2)  0. 39. 2x 2  ax  a 2  0 & 2x 2  2ax  ax  a 2  0 & 2x (x  a)  a (x  a)  0. 40. 4x 2  4bx  (b 2  a 2)  0 & 4x 2  2 (b  a) x  2 (b  a) x  (b 2  a 2)  0. 41. 4x 2  4a 2 x  (a 4  b 4)  0 & 4x 2  2 (a 2  b 2) x  2 (a 2  b 2) x  (a 4  b 4)  0 & 2x [2x  (a 2  b 2)]  (a 2  b 2) [2x  (a 2  b 2)]  0. 42. x 2  5x  (a  3) (a  2)  0 & x 2  (a  3) x  (a  2) x  (a  3) (a  2)  0. 43. x 2  2ax  (a 2  4b 2)  0 & x 2  2ax  (a  2b) (a  2b)  0 & x 2  (a  2b) x  (a  2b) x  (a  2b) (a  2b)  0. 44. x 2  (2b  1) x  (b  5) (b  4)  0 & x 2  (b  5) x  (b  4) x  (b  5) (b  4)  0 & x [x  (b  5)]  (b  4) [x  (b  5)]  0. 45. x 2  6x  (a  4) (a  2)  0 & x 2  (a  4) x  (a  2) x  (a  4) (a  2)  0. 46. abx 2  b 2 x  acx  bc  0 & bx (ax  b)  c (ax  b)  0. 47. x 2  4ax  (2a  b) (2a  b)  0 & x 2  (2a  b) x  (2a  b) x  (4a 2  b 2)  0. 48. 4x 2  2a 2 x  2b 2 x  a 2 b 2  0 & 2x (2x  a 2)  b 2 (2x  a 2)  0. 49. 12abx 2  9a 2 x  8b 2 x  6ab  0 & 3ax (4bx  3a)  2b (4bx  3a)  0. 50. b 2 x (a 2 x  1)  (a 2 x  1)  0. 51. 9x 2  9 (a  b) x  (2a  b) (a  2b)  0 

9x 2  3 {(2a  b)  (a  2b)} x  (2a  b) (a  2b)  0

9x 2  3 (2a  b) x  3 (a  2b) x  (2a  b) (a  2b)  0. 1  1  1 1· 56. 2a  b  2x 2x 2a b 

57.

x (x  3)  (1  x) (x  2) 17 (x 2  3x) (3x  x 2  2) 17   & 4 4 x (x  2) (x 2  2x) 

4 (2x 2  2)  17 (x 2  2x) & 9x 2  34x  8  0 & 9x 2  36x  2x  8  0.

3x  4  1 5 y. Then, y  y  & 2y 2  5y  2  0. 7 2 3x  1  65. Putting y, the given equation becomes 2x  3 2 3y  y  5 & 3y 2  5y  2  0 & 3y 2  6y  y  2  0. 58. Put

69. The given equation is a  b 1m  c   1m  0 c x a xb (a  x  b) (a  x  b)  0  (x  a) (x  b)

Quadratic Equations 

181

1 1 (a  x  b) · <    F  0 (x b) (x a)

2x  (a  b) H 0  (a  x  b) > (x  a) (x  b) (a  x  b) [2x  (a  b)]  0 (a  b) ·  x  (a  b) or x  2 70. The given equation is 

 

b a (   b2  (   a2  0 (ax 1) (bx 1) (a  abx  b) (a  abx  b)  0 (ax  1) (bx  1) 1 (a  abx  b) · c ax  1 

71. 3 x # 3 2  72. 4 x # 4 

1 m  0. bx  1

1  1 10 & 9y  y  10, where y  3 x . 3x

4  2 10 & 2y  y  5, where y  4 x . 4x

73. 2 2x  3 # 2 2 # 2 x  32  0 & y 2  12y  32  0, where 2 x  y & y 2  8y  4y  32  0.

SOLVING QUADRATIC EQUATIONS BY COMPLETING THE SQUARE METHOD

In fact, we can convert any quadratic equation to the form (ax + b) 2  c 2 = 0 and then we can easily find its roots.

SOLVED EXAMPLES EXAMPLE 1

SOLUTION

Solve the equation x 2  10x  2  0 by the method of completing the square. We have x 2  10x  2  0  x 2  10x = 2  x2  2# x # 5  52  2  52  (x  5) 2  (2  25)  27

[adding 5 2 on both sides]



x  5 = ! 27 = !3 3



x  5  3 3 or x  5  3 3 x  (5  3 3 ) or x  (5  3 3 ) .



[taking square root on both sides]

Hence, (5  3 3 ) and (5  3 3 ) are the roots of the given equation.

182 EXAMPLE 2

SOLUTION

Secondary School Mathematics for Class 10

Solve the equation 3x 2  5x  2  0 by the method of completing the square. We have 3x 2  5x  2  0 

9x 2  15x  6  0



9x  15x  6

[multiplying each term by 3]

2

5 2 5 5 2  (3x) 2  2 # 3x #  c m   6  c m 2 2 2 5 2 [adding c m on both sides] 2 2 2 (24  25) 1 25 5 1  c m  c3x  m  c6  m  2 2 4 4 4 5 1  c3x  m  ! [taking square root on both sides] 2 2 5 1 5 1  3x   or 3x   2 2 2 2 1 5 1 5 6 4  3x  c  m   3 or 3x  c  m   2 2 2 2 2 2 2 2  3x  3 or 3x  2 & x  1 or x  · 3 2 Hence, 1 and are the roots of the given equation. 3 EXAMPLE 3

SOLUTION

Solve the equation 2x 2  x  4  0 by the method of completing the square. We have 

2x 2  x  4  0 4x 2  2x  8  0



4x  2x  8

[multiplying both sides by 2]

2

1 2 1 1 2  (2x) 2  2 # 2x #  c 2 m  8  c 2 m 2 1 2 [adding c 2 m on both sides] 2

1 1 2 33  33 d n  c2x  m  c8  m  2 4 4 2 

2x 

1  d 33 n ! 2 2



2x 

1  33 1  33 or 2x   2 2 2 2

[taking square root on both sides]

Quadratic Equations

183

( 33  1) 33 1  n 2 2 2  33 1 ( 33  1)   or 2x  2 2 2   ( 33 1) ( 33  1) ·  x or x  4 4 1  33 1  33 Hence, are the roots of the given and 4 4 equation. 

2x  d

EXAMPLE 4

By using the method of completing the square, show that the equation 4x 2  3x  5  0 has no real roots.

SOLUTION

We have 

4x 2  3x  5  0 4x 2  3x  5

3 2 3 3 2  (2x) 2  2 # 2x #  c m  5  c m 4 4 4 3 2 [adding c m on both sides] 4 2 (80  9)  71 3 9   c2x  m  c5  m   0. 4 16 16 16 3 2 But, c2x  m cannot be negative for any real value of x. 4 So, there is no real value of x satisfying the given equation. Hence, the given equation has no real roots. EXAMPLE 5

SOLUTION

1 Solve the equation 10x  x = 3 by the method of completing the square. We have 1 10x  x  3  10x 2  1  3x  10x 2  3x  1  100x 2  30x  10 [multiplying each side by 10] 3 2 3 3 2  (10x) 2  2 #10x #  c m  10  c m 2 2 2 3 2 [adding c m on both sides] 2 9 3 2 49  7 2 c m  c10x  m  c10  m  2 2 4 4

184

Secondary School Mathematics for Class 10

3 7 [taking square root on both sides] ! 2 2 3 7 3 7  10x   or 10x   2 2 2 2 7 3 4 7 3 10   2  10x  c  m  5 or 10x  c  m  2 2 2 2 2 2 2 1 5 1 ·   10x  5 or 10x  2 & x  or x  5 10 10 2 1 1 Hence, and are the roots of the given equation. 5 2  10x 

EXAMPLE 6

SOLUTION

Solve the equation a 2 x 2  3abx  2b 2  0 by the method of completing the square. We have a 2 x 2  3abx  2b 2  0  a 2 x 2  3abx  2b 2 3b 2 3b 3b 2  (ax) 2  2 #(ax)#  c m  2b 2  c m 2 2 2 3b 2 [adding a k on both sides] 2 2 2 2 2 3b 9b  (8b  9b )  b 2  b 2 c m m  cax  m  c 2b 2  2 2 4 4 4  cax 

3b b m ! [taking square root on both sides] 2 2 b 3b 3b b  cax  m  or cax  m  2 2 2 2  b 3b b  3b 4b  2b  m  ax  c 2b or ax  c  m  b 2 2 2 2 2 2 2b b  x  a or x  a · 2b b Hence, a and a are the roots of the given equation. EXAMPLE 7

SOLUTION

Solve the equation x 2  ( 3  1) x  3  0 by the method of completing the square. We have x 2  ( 3  1) x  3  0 

x 2  ( 3  1) x   3



x 2  2 # x #d

31 31 31 nd n   3 d n 2 2 2 2

2

Quadratic Equations

185

[adding d  (x  

( 3  1) ( 3  1) 2 2 (  32 2 4 2

31 n on both sides] 2 2

2 ( 3  1) 2  4 3 31 d n 4 2

 (x 

( 3  1) ( 3  1) 2 ! 2 2 [taking square root on both sides]



x

( 3  1) ( 3  1) ( 3  1) ( 3  1)   or x  2 2 2 2



x

( 3  1) ( 3  1) 2 3    3 2 2 2

or x  

( 3  1) ( 3  1) 2   1 2 2 2

x  3 or x  1.

Hence, 3 and 1 are the roots of the given equation. f

EXERCISE 4B

Solve each of the following equations by using the method of completing the square:

1. x 2  6x  3  0

2. x 2  4x  1  0

3. x 2  8x  2  0

4. 4x 2  4 3 x  3  0

5. 2x 2  5x  3  0

6. 3x 2  x  2  0

7. 8x 2  14x  15  0

8. 7x 2  3x  4  0 2 5 10. 5x 2  6x  2  0 11. 2  x  2  0 x 13. x 2  ( 2  1) x  2  0 15.

9. 3x 2  2x  1  0 12. 4x 2  4bx  (a 2  b 2)  0 14.

2 x 2  3x  2 2  0

3 x 2  10x  7 3  0

16. By using the method of completing the square, show that the equation 2x 2  x  4  0 has no real roots. ANSWERS (EXERCISE 4B)

1. x  (3  6 ) or x  (3  6 )

2. x  (2  3 ) or x  (2  3 )

3. x  (4  3 2 ) or x  (4  3 2 ) 5. x 

1 or x  3 2

6. x  1 or x 

 3  3 or x  2 2 3 5 7. x  or x  2 4 4. x 

2 3

186

Secondary School Mathematics for Class 10

8. x  1 or x 

4 7

9. x  1 or x 

3  19 3  19 or x  5 5  (a  b) (a  b) 12. x  or x  2 2 1 14. x  or x  2 2 2 10. x 

1 3 11. x  2 or x 

1 2

13. x  2 or x  1 15. x   3 or x 

7 3

QUADRATIC FORMULA [SHRIDHARACHARYA’S RULE] Consider the quadratic equation ax 2 + bx + c = 0, where a, b, c are real = 0. Then, numbers and a Y 

ax 2 + bx + c = 0 ax 2 + bx = c



c b x2 + a $ x = a



b 2 c b 2 b x 2 + a $ x + b 2a l = a + b 2a l



2 c b2 b x + 2ba l = c a + 2 m 4a



2 (b 2  4ac) b x + 2ba l = 4a 2

[dividing throughout by a] b 2 [adding b 2a l on both sides]

! b 2  4ac b m , when (b 2  4ac) $ 0 2a 2a b 2  4ac b  x ! 2a 2a b ! b 2  4ac ·  x 2a This is called the quadratic formula or Shridharacharya’s rule. 

cx 

Thus, ax 2  bx  c  0 has two roots  and , given by 

b  b 2  4ac b  b 2  4ac · and   2a 2a

For the equation ax 2  bx  c  0, the expression D  (b 2  4ac) is called the discriminant.

DISCRIMINANT

AN IMPORTANT NOTE

(b  4ac) $ 0. 2

The roots of ax 2  bx  c  0 are real only when

Quadratic Equations

187

Taking (b 2  4ac)  D, the roots of ax 2  bx  c  0 are given by 

b  D b  D · and   2a 2a

SOLVED EXAMPLES EXAMPLE 1

Show that the equation 9x 2  7x  2  0 has real roots and solve it.

SOLUTION

The given equation is 9x 2  7x  2  0. Comparing it with ax 2  bx  c  0, we get a  9, b  7 and c  2. 

D  (b 2  4ac)  (7 2  4 # 9 #(2)]  121  0.

So, the given equation has real roots. Now, D  121  11.  b  D (7  11)   4  2, 2a 2# 9 18 9 b  D (7  11) 18    1.  2a 2#9 18 2 Hence, the required roots are and –1. 9 



EXAMPLE 2

Show that the equation x 2  6x  6  0 has real roots and solve it.

SOLUTION

The given equation is x 2  6x  6  0. Comparing it with ax 2  bx  c  0, we get a  1, b  6 and c  6. 

D  (b 2  4ac)  (36  4 #1# 6)  12  0.

So, the given equation has real roots. Now, 

D = 12 = 2 3 .

b  D (6  2 3 ) (6  2 3 )    (3  3 ), 2a 2 2 #1 b  D (6  2 3 ) (6  2 3 )    (3  3 ) .  2a 2 2 #1 

Hence, (3  3 ) and (3  3 ) are the roots of the given equation. EXAMPLE 3

Show that the equation 2x 2 + 5 3 x + 6 = 0 has real roots and solve it.

188 SOLUTION

Secondary School Mathematics for Class 10

The given equation is 2x 2 + 5 3 x + 6 = 0. Comparing it with ax 2 + bx + c = 0, we get a  2, b  5 3 and c  6. 

D  (b 2  4ac)  [(5 3 ) 2  4 # 2 # 6]  (75  48)  27  0.

So, the given equation has real roots. Now,

D = 27 = 3 3 .

b  D (5 3  3 3 ) 2 3  3    , 2a 2# 2 2 4 b  D (5 3  3 3 ) 8 3    2 3 .  2a 2# 2 4  3 Hence, and 2 3 are the roots of the given equation. 2 

EXAMPLE 4



Using quadratic formula, solve for x: p 2 x 2 + (p 2  q 2) x  q 2 = 0.

SOLUTION

[CBSE 2014]

The given equation is p 2 x 2 + (p 2  q 2) x  q 2 = 0. Comparing it with ax 2 + bx + c = 0, we get a = p 2, b = (p 2  q 2) and c = q 2 . `

D = (b 2  4ac) = (p 2  q 2) 2  4 # p 2 # (q 2) = (p 2  q 2) 2 + 4p 2 q 2 = (p 2 + q 2) 2 > 0.

So, the given equation has real roots. Now, D  (p 2  q 2) . 

b  D  (p 2  q 2)  (p 2  q 2) 2q 2 q 2   2  2, 2a 2p 2 2p p 2 2 2 2 2 b  D  (p  q )  (p  q ) 2p    1.  2a 2p 2 2p 2 

Hence, EXAMPLE 5

SOLUTION

q2 p2

and –1 are the roots of the given equation.

Using quadratic formula, solve for x: 9x 2  9 (a  b) x  (2a 2  5ab  2b 2)  0.

[CBSE 2009]

The given equation is 9x 2  9 (a  b) x  (2a 2  5ab  2b 2)  0. This is of the form Ax 2  Bx  C  0, where A  9, B  9 (a  b) and C  (2a 2  5ab  2b 2) .

Quadratic Equations



189

D  (B 2  4AC)  81 (a  b) 2  36 (2a 2  5ab  2b 2) = 81 (a 2 + b 2 + 2ab)  36 (2a 2 + 5ab + 2b 2) = 9a 2 + 9b 2  18ab = 9 (a 2 + b 2  2ab) = 9 (a  b) 2 $ 0.

So, the given equation has real roots. Now,

D = 9 (a  b) 2 = 3 (a  b) .

B + D 9 (a + b) + 3 (a  b) 6 (2a + b) (2a + b) = = = 3 , 18 2#9 2A B  D 9 (a + b)  3 (a  b) 6 (a + 2b) (a + 2b) = = = $ = 3 18 2#9 2A (2a + b) (a + 2b) Hence, and are the roots of the given 3 3 equation. 

EXAMPLE 6

SOLUTION

=

Using quadratic formula, solve for x: abx 2 + (b 2  ac) x  bc = 0.

[CBSE 2014]

The given equation is abx + (b  ac) x  bc = 0. 2

2

This is of the form Ax 2 + Bx + C = 0, where A = ab, B = (b 2  ac) and C = bc. 

D = (B 2  4AC) = (b 2  ac) 2 + 4ab 2 c

= b 4 + a 2 c 2  2ab 2 c + 4ab 2 c = b 4 + a 2 c 2 + 2ab 2 c = (b 2 + ac) 2 > 0. So, the given equation has real roots. Now,

D = (b 2 + ac) .

B + D  (b 2  ac) + (b 2 + ac) 2ac c = = = 2A 2ab 2ab b , B  D  (b 2  ac) + (b 2 + ac) 2b 2 b · = = = = a 2A 2ab 2ab b c Hence, and a are the roots of the given equation. b 

EXAMPLE 7 SOLUTION

=

1 1  Solve for x: x  3, x ! 0, 2. (x  2) The given equation may be written as (x  2)  x  3 & 3x (x  2)  2 x (x  2) & 3x 2  6x  2  0.

[CBSE 2010]

… (i)

190

Secondary School Mathematics for Class 10

This equation is of the form ax 2 + bx + c = 0, where a = 3, b = – 6 and c = 2.  D = (b 2  4ac) = (6) 2  4 # 3 # 2 = 36  24 = 12 > 0. So, the given equation has real roots. Now, 

D = 12 = 2 3 .

=

b + D 6 + 2 3 6 + 2 3 3 + 3 = = = 2a 6 3 , 2#3

=

b  D 6  2 3 6  2 3 3  3 · = = = 2a 6 3 2#3

Hence, the required values of x are EXAMPLE 8 SOLUTION

(3 + 3 ) (3  3 ) · and 3 3

x1  x3  1 3 , x ! 2, 4. x2 x4 3 The given equation is x  1  x  3  10 x2 x4 3 (x  1)(x  4)  (x  3)(x  2) 10   3 (x  2)(x  4) Solve for x:

[CBSE 2014]



(x 2  5x  4)  (x 2  5x  6) 10 2x 2  10x  10  10  & 2 3 3 (x  6x  8) x 2  6x  8



3 (2x 2  10x  10)  10 (x 2  6x  8)

 6x 2  30x  30  10x 2  60x  80  4x 2  30x  50  0 & 2x 2  15x  25  0.

… (i)

This equation is of the form ax + bx + c = 0, where a = 2, b = –15 and c = 25. 2

 D = (b 2  4ac) = {(15) 2  4 # 2 # 25} = (225  200) = 25 > 0. So, the given equation has real roots. Now, 

D = 25 = 5.

=

b + D (15 + 5) 20 = = = 5, 2a 4 2#2

=

b  D (15  5) 10 5 = = = $ 2a 2 4 2#2

5 Hence, the required values of x are 5 and 2 $

Quadratic Equations

f

191

EXERCISE 4C

Find the discriminant of each of the following equations:

1.

(i) 2x 2  7x + 6 = 0 (iii) 2x 2  5 2 x + 4 = 0

(ii) 3x 2  2x + 8 = 0 (iv) 3 x 2 + 2 2 x  2 3 = 0 (vi) 1  x = 2x 2

(v) (x  1) (2x  1) = 0

Find the roots of each of the following equations, if they exist, by applying the quadratic formula:

2. x 2  4x  1 = 0 4. 2x 2 + x  4 = 0

3. x 2  6x + 4 = 0 5. 25x 2 + 30x + 7 = 0

6. 16x 2 = 24x + 1 8. 2x 2  2 2 x + 1 = 0

7. 15x 2  28 = x 9. 2 x 2 + 7x + 5 2 = 0

[CBSE 2013, ’17]

10.

3 x + 10x  8 3 = 0

[CBSE 2011]

11.

3x 2 2x2 3 = 0

[CBSE 2015]

2 2

12. 2x + 6 3 x  60 = 0 2

[CBSE 2011, ’15]

13. 4 3 x + 5x  2 3 = 0

[CBSE 2013]

14. 3x  2 6 x + 2 = 0

[CBSE 2012]

2

2

15. 2 3 x  5x + 3 = 0 2

16. x + x + 2 = 0 18. x 2  ( 3 + 1) x + 3 = 0 2

HINT

[CBSE 2011]

17. 2x + ax  a = 0 2

2

[CBSE 2015]

D = ( 3 + 1)  4 3 = ( 3  1) . 2

2

19. 2x 2 + 5 3 x + 6 = 0

20. 3x 2  2x + 2 = 0

1 21. x  x  3, x ! 0

1 1 22. x    3, x ! 0, 2 x 2

1 23. x  x  3, x ! 0 m n 24. n x 2  m  1  2x HINT

[CBSE 2010] [CBSE 2010]

m 2  n   2 2  2  n x 2x b m 1l 0 & m x 2mnx (n mn) 0.

25. 36x 2  12ax + (a 2  b 2) = 0 27. x 2  2ax  (4b 2  a 2) = 0 28. x 2 + 6x  (a 2 + 2a  8) = 0 29. x + 5x  (a 2 + a  6) = 0 30. x 2  4ax  b 2 + 4a 2 = 0 31. 4x 2  4a 2 x + (a 4  b 4) = 0 2

[CBSE 2015]

26. x 2  2ax + (a 2  b 2) = 0 [CBSE 2015] [CBSE 2015] [CBSE 2015] [CBSE 2012] [CBSE 2015]

192

Secondary School Mathematics for Class 10

32. 4x 2 + 4bx  (a 2  b 2) = 0 33. x 2  (2b  1) x + (b 2  b  20) = 0

[CBSE 2015] [CBSE 2015]

34. 3a x  8abx  4b  0, a ! 0 35. a 2 b 2 x 2  (4b 4  3a 4) x  12a 2 b 2  0, a ! 0 and b ! 0 36. 12abx 2  (9a 2  8b 2) x  6ab = 0, where a ! 0 and b ! 0 2

2

2

[CBSE 2006] [CBSE 2009]

ANSWERS (EXERCISE 4C)

1. (i) 1

(ii) –92 (iii) 18 (iv) 32 (v) 1 (vi) 9  2. x (2  5 ) or x  (2  5 ) 3. x = (3 + 5 ) or x = (3  5 ) (1 + 33 ) (1  33 ) or x = 4 4 (3 + 10 ) (3  10 ) 6. x = or x = 4 4 4. x =

8. x =

1 1 , 2 2

11. x = 6 or x =

2 3,

14. x =

5. x =

(3 + 2 ) (3  2 ) or x = 5 5 7

4

7. x = 5 or x = 3

9. x =  2 or x =

5 2

10. x =

2 3 = 4 3 3 or x

 2 2 3 12. x  2 3 or x  5 3 13. x = 4 or x = 3 3 3

2 3

15. x = 2 or x =

a

17. x = 2 or x = a

18. x = 1 or x = 3

20. Do not exist

21. x =

1 3

16. Do not exist 19. x =

 3 = 2 3 2 or x

3+ 5 3 5 or x = 2 2

22. x =

3+ 3 3 3 or x = 3 3

24. x =

 n + mn n  mn or x = m m

3 + 13 3  13 or x = 2 2 (a + b) (a  b) 25. x = 6 or x = 6 23. x =

26. x = (a + b) or x = (a  b)

27. x = (a + 2b) or x = (a  2b)

28. x = (a  2) or x = (4 + a)

29. x  (a  2) or x  (a  3)

30. x = (2a  b) or x = (2a + b)

1 1 31. x = 2 (a 2 + b 2) or x = 2 (a 2  b 2)

1

1

32. x = 2 (a  b) or x =  2 (a + b)

2b

2b

34. x = 3a or x = a

35. x =

33. x = (b + 4) or x = (b  5)

2b 3a 2 3a 4b 2 = 36. x = 2 or x = 3a 4b or x a b2

Quadratic Equations

193

NATURE OF THE ROOTS OF A QUADRATIC EQUATION = 0. Then, the discriminant Let the given equation be ax 2 + bx + c = 0, where a Y 2 is given by D = (b  4ac) . And, the roots of the given equation are = Case I

b + D b  D · and  = 2a 2a

When D > 0

In this case, the roots are real and distinct. These roots are given by = Case II

b + D b  D · and  = 2a 2a

When D = 0

In this case, the roots are real and equal. (b) Each root = 2a · Case III When D < 0

In this case, the roots are imaginary, and we say that the given equation has no real roots. SUMMARY

Value

Nature of roots

D>0

Real and unequal

D=0

Real and equal

D 0.

So, the given equation has real unequal roots. Solving 2x 2  6x + 3 = 0 by quadratic formula, we have x=

6 ! 36  4 # 2 # 3 6 ! 36  24 6 ! 12 = = 4 4 (2 # 2)

6!2 3 3! 3 &x 2 · 4 (3 + 3 ) (3  3 ) So, are the roots of the given equation. and 2 2 

EXAMPLE 3

SOLUTION

x

Show that the equation x 2 + ax  4 = 0 has real and distinct roots for all real values of a. The given equation is x 2 + ax  4 = 0. This is of the form Ax 2 + Bx + C = 0, where A = 1, B = a and C = –4.  D = (B 2  4AC) = {a 2  4 # 1 # ( 4)} = (a 2 + 16) > 0 for all real values of a. Thus, D > 0 for all real values of a. Hence, the given equation has real and distinct roots for all real values of a.

EXAMPLE 4

SOLUTION

Find the nature of the roots of the quadratic equation 3x 2  4 3 x + 4 = 0 and hence solve it. The given equation is 3x 2  4 3 x + 4 = 0. This is of the form ax 2  bx  c  0, where a  3, b  4 3 and c = 4.  D = (b 2  4ac) = {( 4 3 ) 2  4 # 3 # 4} = (48  48) = 0. This shows that the given quadratic equation has real and equal roots. b 4 3 2 3 Each root = 2a = 6 = 3 $ Hence,

2 3 2 3 3 and 3 are the roots of the given equation.

Quadratic Equations EXAMPLE 5

SOLUTION

195

Find the values of k for which the quadratic equation 2x 2  kx  3  0 has two real equal roots. The given equation is 2x 2 + kx + 3 = 0. This is of the form ax 2  bx  c  0, where a  2, b  k and c  3.  D = (b 2  4ac) = (k 2  4 # 2 # 3) = (k 2  24) . For real and equal roots, we much have D  0 & k 2  24  0 & k 2  24 & k  ! 24  !2 6 . Hence, 2 6 and 2 6 are the required values of k.

EXAMPLE 6

SOLUTION

Find the value of k for which the roots of the quadratic equation [CBSE 2017] kx(x  2)  6  0 are equal. The given equation is kx 2  2kx + 6 = 0. This is of the form ax 2  bx  c  0, where a  k, b  2k and c = 6.  D = (b 2  4ac) = (4k 2  4 # k # 6) = (4k 2  24k) . For equal roots, we must have D  0 & 4k 2  24k  0 & 4k (k  6)  0 & k  0 or k  6. Now, k = 0, we get 6 = 0, which is absurd. = 0 and hence k = 6.  kY

EXAMPLE 7

Find the value of k for which the quadratic equation [CBSE 2014] (k + 4) x 2 + (k + 1) x + 1 = 0 has two real equal roots.

SOLUTION

The given equation is (k + 4) x 2 + (k + 1) x + 1 = 0. This is of the form ax 2 + bx + c = 0, where a = (k + 4), b = (k + 1) and c = 1. 

D = (b 2  4ac) = (k + 1) 2  4 # (k + 4) # 1 = (k + 1) 2  4 (k + 4) = (k 2 + 1 + 2k  4k  16) = (k 2  2k  15) .

For equal roots, we must have D  0 & k 2  2k  15  0 & k 2  5k  3k  15  0 & k (k  5)  3 (k  5)  0 & (k  5) (k  3)  0 & k  5  0 or k  3  0 & k  5 or k  3. Hence, the required value of k is 5 or –3. EXAMPLE 8

Find the nonzero value of k for which the quadratic equation kx 2 + 1  2 (k  1) x + x 2 = 0 has equal roots. Hence, find the roots of the equation. [CBSE 2015]

196 SOLUTION

Secondary School Mathematics for Class 10

The given equation is (k + 1) x 2  2 (k  1) x + 1 = 0. This is of the form ax 2  bx  c  0, where a  (k  1), b  2 (k  1) and c  1. 

D = (b 2  4ac) = {4 (k  1) 2  4 # (k + 1) # 1} = 4 (k 2  3k) .

For equal roots, we must have D  0 & 4 (k 2  3k)  0 & 4k (k  3)  0 & k  0 or k  3. 

the required nonzero value of k is 3.

Putting k = 3, the given equation becomes 4x 2  4x  1  0 & (2x  1) 2  0 & (2x  1)  0 & x  Hence, the required roots are EXAMPLE 9

SOLUTION

1· 2

1 1 and · 2 2

If – 4 is a root of the equation x 2 + px  4 = 0 and the equation x 2 + px + q = 0 has equal roots, find the values of p and q. Since – 4 is a root of the equation x 2 + px  4 = 0, we have ( 4) 2  p ( 4)  4  0 & 4p  (16  4)  12 & p  3. Now, the roots of x 2 + px + q = 0 being equal, we have 9 p 2  4q  0 & 3 2  4q  0 & 4q  9 & q  · 4 9 Hence, p = 3 and q  · 4

EXAMPLE 10

If –2 is a root of the equation 3x 2 + 7x + p = 0, find the value of k so that the roots of the equation x 2 + k (4x + k  1) + p = 0 are equal.

SOLUTION

Since –2 is a root of the equation 3x + 7x + p = 0, we have

[CBSE 2015] 2

3 #(2) 2  7 #(2)  p  0 & 12  14  p  0 & p  2. For p = 2, the other given equation becomes x 2 + 4kx + k (k  1) + 2 = 0. This is of the form ax 2 + bx + c = 0, where a = 1, b = 4k and c = (k 2  k + 2) .  D = (b 2  4ac) = {16k 2  4 # 1 # (k 2  k + 2)} = (12k 2 + 4k  8) . For equal roots we must have D  0 & 12k 2  4k  8  0 & 4 (3k 2  k  2)  0 & 3k 2  k  2  0 & 3k 2  3k  2k  2  0 & 3k (k  1)  2 (k  1)  0 & (k  1)(3k  2)  0

Quadratic Equations

197

& k  1  0 or 3k  2  0 & k  1 or k  23 · 2 Hence, the required value of k is –1 or 3 · EXAMPLE 11

Prove that both the roots of the equation (x  a) (x  b) + (x  b) (x  c) + (x  c) (x  a) = 0 are real but they are equal only when a = b = c.

SOLUTION

The given equation may be written as 3x 2  2x (a + b + c) + (ab + bc + ca) = 0. 

D = 4 (a + b + c) 2  12 (ab + bc + ca) = 4 [(a + b + c) 2  3 (ab + bc + ca)] = 4 (a 2 + b 2 + c 2  ab  bc  ca) = 2 (2a 2 + 2b 2 + 2c 2  2ab  2bc  2ca) = 2 [(a  b) 2 + (b  c) 2 + (c  a) 2] $ 0 [a (a  b) 2 $ 0, (b  c) 2 $ 0 and (c  a) 2 $ 0] .

This shows that both the roots of the given equation are real. For equal roots, we must have D = 0. Now, D  0 & (a  b) 2  (b  c) 2  (c  a) 2  0 & (a  b)  0, (b  c)  0 and (c  a)  0 & a  b  c. Hence, the roots are equal only when a = b = c. EXAMPLE 12

SOLUTION

If the roots of the equation (b  c) x 2 + (c  a) x + (a  b) = 0 are equal, prove that 2b = a + c. [CBSE 2002C, ‘06, ’17] Clearly, x = 1 satisfies the given equation. Since its roots are equal, so 1 and 1 are its roots. 

product of roots of the given equation = (1 # 1) = 1.

ab But, product of roots =  $ b c

[B

C product of roots = A ]

ab  1 & a  b  b  c & 2b  a  c. bc Hence, 2b = a + c. 

EXAMPLE 13

Show that the equation 3x 2 + 7x + 8 = 0 is not true for any real value of x.

SOLUTION

The given equation is 3x 2 + 7x + 8 = 0. 

D = (7 2  4 # 3 # 8) = (49  96) =  47 < 0.

198

Secondary School Mathematics for Class 10

So, the given equation has no real roots. Hence, the given equation is not true for any real value of x. EXAMPLE 14

Show that the equation 2 (a 2 + b 2) x 2 + 2 (a + b) x + 1 = 0 has no real = b. roots, when a Y

SOLUTION

The given equation is 2 (a 2 + b 2) x 2 + 2 (a + b) x + 1 = 0 . 

D  4 (a  b) 2  8 (a 2  b 2)  4 (a 2  b 2  2ab)  4 (a  b) 2  0, when a  b ! 0.

So, the given equation has no real roots, when a ! b. EXAMPLE 15

Find the values of k for which the equation x 2 + 5kx + 16 = 0 has no real roots. [CBSE 2013C]

SOLUTION

The given equation is x 2 + 5kx + 16 = 0. This is of the form ax 2  bx  c  0, where a  1, b  5k and c = 16. 

D = (b 2  4ac) = (25k 2  4 # 1 # 16) = (25k 2  64) .

Since, the given equation has no real root, we have D  0 & 25k 2  64  0 & 25k 2  64 8 2 64 & k 2  25 & k 2  c 5 m 8 8 & 5  k  5· 8 8 Hence, the required real values of k are such that k · 5 5 EXAMPLE 16

SOLUTION

Find the values of k for which the given equation has real roots: (i) kx 2  6x  2 = 0 (ii) 3x 2 + 2x + k = 0 (iii) 2x 2 + kx + 2 = 0 (i) The given equation is kx 2  6x  2 = 0. 

D = [( 6) 2  4 # k # ( 2)] = (36 + 8k) .

The given equation will have real roots if D $ 0. 36 9 Now, D $ 0 & 36  8k $ 0 & k $ &k$ 2 · 8 (ii) The given equation is 3x 2 + 2x + k = 0. 

D = (2 2  4 # 3 # k) = (4  12k) .

The given equation will have real roots if D $ 0. 1 Now, D $ 0 & 4  12k $ 0 & 12k # 4 & k # · 3

Quadratic Equations

199

(iii) The given equation is 2x 2 + kx + 2 = 0. 

D = (k 2  4 # 2 # 2) = (k 2  16) .

The given equation will have real roots if D $ 0. Now, D $ 0 & (k 2  16) $ 0 & k 2 $ 16 & k $ 4 or k # 4. EXAMPLE 17

Determine the positive value of p for which the equations x 2 + 2px + 64 = 0 and x 2  8x + 2p = 0 will both have real roots.

SOLUTION

Let D1 and D2 be the discriminants of the first and second given equations respectively.

[CBSE 2013C]

For real roots, we must have D1 $ 0 and D2 $ 0. Now, D1 $ 0 and D2 $ 0 

(4p 2  4 # 64) $ 0 and (64  8p) $ 0



p 2  64 $ 0 and 64  8p $ 0



p 2 $ 64 and 8p # 64



p $ 8 and p # 8 [a p is positive]



p = 8.

Hence, p = 8. f

EXERCISE 4D

1. Find the nature of the roots of the following quadratic equations: (i) 2x 2  8x + 5 = 0 (ii) 3x 2  2 6 x + 2 = 0 (iii) 5x 2  4x + 1 = 0 (v) 12x 2  4 15 x + 5 = 0

(iv) 5x (x  2) + 6 = 0 (vi) x 2  x + 2 = 0

2. If a and b are distinct real numbers, show that the quadratic equation 2 (a 2 + b 2) x 2 + 2 (a + b) x + 1 = 0 has no real roots. 3. Show that the roots of the equation x 2 + px  q 2 = 0 are real for all real values of p and q. 4. For what values of k are the roots of the quadratic equation [CBSE 2008C] 3x 2 + 2kx + 27 = 0 real and equal? 5. For what value of k are the roots of the quadratic equation [CBSE 2013] kx (x  2 5 ) + 10 = 0 real and equal? 6. For what values of p are the roots of the equation 4x 2 + px + 3 = 0 real and equal? [CBSE 2014]

200

Secondary School Mathematics for Class 10

7. Find the nonzero value of k for which the roots of the quadratic equation [CBSE 2014] 9x 2  3kx + k = 0 are real and equal. 8. (i) Find the values of k for which the quadratic equation [CBSE 2014] (3k + 1) x 2 + 2 (k + 1) x + 1 = 0 has real and equal roots. (ii) Find the value of k for which the equation x 2  k (2x  k  1)  2  0 has real and equal roots. [CBSE 2017] 9. Find the values of p for which the quadratic (2p + 1) x 2  (7p + 2) x + (7p  3) = 0 has real and equal roots.

equation [CBSE 2014]

10. Find the values of p for which the quadratic equation = 1 has equal roots. Hence, find (p + 1) x 2  6 (p + 1) x + 3 (p + 9) = 0, p Y the roots of the equation. [CBSE 2015] 2 11. If –5 is a root of the quadratic equation 2x + px  15 = 0 and the quadratic equation p (x 2 + x) + k = 0 has equal roots, find the value of k. [CBSE 2014]

12. If 3 is a root of the quadratic equation x  x + k = 0, find the value of p so that the roots of the equation x 2 + k (2x + k + 2) + p = 0 are equal. 2

[CBSE 2015]

13. If –4 is a root of the equation x + 2x + 4p = 0, find the value of k for which the quadratic equation x 2 + px (1 + 3k) + 7 (3 + 2k) = 0 has equal roots. [CBSE 2015] 2 2 2 2 14. If the quadratic equation (1 + m ) x + 2mcx + c  a = 0 has equal roots, prove that c 2 = a 2 (1 + m 2) . [CBSE 2014, ’17] 2

15. If the roots of the equation (c 2  ab) x 2  2 (a 2  bc) x + (b 2  ac) = 0 are real and equal, show that either a = 0 or (a 3 + b 3 + c 3) = 3abc. [CBSE 2017] 16. Find the values of p for which the quadratic equation 2x 2 + px + 8 = 0 has real roots. 17. Find the value of  for which the equation (  12) x 2 + 2 (  12) x + 2 = 0 has equal roots. [CBSE 2013] 2 + + = 18. Find the value of k for which the roots of 9x 8kx 16 0 are real and equal. 19. Find the values of k for which the given quadratic equation has real and distinct roots: (i) kx 2 + 6x + 1 = 0 (ii) x 2  kx + 9 = 0 (iii) 9x 2 + 3kx + 4 = 0

(iv) 5x 2  kx + 1 = 0

20. If a and b are real and a ! b then show that the roots of the equation (a  b) x 2 + 5 (a + b) x  2 (a  b) = 0 are real and unequal.

Quadratic Equations

201

21. (i) If the roots of the equation (a 2 + b 2) x 2  2 (ac + bd) x + (c 2 + d 2) = 0 are a c equal, prove that = · [CBSE 2017] b d (ii) If ad ! bc then prove that the equation (a 2  b 2) x 2  2(ac  bd) x  (c 2  d 2)  0 has no real roots.

[CBSE 2017]

22. If the roots of the equations ax 2 + 2bx + c = 0 and bx 2  2 ac x + b = 0 are simultaneously real then prove that b 2 = ac. ANSWERS (EXERCISE 4D)

1. (i) Real and unequal

(v) Real and equal 4. k = 9 or k = 9 7. k = 4

(ii) Real and equal (vi) Not real 5. k = 0 or k = 2

8. (i) k = 0 or k = 1

49

10. p = 3 or p = –1

11. k = 28 16. p ≥ 8 or p ≤ –8 17.  = 14 19. (i) k < 9

(iii) Not real

(ii) k > 6 or k < –6

6. p = 4 3 or p =  4 3

4 9. p = 4 or p = 7 10 12. p = 12 13. k = 2 or k = 9 18. k = 3 or k = –3 (ii) k  2

(iii) k > 4 or k < –4

(iv) k > 2 5 or k < 2 5

HINTS TO SOME SELECTED QUESTIONS 2. D  4 (a  b) 2  8 (a 2  b 2)  4 [(a  b) 2  2 (a 2  b 2)]  4 (a 2  b 2  2ab)   4 (a 2  b 2  2ab)   4 (a  b) 2  0. 3. D  (p 2  4q 2) $ 0. 4. D  4k 2  324. For real and equal roots, we must have, D  0. 5. Given equation is kx 2  2 5 kx  10  0. 

D  20k 2  40k  20k (k  2) .

For real and equal roots, we must have D  0. 6. For real and equal roots, we must have D  0. 

D  0 & p 2  48  0 & p 2  48 & p  ! 48 & p  4 3 or  4 3 .

9. D  (7p  2) 2  4 (2p  1) (7p  3)  7p 2  24p  16. 

D  0 & 7p 2  24p  16  0 & 7p 2  28p  4p  16  0 & 7p (p  4)  4 (p  4)  0 & (p  4) (7p  4)  0.

15. D  4 (a

2

bc) 2  4 (c 2  ab) (b 2  ac)

Now, D  0 & (a 2  bc) 2  (c 2  ab)(b 2  ac)  0 & a 4  3a 2 bc  ac 3  ab 3  0

(iv) Not real

202

Secondary School Mathematics for Class 10 & a (a 3  b 3  c 3  3abc)  0 & a  0 or a 3  b 3  c 3  3abc.

16. D  (p

2

64) . So, D $ 0 & p 2 $ 64 & p $ 8 or p # 8.

19. (i) 36  4k  0 & 36  4k & 4k  36 & k  9. (ii) k 2  36  0 & k 2  36 & k  6 or k  6. (iii) 9k 2  144  0 & k 2  16 & k  4 or k  4. (iv) k 2  20  0 & k 2  20 & k  2 5 or k  2 5 . 20. D  25 (a  b) 2  8 (a  b) 2  17 (a  b) 2  8 [(a  b) 2  (a  b) 2]  17 (a  b) 2  16 (a 2  b 2)  0. 21. (i) 4 (ac  bd) 2  4 (a 2  b 2)(c 2  d 2)  0 



(a 2  b 2)(c 2  d 2)  (ac  bd) 2  0









a 2 d 2  b 2 c 2  2abcd  0 & (ad  bc) 2  0 & ad  bc  0 a c ad  bc &  $ b d

22. 4b 2  4ac $ 0 & b 2  ac $ 0 & b 2 $ ac. 4ac  4b 2 $ 0 & ac  b 2 $ 0 & b 2 # ac Hence, b 2  ac.

WORD PROBLEMS ON QUADRATIC EQUATIONS SOLVED EXAMPLES PROBLEMS ON NUMBERS EXAMPLE 1

If the sum of two natural numbers is 27 and their product is 182, find the numbers.

SOLUTION

Let the required numbers be x and (27 – x). Then, x (27  x) = 182 

27x  x 2 = 182  x 2  27x + 182 = 0



x 2  13x  14x + 182 = 0



x (x  13)  14 (x  13) = 0



(x  13)(x  14)  0



x  13 = 0 or x  14 = 0



x = 13 or x = 14.

Hence, the required numbers are 13 and 14.

Quadratic Equations

203

EXAMPLE 2

The sum of the squares of two consecutive odd numbers is 394. Find the numbers. [CBSE 2014]

SOLUTION

Let the required consecutive odd numbers be x and (x + 2). Then, x 2 + (x + 2) 2 = 394  

2x 2 + 4x  390 = 0  x 2 + 2x  195 = 0 x 2 + 15x  13x  195 = 0



x (x + 15)  13 (x + 15) = 0 (x  15)(x  13)  0



x + 15 = 0 or x  13 = 0



x = 15 or x = 13



x = 13.



[rejecting x = –15]

Hence, the required numbers are 13 and 15. EXAMPLE 3

The sum of the squares of two consecutive even numbers is 340. Find the numbers. [CBSE 2014]

SOLUTION

Let the required consecutive even numbers be x and (x + 2). Then, x 2 + (x + 2) 2 = 340  

2x 2 + 4x  336 = 0  x 2 + 2x  168 = 0 x 2 + 14x  12x  168 = 0



x (x + 14)  12 (x + 14) = 0 (x + 14) (x  12) = 0



x + 14 = 0 or x  12 = 0



x = 14 or x = 12



x = 12 [rejecting x = 14]



Hence, the required numbers are 12 and 14. EXAMPLE 4

The sum of the squares of two consecutive multiples of 7 is 637. Find the multiples. [CBSE 2014]

SOLUTION

Let the required consecutive multiples of 7 be 7x and 7 (x + 1) . Then, (7x) 2 + {7 (x + 1)} 2 = 637 

49x 2 + (7x + 7) 2 = 637  98x 2 + 98x  588 = 0



x 2 + x  6 = 0  x 2 + 3x  2x  6 = 0

204

Secondary School Mathematics for Class 10



x (x + 3)  2 (x + 3) = 0  (x + 3) (x  2) = 0



x + 3 = 0 or x  2 = 0



x = 3 or x = 2



x=2

[neglecting x = –3]

Hence, the required numbers are (7 × 2) and (7 × 3), i.e., 14 and 21. EXAMPLE 5

The sum of two natural numbers is 9 and the sum of their reciprocals 1 is 2 · Find the numbers. [CBSE 2012]

SOLUTION

Let the required natural numbers be x and (9 – x). Then, 1+ 1 =1 x (9  x) 2



(9  x) + x 1 = 2 x (9  x) x (9  x) = 18



x  9x + 18 = 0  x 2  6x  3x + 18 = 0



x (x  6)  3 (x  6) = 0  (x  6) (x  3) = 0



x  6 = 0 or x  3 = 0  x = 6 or x = 3.



[by cross multiplication]

2

Hence, the required natural numbers are 6 and 3. EXAMPLE 6

SOLUTION

The difference of two natural numbers is 5 and the difference of their 1 reciprocals is · Find the numbers. [CBSE 2014] 10 Let the required natural numbers be x and (x  5) . Then, 1 1 1 1 x  x5 & x   &   x· x 5 x 5



1 1 = 1 x  5 x 10 x  (x  5) 1 5   1 [by cross multiplication] &  10 (x  5) x (x 5) x 10 (x  5) x = 50  x 2  5x  50 = 0



x 2  10x + 5x  50 = 0  x (x  10) + 5 (x  10) = 0



(x  10) (x + 5) = 0



x  10 = 0 or x + 5 = 0



x = 10 or x = 5



x = 10 [a –5 is not a natural number]

 

Hence, the required natural numbers are 10 and 5.

Quadratic Equations

205

EXAMPLE 7

The difference of squares of two numbers is 180. The square of the smaller number is 8 times the larger number. Find the two numbers.

SOLUTION

Let the required numbers be x and y such that x > y.

[CBSE 2012]

Then, x 2  y 2 = 180.

… (i)

And, y = 8x.

… (ii)

2

From (i) and (ii), we get x 2  8x  180 = 0  x 2  18x  10x  180  0 & x (x  18)  10 (x  18)  0  (x  18)(x  10)  0 & x  18  0 or x  10  0  x = 18 or x = –10. Now, x  18 & y 2  (8 #18)  144 & y  12 or y  12. Also, x  10 & y 2  [8 #(10)]  80, which is not possible. Hence, the numbers are (18 and 12) or (18 and –12). EXAMPLE 8

The numerator of a fraction is 3 less than its denominator. If 2 is added to both of its numerator and denominator then the sum of the 29 new fraction and original fraction is 20 · Find the original fraction. [CBSE 2015]

SOLUTION

Let the denominator of the required fraction be x. Then, its numerator = (x  3) . (x  3) So, the original fraction is x $ (x  3) + 2 (x  3) 29 + =  x 20 x+2 (x  1) (x  3) 29 + =  x 20 (x + 2) x (x  1) + (x  3) (x + 2) 29 =  20 (x + 2) x  

(x 2  x) + (x 2  x  6) 29 = 20 x 2 + 2x 2 2 20 (2x  2x  6) = 29 (x + 2x)



40x 2  40x  120 = 29x 2 + 58x 11x 2  98x  120 = 0



11x 2  110x + 12x  120 = 0



206

Secondary School Mathematics for Class 10

    

11x (x  10) + 12 (x  10) = 0 (x  10) (11x + 12) = 0 x  10 = 0 or 11x + 12 = 0  12 x = 10 or x = 11 x = 10 [a denominator of a fraction is never negative]

denominator = 10 and numerator = (10  3) = 7. 7 Hence, the required fraction is 10 ·



EXAMPLE 9

The denominator of a fraction is one more than twice the numerator. 16 If the sum of the fraction and its reciprocal is 2 21 , find the fraction.

SOLUTION

Let the numerator of the required fraction be x. Then, its denominator = (2x + 1) . (2x + 1) x  fraction = and its reciprocal = $ x (2x + 1) 

(2x + 1) 58 x + = x 21 (2x + 1)



21 # [x 2 + (2x + 1) 2] = 58x (2x + 1)



21 # [5x 2 + 4x + 1] = 116x 2 + 58x

  

11x 2  26x  21 = 0 11x 2  33x  7x  21  0 & 11x (x  3)  7 (x  3)  0 (x  3)(11x  7)  0 & x  3  0 or 11x  7  0 7 x = 3 or x = 11 x = 3 [a numerator cannot be a negative fraction]. x = 3· required fraction = (2x + 1) 7

   EXAMPLE 10

A two-digit number is 5 times the sum of its digits and is also equal to 5 more than twice the product of its digits. Find the number.

SOLUTION

Let the tens and units digits of the required number be x and y respectively. Then, 5x … (i) 10x  y  5 (x  y) & 4y  5x & y  4 5x  5x [using (i)] 10x  y  2xy  5 & 10x  2x #  5 4 4 2 45x 10x & 4  4  5 & 10x 2  45x  20  0 & 2x 2  9x  4  0 & 2x 2  8x  x  4  0

Quadratic Equations

207

& 2x (x  4)  (x  4)  0 & (x  4)(2x  1)  0 & x  4  0 or 2x  1  0 & x  4 or x  12 & x  4 [a a digit cannot be a fraction] . Putting x = 4 in (i), we get y = 5.  x = 4 and y = 5. Hence, the required number is 45. EXAMPLE 11

A two-digit number is such that the product of its digits is 18. When 63 is subtracted from the number, the digits interchange their places. Find the number. [CBSE 2006C]

SOLUTION

Let the tens and units digits of the required number be x and y respectively. Then, 18 … (i) xy  18 & y  x And, (10x + y)  63 = 10y + x 

9x  9y  63 & x  y  7

… (ii)

18 Putting y = x from (i) into (ii), we get 18 x x = 7 

x 2  18  7x & x 2  7x  18  0



x 2  9x  2x  18  0 & x (x  9)  2 (x  9)  0



(x  9)(x  2)  0 & x  9  0 or x  2  0



x = 9 or x = 2



x=9

[a a digit cannot be negative].

Putting x = 9 in (i), we get y = 2. Thus, the tens digit is 9 and the units digit is 2. Hence, the required number is 92. GENERAL PROBLEMS ON MONEY MATTERS EXAMPLE 12

A person on tour has ` 4200 for his expenses. If he extends his tour for 3 days, he has to cut down his daily expenses by ` 70. Find the original duration of the tour. [CBSE 2008C]

208 SOLUTION

Secondary School Mathematics for Class 10

Let the original duration of the tour be x days. Then, 4200  4200 = 70 x (x + 3)



(x  3)  x 1  70 F  70 & (x  3) 4200 x (x  3) 2 x (x  3)  180 & x  3x  180  0



x 2  15x  12x  180  0 & x (x  15)  12 (x  15)  0



(x  15)(x  12)  0 & x  15  0 or x  12  0



x = 15 or x = 12



x = 12



original duration of the tour is 12 days.



1 4200 # 0. ⎝ 4⎠ 16 ⎭ ⎩ So, we may write (i) as 3 21 = 24 ⇒ 2 k 2 + 3 k + 21 = 48 k2 + k + 2 2 ⇒ 2 k 2 + 3 k − 27 = 0 ⇒ 2 k 2 + 9 k − 6 k − 27 = 0 ⇒ k( 2 k + 9) − 3( 2 k + 9) = 0 ⇒ ( 2 k + 9)( k − 3) = 0 9 ⇒ k − 3 = 0 or 2 k + 9 = 9 ⇒ k = 3 or k = − ⋅ 2 9 Hence, k = 3 or k = − ⋅ 2 EXAMPLE 3

If A( 4 , − 6), B( 3 , − 2) and C(5 , 2) are the vertices of a SABC and AD is its median. Prove that the median AD divides SABC into two triangles of equal areas. [CBSE 2014]

SOLUTION

It is being given that A(4, –6), B(3, –2) and C(5, 2) are the vertices of a SABC and AD is its median. Clearly, D is the midpoint of BC. ∴

the coordinates of D are ⎛⎜ 3 + 5 , −2 + 2 ⎞⎟ = ( 4 , 0). ⎝ 2 2 ⎠

For SABD, we have A( x1 = 4 , y1 = −6), B( x2 = 3 , y2 = −2) andD( x3 = 4 , y3 = 0). 1 ar( SABD) = x1 ( y2 − y3 ) + x2 ( y3 − y1 ) + x3 ( y1 − y2 ) 2 1 = 4 ⋅ ( −2 − 0) + 3 ⋅ ( 0 + 6) + 4 ⋅ ( −6 + 2) 2 1 = 4 × ( −2) + 3 × 6 + 4 × ( −4) 2 1 1 = − 8 + 18 − 16 = −6 2 2 1 = ⎛⎜ × 6 ⎞⎟ = 3 sq units. ⎝2 ⎠

Coordinate Geometry

331

For SADC , we have ( x1 = 4 , y1 = −6), ( x2 = 4 , y2 = 0) and ( x3 = 5 , y3 = 2). 1 ∴ ar (SADC ) = x1 ( y2 − y3 ) + x2 ( y3 − y1 ) + x3 ( y1 − y2 ) 2 1 = 4 ⋅ ( 0 − 2) + 4 ⋅ ( 2 + 6) + 5 ⋅ ( −6 − 0) 2 1 = 4 × ( −2) + 4 × 8 + 5 × ( −6) 2 1 1 = − 8 + 32 − 30 = − 6| 2 2 1 = ⎛⎜ × 6 ⎞⎟ = 3 sq units. ⎝2 ⎠ ∴ ar( SABD) = ar( SADC ). Hence, the median AD divides SABC into two triangles of equal areas. EXAMPLE 4

Find the area of the triangle formed by joining the midpoints of the sides of the triangle whose vertices are A( 2 , 2), B( 4 , 4) and C( 2 , 6). [CBSE 2009C]

SOLUTION

Let A( 2 , 2), B( 4 , 4) and C( 2 , 6) be the vertices of the given SABC. Let D, E and F be the midpoints of AB, BC and CA respectively. Then, the coordinates of D, E and F are 2 + 4 2 + 4⎞ 4 + 2 4 + 6⎞ 2 + 2 2 + 6⎞ ⎟ , E ⎛⎜ ⎟ and F ⎛⎜ ⎟, D ⎛⎜ , , , ⎝ 2 ⎝ 2 ⎝ 2 2 ⎠ 2 ⎠ 2 ⎠ i.e., D(3, 3), E(3, 5) and F(2, 4). For SDEF, we have ( x1 = 3 , y1 = 3), ( x2 = 3 , y2 = 5) and ( x3 = 2 , y3 = 4). 1 ∴ ar( SDEF) = x1 ( y2 − y3 ) + x2 ( y3 − y1 ) + x3 ( y1 − y2 ) 2 1 = 3 ⋅ (5 − 4) + 3 ⋅ ( 4 − 3) + 2 ⋅ ( 3 − 5) 2 1 = ( 3 × 1) + ( 3 × 1) + 2 × ( −2) 2 1 1 = 3 + 3 − 4 = ⎛⎜ × 2 ⎞⎟ = 1 sq unit. ⎝2 ⎠ 2 Hence, the area of SDEF is 1 sq unit.

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Secondary School Mathematics for Class 10

EXAMPLE 5

Find the area of a quadrilateral ABCD whose vertices are A( −4 , 8), [CBSE 2015] B( −3 , − 4), C( 0 , − 5) and D(5 , 6).

SOLUTION

Area of quad. ABCD = ar(SABC) + ar(SACD).

For SABC, we have ( x1 = −4 , y1 = 8), ( x2 = −3 , y2 = −4) and ( x3 = 0 , y3 = −5). 1 ∴ ar( SABC ) = x1 ( y2 − y3 ) + x2 ( y3 − y1 ) + x3 ( y1 − y2 ) 2 1 = ( −4) ⋅ ( −4 + 5) + ( −3) ⋅ ( −5 − 8) + 0 ⋅ ( 8 + 4) 2 1 = ( −4) × 1 + ( −3) × ( −13) + 0 2 1 35 sq units. = − 4 + 39 + 0 = 2 2 For SACD, we have ( x1 = −4 , y1 = 8), ( x2 = 0 , y2 = −5) and ( x3 = 5 , y3 = 6). 1 ∴ ar( SACD) = x1 ( y2 − y3 ) + x2 ( y3 − y1 ) + x3 ( y1 − y2 ) 2 1 = ( −4) ⋅ ( −5 − 6) + 0 ⋅ ( 6 − 8) + 5 ⋅ ( 8 + 5) 2 1 = ( −4) × ( −11) + 0 × ( −2) + 5 × 13 2 1 109 sq unit. = 44 + 0 + 65 = 2 2 ∴ ar (quad. ABCD) = ar(SABC) + ar(SACD) 35 109 ⎞ 144 ⎟ sq units = sq units = ⎛⎜ + ⎝ 2 ⎠ 2 2 = 72 sq units. Hence, the area of the given quadrilateral ABCD is 72 sq units. EXAMPLE 6

Find the area of a parallelogram ABCD if three of its vertices are [CBSE 2013] A( 2 , 4), B( 2 + 3 , 5) and C( 2 , 6).

Coordinate Geometry SOLUTION

333

Let ABCD be the given parallelogram, three of whose vertices are A(2, 4), B( 2 + 3 , 5) and C(2, 6).

In SABC, we have ( x1 = 2 , y1 = 4), ( x2 = 2 + 3 , y2 = 5), ( x3 = 2 , y3 = 6). 1 ∴ ar( SABC ) = x1 ( y2 − y3 ) + x2 ( y3 − y1 ) + x3 ( y1 − y2 ) 2 1 = 2(5 − 6) + ( 2 + 3 )( 6 − 4) + 2( 4 − 5) 2 1 = 2 × ( −1) + ( 4 + 2 3 ) − 2 2 1 = ( 2 3 ) = 3 sq unit 2 ⇒ ar(||gm ABCD) = 2 × ar( SABC ) = ( 2 × 3 ) sq units = ( 2 3 ) sq units. Hence, the area of the given parallelogram is 2 3 sq units. EXAMPLE 7

If the points A(1 , − 2), B( 2 , 3), C( −3 , 2) and D( −4 , − 3) are the vertices of a parallelogram ABCD then taking AB as the base, find the height of the parallelogram. [CBSE 2013]

SOLUTION

Let A(1 , −2), B( 2 , 3), C( −3 , 2) and D( −4 , − 3) be the vertices of the given||gm ABCD. Join AC. Then, ar(||gm ABCD) = 2 × ar(AABC ).

In SABC , we have ∴

( x1 = 1 , y1 = −2), ( x2 = 2 , y2 = 3) and ( x3 = −3 , y3 = 2). 1 ar( SABC ) = x1 ( y2 − y3 ) + x2 ( y3 − y1 ) + x3 ( y1 − y2 ) 2 1 = 1 ⋅ ( 3 − 2) + 2 ⋅ ( 2 + 2) − 3 ⋅ ( −2 − 3) 2

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Secondary School Mathematics for Class 10

= ∴

1 1 + 8 + 15 = 12 sq units. 2

ar(||gm ABCD) = 2 × ar( SABC ) = 2 × 12 = 24 sq units.

Base AB = ( 2 − 1) 2 + ( 3 + 2) 2 = 1 2 + 5 2 = 26 units. Let h be the height of the||gm ABCD. Then, ar(||gm ABCD) = (base × height) = ( 26 × h) sq units. ∴

⎧ 24 26 ⎫ 26 × h = 24 ⇒ h = ⎨ × ⎬= 26 ⎭ ⎩ 26

⎛⎜ 24 × 26 ⎞⎟ units ⎝ 26 ⎠

12 61.08 ⇒ h = ⎛⎜ × 5.09 ⎞⎟ = = 4.69 units. ⎝ 13 ⎠ 13 Hence, the height of the parallelogram is 4.69 units. EXAMPLE 8

Show that the points A( −1 , 1), B(5 , 7 ) and C ( 8 , 10) are collinear.

SOLUTION

Let A( −1 , 1), B(5 , 7 ) and C ( 8 , 10) be the given points. Then, ( x1 = −1 , y1 = 1), ( x2 = 5 , y2 = 7 ) and ( x3 = 8 , y3 = 10). ∴

x1 ( y2 − y3 ) + x2 ( y3 − y1 ) + x3 ( y1 − y2 ) = ( −1)(7 − 10) + 5(10 − 1) + 8(1 − 7 )

= ( 3 + 45 − 48) = 0. Hence, the given points are collinear. EXAMPLE 9

Show that the points A( a, b + c), B(b , c + a) and C ( c , a + b) are collinear. [CBSE 2010]

SOLUTION

Let A( a, b + c), B(b , c + a) and C ( c , a + b) be the given points. Then, ( x1 = a, y1 = b + c), ( x2 = b , y2 = c + a) and ( x3 = c , y3 = a + b). ∴

x1 ( y2 − y3 ) + x2 ( y3 − y1 ) + x3 ( y1 − y2 ) = a( c + a − a − b) + b( a + b − b − c) + c(b + c − c − a) = a( c − b) + b( a − c) + c(b − a) = 0.

Hence, the given points are collinear. EXAMPLE 10

If the area of SABC with vertices A( x , y), B(1 , 2) and C( 2 , 1) is 6 sq units then prove that x + y = 15 or x + y + 9 = 0. [CBSE 2013]

SOLUTION

The vertices of SABC are A( x , y), B(1 , 2) and C( 2 , 1). Here, ( x1 = x , y1 = y), ( x2 = 1 , y2 = 2) and ( x3 = 2 , y3 = 1). 1 ∴ ar( SABC ) = x1 ( y2 − y3 ) + x2 ( y3 − y1 ) + x3 ( y1 − y2 ) 2

Coordinate Geometry

335

1 x ⋅ ( 2 − 1) + 1 ⋅ (1 − y) + 2 ⋅ ( y − 2) 2 1 1 = x + 1 − y + 2y − 4 = x+y− 3 . 2 2 =

∴

1 x + y − 3 = 6 ⇒ x + y − 3 = 12 2 ⇒ x + y − 3 = 12 or x + y − 3 = −12 ⇒ x + y = 15 or x + y + 9 = 0.

Hence, x + y = 15 or x + y + 9 = 0. EXAMPLE 11

Find the value of k for which the points A( k + 1 , 2 k), B( 3 k , 2 k + 3) [CBSE 2015, ’17] and C (5 k − 1 , 5 k) are collinear.

SOLUTION

The given points C (5 k − 1 , 5 k).

are

A( k + 1 , 2 k),

B( 3 k , 2 k + 3)

and

Here, ( x1 = k + 1 , y1 = 2 k), ( x2 = 3 k , y2 = 2 k + 3) and ( x3 = 5 k − 1 , y3 = 5 k). Let the given points be collinear. Then, x1 ( y2 − y3 ) + x2 ( y3 − y1 ) + x3 ( y1 − y2 ) = 0 ⇒ ( k + 1)( 2 k + 3 − 5 k) + 3 k(5 k − 2 k) + (5 k − 1)[2 k − ( 2 k + 3)] = 0 ⇒ ( k + 1)( 3 − 3 k) + 3 k × 3 k + (5 k − 1) × ( −3) = 0 ⇒ 3 − 3 k 2 + 9 k 2 − 15 k + 3 = 0 ⇒ 6 k 2 − 15 k + 6 = 0 ⇒ 2k 2 − 5k + 2 = 0 ⇒ 2k 2 − 4k − k + 2 = 0 ⇒ 2 k( k − 2) − ( k − 2) = 0 ⇒ ( k − 2)( 2 k − 1) = 0 1 ⇒ k − 2 = 0 or 2 k − 1 = 0 ⇒ k = 2 or k = ⋅ 2 1 Hence, k = 2 or k = ⋅ 2 EXAMPLE 12

If the points A( −1 , − 4), B(b , c) and C(5 , − 1) are collinear and [CBSE 2014] 2b + c = 4 , find the values of b and c.

SOLUTION

Let A( x1 = −1 , y1 = −4), B( x2 = b , y2 = c) and C ( x3 = 5 , y3 = −1) be the given points. Since these points A, B and C are collinear, we have x1 ( y2 − y3 ) + x2 ( y3 − y1 ) + x3 ( y1 − y2 ) = 0 ⇒ ( −1) ⋅ ( c + 1) + b ⋅ ( −1 + 4) + 5 ⋅ ( −4 − c) = 0 ⇒ − c − 1 + 3b − 20 − 5 c = 0 ⇒ 3b − 6 c = 21 ⇒ b − 2 c = 7 .

... (i)

Also, it is given that 2b + c = 4.

... (ii)

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Secondary School Mathematics for Class 10

On solving (i) and (ii), we get b = 3 and c = −2. Hence, b = 3 and c = −2. EXAMPLE 13

If R ( x , y) is a point on the line segment joining the points P( a, b) and [CBSE 2010] Q(b , a) then prove that x + y = a + b.

SOLUTION

It is being given that the points P( a, b), R ( x , y) and Q(b , a) lie on the same line segment and, therefore, these points are collinear. Let P( x1 = a, y1 = b), R ( x2 = x , y2 = y) and Q( x3 = b , y3 = a) be the given collinear points. Then, x1 ( y2 − y3 ) + x2 ( y3 − y1 ) + x3 ( y1 − y2 ) = 0 ⇒ a( y − a) + x( a − b) + b(b − y) = 0 ⇒ ay − a2 + ax − bx + b 2 − by = 0 ⇒ ( a − b) y − ( a 2 − b 2 ) + ( a − b) x = 0 ⇒ y − ( a + b) + x = 0 ⇒ x + y = a + b. Hence, x + y = a + b.

EXERCISE 6C 1. Find the area of SABC whose vertices are: (i) A(1, 2), B(−2, 3) and C(−3, −4) (ii) A(–5, 7), B(−4, –5) and C(4, 5) (iii) A(3, 8), B(–4, 2) and C(5, –1)

[CBSE 2008] [CBSE 2008C]

(iv) A(10, −6), B(2, 5) and C(−1, 3) 2. Find the area of quadrilateral ABCD whose vertices are A( 3 , − 1), [CBSE 2012] B( 9 , − 5), C(14 , 0) and D( 9 , 19). 3. Find the area of quadrilateral PQRS whose vertices are P( −5 , − 3), [CBSE 2015] Q( −4 , − 6), R( 2 , − 3) and S(1 , 2). 4. Find the area of quadrilateral ABCD whose vertices are A( −3 , −1), [CBSE 2013C] B( −2 , − 4), C( 4 , − 1) and D( 3 , 4). 5. If A(–7, 5), B(– 6, –7), C(–3, –8) and D(2, 3) are the vertices of a quadrilateral ABCD then find the area of the quadrilateral. [CBSE 2017] 6. Find the area of the triangle formed by joining the midpoints of the sides of the triangle whose vertices are A( 2 , 1), B( 4 , 3) and C( 2 , 5). [CBSE 2011]

Coordinate Geometry

337

7. A(7 , − 3), B(5 , 3) and C( 3 , −1) are the vertices of a SABC and AD is its median. Prove that the median AD divides SABC into two triangles of equal areas. 8. Find the area of SABC with A(1 , − 4) and midpoints of sides through A [CBSE 2015] being (2, –1) and (0, –1). 9. A( 6 , 1), B( 8 , 2) and C( 9 , 4) are the vertices of a parallelogram ABCD. If [CBSE 2013C] E is the midpoint of DC, find the area of SADE. 10.

(i) If the vertices of SABC be A(1, –3), B(4, p) and C(–9, 7) and its area [CBSE 2012] is 15 square units, find the values of p. (ii) The area of a triangle is 5 sq units. Two of its vertices are (2, 1) and 7 (3, –2). If the third vertex is ⎛⎜ , y ⎞⎟ , find the value of y. [CBSE 2017] ⎝2 ⎠

11. Find the value of k so that the area of the triangle with vertices [CBSE 2015] A( k + 1 , 1), B( 4 , − 3) and C (7 , − k) is 6 square units. 12. For what value of k( k > 0) is the area of the triangle with vertices ( −2 , 5), [CBSE 2012] ( k , − 4) and ( 2 k + 1 , 10) equal to 53 square units? 13. Show that the following points are collinear: (i) A( 2 , − 2), B( −3 , 8) and C ( −1 , 4) (ii) A( −5 , 1), B(5 , 5) and C (10 , 7 ) (iii) A(5 , 1), B(1 , −1) and C (11 , 4) (iv) A(8, 1), B(3, –4) and C(2, –5) 14. Find the value of x for which the points A( x , 2), B( −3 , − 4) and C(7 , −5) [CBSE 2015] are collinear. 15. For what value of x are the points A( −3 , 12), B(7 , 6) and C ( x , 9) collinear? 16. For what value of y are the points P(1 , 4), Q( 3 , y) and R( −3 , 16) are collinear? 17. Find the value of y for which the points A( −3 , 9), B( 2 , y) and C( 4 , −5) are collinear. 18. For what values of k are the points A( 8 , 1), B( 3 , − 2 k) and C ( k , −5) collinear. [CBSE 2015] 19. Find a relation between x and y, if the points A(2, 1), B( x , y) and C(7 , 5) [CBSE 2009C] are collinear. 20. Find a relation between x and y, if the points A( x , y), B( −5 , 7 ) and [CBSE 2015] C( −4 , 5) are collinear. 1 1 21. Prove that the points A( a, 0), B( 0 , b) and C(1 , 1) are collinear, if + = 1. a b

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22. If the points P( −3 , 9), Q( a, b) and R( 4 , −5) are collinear and a + b = 1, [CBSE 2014] find the values of a and b. 23. Find the area of SABC with vertices A( 0 , − 1), B( 2 , 1) and C( 0 , 3). Also, find the area of the triangle formed by joining the midpoints of its sides. [CBSE 2014] Show that the ratio of the areas of two triangles is 4 : 1. 24. If a ≠ b ≠ c , prove that ( a, a2 ), (b , b 2 ), ( 0 , 0) will not be collinear.

[CBSE 2017]

ANSWERS (EXERCISE 6C)

1. (i) 11 sq units 2. 132 sq units 8. 12 sq units

(ii) 53 sq units

(iii) 37.5 sq units

(iv) 24.5 sq units

3. 28 sq units 4. 28 sq units 5. 77 sq units 6. 1 sq unit 3 13 9. sq unit 10. (i) p = −3 or p = −9 (ii) y = 4 2

15. x = 2 11 19. 4 x − 5 y − 3 = 0 16. y = −2 17. y = −1 18. k = 2 or k = 2 20. 2 x + y + 3 = 0 22. a = 2 , b = −1 23. 4 sq units, 1 sq unit 11. k = 3

12. k = 3

14. x = −63

HINTS TO SOME SELECTED QUESTIONS 8. Let the vertices of SABC be A( 1, − 4 ), B( x2 , y2 ) and C ( x3 , y3 ). Let D( 2 , − 1) and E( 0 , − 1) be the midpoints of AB and AC respectively. Then, −4 + y 2 1 + x2 = 2, = −1 2 2 ⇒ 1 + x 2 = 4 , − 4 + y 2 = −2 x2 = 3 , y2 = 2. −4 + y 3 1 + x3 And, = 0, = −1 2 2 ⇒ 1 + x 3 = 0 , − 4 + y 3 = −2 ⇒

⇒

x3 = −1, y3 = 2.

In SABC, we have A( x1 = 1, y1 = −4 ), B( x2 = 3 , y2 = 2 ) and C ( x3 = −1, y3 = 2 ). Now, find ar( SABC ). 9. Let the fourth vertex be D( x , y ). Midpoint of AC is ⎛⎜ ⎝

6+ 9 4+ , 2 2

1⎞ ⎛ 15 5 ⎞ ⎟ , i.e., ⎜ , ⎟ ⋅ ⎠ ⎝ 2 2⎠

8 + x 2 + y⎞ Midpoint of BD is ⎛⎜ , ⎟⋅ ⎝ 2 2 ⎠

Coordinate Geometry ∴

339

2+ y 5 8 + x 15 = and = ⇒ x = 7 and y = 3. 2 2 2 2

So, we get the point D(7, 3). 7 + 9 3 + 4⎞ ⎛ 7⎞ Midpoint of DC is E ⎛⎜ , ⎟ , i.e., E ⎜ 8 , ⎟ ⋅ ⎝ 2 ⎝ 2⎠ 2 ⎠ 7 Now, A(6, 1), D(7, 3) and E ⎛⎜ 8 , ⎞⎟ are vertices of SADE. Now find its area. ⎝ 2⎠ 10. (i) Taking A( x1 = 1, y1 = −3 ), B( x2 = 4 , y2 = p ) and C ( x3 = −9 , y3 = 7 ), we get ar( SABC ) = 15 sq units ⇒

1 x1 ( y2 − y3 ) + x2 ( y3 − y1 ) + x3 ( y1 − y2 ) = 15 2

⇒

1 ⋅ ( p − 7 ) + 4 ⋅ (7 + 3 ) − 9 ⋅ ( −3 − p ) = 30

⇒

( p − 7 ) + 40 + 27 + 9 p = 30 ⇒ 10 p + 60 = 30

⇒

10 p + 60 = 30 or 10 p + 60 = −30 ⇒ p = −3 or p = −9.

11. Let A( x1 = k + 1, y1 = 1), B( x2 = 4 , y2 = −3 ) and C ( x3 = 7 , y3 = − k ) be the vertices of SABC. Then, 1 ar( SABC ) = x1 ( y2 − y3 ) + x2 ( y3 − y1 ) + x3 ( y1 − y2 ) 2 1 = ( k + 1)( −3 + k ) + 4( − k − 1) + 7 ( 1 + 3 ) 2 1 1 = ( k 2 − 6 k + 21) = ( k 2 − 6 k + 9 ) + 12 2 2 1 1 = ( k − 3 ) 2 + 12 = [( k − 3 ) 2 + 12]. 2 2 1 [( k − 3 ) 2 + 12] = 6 ⇒ ( k − 3 ) 2 + 12 = 12 ⇒ ( k − 3 ) 2 = 0. ∴ 2 Hence, k = 3. 12. Let A( x1 = −2 , y1 = 5 ), B( x2 = k , y2 = −4 ) and C ( x3 = 2 k + 1, y3 = 10 ). Then, ⇒ ⇒ ⇒ ⇒ ⇒ ⇒

ar( SABC ) = 53 sq units 1 x1 ( y2 − y3 ) + x2 ( y3 − y1 ) + x3 ( y1 − y2 ) = 53 2 1 ( −2 )( −4 − 10 ) + k( 10 − 5 ) + ( 2 k + 1)(5 + 4 ) = 53 2 28 + 5 k + 18 k + 9 = 106 23 k + 37 = 106 23 k + 37 = 106 or 23 k + 37 = −106 143 ⋅ k = 3 or k = − 23

14. Here, A( x1 = x , y1 = 2 ), B( x2 = −3 , y2 = −4 ) and C ( x3 = 7 , y3 = −5 ). ∴

S = 0 ⇒ x1 ( y2 − y3 ) + x2 ( y3 − y1 ) + x3 ( y1 − y2 ) = 0

340

Secondary School Mathematics for Class 10 Þ x( -4 + 5 ) - 3( -5 - 2 ) + 7 ( 2 + 4 ) = 0 Þ x + 21 + 42 = 0 Þ x = -63.

20. Here, A( x1 = x , y1 = y ), B( x2 = -5 , y2 = 7 ) and C ( x3 = -4 , y3 = 5 ). \

S = 0 Þ x1 ( y2 - y3 ) + x2 ( y3 - y1 ) + x3 ( y1 - y2 ) = 0 Þ x(7 - 5 ) - 5(5 - y ) - 4( y - 7 ) = 0 Þ 2 x - 25 + 5 y - 4 y + 28 = 0 Þ 2 x + y + 3 = 0.

21. Here, A( x1 = a, y1 = 0 ), B( x2 = 0 , y2 = b ) and C ( x3 = 1, y3 = 1). \

S = 0 Þ x1 ( y2 - y3 ) + x2 ( y3 - y1 ) + x3 ( y1 - y2 ) = 0 Þ a( b - 1) + 0 × ( 1 - 0 ) + 1 × ( 0 - b ) = 0 Þ ab - a - b = 0 Þ a + b = ab 1 1 Þ + = 1. a b

................................................................

EXERCISE 6D Very-Short-Answer Questions 1. Points A( -1 , y) and B(5 , 7 ) lie on a circle with centre O( 2 , - 3 y). Find the values of y. [CBSE 2014] 2. If the point A(0, 2) is equidistant from the points B( 3 , p) and C ( p , 5), [CBSE 2014] find p. 3. ABCD is a rectangle whose three vertices are B(4, 0), C(4, 3) and D(0, 3). [CBSE 2014] Find the length of one of its diagonal. 4. If the point P( k -1 , 2) is equidistant from the points A( 3 , k) and B( k , 5), [CBSE 2014] find the values of k. 5. Find the ratio in which the point P( x , 2) divides the join of A(12 , 5) and [CBSE 2014] B( 4 , - 3). 6. Prove that the diagonals of a rectangle ABCD with vertices A(2, –1), B(5, –1), C(5, 6) and D(2, 6) are equal and bisect each other. [CBSE 2014] 7. Find the lengths of the medians AD and BE of SABC whose vertices are [CBSE 2014] A(7, –3), B(5, 3) and C(3, –1). 8. If the point C ( k , 4) divides the join of A(2, 6) and B(5, 1) in the ratio 2 : 3 [CBSE 2013C] then find the value of k. 9. Find the point on x-axis which is equidistant from points A(–1, 0) and [CBSE 2013C] B(5, 0). æ2 ö æ -8 ö 10. Find the distance between the points ç , 2 ÷ and ç , 2 ÷ × è5 ø è 5 ø

[CBSE 2009C]

Coordinate Geometry

341

11. Find the value of a, so that the point ( 3 , a) lies on the line represented by [CBSE 2009] 2 x − 3 y = 5. 12. If the points A( 4 , 3) and B( x , 5) lie on the circle with centre O( 2 , 3), find [CBSE 2009] the value of x. 13. If P( x , y) is equidistant from the points A(7 , 1) and B( 3 , 5), find the relation between x and y. 14. If the centroid of SABC having vertices A( a, b), B(b , c) and C ( c , a) is the origin, then find the value of ( a + b + c). 15. Find the centroid of SABC whose vertices are A( 2 , 2), B( − 4 , − 4) and C(5 , − 8). 16. In what ratio does the point C( 4 , 5) divide the join of A( 2 , 3) and B(7 , 8)? 17. If the points A( 2 , 3), B( 4 , k) and C( 6 , − 3) are collinear, find the value of k. ANSWERS (EXERCISE 6D)

1. y = 7 or y = −1

2. p = 1

7. AD = 5 units, BE = 5 units 11. a =

1 3

16. 2 : 3

12. x = 2

3. 5 units 16 8. k = 5

13. x − y = 2

4. k = 1 or k = 5 9. P(2, 0)

14. a + b + c = 0

17. k = 0 HINTS TO SOME SELECTED QUESTIONS

1. We know that the radii of a circle are equal. ∴

OA = OB ⇒ OA 2 = OB 2 ⇒ ( 2 + 1) 2 + ( −3 y − y ) 2 = ( 2 − 5 ) 2 + ( −3 y − 7 ) 2 ⇒ 3 2 + 16 y 2 = ( −3 ) 2 + ( 9 y 2 + 49 + 42 y ) ⇒ 7 y 2 − 42 y − 49 = 0 ⇒ y 2 − 6 y − 7 = 0 ⇒ ( y − 7 )( y + 1) = 0 ⇒ y = 7 or y = −1.

3. Diagonal AC = diagonal BD = ( 4 − 0 ) 2 + ( 0 − 3 ) 2 = 25 = 5 units.

5. 3 : 5

10. 2 units −10 ⎞ ⎟ 15. ⎛⎜ 1 , ⎝ 3 ⎠

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Secondary School Mathematics for Class 10

4. PA = PB ⇒ PA 2 = PB 2 ⇒ ( k − 1 − 3)2 + ( 2 − k )2 = ( k − 1 − k )2 + ( 2 − 5)2 ⇒ ( k − 4 ) 2 + ( 2 − k ) 2 = 1 + 9 = 10 ⇒ 2 k 2 − 12 k + 10 = 0 ⇒ k 2 − 6 k + 5 = 0 ⇒ ( k − 1)( k − 5 ) = 0 ⇒ k = 1 or k = 5. 5. Let the required ratio be k : 1. Then, −3 k + 5 3 = 2 ⇒ −3 k + 5 = 2 k + 2 ⇒ 5 k = 3 ⇒ k = ⋅ k+1 5 3 Required ratio = ⎛⎜ : 1⎞⎟ = 3 : 5. ⎝5 ⎠ 6. Diagonal AC = (5 − 2 ) 2 + ( 6 + 1) 2 =

3 2 + 7 2 = 9 + 49 = 58

Diagonal BD = (5 − 2 ) 2 + ( −1 − 6 ) 2 =

3 2 + ( −7 ) 2 = 9 + 49 = 58 .

∴

Diag. AC = Diag. BD. 5 + 2 6 − 1⎞ ⎛ 7 5 ⎞ Midpoint of AC = ⎛⎜ , ⎟ =⎜ , ⎟⋅ ⎝ 2 2 ⎠ ⎝ 2 2⎠

Midpoint of BD = ⎛⎜ ⎝

5 + 2 −1 + 6 ⎞ ⎛ 7 5 ⎞ , ⎟ =⎜ , ⎟⋅ 2 2 ⎠ ⎝ 2 2⎠

5 + 3 3 − 1⎞ 7. Midpoint of BC is D ⎛⎜ , ⎟ , i.e., D(4, 1). ⎝ 2 2 ⎠ 7 + 3 −3 − 1 ⎞ Midpoint of AC is E⎛⎜ , ⎟ , i.e., E(5, –2). ⎝ 2 2 ⎠ AD = (7 − 4 ) 2 + ( −3 − 1) 2 =

3 2 + ( −4 ) 2 = 25 = 5 units.

BE = (5 − 5 ) 2 + ( 3 + 2 ) 2 = 0 + 25 = 25 = 5 units. 8. k =

( 2 × 5 + 3 × 2 ) 16 = ⋅ ( 2 + 3) 5

9. Let the required point be P( x , 0 ). Then, PA 2 = PB 2 ⇒ ( x + 1) 2 + ( 0 − 0 ) 2 = ( x − 5 ) 2 + ( 0 − 0 ) 2 ⇒ x 2 + 1 + 2 x = x 2 + 25 − 10 x ⇒ 12 x = 24 ⇒ x = 2. ∴ the required point is P(2, 0). 2

−8 2 ⎞ 10. Given distance = ⎛⎜ − ⎟ − ( 2 − 2 ) 2 = ( −2 ) 2 − 0 2 = 4 = 2 units. ⎝ 5 5⎠ 11. Since ( 3 , a) lies on the line 2 x − 3 y = 5 , we have 1 2 × 3 − 3a = 5 ⇒ 3a = 1 ⇒ a = ⋅ 3

Coordinate Geometry

343

12. ( x − 2 ) 2 + (5 − 3 ) 2 = ( 4 − 2 ) 2 + ( 3 − 3 ) 2 ⇒

( x − 2)2 + 2 2 = 2 2 + 0 2

⇒

( x − 2 ) 2 = 0 ⇒ x − 2 = 0 ⇒ x = 2.

13. (7 − x ) 2 + ( 1 − y ) 2 = ( 3 − x ) 2 + (5 − y ) 2 ⇒

( 49 + x 2 − 14 x ) + ( 1 + y 2 − 2 y ) = ( 9 + x 2 − 6 x ) + ( 25 + y 2 − 10 y )

⇒

x 2 + y 2 − 14 x − 2 y + 50 = x 2 + y 2 − 6 x − 10 y + 34

⇒

8 x − 8 y = 16 ⇒ x − y = 2.

a + b + c a + b + c⎞ 14. Centroid is ⎛⎜ , ⎟⋅ ⎝ ⎠ 3 3 a+ b + c So, = 0 ⇒ a + b + c = 0. 3 7 k + 2 8k + 3 ⎞ 16. Let the required ratio be k : 1. Then, C is ⎛⎜ , ⎟⋅ ⎝ k+1 k+1 ⎠ 7k + 2 2 = 4 ⇒ 7 k + 2 = 4k + 4 ⇒ 3k = 2 ⇒ k = ⋅ ∴ k+1 3 2 Required ratio is : 1, i.e., 2 : 3. 3 17. Let ( x1 = 2 , y1 = 3 ), ( x2 = 4 , y2 = k ) and ( x3 = 6 , y3 = −3 ). ∴

S = 0 ⇒ x1 ( y2 − y3 ) + x2 ( y3 − y1 ) + x3 ( y1 − y2 ) = 0 ⇒ 2( k + 3 ) + 4( −3 − 3 ) + 6( 3 − k ) = 0 ⇒ − 4 k = 0 ⇒ k = 0.

................................................................

MULTIPLE-CHOICE QUESTIONS (MCQ) Choose the correct answer in each of the following questions: 1. The distance of the point P(–6, 8) from the origin is (a) 8

(b) 2 7

(c) 6

[CBSE 2013C]

(d) 10

2. The distance of the point (–3, 4) from x-axis is (a) 3

(b) –3

(c) 4

[CBSE 2012]

(d) 5

3. The point on x-axis which is equidistant from points A( −1 , 0) and [CBSE 2013] B(5 , 0) is (a) (0, 2)

(b) (2, 0)

(c) (3, 0)

(d) (0, 3)

4. If R(5 , 6) is the midpoint of the line segment AB joining the points [CASE 2014] A( 6 , 5) and B(4, 4) then y equals (a) 5

(b) 7

(c) 12

(d) 6

5. If the point C ( k , 4) divides the join of the points A( 2 , 6) and B(5 , 1) in the ratio 2 : 3 then the value of k is [CBSE 2013C] 28 16 8 (c) (d) (a) 16 (b) 5 5 5

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Secondary School Mathematics for Class 10

6. The perimeter of the triangle with vertices (0, 4), (0, 0) and (3, 0) is [CBSE 2014]

(a) (7 + 5 )

(b) 5

(c) 10

(d) 12

7. If A(1 , 3), B(–1, 2), C(2, 5) and D(x, 4) are the vertices of a||gm ABCD [CBSE 2012] then the value of x is 3 (a) 3 (b) 4 (c) 0 (d) 2 8. If the points A( x , 2), B( −3 , − 4) and C(7 , −5) are collinear then the value [CBSE 2014] of x is (a) –63

(b) 63

(c) 60

(d) –60

9. The area of a triangle with vertices A(5, 0), B(8, 0) and C(8, 4) in square units is [CBSE 2012] (a) 20

(b) 12

(c) 6

(d) 16

10. The area of SABC with vertices A( a, 0), O( 0 , 0) and B( 0 , b) in square units is [CBSE 2011] 1 1 1 (c) a2b 2 (d) b 2 (a) ab (b) ab 2 2 2 a ⎞ ⎛ 11. If P ⎜ , 4 ⎟ is the midpoint of the line segment joining the points ⎝2 ⎠ A( −6 , 5) and B( −2 , 3) then the value of a is (a) –8

(b) 3

(c) –4

[CBSE 2011]

(d) 4

12. ABCD is a rectangle whose three vertices are B(4, 0), C(4, 3) and D(0, 3). The length of one of its diagonals is [CBSE 2014] (a) 5

(b) 4

(c) 3

(d) 25

13. The coordinates of the point P dividing the line segment joining the points A(1 , 3) and B( 4 , 6) in the ratio 2 : 1 is [CBSE 2012] (a) (2, 4)

(b) (3, 5)

(c) (4, 2)

(d) (5, 3)

14. If the coordinates of one end of a diameter of a circle are (2, 3) and the coordinates of its centre are (–2, 5), then the coordinates of the other end [CBSE 2012] of the diameter are (a) (–6, 7)

(b) (6, –7)

(c) (4, 2)

(d) (5, 3)

15. In the given figure P(5 , − 3) and Q( 3 , y) are the points of trisection of the line segment joining A(7 , −2) and B(1, –5). Then, y equals [CBSE 2012] (a) 2 (c) –4

(b) 4 −5 (d) 2

Coordinate Geometry

345

16. The midpoint of segment AB is P(0, 4). If the coordinates of B are (–2, 3), [CBSE 2011] then the coordinates of A are (a) (2, 5)

(b) (–2, –5)

(c) (2, 9)

(d) (–2, 11)

17. The point P which divides the line segment joining the points A(2, –5) and B(5, 2) in the ratio 2 : 3 lies in the quadrant [CBSE 2011] (a) I

(b) II

(c) III

(d) IV

18. If A(–6, 7) and B(–1, –5) are two given points then the distance 2AB is [CBSE 2011]

(a) 13

(b) 26

(c) 169

(d) 238

19. Which point on x-axis is equidistant from the points A(7, 6) and B( −3 , 4)? (a) (0, 4)

(b) (–4, 0)

(c) (3, 0)

(d) (0, 3)

20. The distance of P(3, 4) from the x-axis is (a) 3 units

(b) 4 units

(c) 5 units

(d) 1 unit

21. In what ratio does the x-axis divide the join of A(2, –3) and B(5, 6)? (a) 2 : 3

(b) 3 : 5

(c) 1 : 2

(d) 2 : 1

22. In what ratio does the y-axis divide the join of P(–4, 2) and Q(8, 3)? (a) 3 : 1

(b) 1 : 3

(c) 2 : 1

(d) 1 : 2

23. If P(–1, 1) is the midpoint of the line segment joining A(–3, b) and B(1 , b + 4) then b = ? (a) 1 (b) –1 (c) 2 (d) 0 24. The line 2 x + y − 4 = 0 divides the line segment joining A(2, –2) and B( 3 , 7 ) in the ratio (a) 2 : 5 (b) 2 : 9 (c) 2 : 7 (d) 2 : 3 25. If A( 4 , 2), B( 6 , 5) and C(1 , 4) be the vertices of SABC and AD is a median, then the coordinates of D are 5 7 9 7 (a) ⎛⎜ , 3 ⎞⎟ (b) ⎛⎜ 5 , ⎞⎟ (d) none of these (c) ⎛⎜ , ⎞⎟ ⎝2 ⎠ ⎝ 2 2⎠ ⎝ 2⎠ 26. If A( −1 , 0), B(5 , − 2) and C( 8 , 2) are the vertices of a SABC then its centroid is (a) (12, 0) (b) (6, 0) (c) (0, 6) (d) (4, 0) 27. Two vertices of SABC are A( −1 , 4) and B(5 , 2) and its centroid is G( 0 , − 3). Then, the coordinates of C are (a) (4, 3)

(b) (4, 15)

(c) (–4, –15)

(d) (–15, –4)

28. The points A( − 4 , 0), B( 4 , 0) and C( 0 , 3) are the vertices of a triangle, which is (a) isosceles (b) equilateral (c) scalene (d) right angled

346

Secondary School Mathematics for Class 10

29. The points P( 0 , 6), Q( −5 , 3) and R( 3 , 1) are the vertices of a triangle, which is (a) equilateral (b) isosceles (c) scalene (d) right angled 30. If the points A( 2 , 3), B(5 , k) and C( 6 , 7 ) are collinear then −3 11 (a) k = 4 (b) k = 6 (c) k = (d) k = 2 4 31. If the points A(1 , 2), O( 0 , 0) and C ( a, b) are collinear then (a) a = b

(b) a = 2b

(c) 2a = b

(d) a + b = 0

32. The area of SABC with vertices A(3, 0), B(7, 0) and C(8, 4) is (a) 14 sq units

(b) 28 sq units

(c) 8 sq units

(d) 6 sq units

33. AOBC is a rectangle whose three vertices are A( 0 , 3), O( 0 , 0) and B(5 , 0). The length of each of its diagonals is (a) 5 units

(b) 3 units

(c) 4 units

(d)

34 units

34. If the distance between the points A( 4 , p) and B(1 , 0) is 5 then (a) p = 4 only

(b) p = −4 only

(c) p = ±4

(d) p = 0

ANSWERS (MCQ)

1. (d) 9. (c) 17. (d) 25. (c) 33. (d)

2. (c) 10. (b) 18. (b) 26. (d) 34. (c)

3. (b) 11. (a) 19. (c) 27. (c)

4. (b) 12. (a) 20. (b) 28. (a)

5. (c) 13. (b) 21. (c) 29. (d)

6. (d) 14. (a) 22. (d) 30. (b)

7. (b) 15. (c) 23. (b) 31. (c)

HINTS TO SOME SELECTED QUESTIONS 1. OP 2 = ( −6 ) 2 + 8 2 = 36 + 64 = 100 ⇒ OP = 100 = 10 units. 2. Clearly, the distance of the point P( −3 , 4 ) from x-axis is 4 units.

8. (a) 16. (a) 24. (b) 32. (c)

Coordinate Geometry 3. Let the required point be P( x , 0 ). Then, PA 2 = PB 2 ⇒ ( x + 1) 2 = ( x − 5 ) 2 ⇒ x 2 + 2 x + 1 = x 2 − 10 x + 25 ⇒ 12 x = 24 ⇒ x = 2. So, the required point is P(2, 0). 4.

5+ y = 6 ⇒ 5 + y = 12 ⇒ y = 7 . 2

5. k =

( 2 × 5 ) + ( 3 × 2 ) 16 = ⋅ ( 2 + 3) 5

6. AB = 4 units, BC = 3 units. AC 2 = ( 3 − 0 ) 2 + ( 0 − 4 ) 2 = ( 9 + 16 ) = 25. ⇒

AC = 5 units.

∴

perimeter = (4 + 3 + 5) units = 12 units.

7. Midpoint of BD = midpoint of AC −1 + x 1 + 2 = ⇒ − 1 + x = 3 ⇒ x = 4. ⇒ 2 2

8. Here, ( x1 = x , y1 = 2 ), ( x2 = −3 , y2 = −4 ) and ( x3 = 7 , y3 = −5 ). ∴

x1 ( y2 − y3 ) + x2 ( y3 − y1 ) + x3 ( y1 − y2 ) = 0

⇒

x( −4 + 5 ) − 3( −5 − 2 ) + 7 ( 2 + 4 ) = 0

⇒

x + 21 + 42 = 0 ⇒ x = −63.

9. Here, ( x1 = 5 , y1 = 0 ), ( x2 = 8 , y2 = 0 ) and ( x3 = 8 , y3 = 4 ). 1 ∴ S = x1 ( y2 − y3 ) + x2 ( y3 − y1 ) + x3 ( y1 − y2 ) 2 1 = 5( 0 − 4 ) + 8( 4 − 0 ) + 8( 0 − 0 ) 2 1 1 = − 20 + 32 = ⎛⎜ × 12 ⎞⎟ = 6 sq units. ⎝2 ⎠ 2 10. Here, base = a units and height = b units. 1 ∴ area = ⎛⎜ × base × height ⎞⎟ ⎝2 ⎠ =

1 ab sq units. 2

347

348

11.

Secondary School Mathematics for Class 10 a ( −6 − 2 ) = = −4 ⇒ a = −8. 2 2

12. Diagonal BD = ( 4 − 0 ) 2 + ( 0 − 3 ) 2 = 16 + 9 = 25 = 5 units.

( 2 × 4 + 1 × 1) ( 2 × 6 + 1 × 3 ) ⎞ 13. Coordinates of P are ⎛⎜ , ⎟ = ( 3 , 5 ). ⎝ ⎠ 2+ 1 2+ 1 14. Let C( −2 , 5 ) be the centre of the given circle and A(2, 3) and B( x , y ) be the end points of a diameter ACB. Then, C is the midpoint of AB. 3+ y 2+ x = −2 and =5 ∴ 2 2 ⇒

2 + x = −4 and 3 + y = 10

⇒

x = −6 and y = 7 .

So, the coordinates of B are (–6, 7). 15. Q ( 3 , y ) divides AB in the ratio 2 : 1. 2 × 1 + 1 × 7 2 × ( −5 ) + 1 × ( −2 ) ⎞ So, Q is ⎛⎜ , ⎟ , i.e., (3, –4). ⎝ ⎠ 2+ 1 2+ 1 Hence, y = −4. 16. Let the point A be ( a, b ). Then, a + ( −2 ) b+ 3 = 0 and =4 2 2 ⇒

a − 2 = 0 and b = 8 − 3 ⇒ a = 2 , b = 5.

∴

the point A is (2, 5).

2× 5 + 3× 2 2× 2 − 3× 5⎞ ⎛ −11 ⎞ . 17. The point P is given by P ⎛⎜ , ⎟ = P ⎜ 3, ⎟ ⎝ ⎠ ⎝ 5 ⎠ 2+ 3 2+ 3 So, P lies in IV quadrant.

Coordinate Geometry 18. 2 AB = 2 × ( −1 + 6 ) 2 + ( −5 − 7 ) 2 = 2 × (5 ) 2 + ( −12 ) 2 = 2 × 169 = 2 × 13 = 26. 19. Let the required point be P( x , 0 ). Then, AP 2 = BP 2 ⇒ ( x − 7 ) 2 + ( 0 − 6 ) 2 = ( x + 3 ) 2 + ( 0 − 4 ) 2 ⇒ x 2 − 14 x + 85 = x 2 + 6 x + 25 ⇒ 20 x = 60 ⇒ x = 3. ∴

the required point is P(3, 0).

21. Let the x-axis cut AB at P( x , 0 ) in the ratio k : 1. Then, ∴

6k − 3 1 = 0 ⇒ 6k − 3 − 0 ⇒ 6k = 3 ⇒ k = ⋅ k+1 2

1 required ratio = ⎛⎜ : 1⎞⎟ = 1 : 2. ⎝2 ⎠

22. Let the y-axis cut AB at P( 0 , y ) in the ratio k : 1. Then, 8k − 4 3k + 2 ⎞ 8k − 4 P ⎛⎜ , =0 ⎟ = P( 0 , y ) ⇒ ⎝ k+1 k+1 ⎠ k+1 ⇒ ∴

1 ⋅ 2 1 required ratio = ⎛⎜ : 1⎞⎟ = 1 : 2. ⎝2 ⎠ 8k − 4 = 0 ⇒ k =

23. We have

b + ( b + 4) = 1 ⇒ 2 b + 4 = 2 ⇒ 2 b = −2 ⇒ b = −1. 2

24. Let the required ratio be k : 1. 3k + 2 7 k − 2 ⎞ Then, the point of division is P ⎛⎜ , ⎟. ⎝ k+1 k+1⎠ This point lies on the line 2 x + y − 4 = 0. 2( 3 k + 2 ) (7 k − 2 ) − 4 = 0 ⇒ 6k + 4 + 7 k − 2 − 4k − 4 = 0 + k+1 k+1 2 ⇒ 9k = 2 ⇒ k = ⋅ 9 2 ⎞ ⎛ So, the required ratio is ⎜ : 1⎟ , i. e. , ( 2 : 9 ). ⎝9 ⎠

∴

6 + 1 5 + 4⎞ ⎛7 9⎞ 25. Midpoint of BC is D ⎛⎜ , ⎟ = D⎜ , ⎟. ⎝ 2 ⎝ 2 2⎠ 2 ⎠ x + x2 + x3 y1 + y2 + y3 ⎞ ⎛ −1 + 5 + 8 , 0 − 2 + 2 ⎞ = ( 4 , 0 ). 26. Centroid is G ⎛⎜ 1 , ⎟ ⎟ =G⎜ ⎝ ⎠ ⎝ ⎠ 3 3 3 3 27. Let the vertex C be C ( x , y ). Then,

∴

4+ 2+ y −1 + 5 + x = 0 and = −3 ⇒ x + 4 = 0 and 6 + y = −9 3 3 x = −4 and y = −15.

So, the coordinates of C are (–4, –15).

349

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Secondary School Mathematics for Class 10

28. AB 2 = ( 4 + 4 ) 2 + ( 0 − 0 ) 2 = 8 2 + 0 2 = 64 + 0 = 64 ⇒ AB = 64 = 8 units. BC 2 = ( 0 − 4 ) 2 + ( 3 − 0 ) 2 = ( −4 ) 2 + 3 2 = 16 + 9 = 25 ⇒ BC = 25 = 5 units. AC 2 = ( 0 + 4 ) 2 + ( 3 − 0 ) 2 = 4 2 + 3 2 = 16 + 9 = 25 ⇒ AC = 25 = 5 units. ∴

SABC is isosceles.

29. PQ 2 = ( −5 − 0 ) 2 + ( 3 − 6 ) 2 = ( −5 ) 2 + ( −3 ) 2 = 25 + 9 = 34 , QR 2 = ( 3 + 5 ) 2 + ( 1 − 3 ) 2 = 8 2 + ( −2 ) 2 = 64 + 4 = 68 , PR 2 = ( 3 − 0 ) 2 + ( 1 − 6 ) 2 = 3 2 + ( −5 ) 2 = ( 9 + 25 ) = 34. ∴

PQ 2 + PR 2 = QR 2 .

Hence, SPQR is right-angled. 30. Here, ( x1 = 2 , y1 = 3 ), ( x2 = 5 , y2 = k ) and ( x3 = 6 , y3 = 7 ). Since the given points are collinear, we must have: x1 ( y2 − y3 ) + x2 ( y3 − y1 ) + x3 ( y1 − y2 ) = 0 ⇒

2( k − 7 ) + 5(7 − 3 ) + 6( 3 − k ) = 0 ⇒ 2 k − 14 + 20 + 18 − 6 k = 0

⇒

4 k = 24 ⇒ k = 6.

31. Here, ( x1 = 1, y1 = 2 ), ( x2 = 0 , y2 = 0 ) and ( x3 = a, y3 = b ). Since the given points are collinear, we have x1 ( y2 − y3 ) + x2 ( y3 − y1 ) + x3 ( y1 − y2 ) = 0 ⇒

1 ⋅ ( 0 − b ) + 0 ⋅ ( b − 2 ) + a ⋅ ( 2 − 0 ) = 0 ⇒ − b + 0 + 2 a = 0 ⇒ 2a = b.

32. Here, ( x1 = 3 , y1 = 0 ), ( x2 = 7 , y2 = 0 ) and ( x3 = 8 , y3 = 4 ). 1 ∴ area of S ABC = {x1 ( y2 − y3 ) + x2 ( y3 − y1 ) + x3 ( y1 − y2 )} 2 1 1 = { 3( 0 − 4 ) + 7 ( 4 − 0 ) + 8( 0 − 0 )} = {−12 + 28 + 0} 2 2 1 = ⎛⎜ × 16 ⎞⎟ = 8 sq units. ⎝2 ⎠ 33. Diagonal OC = diagonal AB = (5 − 0 ) 2 + ( 0 − 3 ) 2 = 25 + 9 =

34 units.

34. ( 4 − 1) 2 + ( p − 0 ) 2 = 5 2 ⇒ p 2 = ( 25 − 9 ) = 16 ⇒ p = ±4.

_

Triangles

351

Triangles

7

Two geometric figures which have the same shape and size are known as congruent figures. CONGRUENT FIGURES

Congruent figures are alike in every respect. SIMILAR FIGURES Geometric figures which have the same shape but different sizes are known as similar figures.

Two congruent figures are always similar but two similar figures need not be congruent. Examples (i) Any two line segments are similar. (ii) Any two equivalent triangles are similar. (iii) Any two squares are similar. (iv) Any two circles are similar.

SIMILAR POLYGONS

Two polygons having the same number of sides are said to be similar, if (i) their corresponding angles are equal, and (ii) the lengths of their corresponding sides are proportional. If two polygons ABCDE and PQRST are similar, we write, ABCDE ~ PQRST, where the symbol ‘~’ stands for ‘is similar to’. The constant ratio between the corresponding sides of two similar figures is known as the scale factor, or the representative fraction. Since triangles are also polygons, so the same set of conditions apply for the similarity of triangles. EQUIANGULAR TRIANGLES

Two triangles are said to be equiangular if their corresponding angles are equal. 351

352

Secondary School Mathematics for Class 10

SIMILAR TRIANGLES

Two triangles are said to be similar to each other if (i) their corresponding angles are equal, and (ii) their corresponding sides are proportional.

RESULTS ON SIMILAR TRIANGLES (BASIC-PROPORTIONALITY THEOREM) OR (THALES’ THEOREM) THEOREM 1

GIVEN

If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points then the other two sides are divided in the same ratio. [CBSE 2002C, ’04C, ’05, ’06C, ’07, ’09, ’10]

A 3ABC in which DE  BC and DE intersects AB and AC at D and E respectively.

TO PROVE

AD  AE · DB EC

CONSTRUCTION

Join BE and CD. Draw EL = AB and DM = AC.

PROOF

We have 1 1 ar(3ADE)  2 # AD # EL [a 3  2 # base # height] 1 and ar(3DBE)  2 # DB# EL. 1 ar(3ADE) 2 # AD # EL AD ·    … (i) DB 1 ar(3DBE) # # DB EL 2 1 Again, ar(3ADE)  ar(3AED)  2 # AE # DM 1 and ar(3ECD)  2 # EC # DM. 1 ar(3ADE) 2 # AE # DM AE ·    ar(3ECD) 1 # EC # DM EC 2

… (ii)

Now, 3DBE and 3ECD being on the same base DE and between the same parallels DE and BC, we have … (iii) ar(3DBE)  ar(3ECD) From (i), (ii) and (iii), we have AD  AE · DB EC

Triangles COROLLARY

PROOF

353

In a 3ABC, a line DE  BC intersects AB in D and AC in E, then prove that AB AC AD AE (i) DB  EC (ii) AB  AC ·

(i) From Basic-Proportionality theorem, we have AD  AE AD   AE  DB EC & DB 1 EC 1   AB  AC · & ADDBDB  AEECEC & DB EC (ii) From Basic-Proportionality theorem, we have AD  AE DB  EC DB EC & AD AE DB   EC k a1 AE k & a1  AD (AD  DB) (AE  EC)  & AD AE AB  AC AD  AE · & AD AE & AB AC

SUMMARY

In 3ABC, let DE  BC. Then, AD AE (i) DB  EC (B.P.T.) AB AC (ii) DB  EC AD AE (iii) AB  AC THEOREM 2

(Converse of Thales’ theorem) If a line divides any two sides of

a triangle in the same ratio then the line must be parallel to the third side. GIVEN

A 3ABC and a line l intersecting AB at D and AD AE AC at E, such that DB  EC ·

TO PROVE PROOF

DE  BC.

If possible, let DE not be parallel to BC. Then, there must be another line through D, which is parallel to BC. Let DF  BC. Then, by Thales’ theorem, we have AD  AF · DB FC AD  AE But, DB EC (given).

… (i) … (ii)

354

Secondary School Mathematics for Class 10

From (i) and (ii), we get AF  AE AF   AE  AF  FC  AE  EC FC EC & FC 1 EC 1 & FC EC 1  1 AC  AC & FC EC & FC EC & FC  EC. This is possible only when E and F coincide. Hence, DE < BC.

SOLVED EXAMPLES EXAMPLE 1

In the given figure, MN  AB, BC  7.5 cm, AM  4 cm and MC  2 cm. Find the length of BN. [CBSE 2010]

SOLUTION

In 3ABC, MN  AB. MC NC  AC  BC [by Thales’ theorem] MC  NC  AM  MC BC 2  x  , where NC  x cm 4  2 7.5 2 #7.5  15   x 6 6 2.5  NC  2.5 cm. Hence, BN  BC  NC  (7.5  2.5) cm  5 cm.

EXAMPLE 2

AD 3 In the given figure, DE  BC and DB  5 · If AC  4.8 cm, find the length of AE. [CBSE 2008C]

SOLUTION

Let AE  x cm. Then, EC  (AC  AE)  (4.8  x) cm. Now, in 3ABC, DE  BC. 

AD  AE 3 x DB EC & 5 (4.8  x)



3(4.8  x)  5x  8x  14.4



x  1.8.

Hence, AE  1.8 cm.

Triangles EXAMPLE 3

In the given figure, in 3ABC, DE  BC so that AD  (4x  3) cm, AE  (8x  7) cm, BD  (3x  1) cm and CE  (5x  3) cm. Find the value of x. [CBSE 2002C]

SOLUTION

In 3ABC, DE  BC. AD AE  BD  CE [by Thales’ theorem] 4x  3  8x  7  (4x  3)(5x  3)  (3x  1)(8x  7)  3x  1 5x  3  20x 2  27x  9  24x 2  29x  7  4x 2  2x  2  0  2x 2  x  1  0  2x 2  2x  x  1  0  2x(x  1)  (x  1)  0  (x  1)(2x  1)  0  (x  1)  0

or (2x  1)  0 1  x  1 or x  2 · 1 1 But, x  2 & AD  :4 #a 2 k  3D  5.

1 And, distance can never be negative. So, x ! 2 · Hence, x  1. EXAMPLE 4

SOLUTION

If D and E are points on the sides AB and AC respectively of 3ABC such that AB  5.6 cm, AD  1.4 cm, AC  7.2 cm and AE  1.8 cm, show that DE  BC. Given, AB  5.6 cm, AD  1.4 cm, AC  7.2 cm and AE  1.8 cm. AD  1.4  1 AE 1.8 1  AB 5.6 4 and AC  7.2  4 AD AE  AB  AC · Hence, by the converse of Thales’ theorem, DE  BC.

EXAMPLE 5

In the adjoining figure, MN  QR. Find (i) PN and (ii) PR.

355

356 SOLUTION

Secondary School Mathematics for Class 10

In 3PQR, MN  QR. PM PN  MQ  NR [by Thales’ theorem] 1.9  x  5.7 6.9 , where PN x cm 1.9 # 6.9   x  5.7 2.3. Hence, (i) PN  x cm  2.3 cm and (ii) PR  PN  NR  (2.3  6.9) cm  9.2 cm. 

EXAMPLE 6

AD AE In the given figure, DB  EC and +ADE  +ACB. Prove that 3ABC is an isosceles triangle.

SOLUTION

We have AD  AE DB EC & DE  BC [by the converse of Thales’ theorem]  +ADE  +ABC (corresponding O). But, +ADE  +ACB (given).  +ABC  +ACB. So, AB  AC [sides opposite to equal angles]. Hence, 3ABC is an isosceles triangle.

EXAMPLE 7

M and N are points on the sides AC and BC respectively of a 3ABC. In each of the following cases, state whether MN  AB. (i) CM  4.2 cm, MA  2.8 cm, NB  3.6 cm, CN  5.7 cm (ii) CB  6.92 cm, CN  1.04 cm, CA  1.73 cm, CM  0.26 cm (iii) CM  5.1 cm, CA  6.8 cm, CB  5.6 cm, NB  1.4 cm

SOLUTION

(i) We have CM  4.2  3 CN  5.7  19 · MA 2.8 2 and NB 3.6 12 CM CN Since MA ! NB · So, MN is not parallel to AB. (ii) We have MA  CA  CM  (1.73  0.26) cm  1.47 cm and NB  CB  CN  (6.92  1.04) cm  5.88 cm.

Triangles

357

CM  0.26  26 CN  1.04  26 · MA 1.47 147 and NB 5.88 147 CM CN Clearly, MA  NB and so MN  AB [by the converse of Thales’ theorem].



(iii) We have MA  CA  CM  (6.8  5.1) cm  1.7 cm and CN  CB  NB  (5.6  1.4) cm  4.2 cm. 

CM  5.1  3 CN  4.2  3 · MA 1.7 1 and NB 1.4 1

CM CN Clearly, MA  NB and so MN  AB [by the converse of Thales’ theorem]. EXAMPLE 8

In the given figure, DE  AC and DF  AE. BF BE Prove that FE  EC ·

SOLUTION

[CBSE 2005, ’07]

In 3BAE, DF  AE. 

BD  BF · DA FE

… (i) [by Thales’ theorem]

In 3BAC, DE  AC. 

BD  BE · DA EC

… (ii) [by Thales’ theorem]

From (i) and (ii), we get BF  BE FE EC

:each equal to BD D · DA

EXAMPLE 9

In the figure given along side, DE  OQ and DF  OR. Show that EF  QR.

SOLUTION

In 3POQ, DE  OQ. PD PE  DO  EQ · In 3POR, DF  OR.

… (i) [by Thales’ theorem]

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Secondary School Mathematics for Class 10

PD  PF · DO FR From (i) and (ii), we get PD D · PE  PF : EQ FR each equal to DO



… (ii)

Thus, in 3PQR, E and F are points on PQ and PR respectively PE PF such that EQ  FR · Hence, EF  QR [by the converse of Thales’ theorem]. EXAMPLE 10

In the given figure, LM  CB and LN  CD. AM AN Prove that AB  AD ·

SOLUTION

In 3ALM, LM  CB. AB AC AM AL  AM  AL & AB  AC · In 3ALN, LN  CD. AC AD AL AN  AL  AN & AC  AD · From (i) and (ii), we get AM  AN · AB AD

EXAMPLE 11

… (i)

[by Thales’ theorem]

… (ii)

[by Thales’ theorem]

In the given figure, AB  DE and BD  EF. Prove that DC 2  CF # AC. [CBSE 2004C, ’10]

SOLUTION

In 3ABC, AB  DE. CD CE  DA  EB · In 3CDB, BD  EF. CF CE  FD  EB · From (i) and (ii), we get CD  CF DA FD

… (i) [by Thales’ theorem]

… (ii) [by Thales’ theorem]

Triangles

     EXAMPLE 12

DA  FD [taking reciprocals] DC CF DA   FD  DC 1 CF 1 DA  DC  FD  CF DC CF AC  DC DC CF DC 2  CF # AC.

In the given figure, PQ  AB and PR  AC. Prove that QR  BC.

SOLUTION

359

In 3OAB, PQ  AB. OP OQ  PA  QB · In 3AOC, PR  AC. OP OR  PA  RC · From (i) and (ii), we get

[CBSE 2002, ’05C]

… (i) [by Thales’ theorem]

… (ii) [by Thales’ theorem]

OQ OR  QB RC in 3OBC. Thus, in 3OBC, Q and R are points on OB and OC respectively OQ OR such that QB  RC · Hence, by the converse of Thales’ theorem, QR  BC. EXAMPLE 13

SOLUTION

In the given figure, in 3ABC, +B  +C and BD  CE. Prove that DE  BC.

A 3ABC in which +B  +C and BD  CE. TO PROVE DE  BC. GIVEN

PROOF

In 3ABC, +B  +C & AB  AC [sides opposite equal O are equal].

Now, AB  AC & (AD  BD)  (AE  CE) & AD  AE [a BD  CE (given)]  AE & AD [a BD  CE]. BD CE

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Secondary School Mathematics for Class 10

AD AE Thus, BD  CE · Hence, DE  BC [by the converse of Thales’ theorem]. EXAMPLE 14

In 3ABC, D and E are two points on AB such that AD  BE. If DP  BC and EQ  AC, prove that PQ  AB.

SOLUTION

GIVEN A 3ABC and D, E are two points on AB such that AD  BE. Also, DP  BC, and EQ  AC. TO PROVE

PQ  AB.

In 3ABC, DP  BC. AD  AP ·  DB PC In 3CBA, EQ  AC. PROOF

 

… (i)

[by Thales’ theorem]

BE  BQ [by Thales’ theorem] EA QC AD  BQ · … (ii) DB QC [a BE  AD, EA  (ED  DA)  (DE  EB)  DB]

From (i) and (ii), we get AP  BQ · PC QC 

PQ  AB

[by the converse of Thales’ theorem].

EXAMPLE 15

If three or more parallel lines are intersected by two transversals, prove that the intercepts made by them on the transversals are proportional.

SOLUTION

GIVEN

Three lines l, m, n such that l  m  n.

These lines are cut by the transversals AB and CD in P, Q, R and E, F, G respectively. TO PROVE

PQ EF ·  QR FG

Draw PM  CD, meeting the lines m and n at L and M respectively.

CONSTRUCTION

Triangles PROOF

361

PL  EF and PE  LF & PLFE is a gm

& PL  EF.

… (i) (opp. sides of a gm)

Also, LM  FG and LF  MG & LMGF is a gm

& LM  FG.

… (ii) (opp. sides of a gm)

In 3PRM, QL  RM and therefore, by Thales’ theorem, we have PQ PL EF   QR LM FG [using (i) and (ii)]. 

PQ EF ·  QR FG

EXAMPLE 16

ABCD is a quadrilateral and P, Q, R, S are the points of trisection of the sides AB, BC, CD and DA respectively and are adjacent to A and C. Prove that PQRS is a parallelogram.

SOLUTION

GIVEN A quadrilateral ABCD in which P, Q, R, S are the points of trisection of AB, BC, CD and DA respectively (as shown). TO PROVE

PQRS is a parallelogram.

CONSTRUCTION PROOF



Join AC.

BP BQ 2 In 3BAC, we have PA  QC  1 ·

PQ  AC.

… (i) [by the converse of Thales’ theorem]

Also, in 3DAC, we have



DS  DR  2 · SA RC 1 … (ii) [by the converse of Thales’ theorem] SR  AC.

Thus, PQ  SR [from (i) and (ii)]. Similarly, by joining BD we can prove that SP  RQ. Hence, PQRS is a parallelogram. EXAMPLE 17

ABCD is a trapezium with AB  DC. E and F are points on nonparallel sides AD and BC respectively such that EF  AB. Show that AE  BF · ED FC

362 SOLUTION

Secondary School Mathematics for Class 10 GIVEN A trap. ABCD in which AB  DC. E and F are points on AD and BC respectively such that EF  AB. TO PROVE

AE  BF · ED FC

CONSTRUCTION PROOF

Join AC, intersecting EF at G.

EF  AB and AB  DC & EF  DC.

Now, in 3ADC, EG  DC. AE  AG · ED GC Similarly, in 3CAB, GF  AB. 

CG  CF GA FB AG BF  GC  FC · From (i) and (ii), we get

… (i) [by Thales’ theorem]



[by Thales’ theorem] … (ii)

AE  BF · ED FC EXAMPLE 18

ABCD is a trapezium in which AB  DC and its diagonals intersect each other at the point O. AO BO Prove that OC  OD · [CBSE 2004]

SOLUTION

A trapezium ABCD in which AB  DC and its diagonals AC and BD intersect at O. AO  BO · TO PROVE OC OD CONSTRUCTION Through O, draw EO  AB, meeting AD at E. GIVEN

In 3ADC, EO  DC AE  AO  ED OC In 3DAB, EO  AB. DE DO  EA  OB AE BO  ED  OD · From (i) and (ii), we get AO  BO · OC OD PROOF

[a EO  AB  DC] . … (i) [by Thales’ theorem]

[by Thales’ theorem] … (ii)

Triangles EXAMPLE 19

363

The diagonals of a quadrilateral ABCD intersect each other at the AO BO point O such that OC  OD · Show that ABCD is a trapezium. [CBSE 2005, ’08]

SOLUTION

GIVEN A quadrilateral ABCD whose diagonals AC and BD intersect at a point O such that AO  BO · OC OD TO PROVE ABCD is a trapezium, i.e., AB  DC. CONSTRUCTION

Draw EO  DC, meeting AD at E.

In 3ACD, EO  DC. AO  AE  OC ED [by Thales’ theorem]. AO BO But, OC  OD (given) BO AE DO DE  OD  ED & OB  EA in 3DAB. So, EO  AB [by the converse of Thales’ theorem]. PROOF

But, EO  DC. Hence, AB  DC, i.e., ABCD is a trapezium. EXAMPLE 20

In the given figure, ABCD is a trapezium in which AB  DC and its diagonals intersect at O. If AO  (3x  1) cm, OC  (5x  3) cm, BO  (2x  1) cm and OD  (6x  5) cm, find the value of x.

SOLUTION

We know that AB  DC in trapezium ABCD and its diagonals intersect at O. Then, we have AO  BO 3x  1  2x  1 OC OD & 5x  3 6x  5 & (3x  1)(6x  5)  (2x  1)(5x  3)

& 18x 2  21x  5  10x 2  x  3 & 8x 2  20x  8  0 & 2x 2  5x  2  0 & (x  2)(2x  1)  0 & x  2 or x  12 · 1 1 1 But, x  2 will make OC  (5x  3) cm  a5 # 2  3k cm   2 cm.

364

Secondary School Mathematics for Class 10

And, the distance cannot be negative. 1  x! 2 · Hence, x  2. MIDPOINT THEOREM EXAMPLE 21

Prove that the line segment joining the midpoints of any two sides of a triangle is parallel to the third side.

SOLUTION

A 3ABC in which D and E are the midpoints of AB and AC respectively. GIVEN

DE  BC. Since D and E are the midpoints of AB and AC respectively, we have AD  DB and AE  EC. AD AE  DB  EC [each equal to 1]. Hence, by the converse of Thales’ theorem, DE  BC. TO PROVE PROOF

EXAMPLE 22

Prove that a line drawn through the midpoint of one side of a triangle parallel to another side bisects the third side.

SOLUTION

A 3ABC in which D is the midpoint of AB and DE  BC, meeting AC at E. TO PROVE AE  EC. PROOF Since DE  BC, by Thales’ theorem, we have AE  AD  [a AD  DB (given)] EC DB 1 AE  EC  1 & AE  EC.

EXAMPLE 23

In 3ABC, AD is a median and E is the midpoint of AD. If BE is 1 produced, it meets AC in F. Show that AF  3 AC. [CBSE 2006C]

SOLUTION

A 3ABC in which AD is a median and E is the midpoint of AD. Also, BE is produced to meet AC at F. 1 TO PROVE AF  AC. 3

GIVEN

GIVEN

Triangles CONSTRUCTION PROOF

365

From D, draw DG  EF, meeting AC at G.

In 3BCF, D is the midpoint of BC and DG  BF.

 G is the midpoint of CF. So, FG  GC. In 3ADG, E is the midpoint of AD and EF  DG.  F is the midpoint of AG. So, AF  FG. Thus, AF  FG  GC.  AC  (AF  FG  GC)  3AF. 1 Hence, AF  3 AC. EXAMPLE 24

Prove that the line segments joining the midpoints of the adjacent sides of a quadrilateral form a parallelogram.

SOLUTION

A quadrilateral ABCD in which P, Q, R, S are the midpoints of AB, BC, CD and DA respectively. GIVEN

TO PROVE

PQRS is a parallelogram.

CONSTRUCTION

Join AC.

In 3ABC, P and Q are the midpoints of AB and BC respectively.

PROOF



PQ  AC.

… (i)

[by midpoint theorem]

In3DAC, S and R are the midpoints of AD and CD respectively. 

SR  AC.

… (ii)

[by midpoint theorem]

From (i) and (ii), we get PQ  SR. Similarly, by joining BD, we can prove that PS  QR. Hence, PQRS is a parallelogram. ANGLE-BISECTOR THEOREM EXAMPLE 25

SOLUTION

Prove that the internal bisector of an angle of a triangle divides the opposite side internally in the ratio of the sides containing the angle. GIVEN

A 3ABC in which AD, the bisector of +A, meets BC

in D. BD  AB · DC AC CONSTRUCTION Draw CE  DA, meeting BA produced at E. TO PROVE

366

Secondary School Mathematics for Class 10 PROOF

Since DA  CE, we have

+2  +3

[alternate int. O]

and +1  +4

[corresponding O].

But, +1  +2

[a AD is the bisector of +A]

 +3  +4. So, AE  AC. Now, in 3BCE, DA  CE. BD AB  DC  AE BD AB  DC  AC [a AE  AC] . BD AB Hence, DC  AC · EXAMPLE 26

SOLUTION

BD AB In a 3ABC, let D be a point on BC such that DC  AC · Prove that AD is the bisector of +A. A 3ABC in which D is a point on BC such that BD  AB · DC AC TO PROVE AD is the bisector of +A. GIVEN

Produce BA to E such that  AE AC. Join EC.

CONSTRUCTION

BD  AB DC AC (given) BD AB  DC  AE [a AC  AE]  DA < CE [by the converse of Thales’ theorem]. PROOF

 +2  +3

… (i)

and +1  +4

… (ii) [corresponding O]

Also, AE  AC & +3  +4.

… (iii)

[alternate int. O]

 +1  +2 [from (i), (ii) and (iii)]. Hence, AD is the bisector of +A. EXAMPLE 27

SOLUTION

In the given figure, AD is the bisector of +BAC. If AB  10 cm, [CBSE 2001] AC  6 cm and BC  12 cm, find BD and DC. Let BD  x cm. Then, DC  (BC  BD)  (12  x) cm.

Triangles

367

In 3ABC, AD is the bisector of +BAC. So, by the angle-bisector theorem, we have BD  AB x  10 DC AC & 12  x 6 & 6x  10(12  x)

& 16x  120 & x  7.5. Hence, BD  7.5 cm, and DC  (12  7.5) cm  4.5 cm. EXAMPLE 28

If the bisector of an angle of a triangle bisects the opposite side, prove that the triangle is isosceles. [CBSE 2000, ’01, ’02]

SOLUTION

GIVEN A 3ABC in which AD is the bisector of +BAC such that BD  DC. TO PROVE

AB  AC.

PROOF Since AD is the bisector of +A, by the angle-bisector theorem, we have AB  BD  [a BD  DC] AC DC 1 AB  AC  1 & AB  AC. Hence, 3ABC is an isosceles triangle.

EXAMPLE 29

SOLUTION

If the diagonal BD of a quadrilateral ABCD bisects both +B and +D, AB AD prove that BC  CD · GIVEN A quad. ABCD in which diagonal BD bisects both +B and +D. AB  AD · TO PROVE BC CD CONSTRUCTION Join AC, intersecting BD at E. PROOF In 3CBA, BE is the bisector of +ABC. AE AB  EC  BC · … (i) [by the angle-bisector theorem] In 3ADC, DE is the bisector of +ADC. AE AD  EC  CD · … (ii) [by the angle-bisector theorem] AB AD From (i) and (ii), we get BC  CD ·

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f

EXERCISE 7A

1. D and E are points on the sides AB and AC respectively of a 3ABC such that DE  BC. (i) If AD  3.6 cm, AB  10 cm and AE  4.5 cm, find EC and AC. (ii) If AB  13.3 cm, AC  11.9 cm and EC  5.1 cm, find AD. AD 4 (iii) If DB  7 and AC  6.6 cm, find AE. AD 8 (iv) If AB  15 and EC  3.5 cm, find AE. 2. D and E are points on the sides AB and AC respectively of a 3ABC such that DE  BC. Find the value of x, when (i) AD  x cm, DB  (x  2) cm, AE  (x  2) cm and EC  (x  1) cm. (ii) AD  4 cm, DB  (x  4) cm, AE  8 cm and EC  (3x  19) cm. (iii) AD  (7x  4) cm, AE  (5x  2) cm, DB  (3x  4) cm and EC  3x cm. 3. D and E are points on the sides AB and AC respectively of a 3ABC. In each of the following cases, determine whether DE  BC or not. (i) AD  5.7 cm, DB  9.5 cm, AE  4.8 cm and EC  8 cm. (ii) AB  11.7 cm, AC  11.2 cm, BD  6.5 cm and AE  4.2 cm. (iii) AB  10.8 cm, AD  6.3 cm, AC  9.6 cm and EC  4 cm. (iv) AD  7.2 cm, AE  6.4 cm, AB  12 cm and AC  10 cm. 4. In a 3ABC, AD is the bisector of +A. (i) If AB  6.4 cm, AC  8 cm and BD  5.6 cm, find DC. (ii) If AB  10 cm, AC  14 cm and BC  6 cm, find BD and DC. (iii) If AB  5.6 cm, BD  3.2 cm and BC  6 cm, find AC. (iv) If AB  5.6 cm, AC  4 cm and DC  3 cm, find BC.

[CBSE 2001C] [CBSE 1999C]

Triangles

369

5. M is a point on the side BC of a parallelogram ABCD. DM when produced meets AB produced at N. Prove that DM DC DN AN (i) MN  BN , (ii) DM  DC · 6. Show that the line segment which joins the midpoints of the oblique sides of a trapezium is parallel to the parallel sides. 7. In the adjoining figure, ABCD is a trapezium in which CD  AB and its diagonals intersect at O. If AO  (2x  1) cm, OC  (5x  7) cm, DO  (7x  5) cm and OB  (7x  1) cm, find the value of x. 8. In a 3ABC, M and N are points on the sides AB and AC respectively such that BM  CN. If +B  +C then show that MN  BC. 9. 3ABC and 3DBC lie on the same side of BC, as shown in the figure. From a point P on BC, PQ  AB and PR  BD are drawn, meeting AC at Q and CD at R respectively. Prove that QR  AD. 10. In the given figure, side BC of 3ABC is bisected at D and O is any point on AD. BO and CO produced meet AC and AB at E and F respectively, and AD is produced to X so that D is the midpoint of OX. Prove that AO : AX  AF : AB and show that EF  BC. 11. ABCD is a parallelogram in which P is the midpoint of DC and Q is a point on AC such 1 that CQ  4 AC. If PQ produced meets BC at R, prove that R is the midpoint of BC. 12. In the adjoining figure, ABC is a triangle in which AB  AC. If D and E are points on AB and AC respectively such that AD  AE, show that the points B, C, E and D are concyclic.

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13. In 3ABC, the bisector of +B meets AC at D. A line PQ  AC meets AB, BC and BD at P, Q and R respectively. Show that PR # BQ  QR # BP.

ANSWERS (EXERCISE 7A)

1. (i) EC  8 cm, AC  12.5 cm

(ii) AD  7.6 cm (iii) AE  2.4 cm

(iv) AE  4 cm 2. (i) x  4 3. (i) Yes

(ii) x  11 (iii) x  4 (ii) No

(iii) Yes (iv) No 4. (i) DC  7 cm (ii) BD  2.5 cm, DC  3.5 cm (iii) AC  4.9 cm (iv) BC  7.2 cm 7. x  2 HINTS TO SOME SELECTED QUESTIONS 5. (i) In 3NDA, MB  DA NM NB DM AB DM DC  MD  BA & MN  BN & MN  BN [a AB  DC]. NM NB NM NB NM  MD  NB  BA (ii) MD  BA & MD  1  BA  1 & MD BA DN  AN DN  AN & DM AB & DM DC [a AB  DC] . 6. Let E and F be the midpoints of the sides AD and BC of a trapezium ABCD having AB  CD. Produce AD and BC to meet at P. In 3PAB, DC  AB. 

PD  PC PD  PC PD  PC DA CB & 2DE 2CF & DE CF & DC  EF.

8. +B  +C & AB  AC & AM  BM  AN  CN & AM  AN [a BM  CN] . AM AN Now, AM  AN, BM  CN & BM  CN AM AN & MB  NC & MN  BC.

CP CQ 9. In 3CAB, QP  AB & PB  QA ·

Triangles

371

CP CR In 3CDB, RP  DB & PB  RD · CQ CR  QA  RD & QR  AD (in 3CDA). 10. Join BX and CX. Clearly, BD  DC and OD  DX (given). Thus, the diagonals of quad. OBXC bisect each other. 

OBXC is a parallelogram.



BX  CF and so, OF  BX.

Similarly, OE  XC. In 3ABX, OF  BX. AO  AF ·  AX AB In 3ACX, OE  XC. AO  AE ·  AX AC

… (i)

… (ii)

AF AE From (i) and (ii), we get AB  AC · Hence, FE  BC. 11. Join BD. Suppose it meets AC at S. 1 Since the diagonals of gm bisect each other, CS  2 AC. 1 1 1 Now, CS  2 AC and CQ  4 AC & CQ  2 CS.  Q is the midpoint of CS. So, PQ  DS and therefore, QR  SB. In 3CSB, Q is the midpoint of CS and QR  SB, so R is the midpoint of BC. 12. AB  AC and AD  AE



AB  AD  AC  AE  DB  EC AD  DB [each equal to 1] AE EC [by the converse of Thales‘ theorem] DE  BC

  

+DEC  +ECB  180c



+DEC  +CBD  180c [a AB  AC & +C  +B]



quad. BCED is cyclic.

Hence, the points B, C, E, D are concyclic. 13. In 3BQP, BR is the bisector of +B. 

QR BQ  PR BP

[by angle-bisector theorem].

CRITERIA FOR SIMILARITY OF TWO TRIANGLES Two triangles are similar, if (i) their corresponding angles are equal and (ii) their corresponding sides are proportional.

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Thus, 3ABC +3DEF, if (i) +A  +D, +B  +E, +C  +F AB BC CA and (ii) DE  EF  FD · THEOREM 1

(AAA-similarity) If in two triangles, the corresponding angles are

equal, then their corresponding sides are proportional and hence the triangles are similar. GIVEN

3ABC and 3DEF such that +A  +D, +B  +E and +C  +F.

3ABC +3DEF. CONSTRUCTION Cut DP  AB and DQ  AC. Join PQ. TO PROVE

PROOF

In 3ABC and 3DPQ, we have [by construction] AB  DP [by construction] AC  DQ [given] +A  +D  [by SAS-congruence] 3ABC ,3DPQ  +B  +P  +E  +P [a +B  +E (given)]  [a corresponding O are equal] PQ  EF DP  DQ DE DF AB  CA  [a DP  AB and DQ  AC] DE FD AB BC Similarly, DE  EF · AB  BC  CA ·  DE EF FD AB BC CA Thus, +A  +D, +B  +E, +C  +F and DE  EF  FD · Hence, 3ABC +3DEF. 

IMPORTANT REMARK COROLLARY

Two 3s are similar  they are equiangular.

(AA-similarity) If two angles of one triangle are respectively equal to two angles of another triangle then the two triangles are similar.

Triangles

373

In 3ABC and 3DEF, let +A  +D and +B  +E. Then, 3rd +C  3rd +F.

PROOF

Thus, the two triangles are equiangular and hence similar. REMARK

AA-similarity is the same as AAA-similarity. (SSS-similarity) If the corresponding sides of two triangles are

THEOREM 2

proportional then their corresponding angles are equal, and hence the two triangles are similar. GIVEN

AB BC AC 3ABC and 3DEF in which DE  EF  DF ·

TO PROVE

3ABC +3DEF.

Let us take 3ABC and 3DEF such that AB  BC  AC DE EF DF ( 1). Cut DP  AB and DQ  AC. Join PQ.

CONSTRUCTION

PROOF

AB  AC DP  DQ DE DF & DE DF

[a AB  DP and AC  DQ].

So, by the converse of Thales’ theorem, PQ  EF.  +P  +E [corresponding O] +Q  +F 

[corresponding O]

3DPQ +3DEF [by AA-similarity]

DP  PQ DE EF AB  PQ ·  … (i) [a DP  AB] DE EF AB BC But, DE  EF · … (ii) [given] PQ BC  [from (i) and (ii)]  EF EF  BC  PQ. Thus, AB  DP, AC  DQ and BC  PQ. 

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Secondary School Mathematics for Class 10



3ABC ,3DPQ [by SSS-congruence]



+A  +D, +B  +P  +E and +C  +Q  +F +A  +D, +B  +E and +C  +F.



Thus, the given triangles are equiangular and hence similar. (SAS-similarity) If one angle of a triangle is equal to one angle of the

THEOREM 3

other triangle and the sides including these angles are proportional then the two triangles are similar. GIVEN

AB AC 3ABC and 3DEF in which +A  +D and DE  DF ·

TO PROVE

3ABC +3DEF.

CONSTRUCTION

PROOF

Let us take 3ABC and 3DEF such that AB  AC  DE DF (< 1) and +A +D. Cut DP  AB and DQ  AC. Join PQ.

In 3ABC and 3DPQ, we have [by construction] AB  DP +A  +D AC  DQ  

(given) [by construction]

3ABC ,3DPQ [by SAS-congruence] +A  +D, +B  +P and +C  +Q.

AB AC Now, DE  DF (given) DP  DQ  [a AB  DP and AC  DQ] DE DF  [by the converse of Thales’ theorem] PQ  EF  +P  +E and +Q  +F [corresponding O] +A  +D, +B  +P  +E and +C  +Q  +F. Thus, +A  +D, +B  +E and +C  +F.



So, the given triangles are equiangular and hence similar.

Triangles THEOREM 4

GIVEN

If a perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse then the triangles on both sides of the perpendicular are similar to the whole triangle and also to each other.

A 3ABC in which +BAC  90c and AD = BC.

TO PROVE

PROOF

375

(i) 3DBA +3ABC (ii) 3DAC +3ABC (iii) 3DBA +3DAC. (i) In 3DBA and 3ABC, we have +BDA  +BAC  90c +DBA  +ABC (common)  3DBA +3ABC [by AA-similarity].

(ii) In 3DAC and 3ABC, we have +CDA  +CAB  90c +DCA  +ACB (common)  3DAC +3ABC [by AA-similarity]. (iii) In 3DBA and 3DAC, we have +ADB  +CDA  90c. _b  +B  +BAD  90c bb b `b & +B  +CAD and +C  +BAD. +C  +CAD  90c bb   +BAD +CAD 90c b a Thus, in 3DBA and 3DAC, we have: +ADB  +CDA [each equal to 90] +B  +CAD +BAD  +C  3DBA +3DAC [by AAA-similarity]. SUMMARY

(i) If +A  +D, +B  +E, +C  +F, then 3ABC +3DEF. (AAA-similarity) (ii) If +A  +D, +B  +E, then 3ABC +3DEF. (AA-similarity) AB BC AC (iii) If DE  EF  DF , then 3ABC +3DEF. (SSS-similarity) AB AC (iv) If +A  +D and DE  DF , then 3ABC +3DEF. (SAS-similarity)

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SOME MORE RESULTS THEOREM 1

GIVEN

3ABC and 3DEF in which +A  +D, +B  +E and +C  +F and AL = BC and DM = EF.

TO PROVE PROOF

If two triangles are equiangular, prove that the ratio of their corresponding sides is the same as the ratio of the corresponding altitudes.

BC  AL · EF DM

Since 3ABC and 3DEF are equiangular, 3ABC +3DEF. AB  BC ·  DE EF In 3ALB and 3DME, we have +ALB  +DME  90c and +B  +E (given).

… (i)



3ALB +3DME [by AA-similarity] AB  AL ·  DE DM BC AL From (i) and (ii), we get EF  DM ·

THEOREM 2

GIVEN

If two triangles are equiangular, prove that the ratio of their corresponding sides is the same as the ratio of the corresponding medians.

3ABC and 3DEF in which +A  +D, +B  +E and +C  +F and AL and DM are the medians.

TO PROVE PROOF

… (ii)

BC  AL · EF DM

Since 3ABC and 3DEF are equiangular, we have 3ABC +3DEF.

Triangles

AB  BC · DE EF AB BC 2BL BL But, DE  EF  2EM  EM ·



377

… (i)

Now, in 3ABL and 3DEM, we have AB  BL  DE EM and +B +E (given). 

3ABL +3DEM [by SAS-similarity]



AB  AL · DE DM

… (ii)

BC AL From (i) and (ii), we get EF  DM · THEOREM 3

GIVEN

3ABC and 3DEF in which +A  +D, +B  +E and +C  +F, and AX and DY are the bisectors of +A and +D respectively.

TO PROVE PROOF

If two triangles are equiangular, show that the ratio of the corresponding sides is the same as the ratio of the corresponding angle-bisector segments.

BC  AX · EF DY

Since 3ABC and 3DEF are equiangular, we have 3ABC +3DEF. 

AB  BC · DE EF

… (i)

1 1 Now, +A  +D & 2+A  2+D & +BAX  +EDY. Thus, in 3ABX and 3DEY, we have +BAX  +EDY

(proved)

+B  +E

(given)



3ABX +3DEY [by AA-similarity]



AB  AX · DE DY

BC AX From (i) and (ii), we get EF  DY ·

… (ii)

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Secondary School Mathematics for Class 10

SOLVED EXAMPLES EXAMPLE 1

In the adjoining figure, 3AHK is similar to 3ABC. If AK  10 cm, BC  3.5 cm and HK  7 cm, find AC. [CBSE 2010]

SOLUTION

3AHK +3ABC AK HK 10 7  AC  BC & x  3.5 , where AC  x cm 10 # 3.5   x 5. 7  AC  5 cm.

EXAMPLE 2

SOLUTION

In the given figure, DE  BC, AD  2 cm, BD  2.5 cm, AE  3.2 cm and DE  4 cm. Find AC and BC. [CBSE 2001C] Since DE  BC, we have +ADE  +ABC (corresponding O) and +AED  +ACB (corresponding O).  3ADE +3ABC [by AA-similarity]. So, the corresponding sides of 3ADE and 3ABC are proportional. AD DE AE  AB  BC  AC · … (i) AD DE 2 4 Now, AB  BC & 4.5  BC [a AB  AD  BD  4.5 cm] & BC  a 4 #24.5 k cm  9 cm. DE AE Again, BC  AC [from (i)] 4 3.2  9  AC [a BC  9 cm] 9 # 3.2  AC  a 4 k cm  7.2 cm. Hence, AC  7.2 cm and BC  9 cm.

EXAMPLE 3

In the given figure, AB  CD. Prove that 3AOB +3DOC.

SOLUTION

AB  CD (given).

Triangles

 +OAB  +ODC +OBA  +OCD +AOB  +DOC  3AOB +3DOC EXAMPLE 4

379

(alternate angles) (alternate angles) (vertical opposite O) [by AAA-similarity].

In the given figure, 3AOB +3DOC. Prove that AB  CD.

SOLUTION

3AOB +3DOC. So, the given triangles are equiangular.  +OAB  +ODC. But, these are alternate angles.  AB  CD.

EXAMPLE 5

Find +P in the adjoining figure.

SOLUTION

In 3ABC and 3QRP, we have AB  3.6  1 BC  6  1 CA  3 3  1 QR 7.2 2 , RP 12 2 and PQ 6 3 2 AB BC CA Thus, QR  RP  PQ and so 3ABC +3QRP [by SSS-similarity].  +C  +P [corresponding angles of similar triangles]. But, +C  180c  (+A  +B)  180c  (70c  60c)  50c.  +P  50c.

EXAMPLE 6

In the given figure, 3OQP +3OAB, +OPQ  56c and +BOQ  132c. Find +OAB.

SOLUTION

In 3OPQ, we have +BOQ  +OPQ  +OQP [a the exterior angle is equal to the sum of the two interior opposite angles]

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Secondary School Mathematics for Class 10

 +OQP  +BOQ  +OPQ  132c  56c  76c. Now, 3OQP +3OAB  +OQP  +OAB [corresponding angles of similar triangles].  +OAB  +OQP  76c. EXAMPLE 7

SOLUTION

In the given figure, AP · AR  AS · AQ. Prove that +P  +S and +Q  +R.

We have +PAQ  +SAR.

… (i)

[vertically opposite angles]

Also, AP · AR  AS · AQ (given) AP  AQ · … (ii) AS AR From (i) and (ii), we have 

3PAQ +3SAR [by SAS-similarity]  +P  +S and +Q  +R [corresponding angles of similar triangles]. EXAMPLE 8

A vertical stick which is 15 cm long casts a 12-cm-long shadow on the ground. At the same time, a vertical tower casts a 50-m-long shadow on the ground. Find the height of the tower.

SOLUTION

Let AB be the vertical stick and let AC be its shadow. Then, AB  0.15 m and AC  0.12 m. Let DE be the vertical tower and let DF be its shadow. Then, DF  50 m. Let DE  x m. Now, in 3BAC and 3EDF, we have +BAC  +EDF  90c +ACB  +DFE [angular elevation of the sun at the same time]  3BAC +3EDF AB AC 0.15 0.12  DE  DF & x  50 (0.15 # 50)  62.5.  x 0.12 Hence, the height of the tower is 62.5 m.

Triangles

381

EXAMPLE 9

Prove that the ratio of the perimeters of two similar triangles is the same as the ratio of their corresponding sides.

SOLUTION

GIVEN

3ABC and 3PQR in

which BC  a, CA  b, AB  c and QR  p, RP  q, PQ  r. Also, 3ABC +3PQR. TO PROVE

a  b  c  abc p q r pqr$

Since 3ABC and 3PQR are similar, therefore their corresponding sides are proportional.

PROOF



a bc p q r k (say)



a  kp, b  kq and c  kr.



perimeter of 3ABC a  b  c kp  kq  kr   perimeter of 3PQR p  q  r pqr

… (i)



k (p  q  r)  k. (p  q  r)

… (ii)

From (i) and (ii), we get a  b  c  a  b  c  perimeter of 3ABC p q r p  q  r perimeter of 3PQR

[each equal to k].

EXAMPLE 10

The perimeters of two similar triangles are 25 cm and 15 cm respectively. If one side of the first triangle is 9 cm, find the corresponding side of the second triangle. [CBSE 2002C]

SOLUTION

Let 3ABC +3DEF given in such a way that perimeter of 3ABC  25 cm, perimeter of 3DEF  15 cm and AB  9 cm. Then, we have to find DE. Let DE  x cm. We know that the ratio of the perimeters of two similar triangles is the same as the ratio of their corresponding sides. 

perimeter of 3ABC AB  perimeter of 3DEF DE



25  9  9 #15  15 x & x a 25 k 5.4.



DE  5.4 cm.

Hence, the corresponding side of the second triangle is 5.4 cm.

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Secondary School Mathematics for Class 10

EXAMPLE 11

In the given figure, D is a point on the side BC of 3ABC such that+ADC  +BAC. Prove that CA 2  CB#CD. [CBSE 2004]

SOLUTION

A 3ABC in which D is a point on BC such that +ADC  +BAC. GIVEN

CA 2  CB#CD. PROOF In 3ABC and 3DAC, we have +BAC  +ADC (given) +ACB  +DCA (common)  3ABC +3DAC [by AA-similarity]. TO PROVE

So, the sides of 3ABC and 3DAC are proportional. 

CA  CD · CB CA

Hence, CA 2  CB#CD. EXAMPLE 12

In the given figure, S and T are points on sides PR and QR of 3PQR such that +P  +RTS. Show that 3RPQ +3RTS.

SOLUTION

3RPQ and 3RTS in which +P  +RTS. GIVEN

TO PROVE PROOF

3RPQ +3RTS.

In 3RPQ and 3RTS, we have

+P  +RTS +R  +R  3RPQ +3RTS

(given) (common) [by AA-similarity].

EXAMPLE 13

In the given figure, if 3ABE ,3ACD, show that 3ADE +3ABC.

SOLUTION

3ABE ,3ACD

(given)

AE  AD

… (i)

[cpct]

and AB  AC.

… (ii)

[cpct]



In 3ADE and 3ABC, we have +DAE  +BAC (common) AD AE and AB  AC [using (i) and (ii)].  3ADE +3ABC [by SAS-similarity].

Triangles EXAMPLE 14

383

In the given figure, altitudes AD and CE of 3ABC intersect each other at the point P. Show that: (i) 3AEP +3CDP (ii) 3ABD +3CBE (iii) 3AEP +3ADB (iv) 3PDC +3BEC

SOLUTION

(i) In 3AEP and 3CDP, we have +AEP  +CDP [each equal to 90] +APE  +CPD  3AEP +3CDP

[vertically opposite O] [by AA-similarity].

(ii) In 3ABD and 3CBE, we have +ADB  +CEB  90c [common] +B  +B  3ABD +3CBE

[by AA-similarity].

(iii) In 3AEP and 3ADB, we have +AEP  +ADB  90c +EAP  +DAB

(common)

Hence, 3AEP +3ADB [by AA-similarity]. (iv) In 3PDC and 3BEC, we have +PDC  +BEC  90c +PCD  +BCE  3PDC +3BEC EXAMPLE 15

(common) [by AA-similarity].

Diagonals AC and BD of a trapezium ABCD with AB  DC intersect each other at point O. Using a similarity criterion for two triangles, OA OB show that OC  OD ·

SOLUTION

A trapezium ABCD in which AB  DC. The diagonals AC and BD intersect at O. OA  OB · TO PROVE OC OD PROOF In 3OAB and 3OCD, we have GIVEN

+OAB  +OCD [alternate angles, since AB  DC]

384

Secondary School Mathematics for Class 10

and +OBA  +ODC [alternate angles, since AB  DC].  3OAB +3OCD [by AA-similarity]. OA OB And so, OC  OD · EXAMPLE 16

In 3ABC, AD = BC and AD 2  BD · CD. Prove that +BAC  90c.

SOLUTION

A 3ABC in which AD = BC and AD 2  BD · CD.

GIVEN

TO PROVE

+BAC  90c.

BD AD AD 2  BD · CD & AD  CD · Now, in 3DBA and 3DAC, we have BD AD +BDA  +ADC  90c and AD  CD ·  3DBA +3DAC [by SAS-similarity] PROOF

 +B  +2 and +1  +C  +1  +2  +B  +C & +A  +B  +C

& 2+A  +A  +B  +C  180c & +A  +BAC  90c. EXAMPLE 17

In the given figure PA, QB and RC each is perpendicular to AC such that PA  x, RC  y, QB  z, AB  a and BC  b. 1 1 1 Prove that x  y  z · [CBSE 2000C]

SOLUTION

PA = AC and QB = AC & QB  PA. Thus, in 3PAC, QB  PA. So, 3QBC +3PAC. QB BC z b  PA  AC & x   · … (i) [by the property of similar A] a b In 3RAC, QB  RC. So, 3QBA +3RCA. QB AB z a  RC  AC & y   · … (ii) [by the property of similar A] a b From (i) and (ii), we get zz  b  a  x y aa  b a  b k 1

Triangles



385

1  1  1· zz  x y 1& x y z

1 1 1 Hence, x  y  z · EXAMPLE 18

The side AD of a parallelogram ABCD is produced to a point E. BE intersects CD at F. Show that 3ABE +3CFB.

SOLUTION

We have AD  BC & AE  BC  +AEB  +CBF

[ABCD is a parallelogram] [alternate angles].

In 3ABE and 3CFB, we have +AEB  +CBF

[proved above]

+EAB  +BCF

[a +DAB  +BCD, being opposite angles of a parallelogram]

 3ABE +3CFB EXAMPLE 19

[by AA-similarity].

In the given figure, +ACB  90c and CD = AB. Prove that CD 2  BD · AD.

SOLUTION

GIVEN

[CBSE 2006]

A 3ABC in which

+ACB  90c and CD = AB. TO PROVE PROOF

CD 2  BD · AD.

In right 3ADC, we have +1  +2  90c.

In right 3ACB, we have +1  +3  90c.  +1  +2  +1  +3 & +2  +3. In 3ADC and 3CDB, we have +2  +3 (proved) and +ADC  +CDB  90c.  3ADC +3CDB [by AA-similarity] AD CD  CD  BD · Hence, CD 2  BD · AD.

386 EXAMPLE 20

Secondary School Mathematics for Class 10

In the given figure, 3ABC and 3AMP are right-angled at B and M respectively. Prove that: (i) 3ABC +3AMP CA BC (ii) PA  MP

SOLUTION

3ABC and 3AMP such that +B  90c and +M  90c. CA  BC · TO PROVE (i) 3ABC +3AMP (ii) PA MP PROOF (i) In 3ABC and 3AMP, we have +ABC  +AMP  90c +A  +A (common)  3ABC +3AMP [by AA-similarity]. GIVEN

(ii) Since 3ABC +3AMP, their corresponding sides are proportional. CA BC  PA  MP · EXAMPLE 21

SOLUTION

EXAMPLE 22

In a 3ABC, AB  AC and D is a point on AC such that BC 2  AC # DC. Prove that BD  BC. GIVEN A 3ABC in which AB  AC and D is a point on AC such that BC 2  AC # DC. TO PROVE BD  BC. 2 PROOF BC  AC # DC (given) BC AC  DC  BC · Thus, in 3ABC and 3BDC, we have BC  AC  DC BC and +C +C (common).  3ABC +3BDC [by SAS-similarity] AC AB  BC  BD AC AC  BC  BD [a AB  AC (given)]  BD  BC. Hence, BD  BC.

In the given figure, CD and GH are respectively the bisectors of +ACB and +FGE of 3ABC and 3FEG respectively. If 3ABC +3FEG, prove that: CD AC (a) 3ADC +3FHG (b) 3BCD +3EGH (c) GH  FG

Triangles

SOLUTION

387

3ABC +3FEG (given)  +ACB  +FGE [corresponding angles of similar triangles are equal] 1 1  2+ACB  2+FGE  +ACD  +FGH and +DCB  +HGE. (a) In 3ADC and 3FHG, we have +DAC  +HFG and +ACD  +FGH

[a +A  +F since 3ABC +3FEG] [proved above]

 3ADC +3FHG [by AA-similarity]. (b) In 3BCD and 3EGH, we have +DBC  +HEG and +DCB  +HGE

[a +B  +E since 3ABC +3FEG] [proved above]

 3BCD +3EGH. (c) We have 3ADC +3FHG [proved above]. CD  AC And so, GH FG [corresponding sides of similar triangles are proportional]. EXAMPLE 23

In the given figure, BM and EN are respectively the medians of 3ABC and 3DEF. If 3ABC +3DEF, prove that: (a) 3AMB +3DNE (b) 3CMB +3FNE (c)

BM  AC EN DF

388 SOLUTION

Secondary School Mathematics for Class 10

3ABC +3DEF

(given)

 +A  +D, +B  +E and +C  +F and

AB  BC  CA · DE EF FD

… (i) … (ii)

Since BM and EN are medians, we have CA  2AM  2CM and FD  2DN  2FN. 



from (ii), we have AB  BC  CA  2AM  2CM DE EF FD 2DN 2FN AB  BC  CA  AM  CM · DE EF FD DN FN

… (iii)

(a) In 3AMB and 3DNE, we have +BAM  +EDN [a +A  +D from (i)] AB  AM and [from (iii)]. DE DN  3AMB +3DNE [by SAS-similarity]. (b) In 3CMB and 3FNE, we have +BCM  +EFN [a +C  +F from (i)] BC  CM and [from (iii)]. EF FN  3CMB +3FNE [by SAS-similarity]. (c) As proved above, 3AMB +3DNE and so AB  BM · DE EN From (ii) and (iv), we get

… (iv)

BM  AC · EN FD EXAMPLE 24

In a 3ABC, P and Q are points on AB and AC respectively such that PQ  BC. Prove that the median AD, drawn from A to BC, bisects PQ.

SOLUTION

GIVEN A 3ABC in which P and Q are points on AB and AC respectively such that PQ  BC and AD is the median, cutting PQ at E. TO PROVE

PE  EQ.

Triangles PROOF

389

In 3APE and 3ABD, we have

+PAE  +BAD

[common]

+APE  +ABD [corresponding O]  3APE +3ABD [by AA-similarity]. But, in similar triangles, the corresponding sides are proportional. 

AE  PE · AD BD

… (i)

In 3AEQ and 3ADC, we have +QAE  +CAD [common] +AQE  +ACD [corresponding angles]  3AEQ +3ADC [by AA-similarity]. But, in similar triangles, the corresponding sides are proportional. 

AE  EQ · AD DC

… (ii) AE · PE  EQ :each equal to D BD DC AD [a AD is the median]

From (i) and (ii), we get But, BD  DC  PE  EQ. EXAMPLE 25

SOLUTION

In the given figure, E is a point on side CB produced of an isosceles 3ABC with AB  AC. If AD = BC and EF = AC, prove that 3ABD +3ECF. A 3ABC in which AB  AC and AD = BC. Side CB is produced to E and EF = AC. GIVEN

TO PROVE

3ABD +3ECF.

We know that the angles opposite to equal sides of a triangle are equal. PROOF

 +B  +C [a AB  AC] . Now, in 3ABD and 3ECF, we have +B  +C [proved above] +ADB  +EFC  90c.  3ABD +3ECF [by AA-similarity].

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Secondary School Mathematics for Class 10

EXAMPLE 26

Prove that the line segments joining the midpoints of the sides of a triangle form four triangles, each of which is similar to the original triangle.

SOLUTION

GIVEN A 3ABC in which D, E, F are the midpoints of BC, CA and AB respectively.

3AFE +3ABC, 3FBD +3ABC, 3EDC +3ABC. and 3DEF +3ABC.

TO PROVE

We shall first show that 3AFE +3ABC. Since F and E are the midpoints of AB and AC respectively, so by the midpoint theorem, we have FE  BC.  +AFE  +B [corresponding O] Now, in 3AFE and 3ABC, we have [corresponding O] +AFE  +B and +A  +A (common).  3AFE +3ABC [by AA-similarity]. Similarly, 3FBD +3ABC and 3EDC +3ABC. Now, we shall show that 3DEF +3ABC. In the same manner as above, we can prove that ED  AF and DF  EA.  AFDE is a gm. [opposite angles of a gm].  +EDF  +A Similarly, BDEF is a gm. [opposite angles of a gm].  +DEF  +B Thus, in 3DEF and 3ABC, we have +EDF  +A and +DEF  +B.  3DEF +3ABC [by AA-similarity]. Hence, the result follows. PROOF

EXAMPLE 27

In the given figure, DEFG is a square and +BAC  90c. Prove that (i) 3AGF +3DBG (ii) 3AGF +3EFC (iii) 3DBG +3EFC (iv) DE 2  BD # EC

[CBSE 2009]

Triangles SOLUTION

GIVEN

A 3ABC in which +BAC  90c and DEFG is a square.

TO PROVE

(i) 3AGF +3DBG (iii) 3DBG +3EFC

PROOF

391

(ii) 3AGF +3EFC (iv) DE 2  BD # EC

(i) In 3AGF and 3DBG, we have

+GAF  +BDG  90c +AGF  +DBG [corresponding O] [a GF  BC and AB is the transversal]  3AGF +3DBG. (ii) In 3AGF and 3EFC, we have +FAG  +CEF  90c +GFA  +FCE [corresponding O] [a GF  BC and AC is the transversal]  3AGF +3EFC. (iii) 3DBG +3AGF and 3AGF +3EFC  3DBG +3EFC. BD  DG (iv) 3DBG +3EFC & FE EC BD  DE & DE EC [a DG  DE and FE  DE] . Hence, DE 2  BD # EC. EXAMPLE 28

Two right triangles ABC and DBC are drawn on the same hypotenuse BC and on the same side of BC. If AC and BD intersect at P, prove that [CBSE 2000C] AP # PC  BP # PD.

SOLUTION

Right triangles 3ABC and 3DBC are drawn on the same hypotenuse BC and on the same side of BC. Also, AC and BD intersect at P. TO PROVE AP # PC  BP # PD. GIVEN

PROOF

In 3BAP and 3CDP, we have

+BAP  +CDP  90c +BPA  +CPD (ver. opp. O)  3BAP +3CDP [by AA-similarity] AP  BP  DP CP

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Secondary School Mathematics for Class 10

 AP #CP  BP # DP  AP # PC  BP # PD. Hence, AP # PC  BP # PD. EXAMPLE 29

Through the midpoint M of the side CD of a parallelogram ABCD, the line BM is drawn, intersecting AC in L and AD produced in E. Prove that EL  2BL. [CBSE 2006C, ’08, ’09]

SOLUTION

A gm ABCD and M is the midpoint of CD. Line BM is drawn, intersecting AC in L and AD produced in E. TO PROVE

EL  2BL.

In 3BMC and 3EMD, we have (vert. opp. O) +1  +2 [M is the midpoint of CD] MC  MD +BCM  +EDM [alternate interior O]  3BMC ,3EMD  BC  DE. But, BC  AD [opposite sides of a gm] PROOF



BC  AD  DE & AE  (AD  DE)  2BC.

… (i)

Now, in 3AEL and 3CBL, we have (vert. opp. O) +6  +5  [alternate interior O] +3 +4  3AEL +3CBL [AA-similarity] EL  AE  2BC   2 [using (i)] BL BC BC  EL  2BL. Hence, EL  2BL. EXAMPLE 30

A lamp is 3.3 m above the ground. A boy 110 cm tall walks away from the base of this lamp post at a speed of 0.8 m/s. Find the length of the shadow of the boy after 4 seconds.

SOLUTION

Let AB be the lamp post and PQ be the boy, where P is the position of the boy after 4 seconds.

Triangles

393

AP = distance moved in 4 s at 0.8 m/s  (4 # 0.8) m  3.2 m. PM is the length of shadow of the boy. Let PM  x m. In 3AMB and 3PMQ, we have +MAB  +MPQ  90c [a both the lamp post and the boy stand vertically erect] +AMB  +PMQ (common)  3AMB +3PMQ [by AA-similarity]. AM  AB PM PQ [corres. sides of similar triangles are proportional] 3.2  x 3.3 AP  PM  AB & x  1.1 PM PQ

And so,



[a AB  3.3 m, PQ  110 cm  1.1 m]  

3.2  x  3x & 2x  3.2 & x  1.6. the length of the shadow of the boy after 4 seconds is 1.6 m.

EXAMPLE 31

Sides AB and BC and median AD of a triangle ABC are respectively proportional to sides PQ and QR and median PM of another triangle PQR, as shown in the figure. Prove that 3ABC +3PQR.

SOLUTION

We have AB  BC  AD PQ QR PM 1 BC AB  AD  BC  2  BD ·  PQ PM QR 1 QM QR 2 In 3ABD and 3PQM, we have AB  AD  BD PQ PM QM  3ABD +3PQM

… (i)

[from (i)] [by SSS-similarity].

And so,+B  +Q [corres. angles of similar triangles are equal].

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Secondary School Mathematics for Class 10

Now, in 3ABC and 3PQR, we have +B  +Q AB  BD and PQ QM

[proved above] [from (i)].

 3ABC +3PQR [by SAS-similarity]. EXAMPLE 32

Sides AB and AC and median AD of a triangle ABC are respectively proportional to sides PQ and PR and median PM of another triangle PQR. Prove that 3ABC +3PQR.

SOLUTION

GIVEN AD and PM are medians of 3ABC and 3PQR respectively such that

AB  AC  AD · PQ PR PM 3ABC +3PQR. CONSTRUCTION Produce AD to E such that AD  DE and produce PM to N such that PM  MN. Join EC and NR. TO PROVE

In 3ABD and 3ECD, we have [a D is the midpoint of BC] BD  CD [by construction] AD  ED +BDA  +CDE [vertically opposite angles]  3ABD ,3ECD [by SAS-congruency].

PROOF

Triangles

395

And so, AB  EC. … (i) [cpct] Similarly, 3PQM ,3NRM and so, PQ  NR. … (ii) [cpct] AB  AC  AD Now, (given) PQ PR PM EC  AC  AD  [using (i) and (ii)] NR PR PM EC  AC  2AD  AE  [a 2AD  AE, 2PM  PN] NR PR 2PM PN  3ACE +3PNR [SSS-similarity].  +2  +4 [corresponding angles of similar triangles are equal]. Similarly, +1  +3 [it can be proved by joining BE and QN and showing 3ABE +3PQN ]. … (iii)  +1  +2  +3  +4, i.e., +BAC  +QPR. Now, in 3ABC and 3PQR, we have AB  AC [given] PQ PR +BAC  +QPR [from (iii)]  3ABC +3PQR [by SAS-similarity]. f

EXERCISE 7B

1. In each of the given pairs of triangles, find which pair of triangles are similar. State the similarity criterion and write the similarity relation in symbolic form.

(i)

(ii)

396

Secondary School Mathematics for Class 10

(iii)

(iv)

(v)

2. In the given figure, 3ODC +3OBA, +BOC  115c and +CDO  70c. Find (i) +DOC (ii) +DCO (iii) +OAB (iv) +OBA.

3. In the given figure, 3OAB +3OCD. If AB  8 cm, BO  6.4 cm, OC  3.5 cm and CD  5 cm, find (i) OA (ii) DO.

Triangles

397

4. In the given figure, if +ADE  +B, show that 3ADE +3ABC. If AD  3.8 cm, AE  3.6 cm, BE  2.1 cm and BC  4.2 cm, find DE.

5. The perimeters of two similar triangles ABC and PQR are 32 cm and [CBSE 2001] 24 cm respectively. If PQ  12 cm, find AB. 6. The corresponding sides of two similar triangles ABC and DEF are BC  9.1 cm and EF  6.5 cm. If the perimeter of 3DEF is 25 cm, find the perimeter of 3ABC. 7. In the given figure, +CAB  90c and AD = BC. Show that 3BDA +3BAC. If AC  75 cm, AB  1 m and BC  1.25 m, find AD.

8. In the given figure, +ABC  90c and BD = AC. If AB  5.7 cm, BD  3.8 cm and CD  5.4 cm, find BC.

9. In the given figure, +ABC  90c and BD = AC. If BD  8 cm, AD  4 cm, find CD.

10. P and Q are points on the sides AB and AC respectively of a 3ABC. If AP  2 cm, PB  4 cm, AQ  3 cm and QC  6 cm, show that BC  3PQ.

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Secondary School Mathematics for Class 10

11. ABCD is a parallelogram and E is a point on BC. If the diagonal BD intersects AE at F, prove that AF # FB  EF # FD.

12. In the given figure, DB = BC, DE = AB and AC = BC. Prove that

BE  AC · DE BC

[CBSE 2008]

13. A vertical pole of length 7.5 m casts a shadow 5 m long on the ground and at the same time a tower casts a shadow 24 m long. Find the height of the tower. 14. In an isosceles 3ABC, the base AB is produced both ways in P and Q such that AP # BQ  AC 2 . Prove that 3ACP +3BCQ.

15. In the given figure, +1  +2 and

AC  CB · BD CE

Prove that 3ACB +3DCE.

16. ABCD is a quadrilateral in which AD  BC. If P, Q, R, S be the midpoints of AB, AC, CD and BD respectively, show that PQRS is a rhombus.

17. In a circle, two chords AB and CD intersect at a point P inside the circle. Prove that (a) 3PAC +3PDB (b) PA · PB  PC · PD.

Triangles

399

18. Two chords AB and CD of a circle intersect at a point P outside the circle. Prove that (a) 3PAC +3PDB (b) PA · PB  PC · PD.

19. In a right triangle ABC, right-angled at B, D is a point on hypotenuse such that BD = AC. If DP = AB and DQ = BC then prove that (a) DQ 2  DP · QC (b) DP 2  DQ · AP.

ANSWERS (EXERCISE 7B)

1. (i) 3ABC +3QPR (AAA-similarity)

(ii) not similar (iii) 3CAB +3QRP (SAS-similarity) (iv) 3FED +3PQR (SSS-similarity) (v) 3ABC +3MNR (AA-similarity) 2. (i) +DOC  65c (ii) +DCO  45c (iii) +OAB  45c (iv) +OBA  70c 3. (i) OA  5.6 cm (ii) DO  4 cm 4. 2.8 cm 5. 16 cm 6. 35 cm

7. AD  60 cm

8. BC  8.1 cm

9. CD  16 cm

13. 36 m HINTS TO SOME SELECTED QUESTIONS 1. (i) +A  +Q, +B  +P and +C  +R.    3ABC +3QPR (AAA-similarity). (ii) SAS-similarity is not satisfied as included angles are not equal. (iii) 3CAB +3QRP (SAS-similarity), as

CA  CB and +C  +Q. QR QP

(iv) FE  2 cm, FD  3 cm, ED  2.5 cm; PQ  4 cm, PR  6 cm, QR  5 cm.    3FED +3PQR (SSS-similarity). (v) +B  180c  (+A  +C)  180c  (80c  70c)  30c.    +A  +M and +B  +N, and so 3ABC +3MNR (AA-similarity). 2. +DOC  (180c  115c)  65c, +DCO  180c  (70c  65c)  45c. 3ODC +3OBA & +OAB  +DCO  45c, +OBA  +ODC  70c.

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Secondary School Mathematics for Class 10

3. 3OAB +3OCD &

OA  AB  BO OA  8  6.4 · & 3.5 5 DO OC CD DO

Find OA and DO. 4. +ADE  +B, +A  +A. 3ADE +3ABC &

3.8 AD  DE  x · & AB BC (3.6  2.1) 4.2

Find x. 5.

Perimeter of 3ABC  AB · Perimeter of 3PQR PQ

7. 3BDA +3BAC &

AD  BA · AC BC

8. 3ABC +3BDC &

AB  BC · BD DC

9. In 3DBA and 3DCB, we have +BDA  +CDB and +DBA  +DCB [each  90c  +A] . 

3DBA +3DCB.

BD  AD BD 2 · & CD  CD BD AD AQ AP  2  1  3  1· 10. and AB 6 3 AC 9 3 

In 3APQ and 3ABC, we have AP  AQ · AB AC 3APQ  3ABC [by SAS-similarity].

+A  +A and 

PQ AP 1   & BC  3PQ. BC AB 3 11. 3AFD +3EFB, as +AFD  +EFB (vert. opp. O) and +DAF  +BEF (alt. O). 

AF  FD · EF FB

12. +BED  +ACB  90c +EBD  +CAB  (90c  +B)

3 ; ` 3BED +3ACB.

14. CA  CB & +CAB  +CBA & 180c  +CAB  180c  +CBA & +CAP  +CBQ. Now, AP # BQ  AC 2 &

AP  AC AC BQ

&

AP  BC AC BQ

Thus, +CAP  +CBQ and 

3ACP +3BCQ.

AP  BC · AC BQ

[a AC  BC].

Triangles 15.

AC  CB AC  BD · & BD CE CB CE Also, +2  +1 & BD  DC. Thus,

AC  DC AC  CB & CB CE DC CE

and +ACB  +DCE  +C. 

3ACB +3DCE [SAS-similarity].

16. In 3ABC, P and Q are midpoints of AB and AC respectively. 

PQ  BC

[by midpoint theorem].

And so, 3APQ +3ABC 

PQ 1 AP  AQ  PQ  & PQ  1 BC  1 DA & 2 2 AB AC BC BC 2 :a AP 

1 AB and BC  DAD · 2

1 DA. 2

In 3CDA, RQ  DA and RQ  In 3BDA, SP  DA and SP 

1 DA. 2

In 3CDB, SR  BC and SR 

1 1 BC  DA. 2 2



SP  RQ and PQ  SR and PQ  RQ  SP  SR.

17. In 3PAC and 3PDB, we have +APC  +DPB

[vertically opposite angles]

+PAC  +PDB

[angles in the same segment]



[by AA-similarity]. 3PAC +3PDB PA  PC And so, & PA $ PB  PC $ PD. PD PB

18. Clearly, ABDC is a cyclic quadrilateral. 

+1  +2  180c.

… (i)

In 3PAC and 3PDB, we have: +APC  +DPB [common] +PAC  +PDB [each equal to (180c  +1)] [Note +PAC  +1  180c (linear pair) and +PDB  +1  180c. {using (i)} 

 3PAC +3PDB [by AA-similarity] PA  PC And so, & PA · PB  PC · PD. PD PB 19. (a) AB = BC and DP = AB & DP  BC & DP  BQ BC = AB and DQ = BC & DQ  AB & DQ  PB  BQDP is a rectangle. And so, BQ  DP and BP  DQ In rt. 3BQD: +1  +2  90c

… (i)

401

402

Secondary School Mathematics for Class 10 In rt. 3DQC: +3  +4  90c In rt. 3BDC:

+2  +3  90c

… (ii) … (iii) [a BDC  90c]

From (i) and (iii), we get +1  +3. From (ii) and (iii), we get +2  +4.    3BQD +3DQC [by AA-similarity] And so,

BQ DQ  & DQ 2  BQ · QC & DQ 2  DP · QC [a BQ  DP] DQ QC

(b) By proving 3PDA +3PBD (in a similar way), we get PD  AP & DP 2  BP · AP & DP 2  DQ · AP PB DP

[a BP  DQ].

RATIO OF THE AREAS OF TWO SIMILAR TRIANGLES THEOREM 1

The ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding sides. [CBSE 2002, ’02C, ’04, ’05, ’05C, ’06, ’06C, ’07, ’08, ’08C, ’09C, ’10]

3ABC +3DEF. ar (3ABC) AB 2 AC 2 BC 2 ·    TO PROVE ar (3DEF) DE 2 DF 2 EF 2

GIVEN

CONSTRUCTION

PROOF

Draw AL = BC and DM = EF.

Since 3ABC +3DEF, it follows that they are equiangular and their sides are proportional. 

+A  +D, +B  +E, +C  +F and

AB  BC  AC · DE EF DF

1 Now, ar (3ABC)  a # BC # ALk 2 1 and ar (3DEF)  a # EF # DMk · 2 1 # BC # AL ar (3ABC)  2  BC # AL ·  EF DM ar (3DEF) 1 # EF # DM 2

… (i)

… (ii)

Triangles

Also,

AL  BC DM EF

… (iii)

403

[a in similar triangles, the ratio of the corres. sides is the same as the ratio of corresponding altitudes]

Using (iii) in (ii), we get 2 ar (3ABC)  aBC # BC k  BC2 · EF EF ar (3DEF) EF ar (3ABC) AC 2 ar (3ABC) AB 2 ·   Similarly, and 2 ar (3DEF) DE ar (3DEF) DF 2 ar (3ABC) AB 2 AC 2 BC 2 ·    Hence, ar (3DEF) DE 2 DF 2 EF 2

If the areas of two similar triangles are equal then prove that the triangles are congruent. [CBSE 2010]

THEOREM 2

GIVEN

3ABC +3DEF such that ar (3ABC)  ar (3DEF) .

TO PROVE PROOF

3ABC ,3DEF.

3ABC +3DEF (given) ar (3ABC) AB 2 AC 2 BC 2 ·     ar (3DEF) DE 2 DF 2 EF 2

… (i)

Now, ar (3ABC)  ar (3DEF) [given] 

ar (3ABC)  1. ar (3DEF)

… (ii)

From (i) and (ii), we get



AB 2  AC 2  BC 2  1 DE 2 DF 2 EF 2 AB 2  DE 2, AC 2  DF 2 and BC 2  EF 2 AB  DE, AC  DF and BC  EF



3ABC ,3DEF [by SSS-congruency].



THEOREM 3

Prove that the ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding altitudes.

404 GIVEN

Secondary School Mathematics for Class 10

3ABC +3DEF, AL = BC and DM = EF.

TO PROVE PROOF

2 ar (3ABC)  AL 2 · ar (3DEF) DM

We know that the ratio of the areas of two similar triangles is equal to the ratio of the squares of the corresponding sides. ar (3ABC) AB 2 ·  ar (3DEF) DE 2 Now, in 3ALB and 3DME, we have +ALB  +DME  90c and +B  +E 

 

… (i)

[a 3ABC +3DEF] .

3ALB +3DME [by AA-similarity] AB  AL DE DM

AB 2  AL2 · DE 2 DM 2 From (i) and (ii), we get



… (ii)

2 ar (3ABC)  AL 2 · ar (3DEF) DM

THEOREM 4

Prove that the ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding medians.

3ABC +3DEF, AP and DQ are the medians of 3ABC and 3DEF respectively. ar (3ABC) AP 2 ·  TO PROVE ar (3DEF) DQ 2 GIVEN

PROOF

We know that the ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding sides. ar (3ABC) AB 2 ·   ar (3DEF) DE 2 Now, 3ABC +3DEF AB  BC  2BP  BP  DE EF 2EQ EQ

… (i)

Triangles

   

405

AB  BP and +B  +E [a 3ABC +3DEF] DE EQ 3APB +3DQE [by SAS-similarity] AB  AP DE DQ AB 2  AP 2 · DE 2 DQ 2

… (ii)

From (i) and (ii), we get ar (3ABC) AP 2 ·  ar (3DEF) DQ 2 THEOREM 5

Prove that the ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding angle-bisector segments.

3ABC +3DEF in which AX and DY are the bisectors of +A and +D respectively.

GIVEN

TO PROVE PROOF

ar (3ABC) AX 2 ·  ar (3DEF) DY 2

We know that ratio of the areas of two similar triangles is equal to ratio of the squares of their corresponding sides. 

ar (3ABC) AB 2 ·  ar (3DEF) DE 2

… (i)

3ABC +3DEF  +A  +D 1 1  +A  +D 2 2  +BAX  +EDY. Now, in 3ABX and 3DEY, we have +BAX  +EDY and +B  +E [a 3ABC +3DEF] .  3ABX +3DEY [by AA-similarity] AB  AX DE DY 

AB 2  AX 2 · DE 2 DY 2

From (i) and (ii), we get ar (3ABC) AX 2 ·  ar (3DEF) DY 2

… (ii)

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Secondary School Mathematics for Class 10

SOLVED EXAMPLES EXAMPLE 1

If the areas of two similar triangles are in the ratio 25 : 64, find the ratio of their corresponding sides. [CBSE 2009]

SOLUTION

Let 3ABC and 3DEF be similar. Then, ar (3ABC) AB 2 BC 2 AC 2    ar (3DEF) DE 2 EF 2 DF 2 AB 2  BC 2  AC 2  25  5 2 a k k a k a k DE EF DF 8 64 AB  BC  AC  5 ·  DE EF DF 8 Hence, the ratio of their corresponding sides is 5 : 8.  a

EXAMPLE 2

In the adjoining figure, S and T are points on the sides PQ and PR respectively of 3PQR such that PT  2 cm, TR  4 cm and ST is parallel to QR. Find the ratio of the areas of 3PST and 3PQR. [CBSE 2010]

SOLUTION

ST < QR (given)  +PST  +PQR and +PTS  +PRQ

[corresponding angles] [corresponding angles]. And so, 3PST +3PQR [by AA-similarity].



ar (3PST) PT 2 PT 2   ar (3PQR) PR 2 (PT  TR) 2 

EXAMPLE 3

SOLUTION

22  4  1 ·  (2 4) 2 36 9

The areas of two similar triangles 3ABC and 3PQR are 25 cm 2 and 49 cm 2 respectively. If QR  9.8 cm, find BC. [CBSE 2006] We know that the ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding sides. 

ar (3ABC) BC 2  ar (3PQR) QR 2

 c 

BC 2  ar (3ABC)  25  5 2 m a k 7 QR ar (3PQR) 49

BC  5 BC 5 & 9.8  7 QR 7

Triangles

407

5 # 9.8 cm  5 #1.4 cm  7 cm. 7 Hence, BC  7 cm.  BC 

EXAMPLE 4

The areas of two similar triangles are 81 cm 2 and 49 cm 2 respectively. If the altitude of the bigger triangle is 4.5 cm, find the corresponding altitude of the smaller triangle. [CBSE 2002]

SOLUTION

Let the given triangles be 3ABC and 3DEF such that ar (3ABC)  81 cm 2 and ar (3DEF)  49 cm 2 . Let AL and DM be the corresponding altitudes of 3ABC and 3DEF respectively. Then, AL  4.5 cm. We know that the ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding altitudes. 2 ar (3ABC)  AL 2  ar (3DEF) DM AL 2  ar (3ABC)  81  9 2  a a k k 7 DM ar (3DEF) 49 4.5  9 AL  9  & DM 7 DM 7 4.5 #7 7  DM  cm  cm  3.5 cm. 9 2 Hence, the altitude of the smaller triangle is 3.5 cm.

EXAMPLE 5

The areas of two similar triangles are 121 cm 2 and 64 cm 2 respectively. If the median of first triangle is 12.1 cm, find the corresponding median of the other. [CBSE 2001]

SOLUTION

Let 3ABC +3DEF such that ar (3ABC)  121 cm 2 and ar (3DEF)  64 cm 2 . Let AP and DQ be the corresponding medians of 3ABC and 3DEF respectively such that AP  12.1 cm. We know that the ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding medians. ar (3ABC) AP 2   ar (3DEF) DQ 2 AP 2  ar (3ABC)  121  11 2 m  c a k 8 64 DQ ar (3DEF) AP  11 .1  11  & 12 [a AP  12.1 cm] 8 DQ 8 DQ

408

Secondary School Mathematics for Class 10

(12.1# 8) cm  (1.1# 8) cm  8.8 cm. 11 Hence, the corresponding median is 8.8 cm.  DQ 

EXAMPLE 6

3ABC +3DEF in which AX and DY are the bisectors of +A and +D respectively. If AX  6.5 cm and DY  5.2 cm, find the ratio of the areas of 3ABC and 3DEF.

SOLUTION

We know that the ratio of the areas of two similar triangles is equal to the ratio of the squares of the corresponding anglebisector segments.  

2 2 ar (3ABC) AX 2 (6.5) 2    a6.5 k  a 5 k  25 · 5.2 16 4 ar (3DEF) DY 2 (5.2) 2 ar (3ABC) : ar (3DEF)  25 : 16.

EXAMPLE 7

In the given figure, the line segment XY is parallel to side AC of 3ABC and it divides the triangle into two parts of equal area. Prove that AX : AB  (2  2 ) : 2.

SOLUTION

Since XY  AC, we have +A  +BXY and +C  +BYX [corres. O].  3ABC +3XBY ar (3ABC) AB 2 ·   ar (3XBY) XB 2 But, ar (3ABC)  2 # ar (3XBY) [given] 

ar (3ABC)  2. ar (3XBY)

From (i) and (ii), we get

   

AB 2  AB 2 2&a k 2 2 XB XB AB  2 & AB  2 (XB) XB AB  2 (AB  AX) 2 AX  ( 2  1) AB 2 (2  2 ) AX  ( 2  1) ·  # 2 AB 2 2

Hence, AX : AB  (2  2 ) : 2.

… (i)

… (ii)

Triangles EXAMPLE 8

In the given figure, 3ABC and 3DBC are on the same base BC. If AD intersects BC at O, prove that ar (3ABC) AO ·  ar (3DBC) DO

SOLUTION

409

[CBSE 2000, ’05, ’07C]

GIVEN 3ABC and 3DBC are on the same base BC and AD intersects BC at O.

ar (3ABC) AO ·  ar (3DBC) DO

TO PROVE

CONSTRUCTION

Draw AL = BC and

DM = BC. PROOF

In 3ALO and 3DMO, we

have +ALO  +DMO  90c and +AOL  +DOM (vert. opp. O).  3ALO +3DMO [by AA-similarity] 

AL  AO · DM DO



1 # BC # AL ar (3ABC)  2  AL  AO DM DO ar (3DBC) 1 # BC # DM 2

Hence, EXAMPLE 9

SOLUTION

… (i) [using (i)]

ar (3ABC) AO ·  ar (3DBC) DO

ABCD is a trapezium in which AB  DC and AB  2DC. If the diagonals of the trapezium intersect each other at a point O, find the ratio of the areas of 3AOB and 3COD. A trapezium ABCD in which AB  DC and AB  2DC. Its diagonals intersect each other at the point O. GIVEN

TO FIND

ar (3AOB) · ar (3COD)

METHOD OF SOLUTION

In 3AOB and 3COD, we have

+AOB  +COD [vert. opp. O]

410

Secondary School Mathematics for Class 10

+OAB  +OCD [alt. int. O]  3AOB +3COD [by AA-similarity]. We know that the ratio of the areas of two similar triangles is equal to the ratio of the squares of the corresponding sides. 

ar (3AOB) AB 2 (2 # DC) 2   ar (3COD) DC 2 DC 2

[a AB  2DC]

2  4 # DC  4· 2 1 DC Hence, ar (3AOB) : ar (3COD)  4 : 1.

EXAMPLE 10

In a trapezium ABCD, O is the point of intersection of AC and BD, AB  CD and AB  2#CD. If the area of 3AOB  84 cm 2, find the area of 3COD. [CBSE 2005, ’09]

SOLUTION

In 3AOB and 3COD, we have +OAB  +OCD [alt. int. O] and +OBA  +ODC [alt. int. O].  3AOB +3COD [by AA-similarity] 

ar (3AOB) AB 2 (2CD) 2   ar (3COD) CD 2 CD 2

[a AB  2 #CD]

2  4 #CD 4 CD 2 1 1  ar (3COD)  # ar (3AOB)  a # 84k cm 2  21 cm 2 . 4 4

Hence, the area of 3COD is 21 cm 2. EXAMPLE 11

D, E and F are respectively the midpoints of sides AB, BC and CA of 3ABC. Find the ratio of the areas of 3DEF and 3ABC.

SOLUTION

In 3ABC, D and F are the midpoints of sides AB and CA respectively. 

DF  BC [by midpoint theorem]

 DF  BE. Similarly, EF  BD. 

BEFD is a parallelogram 1  +B  +EFD, EF  BD  AB 2

Triangles

411

1 BC. 2 Also, ECFD is a parallelogram and DF  BE 

 +EDF  +C. Now, in 3DEF and 3CAB, we have +EFD  +B and +EDF  +C.  3DEF +3CAB

[by AA-similarity].

2 1 ar (3DEF) ar (3DEF) DF 2 a 2 BCk 1     · And so, 4 ar (3ABC) ar (3CAB) BC 2 BC 2

EXAMPLE 12

Prove that the area of an equilateral triangle described on one side of a square is equal to half the area of an equilateral triangle described on one of its diagonals. [CBSE 2005C]

SOLUTION

A square ABCD and equilateral A BCE and ACF have been described on side BC and diagonal AC respectively. GIVEN

TO PROVE

ar (3BCE) 

1 ar (3ACF) . 2

Since each of the 3BCE and 3ACF is an equilateral triangle, so each angle of each one of them is 60. So, the triangles are equiangular, and hence similar. PROOF

 3BCE +3ACF. We know that the ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding sides. 2 ar (3BCE) BC 2   BC 2 [a AC  2 BC] 2 ar (3ACF) AC 2(BC)  1· 2 1 Hence, ar (3BCE)  # ar (3ACF) . 2



412 EXAMPLE 13

SOLUTION

Secondary School Mathematics for Class 10

Prove that the area of an equilateral triangle described on a side of a right-angled isosceles triangle is half the area of the equilateral triangle described on its hypotenuse. [CBSE 2006] A 3ABC in which +ABC  90c and AB  BC. 3ABD and 3CAE are equilateral triangles. GIVEN

TO PROVE PROOF



ar (3ABD) 

1 # ar (3CAE) . 2

Let AB  BC  x units.

hyp. CA  x 2  x 2  x 2 units.

Each of the AABD and CAE being equilateral, each angle of each one of them is 60.  3ABD +3CAE [by AAA-similarity]. But, the ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding sides. 

ar (3ABD) AB 2 x2  x2  1 ·   ar (3CAE) CA 2 (x 2 ) 2 2x 2 2

Hence, ar (3ABD)  EXAMPLE 14

SOLUTION

1 # ar (3CAE) . 2

If D is a point on the side AB of 3ABC such that AD : DB  3 : 2 and E is a point on BC such that DE  AC, find the ratio of the areas of 3ABC and [CBSE 2008C] 3DBE. Let AD  3x cm and DB  2x cm. Then, AB  (AD  DB)  (3x  2x) cm  5x cm. In 3ABC and 3DBE, we have +CAB  +EDB

[corresponding O]

and +ACB  +DEB

[corresponding O].

 3ABC +3DBE [by AA-similarity] ar (3ABC) (AB) 2 (5x) 2 25x 2 25     4 ar (3DBE) (DB) 2 (2x) 2 4x 2  ar (3ABC) : ar (3DBE)  25 : 4.



Triangles EXAMPLE 15

413

In the given figure, DE  BC and AD : DB  5 : 4. Find the ratio ar (3DFE) : ar (3CFB) . [CBSE 2000]

SOLUTION

Let AD  5x cm and DB  4x cm. Then, AB  AD  DB  5x cm  4x cm  9x cm. In 3ADE and 3ABC, we have +ADE  +ABC [corres. O] +AED  +ACB [corres. O].  3ADE +3ABC [by AA-similarity] DE  AD  5x  5 ·  BC AB 9x 9 In 3DFE and 3CFB, we have +EDF  +BCF [alt. int. O] and +DEF  +CBF [alt. int. O].  3DFE +3CFB 2 2 ar (3DFE) DE 2 DE 2    aDE k  a 5 k  25  2 2 9 81 BC ar (3CFB) CB BC  ar (3DFE) : ar (3CFB)  25 : 81.

f

… (i)

EXERCISE 7C

1. 3ABC +3DEF and their areas are respectively 64 cm2 and 121 cm2. If EF  15.4 cm, find BC. 2. The areas of two similar triangles ABC and PQR are in the ratio 9 : 16. If [CBSE 2004] BC  4.5 cm, find the length of QR. 3. 3ABC +3PQR and ar (3ABC)  4ar (3PQR) . If BC  12 cm, find QR. 4. The areas of two similar triangles are 169 cm2 and 121 cm 2 respectively. If the longest side of the larger triangle is 26 cm, find the longest side of the smaller triangle. 5. 3ABC +3DEF and their areas are respectively 100 cm 2 and 49 cm 2 . If the altitude of 3ABC is 5 cm, find the corresponding altitude of 3DEF. [CBSE 2002]

6. The corresponding altitudes of two similar triangles are 6 cm and 9 cm respectively. Find the ratio of their areas.

414

Secondary School Mathematics for Class 10

7. The areas of two similar triangles are 81 cm2 and 49 cm2 respectively. If the altitude of the first triangle is 6.3 cm, find the corresponding altitude of the other. [CBSE 2001] 8. The areas of two similar triangles are 100 cm2 and 64 cm2 respectively. If a median of the smaller triangle is 5.6 cm, find the corresponding median of the other. 9. In the given figure, ABC is a triangle and PQ is a straight line meeting AB in P and AC in Q. If AP = 1 cm, PB = 3 cm, AQ = 1.5 cm, 1 QC = 4.5 cm, prove that area of 3APQ is 16 of the area of 3ABC. [CBSE 2005] 10. In the given figure, DE  BC. If DE  3 cm, BC  6 cm and ar (3ADE)  15 cm 2, find the area of 3ABC.

11. 3ABC is right-angled at A and AD = BC. If BC  13 cm and AC  5 cm, find the ratio of the areas of 3ABC and 3ADC. [CBSE 2000C]

12. In the given figure, DE  BC and DE : BC  3 : 5. Calculate the ratio of the areas of 3ADE and the trapezium BCED.

13. In 3ABC, D and E are the midpoints of AB and AC respectively. Find the ratio of the areas of 3ADE and 3ABC.

Triangles

415

ANSWERS (EXERCISE 7C)

1. 11.2 cm

2. 6 cm

3. 6 cm

7. 4.9 cm

8. 7 cm

10. 60 cm

4. 22 cm 2

5. 3.5 cm

11. 169 : 25 12. 9 : 16

HINTS TO SOME SELECTED QUESTIONS 6. Required ratio  9.

(9) 2

 36  4  4 : 9. 81 9

1.5 AP  1  1 AQ   1.5  1 · , 6 4 AB (1  3) 4 AC (1.5  4.5)   

10.

(6) 2

3APQ +3ABC [by SAS-similarity] 2 2 ar (3APQ) AP 2   a AP k  a1 k  1 16 4 AB ar (3ABC) AB 2 1 ar (3APQ)  $ ar (3ABC) . 16

ar (3ADE) DE 2 3 2    9 1 ar (3ABC) BC 2 6 2 36 4 

[a 3ADE +3ABC]

ar (3ABC)  4 # ar (3ADE)  (4 #15) cm 2  60 cm 2 .

11. In 3BAC and 3ADC, we have +BAC  +ADC  90c and +ACB  +DCA  +C. 

3BAC +3ADC. [by AA-similarity].



ar (3ABC) ar (3BAC) BC 2 ·   ar (3ADC) ar (3ADC) AC 2

12. 3ADE +3ABC. 2 2 ar (3ADE) DE 2   aDE k  a 3 k  9 ·  5 25 BC ar (3ABC) BC 2 Let ar (3ADE)  9x sq units. Then, ar (3ABC)  25x sq units. 

ar (trap. BCED)  ar (3ABC)  ar (3ADE)



ar (3ADE)  9x  9 · ar (trap. BCED) 16x 16

 (25x  9x) sq units  (16x) sq units.

13. Clearly, DE  BC.  

3ADE +3ABC. 2 ar (3ADE) AD 2   AD  1 · ar (3ABC) AB 2 (2AD) 2 4

6. 4 : 9 13. 1 : 4

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Secondary School Mathematics for Class 10

PYTHAGORAS’ THEOREM We have proved earlier in this chapter that: ”If a perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse then the triangles on both sides of the perpendicular are similar to the whole triangle and also to each other.” Now, we shall use this theorem to prove the Pythagoras’ theorem. THEOREM 1

(Pythagoras’ theorem) In a right triangle, the square of the

hypotenuse is equal to the sum of the squares of the other two sides. [CBSE 2001, ’02, ’03, ’04, ’04C, ’05, ’06, ’06C, ’07, ’07C, ’09] GIVEN

A 3ABC in which +ABC  90c.

TO PROVE

AC 2  AB 2  BC 2 .

CONSTRUCTION PROOF

Draw BD = AC.

In 3ADB and 3ABC, we have (common) +A  +A  +ADB +ABC [each equal to 90]  3ADB +3ABC [by AA-similarity] AD  AB  AB AC  AD # AC  AB 2.

… (i)

In 3BDC and 3ABC, we have (common) +C  +C  +BDC +ABC [each equal to 90]  3BDC +3ABC [by AA-similarity] DC  BC  BC AC  DC # AC  BC 2.

… (ii)

From (i) and (ii), we get AD # AC  DC # AC  AB 2  BC 2  (AD  DC)# AC  AB 2  BC 2  THEOREM 2

AC # AC  AB 2  BC 2  AC 2  AB 2  BC 2 . (Converse of Pythagoras’ theorem) In a triangle, if the square of one side is equal to the sum of the squares of the other two sides then the angle opposite to the first side is a right angle. [CBSE 2001, ’03, ’05C, ’06, ’06C, ’07, ’09, ’09C]

Triangles

417

A 3ABC in which AC 2  AB 2  BC 2 . TO PROVE +B  90c. GIVEN

CONSTRUCTION

PROOF

Draw a 3DEF such that DE  AB, EF  BC and +E  90c.

In 3DEF, we have +E  90c. So, by Pythagoras’ theorem, we have DF 2  DE 2  EF 2 DF 2  AB 2  BC 2. But, AC 2  AB 2  BC 2.

… (i) [a DE  AB and EF  BC]



… (ii) [given]  From (i) and (ii), we get AC DF 2 & AC  DF. 2

Now, in 3ABC and 3DEF, we have AB  DE, BC  EF and AC  DF. 

3ABC ,3DEF. Hence, +B  +E  90c.

SOME IMPORTANT RESULTS BASED UPON PYTHAGORAS’ THEOREM THEOREM 1

In a 3ABC, AD is perpendicular to BC. Prove that (AB 2  CD 2)  (AC 2  BD 2) . [CBSE 2003, ’05C, ’09]

GIVEN

A 3ABC in which AD = BC. (AB 2  CD 2)  (AC 2  BD 2).

TO PROVE PROOF

From right 3ADB, we have 

AB 2  AD 2  BD 2 [by Pythagoras’ theorem] (AB 2  BD 2)  AD 2.

… (i)

From right 3ADC, we have 

AC 2  AD 2  CD 2 (AC 2  CD 2)  AD 2.

… (ii)

From (i) and (ii), we get (AB  BD )  (AC  CD ) . Hence, (AB 2  CD 2)  (AC 2  BD 2) . 2

2

2

2

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Secondary School Mathematics for Class 10

THEOREM 2

Given a 3ABC in which +A  90c and AD = BC. Prove that AD 2  BD · CD.

[CBSE 2004C, ’06, ’09]

A 3ABC in which +A  90c and AD = BC. 2 TO PROVE AD  BD · CD. GIVEN

PROOF

In 3BAC, +A  90c.  BC 2  AB 2  AC 2

… (i)

[by Pythagoras’ theorem]. In 3ADB, +ADB  90c. 

AB 2  AD 2  BD 2.

… (ii) [by Pythagoras’ theorem]

In 3ADC, +ADC  90c. 

AC 2  AD 2  CD 2.

… (iii) [by Pythagoras’ theorem]

From (ii) and (iii), we get (AB 2  AC 2)  2AD 2  BD 2  CD 2    

[using (i)] BC 2  2AD 2  BD 2  CD 2 (BD  CD) 2  2AD 2  BD 2  CD 2 BD 2  CD 2  2BD · CD  2AD 2  BD 2  CD 2 2AD 2  2BD · CD  AD 2  BD · CD.

Hence, AD 2  BD · CD. THEOREM 3

In 3ABC, AD = BC such that AD 2  BD · CD. Prove that 3ABC is right-angled at A. [CBSE 2006]

A 3ABC in which AD = BC and AD 2  BD · CD. TO PROVE +A  90c. GIVEN

PROOF

In right 3ADB, +ADB  90c.  AB 2  AD 2  BD 2.

… (i)

[by Pythagoras’ theorem] In right 3ADC, +ADC  90c.  … (ii) [by Pythagoras’ theorem] AC 2  AD 2  CD 2. Adding (i) and (ii), we get (AB 2  AC 2)  BD 2  CD 2  2AD 2  BD 2  CD 2  2BD · CD [a AD 2  BD · CD]  (BD  CD) 2  BC 2 . Thus, (AB 2  AC 2)  BC 2 . Hence, 3ABC is right-angled at A.

Triangles THEOREM 4

419

In a 3ABC, +ABC  90c (i.e., +B is acute) and AD = BC. Prove that AC 2  AB 2  BC 2  2BC · BD.

A 3ABC in which +ABC  90c and AD = BC. 2 2 2 TO PROVE AC  AB  BC  2BC · BD. GIVEN

PROOF

In 3ADB, +ADB  90c.  AB 2  AD 2  BD 2.

… (i)

In 3ADC, +ADC  90c.  AC 2  AD 2  CD 2  AD  (BC  BD) 2

[by Pythagoras’ theorem] [by Pythagoras’ theorem]

2

 AD 2  BC 2  BD 2  2BC · BD  (AD 2  BD 2)  BC 2  2BC · BD  AB 2  BC 2  2BC · BD [using (i)]. Hence, AC 2  AB 2  BC 2  2BC · BD. Note

BD is known as the projection of AB on BC. So, this theorem is stated as:

”In an acute-angled triangle, the square of the side opposite to an acute angle is equal to the sum of the squares of the other two sides minus twice the product of one side and the projection of the other on the first.” THEOREM 5

GIVEN

In a 3ABC, +ABC  90c (i.e., +B is obtuse) and AD = (CB produced). Prove that AC 2  AB 2  BC 2  2BC · BD.

A 3ABC in which +ABC  90c and

AD = (CB produced). TO PROVE PROOF

AC 2  AB 2  BC 2  2BC · BD.

In 3ADB, +ADB  90c.  … (i) [by Pythagoras’ theorem] AB 2  AD 2  BD 2. In 3ADC, +ADC  90c.  [by Pythagoras’ theorem] AC 2  AD 2  CD 2  AD 2  (BC  BD) 2 [a CD  BC  BD]  AD 2  BC 2  BD 2  2BC · BD  (AD 2  BD 2)  BC 2  2BC · BD  AB 2  BC 2  2BC · BD [using (i)].

Note

BD is the projection of AB on BC. So, this theorem is stated as:

”In an obtuse-angled triangle, the square of the side opposite to the obtuse angle is equal to the sum of the squares of the other two sides plus twice the product of one side and the projection of the other on the first.”

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Secondary School Mathematics for Class 10

THEOREM 6

In 3ABC, if AD is the median, then prove that (AB 2  AC 2)  2(AD 2  BD 2) .

GIVEN

A 3ABC in which AD is the median. (AB 2  AC 2)  2(AD 2  BD 2) .

TO PROVE

Draw AL = BC. In 3ALD, +ALD  90c.

CONSTRUCTION PROOF



+ADL  90c and therefore, +ADB  90c.

Thus, in 3ADB, +ADB  90c and AL = (BD produced).  AB 2  AD 2  BD 2  2BD · DL.

… (i)

In 3ADC, +ADC  90c and AL = DC.  AC 2  AD 2  CD 2  2CD · DL … (ii) [a CD  BD] AC 2  AD 2  BD 2  2BD · DL. 2 2 Adding (i) and (ii), we get (AB  AC )  2(AD 2  BD 2) .  Note

This theorem can be stated as: ”In any triangle, the sum of the squares of any two sides is equal to twice the square of half of the third side together with twice the square of the median which bisects the third side.”

SOLVED EXAMPLES EXAMPLE 1

Sides of some triangles are given below. Determine which of them are right triangles. (i) 8 cm, 15 cm, 17 cm (ii) 9 cm, 11 cm, 6 cm (iii) (2a  1) cm, 2 2a cm and (2a  1) cm

SOLUTION

For the given triangle to be right-angled, the sum of the squares of the two smaller sides must be equal to the square of the largest side. (i) Let a = 8 cm, b = 15 cm and c = 17 cm. Then, (a 2  b 2)  {(8) 2  (15) 2} cm 2  (64  225) cm 2  289 cm 2 and c 2  (17) 2 cm 2  289 cm 2 .  (a 2  b 2)  c 2 . Hence, the given triangle is right-angled. (ii) Let a = 9 cm, b = 6 cm and c = 11 cm. Then, (a 2  b 2)  {(9) 2  (6) 2} cm 2  (81  36) cm 2  117 cm 2

Triangles

421

and c 2  (11) 2 cm 2  121 cm 2 .  (a 2  b 2) ! c 2 . Hence, the given triangle is not right-angled. (iii) Let p  (2a  1) cm, q  2 2a cm and r  (2a  1) cm. Then, (p 2  q 2)  (2a  1) 2 cm 2  (2 2a ) 2 cm 2  {(4a 2  1  4a)  8a} cm 2  (4a 2  4a  1) cm 2  (2a  1) 2 cm 2  r 2 .  (p  q )  r 2 . 2

2

Hence, the given triangle is right-angled. EXAMPLE 2

A man goes 15 m due west and then 8 m due north. How far is he from the starting point?

SOLUTION

Starting from A, let the man go from A to B and then from B to C, as shown in the figure. Then, AB  15 m, BC  8 m and +ABC  90c. From right 3ABC, we have AC 2  AB 2  BC 2  {(15) 2  (8) 2} m 2  (225  64) m 2  289 m 2 

AC  289 m = 17 m.

Hence, the man is 17 m away from the starting position. EXAMPLE 3

A ladder 25 m long just reaches the top of a building 24 m high from the ground. Find the distance of the foot of the ladder from the building.

SOLUTION

Let AB be the building and CB be the ladder. Then, AB  24 m, CB  25 m and +CAB  90c. By Pythagoras’ theorem, we have CB 2  AB 2  AC 2  AC 2  CB 2  AB 2  [(25) 2  (24) 2] m 2  (625  576) m 2  49 m 2 

AC  49 m  7 m.

Hence, the distance of the foot of the ladder from the building is 7 m.

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Secondary School Mathematics for Class 10

EXAMPLE 4

A ladder 15 m long reaches a window which is 9 m above the ground on one side of a street. Keeping its foot at the same point, the ladder is turned to the other side of the street to reach a window 12 m high. Find the width of the street.

SOLUTION

Let AB be the street and let C be the foot of the ladder. Let D and E be the given windows such that AD  9 m and BE  12 m. Then, CD and CE are the two positions of the ladder. Clearly, +CAD  90c, +CBE  90c and CD  CE  15 m. From right 3CAD, we have CD 2  AC 2  AD 2 [by Pythagoras’ theorem] 

AC 2  CD 2  AD 2  [(15) 2  (9) 2] m 2  (225  81) m 2  144 m 2 .



AC  144 m  12 m.

From right 3CBE, we have CE 2  CB 2  BE 2 [by Pythagoras’ theorem]  CB 2  CE 2  BE 2  [(15) 2  12) 2] m 2  (225  144) m 2  81 m 2  CB  81 m  9 m. Width of the street  AC  CB  12 m  9 m  21 m. EXAMPLE 5

Two poles of heights 6 metres and 11 metres stand vertically on a plane ground. If the distance between their feet is 12 metres, find the distance between their tops. [CBSE 2002]

SOLUTION

Let AB and CD be the given vertical poles. Then, AB = 6 m, CD = 11 m and AC = 12 m. Draw BE  AC. Then, CE = AB = 6 m, BE = AC = 12 m. 

DE  CD  CE  11 m  6 m  5 m.

In right 3BED, we have: BD 2  BE 2  DE 2  {(12) 2  (5) 2} m 2  (144  25) m 2  169 m 2

Triangles

423

 BD  169 m  13 m. Hence, the distance between the tops of the poles = 13 m. EXAMPLE 6

In a rhombus of side 10 cm, one of the diagonals is 12 cm long. Find the length of the second diagonal. [CBSE 2001]

SOLUTION

Let ABCD be the given rhombus whose diagonals intersect at O. Then, AB  10 cm. Let AC = 12 cm and BD = 2x cm. We know that the diagonals of a rhombus bisect each other at right angles. 

OA 

1 1 AC  6 cm, OB  BD  x cm, and +AOB  90c. 2 2

From right 3AOB, we have AB 2  OA 2  OB 2  OB 2  AB 2  OA 2  {(10) 2  (6) 2} cm 2  (100  36) cm 2  64 cm 2 

x 2  64 & x  64  8.

 OB  8 cm.  BD  2 #OB  2 # 8 cm  16 cm. Hence, the length of the second diagonal is 16 cm. EXAMPLE 7

3ABD is a right triangle in which +A  90c and AC = BD. Prove that: (i) AB 2  BC · BD

SOLUTION

(ii) AC 2  BC · DC

We know that: If a perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse then the triangles on both sides of the perpendicular are similar to the whole triangle and also to each other.  3ABC +3DBA 3ABC +3DAC 3DBA +3DAC.

(iii) AD 2  BD · CD

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Secondary School Mathematics for Class 10

(i) 3ABC +3DBA AB  BC AB  BC  & BD & AB 2  BC · BD. DB BA AB (ii) 3ABC +3DAC AC  BC  & AC 2  BC · DC. DC AC (iii) 3DBA +3DAC AD  BD  & AD 2  BD · CD. CD AD EXAMPLE 8

BL and CM are medians of a 3ABC, right-angled at A. Prove that 4(BL2  CM 2)  5BC 2 .

SOLUTION

[CBSE 2006C, ’10]

A 3ABC in which BL and CM are medians and +A  90c. GIVEN

TO PROVE

In 3BAC, +A  90c.

PROOF



4(BL2  CM 2)  5BC 2 .

BC  AB 2  AC 2. 2

… (i) [by Pythagoras’ theorem]

In 3BAL, +A  90c. 

BL2  AL2  AB 2

[by Pythagoras’ theorem]

1 1  BL2  a ACk  AB 2  BL2  AC 2  AB 2 2 4 2



4BL2  AC 2  4AB 2.

… (ii)

In 3CAM, +A  90c. 

CM 2  AM 2  AC 2

2 1 1  CM 2  a ABk  AC 2  CM 2  AB 2  AC 2 2 4



4CM 2  AB 2  4AC 2.

… (iii)

On adding (ii) and (iii), we get 4(BL  CM )  5(AB  AC 2) . 2

Hence, 4(BL2  CM 2)  5BC 2 [using (i)]. EXAMPLE 9

In the given figure, the perpendicular from A on side BC of a 3ABC, intersects BC at D such that DB  3CD. Prove that 2AB 2  2AC 2  BC 2 . [CBSE 2003, ’05, ’09]

2

2

Triangles SOLUTION

GIVEN

A 3ABC in which AD = BC and BD  3CD.

TO PROVE PROOF



425

2AB 2  2AC 2  BC 2 .

We have BD  3CD.

BC  BD  CD  3CD  CD  4CD

 CD 

1 BC. 4

… (i)

In 3ADB, +ADB  90c. AB 2  AD 2  BD 2. In 3ADC, +ADC  90c.

… (ii) [by Pythagoras‘ theorem]

AC 2  AD 2  CD 2 .

… (iii) [by Pythagoras’ theorem]

 

On subtracting (iii) from (ii), we get AB 2  AC 2  BD 2  CD 2  [(3CD) 2  (CD 2)]  8CD 2 [a BD  3CD]  8 # 1 BC 2  1 BC 2 2 16 

[using (i)].

2AB 2  2AC 2  BC 2 .

Hence, 2AB 2  2AC 2  BC 2 . EXAMPLE 10

In 3ABC, +B  90c and D is the midpoint of BC. Prove that AC 2  AD 2  3CD 2 .

SOLUTION

GIVEN

A 3ABC in which +B  90c

and D is the midpoint of BC. TO PROVE

AC 2  AD 2  3CD 2 . Join AD.

CONSTRUCTION PROOF



In 3ABC, +B  90c.

AC 2  AB 2  BC 2.

… (i) [by Pythagoras’ theorem]

In 3ABD, +B  90c. 

AD 2  AB 2  BD 2



AB  AD 2  BD 2 .



AC 2  (AD 2  BD 2)  BC 2 [using (i)]



AC 2  AD 2  CD 2  (2CD) 2



AC  AD  3CD .

… (ii) [by Pythagoras’ theorem]

2

2

2

2

Hence, AC 2  AD 2  3CD 2 .

[a BD  CD and BC  2CD]

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Secondary School Mathematics for Class 10

EXAMPLE 11

3ABC is right-angled at B and D is the midpoint of BC. Prove that [CBSE 2008C, ’10] AC 2  (4AD 2  3AB 2) .

SOLUTION

GIVEN

A 3ABC in which+B  90c and D is the midpoint of BC.

TO PROVE PROOF



AC 2  (4AD 2  3AB 2) .

In 3ABC, +B  90c.

AC  AB 2  BC 2 [by Pythagoras’ theorem] 2

 AB 2  (2BD) 2 [a BC  2BD]  AB 2  4BD 2  AB 2  4 (AD 2  AB 2) [a AB 2  BD 2  AD 2]  (4AD 2  3AB 2) . Hence, AC 2  (4AD 2  3AB 2) . EXAMPLE 12

In an isosceles 3ABC, AB  AC and BD = AC. Prove that (BD 2  CD 2)  2CD · AD.

SOLUTION

A 3ABC in which AB  AC and BD = AC.

GIVEN

TO PROVE PROOF

(BD 2  CD 2)  2CD · AD.

From right 3ADB, we have

AB  AD 2  BD 2 [by Pythagoras’ theorem] 2

 AC 2  AD 2  BD 2 [a AB  AC]  (CD  AD) 2  AD 2  BD 2 [a AC  CD  AD]  CD 2  AD 2  2CD · AD  AD 2  BD 2  (BD 2  CD 2)  2CD · AD. Hence, (BD 2  CD 2)  2CD · AD. EXAMPLE 13

3ABC is an isosceles triangle, right-angled at C. Prove that AB 2  2AC 2 .

SOLUTION

3ABC is an isosceles triangle, rightangled at C … (i)  BC  AC Now, by Pythagoras’ theorem, AB 2  BC 2  AC 2

Triangles

EXAMPLE 14

SOLUTION

427



AB 2  AC 2  AC 2 [using (i)]



AB 2  2AC 2 .

3ABC is an isosceles triangle with AC  BC. If AB 2  2AC 2, prove that 3ABC is a right triangle. [CBSE 2000C] In 3ABC, we have AC  BC.

… (i)

Now, AB 2  2AC 2

(given)



AB  AC  AC



AB 2  AC 2  BC 2

2

2

(given)

2

[using (i)]

 +C  90c [by converse of Pythagoras’ theorem] Hence, 3ABC is a right triangle. EXAMPLE 15

3ABC is a right triangle in which +C  90c and CD = AB. If BC  a, CA  b, AB  c and CD  p then prove that (i) cp  ab

SOLUTION

(ii)

1  1  1· p2 a2 b2

[CBSE 1997C, ’98, ’99, ’02]

(i) We have ar (3ABC) 

1 1 # AB#CD  cp 2 2 [taking AB as base]

and ar (3ABC) 

1 1 # BC # AC  ab 2 2 [taking BC as base].

1 1 cp ab & cp  ab. 2 2 Hence, cp  ab. 

c 1 (ii) cp  ab & p  · ab 

1  c2  b2  a2 p2 a2 b2 a2 b2

2 2  d b2 2  a2 2 n  c 12  12 m · a b a b a b 1  1  1· Hence, 2 p a2 b2

[a AB 2  AC 2  BC 2]

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Secondary School Mathematics for Class 10

EXAMPLE 16

In an equilateral triangle, prove that three times the square of one side is equal to four times the square of one of its altitudes.

SOLUTION

GIVEN

[CBSE 2002, ’07C]

A 3ABC in which AB  BC  CA and AD = BC.

3AB 2  4AD 2 . PROOF In 3ADB and 3 ADC, we have AB  AC (given), +B  +C  60c and +ADB  +ADC  90c.  3ADB ,3ADC [AAS-congruence] 1  BD  DC  BC. 2 From right 3ADB, we have TO PROVE

AB 2  AD 2  BD 2

[by Pythagoras’ theorem]



 AD 2  a1 BCk  AD 2  1 BC 2 2 4 4AB 2  4AD 2  BC 2



3AB 2  4AD 2 [a BC  AB].

2

Hence, 3AB 2  4AD 2 . EXAMPLE 17

In an equilateral triangle with side a, prove that (i) altitude 

SOLUTION

3 3 2 a (ii) area  a . 2 4

[CBSE 1997, ’99, ’01C, ’02C]

Let 3ABC be an equilateral triangle with side a. Then, AB  AC  BC  a. Draw AD = BC. In 3ADB and 3ADC, we have AB  AC (given), +B  +C  60c and +ADB  +ADC  90c.  3ADB ,3ADC. a  BD  DC  · 2 (i) From right 3ADB, we have AB 2  AD 2  BD 2 [by Pythagoras’ theorem]  AD  AB 2  BD 2 

a 2 a2  ` j  2

Hence, altitude 

a2  3 a. 2

a2  4

3a 2  3 a. 2 4

Triangles

429

1 1 # base # altitude  a # BC # ADk 2 2 3  d1 # a# an [using (i)] 2 2

(ii) Area of 3ABC 

d

3 2 a n sq units. 4

Hence, area(3ABC)  d EXAMPLE 18

3 2 a n sq units. 4

O is a point in the interior of 3ABC, OD = BC, OE = AC and OF = AB, as shown in the figure.

Prove that: (i) OA 2  OB 2  OC 2  OD 2  OE 2  OF 2  AF 2  BD 2  CE 2 (ii) AF 2  BD 2  CE 2  AE 2  BF 2  CD 2 SOLUTION

(i) Using Pythagoras’ theorem for each of the right triangles namely 3OFA, 3ODB and 3OEC, we get … (i) OA 2  OF 2  AF 2 … (ii) OB 2  OD 2  BD 2 … (iii) OC 2  OE 2  CE 2 Adding (i), (ii) and (iii), we get OA 2  OB 2  OC 2  OD 2  OE 2  OF 2  AF 2  BD 2  CE 2 . 2 Hence, OA  OB 2  OC 2  OD 2  OE 2  OF 2  AF 2  BD 2  CE 2 . (ii) Using Pythagoras’ theorem for each of the right triangles, namely 3ODB and 3ODC, we get OB 2  OD 2  BD 2 and OC 2  OD 2  CD 2 .  OB 2  OC 2  BD 2  CD 2 … (iv) Similarly, we have OC 2  OA 2  CE 2  AE 2 and OA 2  OB 2  AF 2  BF 2.

… (v) … (vi)

430

Secondary School Mathematics for Class 10

Adding the corresponding sides of (iv), (v) and (vi), we get AF 2  BD 2  CE 2  AE 2  BF 2  CD 2  0. Hence, AF 2  BD 2  CE 2  AE 2  BF 2  CD 2 . EXAMPLE 19

O is any point inside a rectangle ABCD. Prove that OB 2  OD 2  OA 2  OC 2 . DEDUCTION In the given figure, O is a point inside a rectangle ABCD such that OB  6 cm, OD  8 cm and OA  5 cm, find the length of OC. [CBSE 2009C]

SOLUTION

GIVEN

[CBSE 2006C]

O is a point inside a rectangle

ABCD. TO PROVE

OB 2  OD 2  OA 2  OC 2 .

Through O, draw POQ  BC so that P lies on AB and Q lies on DC.

CONSTRUCTION

PROOF

we have

POQ  BC & PQ = AB and QP = DC  +BPQ  90c and +CQP  90c. 

BPQC and APQD are both rectangles.



BP  CQ

[opposite sides of a rectangle]

DQ  AP

[opposite sides of a rectangle]

From right 3OPB, we have OB 2  OP 2  BP 2 . From right 3OQD, we have OD 2  OQ 2  DQ 2 . From right 3OPA, we have OA 2  OP 2  AP 2 . From right 3OQC, we have OC 2  OQ 2  CQ 2 . 

OB 2  OD 2  OP 2  OQ 2  BP 2  DQ 2  OP 2  OQ 2  CQ 2  AP 2 [a BP  CQ and DQ  AP]  (OP  AP )  (OQ 2  CQ 2) 2

2

 OA 2  OC 2 . Hence, OB 2  OD 2  OA 2  OC 2 .

Triangles DEDUCTION

431

Let OC  x cm. Then,

OB 2  OD 2  OA 2  OC 2  62  82  52  x2 

x 2  36  64  25  75



x  75  5 3  (5 #1.732)  8.66 & OC  8.66 cm.

EXAMPLE 20

Prove that the sum of the squares on the sides of a rhombus is equal to the sum of the squares on its diagonals. [CBSE 2005, ’06, ’08C]

SOLUTION

GIVEN A rhombus ABCD whose diagonals AC and BD intersect at O.

(AB 2  BC 2  CD 2  DA 2)

TO PROVE

 (AC 2  BD 2) . We know that the diagonals of a rhombus bisect each other at right angles. PROOF

 +AOB  +BOC  +COD  +DOA  90c, OA 

1 1 AC and OB  BD. 2 2

From right 3AOB, we have AB 2  OA 2  OB 2 [by Pythagoras’ theorem] 2 2  a1 ACk  a1 BDk  1 (AC 2  BD 2) 2 2 4



4AB 2  (AC 2  BD 2) .

… (i)

Similarly, we have: 4BC 2  (AC 2  BD 2)

… (ii)

4CD  (AC 2  BD 2)

… (iii)

4DA  (AC 2  BD 2)

… (iv)

2

2

On adding (i), (ii), (iii) and (iv), we get (AB 2  BC 2  CD 2  DA 2)  (AC 2  BD 2) . REMARK

In a rhombus ABCD, we have AB  BC  CD  DA, so the above result may be given as 4AB 2  (AC 2  BD 2) .

EXAMPLE 21

P and Q are points on the sides CA and CB of a 3ABC, right-angled at C. Prove that (AQ 2  BP 2)  (AB 2  PQ 2) . [CBSE 2007, ’08]

432 SOLUTION

Secondary School Mathematics for Class 10

A 3ABC in which +C  90c. P and Q are points on CA and CB respectively. GIVEN

TO PROVE PROOF

(AQ 2  BP 2)  (AB 2  PQ 2) .

From right 3ACQ, we have

AQ  (AC 2  CQ 2). 2

… (i) [by Pythagoras’ theorem]

From right 3BCP, we have BP 2  (BC 2  CP 2) .

… (ii) [by Pythagoras’ theorem]

From right 3ACB, we have AB 2  AC 2  BC 2.

… (iii) [by Pythagoras’ theorem]

From right 3PCQ, we have PQ 2  (CQ 2  CP 2).

… (iv) [by Pythagoras’ theorem]

From (i) and (ii), we get (AQ 2  BP 2)  (AC 2  BC 2)  (CQ 2  CP 2)  (AB 2  PQ 2) [using (iii) and (iv)]. Hence, (AQ 2  BP 2)  (AB 2  PQ 2) . EXAMPLE 22

In the figure given below, 3PQR is right-angled at Q and the points S and T trisect the side QR. Prove that 8PT 2  3PR 2  5PS 2 . [CBSE 2006C]

SOLUTION

A 3PQR in which +PQR  90c, S and T are the points of trisection of QR. 2 2 2 TO PROVE 8PT  3PR  5PS . PROOF Let QS  ST  TR  x. Then, QS  x, QT  2x and QR  3x. GIVEN

From right triangles PQS, PQT and PQR, by Pythagoras’ theorem, we have PS 2  PQ 2  QS 2, PT 2  PQ 2  QT 2 and PR 2  PQ 2  QR 2 . 

3PR 2  5PS 2  8PT 2  3 (PQ 2  QR 2)  5 (PQ 2  QS 2)  8 (PQ 2  QT 2)  3QR 2  5QS 2  8QT 2  3 #(3x) 2  5 (x) 2  8 #(2x) 2 [a QR  3x, QS  x and QT  2x]  (27x 2  5x 2  32x 2)  0.

Triangles

433

Thus, 3PR 2  5PS 2  8PT 2  0. Hence, 8PT 2  3PR 2  5PS 2 . EXAMPLE 23

In an isosceles 3ABC, AB  AC and D is a point on BC. Prove that AB 2  AD 2  BD · CD.

SOLUTION

GIVEN

A 3ABC in which AB  AC and D is a point on BC.

TO PROVE

(AB 2  AD 2)  BD $ CD. Draw AL = BC.

CONSTRUCTION PROOF

In right A ALB and ALC, we have:

hyp. AB = hyp. AC (given) AL  AL

(common)

 3ALB ,3ALC 

[by RHS-congruence]

BL  CL.

From right A ALB and ALD, by Pythagoras’ theorem, we have: AB 2  AL2  BL2

… (i)

AD  AL  DL 2



2

… (ii)

2

AB  AD  BL  DL 2

2

2

2

 (BL  DL)(BL  DL)  BD · (CL  DL)

[a BL  CL]

 BD · (CL  DL) [a BL  DL  BD and BL  CL]  BD · CD. Hence, AB  AD 2  BD · CD. 2

EXAMPLE 24

In an equilateral3ABC, D is a point on side BC such that BD  Prove that 9AD 2  7AB 2 .

SOLUTION

GIVEN

A 3ABC in which AB  BC  CA

and D is a point on BC such that 1 BD  BC. 3 TO PROVE

9AD 2  7AB 2 .

CONSTRUCTION

Draw AL = BC.

1 BC. 3

434

Secondary School Mathematics for Class 10 PROOF

In right triangles ALB and ALC, we have

AB  AC (given) and AL  AL (common).  3ALB ,3ALC

[by RHS axiom]

So, BL  CL. 1 1 Thus, BD  BC and BL  BC. 3 2 In 3ALB, +ALB  90c. 

AB 2  AL2  BL2.

… (i) [by Pythagoras’ theorem]

In 3ALD, +ALD  90c. 

AD 2  AL2  DL2

[by Pythagoras’ theorem]

 AL2  (BL  BD) 2  AL2  BL2  BD 2  2BL · BD  (AL2  BL2)  BD 2  2BL · BD  AB 2  BD 2  2BL · BD

[using (i)]

 BC 2  a1 BCk  2 a1 BCk · 1 BC 3 2 3 2

:a AB  BC, BD 

1 1 BC and BL  BCD 3 2

 BC 2  1 BC 2  1 BC 2 9 3 7 7 2 2  BC  AB [a BC  AB] . 9 9 Hence, 9AD 2  7AB 2 . EXAMPLE 25

SOLUTION

In a quadrilateral ABCD, +B  90c. If AD 2  AB 2  BC 2  CD 2, prove that +ACD  90c. GIVEN

A quad. ABCD in which

+B  90c and AD 2  AB 2  BC 2  CD 2 . TO PROVE +ACD  90c. CONSTRUCTION PROOF



Join AC.

In 3ABC, +B  90c.

AC  AB 2  BC 2.

… (i) [by Pythagoras’ theorem]

2

Now, AD  AB  BC  CD (given) 2



2

2

2

AD 2  AC 2  CD 2 [using (i)].

Triangles

435

Thus, in 3ACD, we have AD 2  AC 2  CD 2 . Hence, +ACD  90c [by converse of Pythagoras’ theorem]. EXAMPLE 26

SOLUTION

Equilateral triangles are drawn on the sides of a right triangle. Prove that the area of the triangle on the hypotenuse is equal to the sum of the areas of the triangles on the other two sides. [CBSE 2002] GIVEN A 3ABC in which +B  90c. Equilateral triangles 3BCD, 3CAE and 3ABF are drawn on the sides BC, CA and AB respectively. TO PROVE

ar (3CAE)  ar (3BCD)  ar (3ABF).

PROOF 3BCD, 3CAE and 3ABF are equiangular and hence similar.

We have



ar (3BCD) ar (3ABF) BC 2 AB 2    ar (3CAE) ar (3CAE) CA 2 CA 2 ar (3BCD)  ar (3ABF) BC 2  AB 2 CA 2   1 ar (3CAE) CA 2 CA 2 [by Pythagoras’ theorem]

 ar (3BCD)  ar (3ABF)  ar (3CAE). EXAMPLE 27

Given a right-angled 3ABC. The lengths of the sides containing the right angle are 6 cm and 8 cm. A circle is inscribed in 3ABC. Find the radius of the circle.

SOLUTION

In 3ABC, we have +B  90c, AB  6 cm and BC  8 cm. A circle is inscribed in 3ABC. Let O be its centre and M, N and P be the points where it touches the sides AB, BC and CA respectively. Then, OM = AB, ON = BC, OP = CA. Let r cm be the radius of the circle. Then, OM  ON  OP  r cm. Now, AB 2  BC 2  CA 2

[by Pythagoras’ theorem]

 6 cm  8 cm  CA 2 & CA  10 cm. 2

2

2

2

436

Secondary School Mathematics for Class 10

Now, ar (3ABC)  ar (3AOB)  ar (3BOC)  ar (3COA) 1 1 1 # AB# BC  a # AB #OMk  a # BC #ONk  2 2 2  a1 #CA#OPk 2 1 1 1 1 # 6 # 8  a # 6 # rk  a # 8 # rk  a #10 # rk  2 2 2 2  r  2 & radius  2 cm. EXAMPLE 28

Prove that the sum of the squares of the diagonals of a parallelogram is equal to the sum of the squares of its sides.

SOLUTION

GIVEN

A parallelogram PQRS.

TO PROVE

PR 2  QS 2  PQ 2  QR 2  RS 2  SP 2 .

We have proved earlier that if AD is a median of a 3ABC, then PROOF

AB 2  AC 2  2(AD 2  BD 2) 1 i.e., AB 2  AC 2  2AD 2  BC 2. 2 Now, let O be the point of intersection of the diagonal PR and QS.

… (i)

The diagonals of a parallelogram bisect each other.  O is the midpoint of PR as well as QS. Applying result (i) to 3PQR and 3RSP, we get 1 PQ 2  QR 2  2OQ 2  PR 2 2 1 and RS 2  SP 2  2OS 2  PR 2. 2 Adding (ii) and (iii), we get 2 2 1 1 PQ 2  QR 2  RS 2  SP 2  2 a QSk  2 a QSk  PR 2 2 2  PQ 2  QR 2  RS 2  SP 2  PR 2  QS 2 . EXAMPLE 29

SOLUTION

… (ii) … (iii)

Given a 3ABC in which +B  90c and AB  3 BC. Prove that +C  60c. Let D be the midpoint of the hypotenuse AC. Join BD.

Triangles

437

We have AC 2  AB 2  BC 2 [by Pythagoras’ theorem] 

AC 2  ( 3 BC) 2  BC 2



AC  4BC & AC  2BC



2CD  2BC [a D is the midpoint of AC]

[a AB  3 BC (given)] 2

2

 CD  BC.

… (i)

Also, we know that the midpoint of the hypotenuse of a right triangle is equidistant from the vertices. 

BD  CD

… (ii)

From (i) and (ii), we get BC  BD  CD.  3BCD is equilateral and hence +C  60c. f

EXERCISE 7D

1. The sides of certain triangles are given below. Determine which of them are right triangles. (i) 9 cm, 16 cm, 18 cm (iii) 1.4 cm, 4.8 cm, 5 cm (v) (a  1) cm, 2 a cm, (a  1) cm

(ii) 7 cm, 24 cm, 25 cm (iv) 1.6 cm, 3.8 cm, 4 cm

2. A man goes 80 m due east and then 150 m due north. How far is he from the starting point? 3. A man goes 10 m due south and then 24 m due west. How far is he from the starting point? 4. A 13-m-long ladder reaches a window of a building 12 m above the ground. Determine the distance of the foot of the ladder from the building. 5. A ladder is placed in such a way that its foot is at a distance of 15 m from a wall and its top reaches a window 20 m above the ground. Find the length of the ladder. 6. Two vertical poles of height 9 m and 14 m stand on a plane ground. If the distance between their feet is 12 m, find the distance between their tops.

438

Secondary School Mathematics for Class 10

7. A guy wire attached to a vertical pole of height 18 m is 24 m long and has a stake attached to the other end. How far from the base of the pole should the stake be driven so that the wire will be taut? 8. In the given figure, O is a point inside a 3PQR such that +POR  90c, OP  6 cm and OR  8 cm. If PQ  24 cm and QR  26 cm, prove that 3PQR is right-angled. [CBSE 2006, ’09C]

9. 3ABC is an isosceles triangle with AB  AC  13 cm. The length of altitude from A on BC is 5 cm. Find BC. [CBSE 2000C] 10. Find the length of altitude AD of an isosceles 3ABC in which AB  AC  2a units and BC  a units. 11. 3ABC is an equilateral triangle of side 2a units. Find each of its altitudes. 12. Find the height of an equilateral triangle of side 12 cm. 13. Find the length of a diagonal of a rectangle whose adjacent sides are 30 cm and 16 cm. 14. Find the length of each side of a rhombus whose diagonals are 24 cm and 10 cm long. 15. In 3ABC, D is the midpoint of BC and AE = BC. If AC  AB, show that AB 2  AD 2  BC · DE 

1 BC 2 . 4

[CBSE 2006C]

16. In the given figure, +ACB  90c and CD = AB. Prove that

BC 2  BD · AC 2 AD

17. In the given figure, D is the midpoint of side BC and AE = BC. If BC  a, AC  b, AB  c, ED  x, AD  p and AE  h, prove that (i) b 2  p 2  ax 

a2 4

(ii) c 2  p 2  ax 

a2 4

1 (iii) (b 2  c 2)  2p 2  a 2 2 2 2 (iv) (b  c )  2ax

Triangles

439

18. In 3ABC, AB  AC. Side BC is produced to D. Prove that (AD 2  AC 2)  BD · CD.

19. ABC is an isosceles triangle, rightangled at B. Similar triangles ACD and ABE are constructed on sides AC and AB. Find the ratio between the areas of [CBSE 2002] 3ABE and 3ACD.

20. An aeroplane leaves an airport and flies due north at a speed of 1000 km per hour. At the same time, another aeroplane leaves the same airport and flies due west at a speed of 1200 km per hour. How far apart will be 1 the two planes after 1 hours? 2 21. In a 3ABC, AD is a median and AL = BC. Prove that: (a) AC 2  AD 2  BC · DL  a

BC 2 k 2 BC 2 (b) AB 2  AD 2  BC · DL  a k 2 1 2 2 2 (c) AC  AB  2AD  BC 2 2 22. Naman is doing fly-fishing in a stream. The tip of his fishing rod is 1.8 m above the surface of the water and the fly at the end of the string rests on the water 3.6 m away from him and 2.4 m from the point directly under the tip of the rod. Assuming that the string (from the tip of his rod to the fly) is taut, how much string does he have out (see the adjoining figure)? If he pulls in the string at the rate of 5 cm per second, what will be the horizontal distance of the fly from him after 12 seconds?

440

Secondary School Mathematics for Class 10 ANSWERS (EXERCISE 7D)

1. (ii), (iii), (v)

2. 170 m

3. 26 m

6. 13 m

7. 6 7 m

9. 24 m

12. 6 3 cm

13. 34 m

11. a 3 units

4. 5 m 10.

a 15 units 2

14. 13 cm

20. 300 61 km 22. 2.8 m (approx.) HINTS TO SOME SELECTED QUESTIONS 7. Let AB be the vertical pole and C be the position of the stake so that BC is a taut wire. In rt. 3ABC, +A  90c, AB  18 m, BC  24 m. Now, AC 2  AB 2  BC 2 & AC 2  (24 2  18 2) m 2  252 m 2 & AC  252 m  6 7 m. 8. PR  OP 2  OR 2  6 2  8 2  10. Now, 24 2  10 2  26 2, i.e., PQ 2  PR 2  QR 2 & +QPR  90c. 10. In 3ABC, AB  AC  2a and BC  a. Draw AD = BC. Then, D is the midpoint of BC a and so BD  $ 2 a 15 a 2  15a 2 AD 2  AB 2  BD 2  4a 2  & AD  $ 2 4 4

15. In 3AEB, +AEB  90c. 

AB 2  AE 2  BE 2

… (i)

In 3AED, +AED  90c. 

AD 2  (AE 2  DE 2)



AE 2  (AD 2  DE 2).



AB 2  (AD 2  DE 2)  BE 2  (AD

2

[using (i)]

DE )  (BD  DE) 2 2

2

 (AD 2  DE 2)  a1 BC  DEk 2  AD 2  1 BC 2  BC $ DE. 4 16. 3ABC +3CBD BC  AB  & BC 2  AB $ BD. BD CB

… (i)

5. 25 m

19. 1 : 2

Triangles

441

3ABC +3ACD AC  AB  & AC 2  AB · AD. … (ii) AD AC Dividing (i) by (ii), we get

BC 2  BD · AC 2 AD

17. (i) AC 2  AE 2  EC 2 



a2 a2 b 2  h 2  ax  k  (h 2  x 2)  ax  2 4 2

 p 2  ax  a · 4 (ii) AB 2  AE 2  BE 2 2 a a2  c 2  h 2  a  xk  (h 2  x 2)  ax  2 4 2 a 2  p  ax  · 4 (iii) Add (i) and (ii). (iv) Subtract (ii) from (i). 18. Draw AE = BC. Then, BE  CE. AD 2  AE 2  DE 2 and AC 2  AE 2  CE 2 

(AD 2  AC 2)  DE 2  CE 2  (DE  CE)(DE  CE)  (DE  BE)(DE  CE) [a CE  BE]  BD · CD.

2 ar (3ABC) AB 2   AB  1 19. ar (3ACD) AC 2 2AB 2 2

[a AC 2  AB 2  BC 2  2AB 2].

20. Distance covered by first plane in 1  OP  1500 km (north).

1 hours 2

Distance covered by second plane in 1  OQ  1800 km (west). Distance between the two planes after 1

1 hours 2

1 hours 2

 PQ  OP 2  OQ 2  300 61 km. 21. (a) In right 3ACL, AC 2  AL2  LC 2.

… (i)

In right 3 ALD, AL  AD 2  DL2. 2



… (ii)

AC 2  (AD 2  DL2)  LC 2 [from (i) and (ii)]  (AD 2  DL2)  (DL  DC) 2 2

 (AD 2  DL2)  aDL  1 BCk 2

2

 AD 2  BC · DL  1 BC 2  AD 2  BC · DL  aBC k · 2 4

442

Secondary School Mathematics for Class 10 (b) In right 3ABL, AB 2  AL2  LB 2.

… (iii)

In right 3ALD, AL  AD 2  DL2 .. 2



AB 2  (AD 2  DL2)  LB 2

… (iv)

[from (iii) and (iv)]

 (AD 2  DL2)  (BD  DL) 2 2

2

 (AD 2  DL2)  a1 BC  DLk  AD 2  BC · DL  aBC k · 2 2 (c) Adding the results of (a) and (b), we get 1 AC 2  AB 2  2AD 2  BC 2 . 2 22. Let of the string that was out of the rod  BC  BM 2  CM 2  (1.8) 2  (2.4) 2 m  3.24  5.76 m  9 m  3 m. He pulls the string at a rate of 5 cm per second. 

length of string pulled in 12 s  (5 #12) cm  60 cm  0.6 m.

So, after 12 s, we have BCl  (3  0.6) m  2.4 m and BM  1.8 m. 

ClM  (BCl) 2  BM 2  (2.4) 2  (1.8) 2 m  2.52 m . 1.6 m.

Horizontal distance of the fly from him after 12 s  ClA  ClM  MA  (1.6  1.2) m  2.8 m.

f

EXERCISE 7E

Very-Short-Answer and Short-Answer Questions: 1. State the two properties which are necessary for given two triangles to be similar. 2. State the basic proportionality theorem. 3. State the converse of Thales’ theorem. 4. State the midpoint theorem. 5. State the AAA-similarity criterion. 6. State the AA-similarity criterion. 7. State the SSS-criterion for similarity of triangles. 8. State the SAS-similarity criterion. 9. State Pythagoras’ theorem. 10. State the converse of Pythagoras’ theorem. 11. If D, E and F are respectively the midpoints of sides AB, BC and CA of 3ABC then what is the ratio of the areas of 3DEF and 3ABC?

Triangles

443

12. Two triangles ABC and PQR are such that AB  3 cm, AC  6 cm, +A  70c, PR  9 cm, +P  70c and PQ  4.5 cm. Show that 3ABC +3PQR and state the similarity criterion. 13. If 3ABC +3DEF such that 2AB  DE and BC  6 cm, find EF. 14. In the given figure, DE  BC such that AD  x cm, DB  (3x  4) cm, AE  (x  3) cm and EC  (3x  19) cm. Find the value of x. 15. A ladder 10 m long reaches the window of a house 8 m above the ground. Find the distance of the foot of the ladder from the base of the wall. 16. Find the length of the altitude of an equilateral triangle of side 2a cm. 17. 3ABC +3DEF such that ar (3ABC)  64 cm 2 and ar (3DEF)  169 cm 2 . If BC  4 cm, find EF. 18. In a trapezium ABCD, it is given that AB  CD and AB  2CD. Its diagonals AC and BD intersect at the point O such that ar (3AOB)  84 cm 2 . Find ar (3COD) . 19. The corresponding sides of two similar triangles are in the ratio 2 : 3. If the area of the smaller triangle is 48 cm2, find the area of the larger triangle. 20. In an equilateral triangle with side a, prove that area 

3 2 a . 4

21. Find the length of each side of a rhombus whose diagonals are 24 cm and 10 cm long. 22. Two triangles DEF and GHK are such that +D  48c and +H  57c. If 3DEF +3GHK then find the measure of +F. 23. In the given figure, MN  BC and AM : MB  1 : 2. Find

area (3AMN) · area (3ABC)

24. In triangles BMP and CNR it is given that PB  5 cm, MP  6 cm, BM  9 cm and NR  9 cm. If 3BMP +3CNR then find the perimeter of 3CNR. 25. Each of the equal sides of an isosceles triangle is 25 cm. Find the length of its altitude if the base is 14 cm.

444

Secondary School Mathematics for Class 10

26. A man goes 12 m due south and then 35 m due west. How far is he from the starting point? 27. If the lengths of the sides BC, CA and AB of a 3ABC are a, b and c respectively and AD is the bisector of +A then find the lengths of BD and DC. 28. In the given figure, +AMN  +MBC  76c. If p, q and r are the lengths of AM, MB and BC respectively then express the length of MN in terms of p, q and r.

29. The lengths of the diagonals of a rhombus are 40 cm and 42 cm. Find the length of each side of the rhombus. For each of the following statements state whether true (T) or false (F): 30.

(i) Two circles with different radii are similar. (ii) Any two rectangles are similar. (iii) If two triangles are similar then their corresponding angles are equal and their corresponding sides are equal. (iv) The length of the line segment joining the midpoints of any two sides of a triangle is equal to half the length of the third side. (v) In a 3ABC, AB  6 cm, +A  45c and AC  8 cm and in a 3DEF, DF  9 cm, +D  45c and DE  12 cm, then 3ABC +3DEF. (vi) The polygon formed by joining the midpoints of the sides of a quadrilateral is a rhombus. (vii) The ratio of the areas of two similar triangles is equal to the ratio of their corresponding angle-bisector segments.

(viii) The ratio of the perimeters of two similar triangles is the same as the ratio of their corresponding medians. (ix) If O is any point OA 2  OC 2  OB 2  OD 2 .

inside

a

rectangle

ABCD

then

(x) The sum of the squares on the sides of a rhombus is equal to the sum of the squares on its diagonals.

Triangles

445

ANSWERS (EXERCISE 7E)

11. 1 : 4

12. SAS-similarity

13. 12 cm

14. x  2

16.

17. 6.5 cm

18. 21 cm

19. 108 cm

1 23. 9

24. 30 cm

3 a cm

22. 75 27. BD 



ab ac ; DC  bc bc

28. MN 

2

25. 24 cm

ac ab

15. 6 m 2

21. 13 cm 26. 37 cm

29. 29 cm

30. (i) T (ii) F (iii) F (iv) T (v) F (vi) F (vii) T (viii) T (ix) T (x) T

HINTS TO SOME SELECTED QUESTIONS 14.

AD  AE [by Thales’ theorem] DB EC

17.

2 ar (3AOB) AB 2   a AB k  4 1 CD ar (3COD) CD 2



ar (3COD) 

[a 3AOB +3COD]

1 # ar (3AOB). 4

22. +D  48c, +E  +H  57c [a 3DEF +3GHK] 

+F  180c  (+D  +E)  180c  (48c  57c)  75c.

23. MN  BC & 3AMN +3ABC. 

24.

2 2 area (3AMN) AM 2   c AM m  a x k  1 x  2x 9 AM  MB area (3ABC) AB 2  [AM : MB 1 : 2, so, let AM  x, then MB  2x].

Perimeter of 3BMP  MP Perimeter of 3CNR NR 

perimeter of 3CNR 

[a 3BMP +3CNR] NR NR # perimeter of 3BMP  #(PB  MP  BM) MP MP  9 #(5  6  9) cm  30 cm. 6

25. Let the given triangle be 3ABC having AB  AC  25 cm, base BC  14 cm. Let AD = BC. Then, D is the midpoint of BC. Altitude AD 

AC 2  DC 2  25 2  7 2 cm  24 cm.

446

Secondary School Mathematics for Class 10

27. Let BD  x. Then, DC  BC  BD  a  x. BD  AB [by angle-bisector theorem] DC AC ab · x c ac and so, a  x  &x ax b bc bc

Now, 

28. +AMN  +MBC & MN < BC & 3AMN +3ABC. 

AM  MN a  MN  ac · & c & MN a  b AB BC ab

29. The diagonals of a rhombus intersect at right angles and bisect each other. In the figure, 40 42 OA  cm  20 cm, OB  cm  21 cm. 2 2 AB  OA 2  OB 2  20 2  21 2 cm  841 cm  29 cm. 30. (ii) Two rectangles are similar only if their corresponding sides are proportional. (iii) If two triangles are similar, then (a) their corresponding angles are equal (b) their corresponding sides are proportional (but not necessarily equal) (iv) Let D and E be the midpoints of sides AB and AC respectively of a 3ABC. Then, by the midpoint theorem, DE < BC & 3ADE +3ABC. 

AD  DE 1 DE 1 &  & DE  BC. 2 BC 2 AB BC

(v) We have Clearly, Note

AB  6  1 AC  8 · , DE 12 2 DF 9

AB AC and so 3ABC is not similar to 3DEF. ! DE DF

Here 3ABC +3DFE.

(vi) The polygon formed by joining the midpoints of the sides of a quadrilateral is a parallelogram (not necessarily a rhombus). (vii) The ratio of the perimeters of two similar triangles is the same as the ratio of their corresponding sides which is the same as the ratio of the corresponding medians.

Triangles

447

MULTIPLE-CHOICE QUESTIONS (MCQ) Choose the correct answer in each of the following questions: 1. A man goes 24 m due west and then 10 m due north. How far is he from the starting point? (a) 34 m

(b) 17 m

(c) 26 m

(d) 28 m

2. Two poles of height 13 m and 7 m respectively stand vertically on a plane ground at a distance of 8 m from each other. The distance between their tops is (a) 9 m

(b) 10 m

(c) 11 m

(d) 12 m

3. A vertical stick 1.8 m long casts a shadow 45 cm long on the ground. At the same time, what is the length of the shadow of a pole 6 m high? (a) 2.4 m

(b) 1.35 m

(c) 1.5 m

(d) 13.5 m

4. A vertical pole 6 m long casts a shadow of length 3.6 m on the ground. What is the height of a tower which casts a shadow of length 18 m at the same time? (a) 10.8 m

(b) 28.8 m

(c) 32.4 m

(d) 30 m

5. The shadow of a 5-m-long stick is 2 m long. At the same time the length of the shadow of a 12.5-m-high tree (in m) is [CBSE 2011] (a) 3.0

(b) 3.5

(c) 4.5

(d) 5.0

6. A ladder 25 m long just reaches the top of a building 24 m high from the ground. What is the distance of the foot of the ladder from the building? (a) 7 m

(b) 14 m

(c) 21 m

(d) 24.5 m

7. In the given figure, O is a point inside a 3MNP such that +MOP  90c, and If OM  16 cm OP  12 cm. MN  21 cm and +NMP  90c then NP  ? (a) 25 cm

(b) 29 cm

(c) 33 cm

(d) 35 cm

8. The hypotenuse of a right triangle is 25 cm. The other two sides are such that one is 5 cm longer than the other. The lengths of these sides are (a) 10 cm, 15 cm

(b) 15 cm, 20 cm

(c) 12 cm, 17 cm

(d) 13 cm, 18 cm

9. The height of an equilateral triangle having each side 12 cm, is (a) 6 2 cm

(b) 6 3 cm

(c) 3 6 cm

(d) 6 6 cm

448

Secondary School Mathematics for Class 10

10. 3ABC is an isosceles triangle with AB  AC  13 cm and the length of altitude from A on BC is 5 cm. Then, BC  ? (a) 12 cm (b) 16 cm (c) 18 cm (d) 24 cm 11. In a 3ABC it is given that AB  6 cm, AC  8 cm and AD is the bisector of +A. Then, BD : DC  ? (a) 3 : 4 (b) 9 : 16 (c) 4 : 3

(d)

3 :2

12. In a 3ABC it is given that AD is the internal bisector of +A. If BD  4 cm, DC  5 cm and AB  6 cm, then AC  ? (a) 4.5 cm (b) 8 cm (c) 9 cm (d) 7.5 cm 13. In a 3ABC, it is given that AD is the internal bisector of +A. If AB  10 cm, AC  14 cm and BC  6 cm, then CD  ? (a) 4.8 cm (b) 3.5 cm (c) 7 cm (d) 10.5 cm 14. In a triangle, the perpendicular from the vertex to the base bisects the base. The triangle is (a) right-angled (b) isosceles (c) scalene (d) obtuse-angled 15. In an equilateral triangle ABC, if AD = BC then which of the following is true? (a) 2AB 2  3AD 2 (b) 4AB 2  3AD 2 (c) 3AB 2  4AD 2

(d) 3AB 2  2AD 2

16. In a rhombus of side 10 cm, one of the diagonals is 12 cm long. The length of the second diagonal is (a) 20 cm (b) 18 cm (c) 16 cm (d) 22 cm 17. The lengths of the diagonals of a rhombus are 24 cm and 10 cm. The length of each side of the rhombus is (a) 12 cm (b) 13 cm (c) 14 cm (d) 17 cm

Triangles

449

18. If the diagonals of a quadrilateral divide each other proportionally then it is a (a) parallelogram

(b) trapezium

(c) rectangle

(d) square

19. In the given figure, ABCD is a trapezium whose diagonals AC and BD intersect at O such that OA  (3x  1) cm, OB  (2x  1) cm, OC  (5x  3) cm and OD  (6x  5) cm. Then, x? (a) 2

(b) 3

(c) 2.5

(d) 4

20. The line segments joining the midpoints of the adjacent sides of a quadrilateral form (a) a parallelogram

(b) a rectangle

(c) a square

(d) a rhombus

21. If the bisector of an angle of a triangle bisects the opposite side then the triangle is (a) scalene

(b) equilateral

(c) isosceles

(d) right-angled

22. In 3ABC it is given that +C  50c then +BAD  ?

AB  BD · If +B  70c and AC DC

(a) 30

(b) 40

(c) 45

(d) 50

23. In 3ABC, DE  BC so that AD  2.4 cm, AE  3.2 cm and EC  4.8 cm. Then, AB  ? (a) 3.6 cm (c) 6.4 cm

(b) 6 cm (d) 7.2 cm

24. In a 3ABC, if DE is drawn parallel to BC, cutting AB and AC at D and E respectively such that AB  7.2 cm, AC  6.4 cm and AD  4.5 cm. Then, AE  ? (a) 5.4 cm (c) 3.6 cm

(b) 4 cm (d) 3.2 cm

450

Secondary School Mathematics for Class 10

25. In 3ABC, DE  BC so that AD  (7x  4) cm, AE  (5x  2) cm, DB  (3x  4) cm and EC  3x cm. Then, we have (a) x  3 (b) x  5 (c) x  4 (d) x  2.5 26. In 3ABC, DE < BC such that

AD  3 · If AC  5.6 cm DB 5

then AE  ? (a) 4.2 cm

(b) 3.1 cm

(c) 2.8 cm

(d) 2.1 cm

27. 3ABC +3DEF and the perimeters of 3ABC and 3DEF are 30 cm and 18 cm respectively. If BC  9 cm then EF  ? (a) 6.3 cm

(b) 5.4 cm

(c) 7.2 cm

(d) 4.5 cm

28. 3ABC +3DEF such that AB  9.1 cm and DE  6.5 cm. If the perimeter of 3DEF is 25 cm, what is the perimeter of 3ABC? (a) 35 cm

(b) 28 cm

(c) 42 cm

(d) 40 cm

29. In 3ABC, it is given that AB  9 cm, BC  6 cm and CA  7.5 cm. Also, 3DEF is given such that EF  8 cm and 3DEF +3ABC. Then, perimeter of 3DEF is (a) 22.5 cm

(b) 25 cm

(c) 27 cm

(d) 30 cm

30. ABC and BDE are two equilateral triangles such that D is the midpoint of BC. Ratio of the areas of triangles ABC and BDE is (a) 1 : 2

(b) 2 : 1

(c) 1 : 4

(d) 4 : 1

31. It is given that 3ABC +3DFE. If +A  30c, +C  50c, AB  5 cm, AC  8 cm and DF  7.5 cm then which of the following is true? (a) DE  12 cm, +F  50c (b) DE  12 cm, +F  100c (c) EF  12 cm, +D  100c

(d) EF  12 cm, +D  30c

32. In the given figure, +BAC  90c and AD = BC. Then, (a) BC $ CD  BC 2 (b) AB $ AC  BC 2 (c) BD $ CD  AD 2 (d) AB $ AC  AD 2 33. In 3ABC, AB  6 3 cm, AC  12 cm and BC  6 cm. Then, +B is (a) 45c

(b) 60c

(c) 90c

(d) 120c

Triangles

34. In 3ABC and 3DEF, it is given that (a) +B  +E

451

AB  BC then DE FD

(b) +A  +D

(c) +B  +D (d) +A  +F 35. In 3DEF and 3PQR, it is given that +D  +Q and +R  +E, then which of the following is not true? EF  DF EF  DE DE  EF DE  DF (a) (b) (c) (d) PR PQ RP QR PQ RP QR PQ 36. If 3ABC +3EDF and 3ABC is not similar to 3DEF then which of the following is not true? (a) BC · EF  AC · FD (b) AB · EF  AC · DE (c) BC · DE  AB · EF (d) BC · DE  AB · FD 37. In 3ABC and 3DEF, it is given that +B  +E, +F  +C and AB  3DE, then the two triangles are (a) congruent but not similar (c) neither congruent nor similar 38. If in 3ABC and 3PQR, we have (a) 3PQR +3CAB (c) 3CBA +3PQR

(b) similar but not congruent (d) similar as well as congruent

AB  BC  CA then QR PR PQ (b) 3PQR +3ABC (d) 3BCA +3PQR

39. In the given figure, two line segments AC and BD intersect each other at the point P such that PA  6 cm, PB  3 cm, PC  2.5 cm, PD  5 cm, +APB  50c and +CDP  30c then +PBA  ? (a) 50

(b) 30

(c) 60

(d) 100

40. Corresponding sides of two similar triangles are in the ratio 4 : 9. Areas of these triangles are in the ratio (a) 2 : 3

(d) 16 : 81 ar (3PQR) BC  2 ? 41. It is given that 3ABC +3PQR and then QR 3 ar (3ABC) (a)

2 3

(b) 4 : 9

(b)

3 2

(c) 9 : 4

(c)

4 9

42. In an equilateral 3ABC, D is the midpoint of AB and E is the midpoint of AC. Then, ar (3ABC) : ar (3ADE)  ? (a) 2 : 1

(b) 4 : 1

(c) 1 : 2

(d) 1 : 4

(d)

9 4

452

Secondary School Mathematics for Class 10

43. In 3ABC and 3DEF, we have ar (3ABC) : ar (3DEF)  ? (a) 5 : 7

(b) 25 : 49

(a) 36 : 49

(b) 6 : 7

AB  BC  AC  5 , then DE EF DF 7

(c) 49 : 25 (d) 125 : 343 2 44. 3ABC +3DEF such that ar(3ABC)  36 cm and ar (3DEF)  49 cm 2 . Then, the ratio of their corresponding sides is (c) 7 : 6

(d)

6: 7

45. Two isosceles triangles have their corresponding angles equal and their areas are in the ratio 25 : 36. The ratio of their corresponding heights is (a) 25 : 36

(b) 36 : 25

(c) 5 : 6

(d) 6 : 5

46. The line segments joining the midpoints of the sides of a triangle form four triangles, each of which is (a) congruent to the original triangle (b) similar to the original triangle (c) an isosceles triangle (d) an equilateral triangle 47. If 3ABC +3QRP, PR  ? (a) 8 cm

ar (3ABC) 9  , AB  18 cm and BC  15 cm then ar (3PQR) 4 (b) 10 cm

(c) 12 cm

(d)

20 cm 3

48. In the given figure, O is the point of intersection of two chords AB and CD such that OB  OD and +AOC  45c. Then, 3OAC and 3ODB are (a) (b) (c) (d)

equilateral and similar equilateral but not similar isosceles and similar isosceles but not similar 49. In an isosceles 3ABC, if AC  BC and AB 2  2AC 2 then +C  ? (a) 30

(c) 60 (d) 90 50. In 3ABC, if AB  16 cm, BC  12 cm and AC  20 cm, then 3ABC is (a) acute-angled (c) obtuse-angled

(b) 45

(b) right-angled (d) not possible

True/False Type

51. Which of the following is a true statement? (a) Two similar triangles are always congruent. (b) Two figures are similar if they have the same shape and size.

Triangles

453

(c) Two triangles are similar if their corresponding sides are proportional. (d) Two polygons are similar if their corresponding sides are proportional. 52. Which of the following is a false statement? (a) If the areas of two similar triangles are equal then the triangles are congruent. (b) The ratio of the areas of two similar triangles is equal to the ratio of their corresponding sides. (c) The ratio of the areas of two similar triangles is equal to the ratio of squares of their corresponding medians. (d) The ratio of the areas of two similar triangles is equal to the ratio of squares of their corresponding altitudes. Matching of columns 53. Match the following columns: Column I

Column II

(a) In a given 3ABC, DE  BC and (p) 6 AD  3 · If AC  5.6 cm then DB 5 AE  …… cm. (b) If 3ABC +3DEF such that (q) 4 2AB  3DE and BC  6 cm then EF  …… cm. (c) If 3ABC +3PQR such that ar (3ABC) : ar (3PQR)  9 : 16 and BC  4.5 cm then QR  …… cm.

(r) 3

(d) In the given figure, AB  CD and (s) 2.1 OA  (2x  4) cm, OB  (9x  21) cm, OC  (2x  1) cm and OD  3 cm. Then x  ?

The correct answer is (a) —……, (b) —……,

(c) —……,

(d) —……

454

Secondary School Mathematics for Class 10

54. Match the following columns: Column I

Column II

(a) A man goes 10 m due east and then (p) 25 3 20 m due north. His distance from the starting point is …… m. (b) In an equilateral triangle with each (q) 5 3 side 10 cm, the altitude is …… cm. (c) The area of an equilateral triangle having each side 10 cm is …… cm2.

(r) 10 5

(d) The length of diagonal of a rectangle having length 8 m and breadth 6 m is …… m.

(s) 10

The correct answer is (a) —……,

(b) —……,

(c) —……,

(d) —…… ANSWERS (MCQ)

1. (c)

2. (b)

6. (a)

7. (b)

10. (d) 11. (a)

12. (d) 13. (b) 14. (b) 15. (c)

16. (c)

17. (b) 18. (b)

19. (a)

20. (a)

21. (c)

23. (b) 24. (b) 25. (c)

26. (d) 27. (b)

28. (a)

29. (d) 30. (d) 31. (b) 32. (c)

37. (b) 38. (a)

3. (c)

4. (d) 22. (a)

5. (d)

33. (c)

34. (c)

9. (b)

35. (b) 36. (c)

39. (d) 40. (d) 41. (d) 42. (b) 43. (b) 44. (b) 45. (c)

46. (b) 47. (b) 48. (c)

49. (d) 50. (b) 51. (c)

53. (a)–(s), (b)–(q), (c)–(p), (d)–(r)

52. (b)

54. (a)–(r), (b)–(q), (c)–(p), (d)–(s)

HINTS TO SOME SELECTED QUESTIONS 5. Let AB be the stick and AC be its shadow. Let DE be the tree and DF be its shadow. 3ABC +3DEF.

11.

8. (b)



5 2 AB  AC & DE DF 12.5 x



12.5 # 2  xa k 5. 5

BD  AB  6  3 DC AC 8 4

[by angle-bisector theorem]

Triangles 15. AB 2  BD 2  AD 2 2  a1 ABk  AD 2  1 AB 2  AD 2 2 4

3 AB 2  AD 2 & 3AB 2  4AD 2 . 4



19. We know that the diagonals of a trapezium divide each other proportionally. OA  OB 3x  1  2 x  1  & 5x  3 6x  5 OC OD  (3x  1)(6x  5)  (5x  3)(2x  1) 

18x 2  21x  5  10x 2  x  3 & 8x 2  20x  8  0



2x 2  5x  2  0 & 2x 2  4x  x  2  0 & 2x (x  2)  (x  2)  0



(x  2)(2x  1)  0 & x  2 or x 

But, x  

1 $ 2

1 gives (6x  5)  0 and the distance cannot be negative. 2

x  2.

21. Let AD be the bisector of +A of 3ABC such that BD  DC. AB  BD  1 & AB  AC. AC DC So, the given triangle is isosceles. Then,

22. +A  180c  (70c  50c)  60c. Since 

BD  AB , it means AD is the bisector of +A. DC AC

1 +BAD  a # 60ck  30c. 2

30. 3ABC +3BDE [£ both are equilateral]. 1 1 BC  AB [£ D is the midpoint of BC]. 2 2 2 ar (3ABC) AB 2   AB 2  4 · Now, 1 ar (3BDE) BD 2 1 a ABk 2 Also, BD 

31. +B  180c  (30c  50c)  100c. Since 3ABC +3DFE, we have +D  +A  30c, +F  +B  100c and +E  +C  50c. Let DE  x cm. Then, 5 8 AB  AC & DF DE 7.5 x 8 #7.5   5x  8 #7.5 & x  12. 5 Hence, DE  12 cm and +F  100c.

455

456

Secondary School Mathematics for Class 10

32. In 3DBA and 3DAC, we have +ADB  +CDA  90c, +ABD  +CAD  90c  +C and +BAD  +ACD  90c  +B. BD  AD  3DBA +3DAC & AD CD & BD $ CD  AD 2 . 33. In 3ABC, AC is the longest side. AB 2  BC 2  {(6 3 ) 2  6 2} cm 2  (108  36) cm 2  144 cm 2  (12 cm) 2  AC 2 . 

by the converse of Pythagoras’ theorem, we have +B  90c.

34. Clearly, B ) D, A ) E and C ) F 

+B  +D.

35. +D  +Q, +E  +R and +F  +P. DE  DF  EF · QR PQ RP



3DEF +3QRP &



DE  EF is not true. PQ RP

36. Since 3ABC +3EDF and 3ABC is not similar to 3DEF, so AB BC ! & BC $ DE  AB $ EF is not true. DE EF 37. 3ABC +3DEF

[by AA-similarity].

But 3ABC and 3DEF are not congruent since their corresponding sides AB and DE are not equal. 38.

AB  BC  CA & B ) R, A ) Q and C ) P. QR PR PQ 

3PQR +3CAB.

ar (3PQR) QR 2 QR 2 3 2 9 m a k  ·  c 2 2 4 BC ar (3ABC) BC AD  AE  1 42. Clearly, and +A  +A (common). AB AC 2 41.



3ABC +3ADE [by SAS-similarity].



ar (3ABC) : ar (3ADE)  a

44. a

AB 2  2 2  4  4 : 1. k a k 1 1 AD

AB 2  ar (3ABC)  36  6 2 AB  6 · a k & k 7 DE DE 7 ar (3DEF) 49



the ratio of the corresponding sides is 6 : 7.

45. The two triangles have corresponding angles equal and so they are similar. 

the ratio of their areas is equal to the ratio of the squares of their corresponding sides but the ratio of their corresponding sides is equal to the ratio of their corresponding altitudes (or heights).

Triangles

457

So, the ratio of their areas is equal to the ratio of the squares of their heights. 25  5 2 36 6 2 5  ratio of their heights  · 6 46. ar (3QRP)  ar (3PQR) . Now, ratio of their areas 

ar (3ABC) 9 3 2 BC  3   & [a 3ABC +3QRP] RP 2 ar (3QRP) 4 2 2 2 2 & PR  # BC  #15 cm  10 cm. 3 3 48. In 3OAC and 3ODB, we have 

+AOC  +DOB (ver. opp. O) and +OAC  +ODB (O in the same segment). OC  OA  AC  3OAC +3ODB & OB OD BD OC  OB   1 & OC  OA [a OB  OD (given)]. OA OD Clearly, OA ! OD. OA AC !1 & ! 1 & AC ! BD. BD OD

 

3OAC and 3ODB are isosceles and similar.

49. AB  2AC 2  AC 2  AC 2  BC 2  AC 2 [a AC  BC] . 2



by converse of Pythagoras’ theorem, we have +C  90c.

50. AC is the longest side of 3ABC. AB 2  BC 2  16 2  12 2  256  144  400  20 2  AC 2. 

+B  90c [by the converse of Pythagoras’ theorem].

51. (a) Two similar triangles need not be congruent. (b) Similar figures need not be of the same size. (c) It is clearly true. (d) Two polygons are similar only when their corresponding angles are equal and their corresponding sides are proportional. 53. (a) Let AE  x cm. Then, EC  (5.6  x) cm.

 

3 x AD  AE &  DB EC 5 5.6  x 3(5.6  x)  5x & 8x  3 # 5.6 3 # 5.6  16.8  x 2.1. 8 8

3 6 AB  BC &  x & 3x  12 & x  4. 2 DE EF ar (3ABC) BC 2 9  BC 2 32 BC 2  m (c) & &a k c 16 QR 2 4 QR ar (3PQR) QR 2

(b)



BC  3 4 4 & QR  # BC  a # 4.5k cm  6 cm. 3 3 QR 4

458

Secondary School Mathematics for Class 10 (d) 3OAB +3OCD &

OA  OB 2x  4  9x  21 & 3 2x  1 OC OD



6x  12  18x 2  51x  21 & 18x 2  57x  9  0



6x 2  19x  3  0 & (x  3)(6x  1)  0 & x  3 or x 

1· 6

1 makes (2x  1)  0. So, we reject it. 6 x  3.

But, x  

The correct answer is: (a)–(s), (b)–(q), (c)–(p), (d)–(r). 54. (a) Let OA  10 m and AB  20 m. Then, OB 2  OA 2  AB 2  {(10) 2  (20) 2} m 2  500 m 2 

OB  500 m  5 #100 m  10 5 m.

(b) Altitude  (c) Area 

3 3 a  d #10n cm  5 3 cm. 2 2

3 2 3 a  d #10 #10n cm 2  25 3 cm 2 . 4 4

(d) d 2  (8 2  6 2) m 2  (64  36) m 2  100 m 2 & d  100 m  10 m. Then, the correct answer is: (a)–(r), (b)–(q), (c)–(p), (d)–(s).

TEST YOURSELF MCQ 1. 3ABC +3DEF and their perimeters are 32 cm and 24 cm respectively. If AB  10 cm then DE  ? (a) 8 cm

(b) 7.5 cm

(c) 15 cm

(d) 5 3 cm

2. In the given figure, DE  BC. If DE  5 cm, BC  8 cm and AD  3.5 cm then AB  ? (a) 5.6 cm

(b) 4.8 cm

(c) 5.2 cm

(d) 6.4 cm

3. Two poles of height 6 m and 11 m stand vertically upright on a plane ground. If the distance between their feet is 12 m then the distance between their tops is (a) 12 m

(b) 13 m

(c) 14 m

(d) 15 m

cm2

4. The areas of two similar triangles are 25 and 36 cm2 respectively. If the altitude of the first triangle is 3.5 cm then the corresponding altitude of the other triangle is (a) 5.6 cm

(b) 6.3 cm

(c) 4.2 cm

(d) 7 cm

Triangles

459

Short-Answer Questions 5. If 3ABC +3DEF such that 2AB  DE and BC  6 cm, find EF. 6. In the given figure, DE  BC such that AD  x cm, DB  (3x  4) cm, AE  (x  3) cm and EC  (3x  19) cm. Find the value of x.

7. A ladder 10 m long reaches the window of a house 8 m above the ground. Find the distance of the foot of the ladder from the base of the wall. 8. Find the length of the altitude of an equilateral triangle of side 2a cm. 9. 3ABC +3DEF such that ar (3ABC)  64 cm 2 and ar (3DEF)  169 cm 2 . If BC  4 cm, find EF. 10. In a trapezium ABCD, it is given that AB  CD and AB  2CD. Its diagonals AC and BD intersect at the point O such that ar (3AOB)  84 cm 2 . Find ar (3COD) . 11. The corresponding sides of two similar triangles are in the ratio 2 : 3. If the area of the smaller triangle is 48 cm2, find the area of the larger triangle. 12. In the given figure, LM  CB and LN  CD. Prove that

AM  AN · AB AD

13. Prove that the internal bisector of an angle of a triangle divides the opposite side internally in the ratio of the sides containing the angle. 14. In an equilateral triangle with side a, prove that area 

3 2 a . 4

15. Find the length of each side of a rhombus whose diagonals are 24 cm and 10 cm long. 16. Prove that the ratio of the perimeters of two similar triangles is the same as the ratio of their corresponding sides.

460

Secondary School Mathematics for Class 10

Long-Answer Questions 17. In the given figure, 3ABC and 3DBC have the same base BC. If AD and BC intersect at ar (3ABC) AO ·  O, prove that ar (3DBC) DO

18. In the given figure, XY  AC and XY divides 3ABC into two regions, equal in area. Show that

AX  (2  2 ) · 2 AB

19. In the given figure, 3ABC is an obtuse triangle, obtuse-angled at B. If AD = CB (produced) prove that AC 2  AB 2  BC 2  2BC · BD.

20. In the given figure, each one of PA, QB and RC is perpendicular to AC. If AP  x, QB  z, RC  y, AB  a 1  1  1· x y z

and

BC  b,

show

that

ANSWERS (TEST YOURSELF)

1. (b) 8.

2. (a)

3 a cm

3. (b)

4. (c)

9. 6.5 cm

5. 12 cm

6. x  2

10. 21 cm2 11. 108 cm2



7. 6 m 15. 13 cm

Circles

461

Circles

8

A circle is a collection of all points in a plane which are at a constant distance from a fixed point. The constant distance is called the radius and the fixed point is called the centre of the circle.

CIRCLE

SECANT A line which intersects a circle in two distinct points is called a secant to the circle.

A line meeting a circle only in one point is called a tangent to the circle at that point.

TANGENT

The point at which the tangent line intersects the circle is called the point of contact. NOTE

The tangent to a circle is a special case of the secant, when the two end points of its corresponding chord coincide.

Consider a line l which is a secant to a circle with centre O. Let PQ be the corresponding chord of this secant. If we rotate the secant anticlockwise about the point P then point Q comes closer to the point P. Let Q1, Q2, Q3 etc., be the positions of Q as the secant goes on rotating then we find these positions getting closer to P sequentially. Finally, as the position of Q coincides with P, the secant is reduced to a tangent at point P. NUMBER OF TANGENTS TO A CIRCLE

(i) There is no tangent passing through a point lying inside the circle. (ii) There is one and only one tangent passing through a point lying on a circle. (iii) There are exactly two tangents through a point lying outside a circle. In the figure, PT and PTl are two tangents from a point P lying outside the circle. 461

462

Secondary School Mathematics for Class 10

LENGTH OF A TANGENT The length of the segment of the tangent from the external point P to the point of contact with the circle is called the length of the tangent from the point P to the circle.

RESULTS ON TANGENTS

The tangent at any point of a circle is perpendicular to the radius through the point of contact. [CBSE 2007, ’09, ’11, ’12, ’13, ’14, ’15]

THEOREM 1

GIVEN

A circle with centre O and a tangent AB at a point P of the circle. OP = AB.

TO PROVE

CONSTRUCTION

Take a point Q, other than P, on AB. Join OQ.

Q is a point on the tangent AB, other than the point of contact P.

PROOF



Q lies outside the circle.

Let OQ intersect the circle at R. Then, OR  OQ But, OP  OR

[a part is less than the whole].

… (i)

[radii of the same circle].

… (ii)



[from (i) and (ii)].

OP  OQ

Thus, OP is shorter than any other line segment joining O to any point of AB, other than P. In other words, OP is the shortest distance between the point O and the line AB. But, the shortest distance between a point and a line is the perpendicular distance. 

OP = AB.

(i) From this theorem we also conclude that at any point on a circle, one and only one tangent can be drawn to the circle.

REMARKS

(ii) The line containing the radius through the point of contact is called the ’normal‘ to the circle at the point of contact. THEOREM 2

GIVEN

(Converse of Theorem 1) A line drawn through the end of a radius and perpendicular to it is a tangent to the circle.

A circle with centre O in which OP is a radius and AB is a line through P such that OP = AB.

TO PROVE

AB is a tangent to the circle at the point P.

Circles CONSTRUCTION PROOF

463

Take a point Q, different from P, on AB. Join OQ.

We know that the perpendicular distance from a point to a line is the shortest distance between them. 

OP = AB



OP  OQ.



Q lies outside the circle [a OP is the radius and OP  OQ].

& OP is the shortest distance from O to AB.

Thus, every point on AB, other than P, lies outside the circle. 

AB meets the circle at the point P only.

Hence, AB is the tangent to the circle at the point P. REMARK

This theorem gives us a method of constructing a tangent at a point P on the circle. We first draw the radius OP and then draw PT = OP. Then, PT is the tangent at P.

THEOREM 3

The lengths of tangents drawn from an external point to a circle are equal. [CBSE 2007, ’08, ’08C, ’09, ’09C, ’10, ’11, ’12, ’13, ’13C, ’14, ’15, ’17]

Two tangents AP and AQ are drawn from a point A to a circle with centre O. TO PROVE AP  AQ.

GIVEN

CONSTRUCTION PROOF

Join OP, OQ and OA.

AP is a tangent at P and OP is the radius through P. 

OP = AP.

Similarly, OQ = AQ. In the right AOPA and OQA, we have: [radii of the same circle] OP  OQ OA  OA

[common]



3OPA ,3OQA [by RHS-congruence]. Hence, AP  AQ [cpct]. REMARK

The above theorem can also be proved by using the Pythagoras’ theorem: AP 2  OA 2  OP 2  OA 2  OQ 2  AQ 2 [a OP  OQ  radius] 

THEOREM 4

AP  AQ. If two tangents are drawn from an external point then (i) they subtend equal angles at the centre, and (ii) they are equally inclined to the line segment joining the centre to that point.

464

Secondary School Mathematics for Class 10

A circle with centre O and a point A outside it. Also, AP and AQ are the two tangents to the circle. TO PROVE +AOP  +AOQ and +OAP  +OAQ. GIVEN

PROOF

In 3AOP and 3AOQ, we have [tangents from an external point are equal] AP  AQ OP  OQ OA  OA 

REMARK

[radii of the same circle] [common]

3AOP ,3AOQ [by SSS-congruence].

Hence, +AOP  +AOQ and +OAP  +OAQ. Since +OAP  +OAQ, i.e., AO is the bisector of +PAQ, so the centre lies on the bisector of the angle between the two tangents.

THEOREM 5

Prove that in two concentric circles, the chord of the larger circle, which touches the smaller circle, is bisected at the point of contact. [CBSE 2009, ’12]

Two circles with the same centre O and AB is a chord of the larger circle touching the smaller circle at C. TO PROVE AC  BC. GIVEN

CONSTRUCTION PROOF

Join OC.

AB is a tangent to the smaller circle at the point C and OC is the radius through C. 

OC = AB.

But, the perpendicular drawn from the centre of a circle to a chord bisects the chord. 

OC bisects AB Hence, AC  BC.

THEOREM 6

GIVEN

Prove that the tangents drawn at the ends of a diameter of a circle are parallel. [CBSE 2011, ’12, ’14, ’17]

CD and EF are the tangents at the end points A and B of the diameter AB of a circle with centre O.

TO PROVE PROOF

[a AB is a chord of larger circle].

CD  EF.

CD is the tangent to the circle at the point A.

Circles

465

 CD = OA & +OAD  90c & +BAD  90c. EF is the tangent to the circle at the point B.  EF = OB & +OBE  90c & +ABE  90c. Thus, +BAD  +ABE (each equal to 90). But these are alternate interior angles.  THEOREM 7

GIVEN

CD  EF. Prove that the line segment joining the points of contact of two parallel tangents to a circle is a diameter of the circle.

CD and EF are two parallel tangents at the points A and B of a circle with centre O.

TO PROVE

AOB is a diameter of the circle.

CONSTRUCTION PROOF

Join OA and OB. Draw OG  CD.

OG  CD and AO cuts them.  +CAO  +GOA  180c  +90c  +GOA  180c [a OA = CD]  +GOA  90c. Similarly, +GOB  90c. 

+GOA  +GOB  90c  90c  180c



AOB is a straight line.

Hence, AOB is a diameter of the circle with centre O. THEOREM 8

Prove that the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line segments joining the points of contact to the centre. [CBSE 2008C, ’14]

PA and PB are the tangents drawn from a point P to a circle with centre O. Also, the line segments OA and OB are drawn. TO PROVE +APB  +AOB  180c. GIVEN

PROOF

We know that the tangent to a circle is perpendicular to the radius through the point of contact.  PA = OA & +OAP  90c, and 

PB = OB & +OBP  90c. +OAP  +OBP  90c  90c  180c.

… (i)

But, we know that the sum of all the angles of a quadrilateral is 360.

466

Secondary School Mathematics for Class 10



+OAP  +OBP  +APB  +AOB  360c.

… (ii)

From (i) and (ii), we get +APB  +AOB  180c. THEOREM 9

PROOF

Prove that there is one and only one tangent at any point on the circumference of a circle.

Let P be a point on the circumference of a circle with centre O. If possible, let PT and PTl be two tangents at a point P of the circle. Now, the tangent at any point of a circle is perpendicular to the radius through the point of contact.  

OP = PT and similarly, OP = PTl +OPT  90c and +OPTl  90c.



+OPT  +OPTl

This is possible only when PT and PTl coincide. Hence, there is one and only one tangent at any point on the circumference of a circle. THEOREM 10

PROOF

Prove that the perpendicular at the point of contact of the tangent to a circle passes through the centre.

Let PT be a tangent to a circle with centre O, where P is the point of contact. Let PQ = PT, where Q lies on the circle, i.e., +QPT  90c. If possible, let PQ not pass through the centre O. Join PO and produce it to meet the circle at R. Then PO being the radius through the point of contact, we have



PO = PT [a the tangent is perpendicular to the radius through the point of contact] +OPT  90c & +RPT  90c.

Thus, we have +QPT  +RPT  90c. This is possible only if P, Q and R are collinear. But a straight line cuts a circle in at the most two points. So, the points Q and R coincide. Hence, PQ passes through the centre O, i.e., the perpendicular at the point of contact to the tangent passes through the centre.

Circles

467

SOLVED EXAMPLES EXAMPLE 1

From a point P, 10 cm away from the centre of a circle, a tangent PT of length 8 cm is drawn. Find the radius of the circle. [CBSE 2002]

SOLUTION

Let O be the centre of the given circle. Then, OP  10 cm. Also, PT  8 cm. Join OT. Now, PT is a tangent at T and OT is the radius through the point of contact T. 

OT = PT.

In the right 3OTP, we have OP 2  OT 2  PT 2

[by Pythagoras’ theorem]

 OT  OP  PT  (10) 2  (8) 2 cm  36 cm  6 cm. 2

2

Hence, the radius of the circle is 6 cm. EXAMPLE 2

SOLUTION

A tangent PQ at a point P of a circle of radius 5 cm meets a line through the centre O at a point Q so that OQ  13 cm. Find the length of PQ. [CBSE 2010] We have OP  radius = 5 cm, OQ  13 cm. PQ is a tangent at P and OP is the radius through the point of contact P.  OP = PQ. In right 3 OPQ, we have OQ 2  OP 2  PQ 2 

EXAMPLE 3

SOLUTION

[by Pythagoras’ theorem]

PQ  OQ  OP  13 2  5 2 cm  144 cm  12 cm. 2

2

In the given figure, AB is the diameter of a circle with centre O and AT is a tangent. If +AOQ  58c, find +ATQ. [CBSE 2015] +AOQ  58c  +ABQ 

1 +AOQ  29c 2 [a the angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle]

468

Secondary School Mathematics for Class 10

 +ABT  29c. Now, AT is a tangent at A and OA is the radius through the point of contact A. 

OA = AT, i.e., +OAT  90c & +BAT  90c.

In 3BAT, we have +BAT  +ABT  +ATB  180c 

90c  29c  +ATB  180c & +ATB  61c.

 +ATQ  +ATB  61c. EXAMPLE 4

Tangents PA and PB are drawn from an external point P to two concentric circles with centre O and radii 8 cm and 5 cm respectively, as shown in the figure. If AP  15 cm then find the length of BP.

SOLUTION

We have OA = AP and OB = BP [a the tangent at any point of a circle is perpendicular to the radius through the point of contact]. Join OP.

[CBSE 2012]

In right 3OAP, we have OA  8 cm, AP  15 cm. 

OP 2  OA 2  AP 2 [by Pythagoras’ theorem]

 OP  OA 2  AP 2  8 2  15 2 cm  289 cm  17 cm. In right 3OBP, we have OB  5 cm, OP  17 cm. OP 2  OB 2  BP 2 [by Pythagoras’ theorem]  BP  OP 2  OB 2  17 2  5 2 cm  264 cm. Thus, the length of BP  264 cm  16.25 cm (approx).



EXAMPLE 5

In the given figure, two circles touch each other at the point C. Prove that the common tangent to the circles at C, bisects the common tangent at P and Q. [CBSE 2013]

Circles SOLUTION

469

In the given figure, PR and CR are both tangents drawn to the same circle from an external point R. 

PR  CR.

… (i)

Also, QR and CR are both tangents drawn to the same circle (second circle) from an external point R. 

QR  CR.

… (ii)

From (i) and (ii), we get PR  QR [each equal to CR]. 

R is the midpoint of PQ,

i.e., the common tangent to the circles at C, bisects the common tangent at P and Q. EXAMPLE 6

Two concentric circles of radii a and b (a  b) are given. Find the length of the chord of the larger circle which touches the smaller circle. [CBSE 2015]

SOLUTION

Let O be the common centre of the two circles and AB be the chord of the larger circle which touches the smaller circle at C. Join OA and OC. Then, OA  a and OC  b. Now, OC = AB and OC bisects AB [a the chord of the larger circle touching the smaller circle, is bisected at the point of contact]. In right 3ACO, we have OA 2  OC 2  AC 2

[by Pythagoras’ theorem]



AC  OA  OC  a 2  b 2 .



AB  2AC  2 a 2  b 2

2

2

[a C is the midpoint of AB]

i.e., required length of the chord AB  2 a 2  b 2 . EXAMPLE 7

Two concentric circles are of radii 7 cm and r cm respectively, where r  7. A chord of the larger circle of length 46 cm, touches the smaller circle. Find the value of r. [CBSE 2011]

SOLUTION

Let O be the common centre of the two circles and AB be the chord of the larger circle which touches the smaller circle at C. Then, AB  46 cm.

470

Secondary School Mathematics for Class 10

Join OA and OC. Then, OA  r cm and OC  7 cm. Now, OC = AB and OC bisects AB.

[See Theorem 5.]

In right 3ACO, we have OA 2  OC 2  AC 2

[by Pythagoras’ theorem]

 OA  OC 2  AC 2 

2 1 OC 2  a ABk 2 [a C is the midpoint of AB]

 r cm  7 2  23 2 cm  578 cm  17 2 & r  17 2 cm. EXAMPLE 8

SOLUTION

In the given figure, the radii of two concentric circles are 13 cm and 8 cm. AB is a diameter of the bigger circle and BD is a tangent to the smaller circle touching it at D. Find the length of AD. [HOTS] [CBSE 2014] We have +AEB  90c [angle in a semicircle]. Also, OD = BE and OD bisects BE.

[See Theorem 5.]

In right 3OBD, we have OB 2  OD 2  BD 2

[by Pythagoras’ theorem]

 BD  OB 2  OD 2  13 2  8 2 cm  105 cm [a OB  13 cm, OD  8 cm].  BE  2BD  2 105 cm In right 3AEB, we have AB 2  AE 2  BE 2 

[a D is the midpoint of BE].

[by Pythagoras’ theorem]

AE  AB  BE  26 2  (2 105 ) 2 cm  256 cm  16 cm 2

2

[a AB  diameter  2 #OB  2 #13 cm  26 cm] . In right 3AED, we have AD 2  AE 2  DE 2 

[by Pythagoras’ theorem]

AD  AE  DE  16 2  ( 105 ) 2 cm 2

2

 19 cm [a DE  BD  105 cm] . [Note We can also find AE by using midpoint theorem, since in 3ABE, O is the midpoint of AB and D is the midpoint of BE and so OD  AE and AE  2 #OD  16 cm.] EXAMPLE 9

From a point P outside a circle with centre O, tangents PA and PB are drawn to the circle. Prove that OP is the right bisector of the line segment AB. [CBSE 2015]

Circles SOLUTION

471

GIVEN PA and PB are tangents to a circle with centre O from an external point P. TO PROVE

OP is the right bisector

of AB. Join AB. Let AB intersect OP at M. CONSTRUCTION

In 3MAP and 3MBP, we have PA  PB [a tangents to a circle from an external point are equal]  MP MP [common]

PROOF

+MPA  +MPB [a tangents from an external point are equally inclined to the line segment joining the centre to that point, i.e., +OPA  +OPB]  3MAP ,3MBP [by SAS-congruence]. And so, MA  MB [cpct] and +AMP  +BMP But, +AMP  +BMP  180c

[cpct]. [linear pair]

 +AMP  +BMP  90c. Hence, OP is the right bisector of AB. EXAMPLE 10

Prove that the tangents at the extremities of any chord of a circle, make equal angles with the chord. [CBSE 2014, ’17]

SOLUTION

AB is any chord of a circle with centre O. Tangents at the extremities A and B of this chord meet at an external point P. Chord AB intersects the line segment OP at M. GIVEN

TO PROVE PROOF

+MAP  +MBP.

In 3MAP and 3MBP, we have

PA  PB [a tangents from an external point are equal] MP  MP [common] +MPA  +MPB [a +OPA  +OPB since tangents from an external point are equally inclined to the line segment joining the point to the centre]

472

Secondary School Mathematics for Class 10

 3MAP , 3MBP [by SAS-congruence]. And so, +MAP  +MBP [cpct]. EXAMPLE 11

SOLUTION

Prove that the tangent drawn at the midpoint of an arc of a circle is parallel to the chord joining the end points of the arc. [CBSE 2015] % GIVEN Point P is the midpoint of arc QR of a circle with centre O. ST is the tangent to the circle at point P. TO PROVE

Chord QR  ST.

% P is the midpoint of QR % %  QP  PR  chord QP  chord PR [a in a circle, if two arcs are equal, then their corresponding chords are equal] PROOF

 +PQR  +PRQ  +TPR  +PRQ [Note +PQR  +TPR, angles in alternate segments]  QR  ST. [a +TPR and +PRQ are alternate int. O] EXAMPLE 12

In the given figure, AB is a chord of length 9.6 cm of a circle with centre O and radius 6 cm. The tangents at A and B intersect at P. Find the length of PA. [CBSE 2009C, ’13C, ’15]

SOLUTION

A circle with centre O and radius 6 cm. AB is a chord of length 9.6 cm. The tangents at A and B intersect at P. GIVEN

TO FIND

The length PA.

CONSTRUCTION

Join OP and OA. Let OP and AB intersect at M.

Let PA  x cm and PM  y cm. Now, PA  PB [a tangents from an external point are equal] and OP is the bisector of +APB [a two tangents to a circle from an external point are equally inclined to the line segment joining the centre to that point]. Also, OP = AB and OP bisects AB at M [a OP is the right bisector of AB].

Circles



AM  MB 

473

9.6 cm  4.8 cm. 2

In right 3AMO, we have OA  6 cm and AM  4.8 cm. 

OM  OA 2  AM 2  6 2  4.8 2  12.96  3.6 cm.

In right 3AMP, we have AP 2  PM 2  AM 2 & x 2  y 2  (4.8) 2 

x 2  y 2  23.04.

… (i)

In right 3PAO, we have OP 2  PA 2  OA 2

[Note +PAO  90c, since AO is the radius at the point of contact]

 (y  3.6) 2  x 2  6 2

[a OP  PM  MO  (y  3.6) cm]



y  7.2y  12.96  x  36 & 7.2y  46.08



y  6.4 cm.

2

2

[using (i)]

Putting this value of y in (i), we get x 2  (6.4) 2  23.04  40.96  23.04  64  x  64  8. 

PA  8 cm.

EXAMPLE 13

Two tangents PA and PB are drawn to a circle with centre O from an external point P. Prove that +APB  2+OAB. [CBSE 2009, ’14]

SOLUTION

GIVEN

A circle with centre O and PA, PB are the tangents on it from a point P outside it.

TO PROVE PROOF

+APB  2+OAB.

Let +APB  xc.

We know that the tangents to a circle from an external point are equal. So, PA  PB. Since the angles opposite to the equal sides of a triangle are equal, so PA  PB & +PBA  +PAB. Also, the sum of the angles of a triangle is 180.  +APB  +PAB  +PBA  180c 

xc  2+PAB  180c

[a +PBA  +PAB]

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Secondary School Mathematics for Class 10

 +PAB 

1 1 (180c  xc)  a90c  xck · 2 2

But, PA is a tangent and OA is the radius of the given circle.  +OAB  +PAB  90c 1 1 1  +OAB  90c  a90c  xck & +OAB  xc  +APB 2 2 2  +APB  2+OAB. EXAMPLE 14

In the given figure, the incircle of 3ABC touches the sides BC, CA and AB at P, Q and R respectively. Prove that (AR  BP  CQ)  (AQ  BR  CP)  1 (perimeter of 3ABC). 2

SOLUTION

We know that the lengths of tangents from an exterior point to a circle are equal. 



AR  AQ,

… (i)

[tangents from A]

BP  BR,

… (ii)

[tangents from B]

CQ  CP.

… (iii)

[tangents from C]

(AR  BP  CQ)  (AQ  BR  CP)  k (say).

Perimeter of 3ABC  (AB  BC  CA)  (AR  BR)  (BP  CP)  (CQ  AQ)  (AR  BP  CQ)  (AQ  BR  CP)  (k  k)  2k 1 (perimeter of 3ABC) . 2



k



(AR  BP  CQ)  (AQ  BR  CP)  1 (perimeter of 3ABC) . 2

EXAMPLE 15

In the given figure, a circle is inscribed in a triangle PQR. If PQ  10 cm, QR  8 cm and PR  12 cm, find the lengths of QM, RN and PL. [CBSE 2012]

Circles SOLUTION

475

We know that the lengths of the tangents drawn from an external point to a circle are equal. Let PL  PN  x; QL  QM  y; RM  RN  z. Now, PL  QL  PQ & x  y  10,

… (i)

QM  RM  QR & y  z  8,

… (ii)

RN  PN  PR & z  x  12.

… (iii)

Subtracting (ii) from (iii), we get x  y  4.

… (iv)

Solving (i) and (iv), we get x  7, y  3. Substituting y  3 in (ii), we get z  5. 

QM  y  3 cm, RN  z  5 cm, PL  x  7 cm.

EXAMPLE 16

A circle is inscribed in a 3ABC, touching BC, CA and AB at P, Q and R respectively, as shown in the given figure. If AB  10 cm, AQ  7 cm and CQ  5 cm then find the length of BC. [CBSE 2009C]

SOLUTION

We know that the lengths of tangents drawn from an external point to a circle are equal. 

AR  AQ  7 cm. BR  (AB  AR)  (10  7) cm  3 cm.



BP  BR  3 cm, CP  CQ  5 cm.

 EXAMPLE 17

BC  (BP  CP)  (3  5) cm  8 cm.

A circle is touching the side BC of 3ABC at P and touching AB and AC produced at Q and R respectively. Prove that 1 AQ  (perimeter of 3ABC). 2 [CBSE 2001, ’02, ’06, ’14, ’17]

SOLUTION

We know that the lengths of tangents drawn from an external point to a circle are equal. 

AQ  AR,

… (i)

[tangents from A]

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Secondary School Mathematics for Class 10

BP  BQ,

… (ii) [tangents from B]

CP  CR.

… (iii) [tangents from C]

Perimeter of 3ABC  AB  BC  AC  AB  BP  CP  AC  AB  BQ  CR  AC [using (ii) and (iii)]  AQ  AR  2AQ [using (i)]. 

AQ 

1 (perimeter of 3ABC). 2

EXAMPLE 18

PA and PB are tangents to the circle with centre O from an external point P, touching the circle at A and B respectively. Show that the quadrilateral AOBP is cyclic. [CBSE 2014]

SOLUTION

GIVEN PA and PB are tangents to the circle with centre O from an external point P. TO PROVE

Quadrilateral AOBP is

cyclic. We know that the tangent at any point of a circle is perpendicular to radius through the point of contact. PROOF



PA = OA, i.e., +OAP  90c

… (i)

and PB = OB, i.e., +OBP  90c

… (ii)

Now, the sum of all the angles of a quadrilateral is 360.  +AOB  +OAP  +APB  +OBP  360c  +AOB  +APB  180c [using (i) and (ii)] 

EXAMPLE 19

quadrilateral OAPB is cyclic

In the given figure, tangents PQ and PR are drawn from an external point P to a circle with centre O, such that +RPQ  30c. A chord RS is drawn parallel to the tangent PQ. Find +RQS.

[since both pairs of opposite angles have the sum 180.].

Circles SOLUTION

477

Join OQ and OR. Also, produce PQ and PR to M and N respectively. We know that the angle between two tangents from an external point is supplementary to the angle subtended by the radii at the points of contact.  +RPQ  +ROQ  180c  +ROQ  180c  +RPQ  180c  30c  150c. 1 1 +ROQ  #150c  75c 2 2 [a the angle subtended by an arc at the centre is twice the angle subtended on the remaining part of the circle].

Now, +RSQ 

 +SQM  +RSQ  75c

[alternate int. O; since RS  PQ]

Also, +PQR  +RSQ  75c [angles in alternate segments]. Now, +SQM  +RQS  +PQR  180c [angles on a straight line]  +RQS  180c  (+SQM  +PQR)  180c  (75c  75c)  30c. EXAMPLE 20

In the given figure, the sides AB, BC and CA of a triangle ABC touch a circle with centre O and radius r at P, Q and R respectively. Prove that (a) AB  CQ  AC  BQ (b) area (3ABC) 

SOLUTION

1 (perimeter of 3ABC)# r. 2

[CBSE 2013]

We know that the lengths of tangents from an exterior point to a circle are equal. 

AP  AR,

… (i)

[tangents from A]

BP  BQ,

… (ii)

[tangents from B]

CQ  CR.

… (iii)

[tangents from C]

(a) AB  CQ  AP  BP  CQ  AR  BQ  CR

[using (i), (ii) and (iii)]

 (AR  CR)  BQ  AC  BQ.

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Secondary School Mathematics for Class 10

(b) Join OA, OB and OC. Area (3ABC)  area (3OAB)  area (3OBC)  area (3OCA)  a1 # AB#OPk 2  a1 # BC #OQk 2 1  a #CA#OR)k 2 1  a # AB# rk  a1 # BC # rk  a1 #CA# rk 2 2 2 1  (AB  BC  CA)# r 2 1  (perimeter of 3ABC)# r. 2 EXAMPLE 21

In the given figure, ABC is a right-angled triangle with AB  6 cm and AC  8 cm. A circle with centre O has been inscribed inside the triangle. Calculate the value of r, the radius of the inscribed circle. [CBSE 2006C, ’13]

SOLUTION

Join OA, OB and OC. Draw OD = AB, OE = BC and OF = CA. Then, OD  OE  OF  r cm. 

ar (3ABC) 

1 1 # AB# AC  a # 6 # 8k cm 2  24 cm 2 . 2 2

Now, ar (3ABC) 

1 #(perimeter of 3ABC)# r 2



24 

1 #(AB  BC  CA)# r 2



24 

1 #(6  10  8)# r & r  2 2 [a BC 2  AB 2  AC 2 & BC  6 2  8 2  10] .

Hence, the radius of the inscribed circle is 2 cm.

Circles

479

EXAMPLE 22

A triangle ABC is drawn to circumscribe a circle of radius 4 cm such that the segments BD and DC into which BC is divided by the point of contact D are of lengths 6 cm and 8 cm respectively. Find the lengths of the sides AB and AC. [HOTS] [CBSE 2014]

SOLUTION

We know that the lengths of tangents drawn from an exterior point to a circle are equal.  AE  AF  x cm (say), BD  BF  6 cm, CD  CE  8 cm. And so, AB  AF  BF  (x  6) cm, BC  BD  CD  14 cm, AC  CE  AE  (x  8) cm. Perimeter, 2s  AB  BC  AC  [(x  6)  14  (x  8)] cm  (2x  28) cm  s  (x  14) cm. 

ar(3ABC)  s(s  AB)(s  BC)(s  AC)

 (x  14){(x  14)  (x  6)}{(x  14)  14}{(x  14)  (x  8)} cm 2  48x (x  14) cm 2. Join OE and OF and also OA, OB and OC.  ar (3ABC)  ar (3OAB)  ar (3OBC)  ar (3OCA)

… (i)

 a1 # AB#OFk 2  a1 # BC #ODk 2 1  a # AC #OEk 2 1  : #(x  6)# 4D  :1 #14 # 4D  :1 #(x  8)# 4D 2 2 2  2 [(x  6)  14  (x  8)]  4 (x  14) cm 2. … (ii) From (i) and (ii), we get 48x (x  14)  4(x  14) 48x (x  14)  16 (x  14) 2 [on squaring both sides] 16 #14   48x  16 (x  14) & x  7. 32  AB  (x  6) cm  (7  6) cm  13 cm and AC  (x  8) cm  (7  8) cm  15 cm. 

480 EXAMPLE 23

Secondary School Mathematics for Class 10

In the given figure, a triangle ABC is drawn to circumscribe a circle of radius 3 cm, such that the segments BD and DC into which BC is divided by the point of contact D are of lengths 6 cm and 8 cm respectively. Find the side AB, if the area of 3ABC is 63 cm 2 . [CBSE 2010, ’11, ’15]

SOLUTION

We know that the lengths of tangents drawn from an exterior point to a circle are equal. AE  AF  x cm (say); BD  BF  6 cm; CD  CE  8 cm. And so, AB  AF  BF  (x  6) cm; BC  BD  CD  14 cm; CA  CE  AE  (x  8) cm.



Join OE and OF and also OA, OB and OC. ar (3ABC)  ar (3OAB)  ar (3OBC)  ar (3OCA) 1 1 1  63  a # AB#OFk  a # BC #ODk  a #CA#OEk 2 2 2 1 1 1  63  & #(x  6)# 30  a #14 # 3k  & #(x  8)# 30 2 2 2 3  63  #(2x  28) & x  7. 2  AB  (x  6) cm  (7  6) cm  13 cm.



EXAMPLE 24

In the given figure, XP and XQ are two tangents to the circle with centre O, drawn from an external point X. ARB is another tangent, touching the circle at R. Prove that [CBSE 2014] XA  AR  XB  BR.

SOLUTION

We know that the lengths of tangents drawn from an exterior point to a circle are equal. 

XP  XQ, AP  AR, BR  BQ .

… (i) … (ii) … (iii)

[tangents from X] [tangents from A] [tangents from B]

Circles

481

Now, XP  XQ & XA  AP  XB  BQ  EXAMPLE 25

XA  AR  XB  BR [using (ii) and (iii)].

If from an external point P of a circle with centre O, two tangents PQ and PR are drawn such that +QPR  120c, prove that 2PQ  PO. [HOTS] [CBSE 2014]

SOLUTION

In 3OPQ, we have +PQO  90c [a the tangent at any point is perpendicular to the radius through the point of contact] and +QPO 

1 1 #+QPR  #120c  60c. 2 2 [a the two tangents drawn from an external point are equally inclined to the line segment joining the centre to that point and so +QPO  +RPO]

In right 3OPQ, we have PQ cos (+QPO)  PO PQ PQ  cos 60c  & 12  PO & 2PQ  PO. PO EXAMPLE 26

A quadrilateral ABCD is drawn to circumscribe a circle, as shown in the figure. Prove that AB  CD  AD  BC. [CBSE 2008, ’08C, ’09, ’12, ’13, ’14, ’17]

SOLUTION

We know that the lengths of tangents drawn from an exterior point to a circle are equal. … (i) [tangents from A] AP  AS, … (ii) [tangents from B] BP  BQ, … (iii) [tangents from C] CR  CQ, … (iv) [tangents from D] DR  DS.  AB  CD  (AP  BP)  (CR  DR)  (AS  BQ)  (CQ  DS) [using (i), (ii), (iii), (iv)]  (AS  DS)  (BQ  CQ)  AD  BC. Hence, AB  CD  AD  BC.



482 EXAMPLE 27

SOLUTION

Secondary School Mathematics for Class 10

In the given figure, ABCD is a quadrilateral such that +D  90c. A circle with centre O and radius r, touches the sides AB, BC, CD and DA at P, Q, R and S respectively. If BC  40 cm, CD  25 cm and BP  28 cm, find r. It is given that +D  90c. Also,+ORD  +OSD  90c. [a tangent at a point is perpendicular to the radius through the point of contact]  +ROS  180c  +D  90c. [a angle between the tangents from an external point is supplementary to the angle subtended by the line segments joining the points of contact to the centre] And, OR  OS  r. 

ROSD is a square and so OR  DR, i.e., r  DR.

… (i)

We know that the lengths of tangents drawn from an exterior point to a circle are equal. 

BP  BQ and CQ  CR.

Now, CQ  BC  BQ  BC  BP  (40  28) cm  12 cm. And so, DR  CD  CR  CD  CQ  (25  12) cm  13 cm.  EXAMPLE 28

r  DR  13 cm

[using (i)].

Prove that the parallelogram circumscribing a circle is a rhombus. [CBSE 2008, ’09, ’10, ’12, ’13, ’14]

SOLUTION

A parallelogram ABCD circumscribes a circle with centre O. TO PROVE AB  BC  CD  AD. PROOF We know that the lengths of tangents drawn from an exterior point to a circle are equal.  AP  AS, … (i) [tangents from A] … (ii) [tangents from B] BP  BQ, … (iii) [tangents from C] CR  CQ,  … (iv) [tangents from D] DR DS. GIVEN

Circles



483

AB  CD  AP  BP  CR  DR  AS  BQ  CQ  DS [from (i), (ii), (iii) and (iv)]  (AS  DS)  (BQ  CQ)  AD  BC.

Thus, AB  CD  AD  BC 

2AB  2AD [a opposite sides of a gm are equal]



AB  AD.



CD  AB  AD  BC.

Hence, ABCD is a rhombus. EXAMPLE 29

Prove that the opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle. [CBSE 2012, ’13C, ’14, ’17]

SOLUTION

GIVEN A quad. ABCD circumscribes a circle with centre O. TO PROVE

+AOB  +COD  180c,

and +AOD  +BOC  180c. CONSTRUCTION

Join OP, OQ, OR and OS.

We know that the tangents drawn from an external point of a circle subtend equal angles at the centre. PROOF

 +1  +2, +3  +4, +5  +6 and +7  +8. And, +1  +2  +3  +4  +5  +6  +7  +8  360c [O at a point]  2(+2  +3)  2(+6  +7)  360c, and 2(+1  +8)  2(+4  +5)  360c  +2  +3  +6  +7  180c and +1  +8  +4  +5  180c  +AOB  +COD  180c and +AOD  +BOC  180c. EXAMPLE 30

In the given figure, PA is a tangent from an external point P to a circle with centre O. If +POB  115c, find [CBSE 2009C] +APO.

SOLUTION

We know that the tangent at a point to a circle is perpendicular to the radius passing through the point of contact.

484

Secondary School Mathematics for Class 10

 +OAP  90c. Now, +AOP  +BOP  180c  +AOP  180c  +BOP  180c  115c  65c. Now, +OAP  +AOP  +APO  180c [sum of angles of a triangle is 180]  +APO  180c  (+OAP  +AOP)  180c  (90c  65c)  25c. EXAMPLE 31

From a point P, two tangents PA and PB are drawn to a circle C(O, r) . If OP  2r, show that 3APB is equilateral. [CBSE 2008, ’11, ’12]

SOLUTION

Let OP meet the circle at Q. Join OA and AQ. Clearly, OA = AP & +OAP  90c [radius through the point of contact is perpendicular to the tangent]. Now, OQ  QP  r. Thus, Q is the midpoint of the hypotenuse OP of 3OAP. So, Q is equidistant from O, A and P.  QA  OQ  QP  r  OA  OQ  QA  r  3AOQ is equilateral  +AOQ  60c [a each angle of an equilateral triangle is 60]  +AOP  60c  +APO  30c

[a +AOP  +OAP  +APO  180c]

 +APB  2+APO  60c. Also, PA  PB & +PAB  +PBA  60c. Hence, 3PAB is an equilateral triangle. EXAMPLE 32

In the given figure, XY and XlYl are two parallel tangents to a circle with centre O and another tangent AB with point of contact C intersects XY at A and XlYl at B. Prove that +AOB  90c. [HOTS] [CBSE 2012, ’13, ’14, ’17]

Circles SOLUTION

485

In quad. APQB, we have +APO  90c and +BQO  90c [a tangent at any point is perpendicular to the radius through the point of contact]. Now, +APO  +BQO  +QBC  +PAC  360c  +PAC  +QBC  360c  (+APO  +BQO)  180c.

… (i)

We have +CAO 

1 1 +PAC and +CBO  +QBC 2 2 [a tangents from an external point are equally inclined to the line segment joining the centre to that point].

 +CAO  +CBO 

1 1 (+PAC  +QBC)  #180c  90c. … (ii) 2 2

In 3AOB, we have +CAO  +AOB  +CBO  180c  +AOB  180c  (+CAO  +CBO)  90c EXAMPLE 33

SOLUTION

f

[using (ii)].

The incircle of an isosceles triangle ABC, with AB  AC, touches the sides AB, BC, CA at D, E and F respectively. Prove that E bisects BC. [CBSE 2008, ’12, ’13C, ’14] We know that the tangents drawn from an external point to a circle are equal.  AD  AF, … (i) [tangents from A] BD  BE, … (ii) [tangents from B] CE  CF. … (iii) [tangents from C] [given] Now, AB  AC  AD  BD  AF  CF  BD  CF  BE  CE [using (ii) and (iii)]  E bisects BC.

EXERCISE 8A

1. A point P is at a distance of 29 cm from the centre of a circle of radius 20 cm. Find the length of the tangent drawn from P to the circle. [CBSE 2017]

486

Secondary School Mathematics for Class 10

2. A point P is 25 cm away from the centre of a circle and the length of tangent drawn from P to the circle is 24 cm. Find the radius of the circle. 3. Two concentric circles are of radii 6.5 cm and 2.5 cm. Find the length of the chord of the larger circle which touches the smaller circle. [CBSE 2011]

4. In the given figure, a circle inscribed in a triangle ABC, touches the sides AB, BC and AC at points D, E and F respectively. If AB  12 cm, BC  8 cm and AC  10 cm, find the lengths of AD, BE and CF. [CBSE 2013] 5. In the given figure, PA and PB are the tangent segments to a circle with centre O. Show that the points A, O, B and P are concyclic.

6. In the given figure, the chord AB of the larger of the two concentric circles, with centre O, touches the smaller circle at C. Prove that AC  CB.

7. From an external point P, tangents PA and PB are drawn to a circle with centre O. If CD is the tangent to the circle at a point E and PA  14 cm, find the perimeter of 3PCD. [CBSE 2002] 8. A circle is inscribed in a 3ABC touching AB, BC and AC at P, Q and R respectively. If AB  10 cm, AR  7 cm and CR  5 cm, find the length of BC. [CBSE 2009C]

Circles

487

9. In the given figure, a circle touches all the four sides of a quadrilateral ABCD whose three sides are AB  6 cm, BC  7 cm and CD  4 cm. Find AD.

10. In the given figure, an isosceles triangle ABC, with AB  AC, circumscribes a circle. Prove that the point of contact P bisects the base BC. [CBSE 2012, ’17]

11. In the given figure, O is the centre of two concentric circles of radii 4 cm and 6 cm respectively. PA and PB are tangents to the outer and inner circle respectively. If PA  10 cm, find the length of PB up to one place of decimal. 12. In the given figure, a triangle ABC is drawn to circumscribe a circle of radius 3 cm such that the segments BD and DC into which BC is divided by the point of contact D, are of lengths 6 cm and 9 cm respectively. If the area of 3ABC  54 cm 2 then find the lengths of sides AB and AC. [CBSE 2011, ’15]

13. PQ is a chord of length 4.8 cm of a circle of radius 3 cm. The tangents at P and Q intersect at a point T as shown in the figure. Find the length of TP. [CBSE 2013C]

14. Prove that the line segment joining the points of contact of two parallel tangents of a circle, passes through its centre. [CBSE 2014]

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Secondary School Mathematics for Class 10

15. In the given figure, a circle with centre O, is inscribed in a quadrilateral ABCD such that it touches the side BC, AB, AD and CD at points P, Q, R and S respectively. If AB  29 cm, AD  23 cm, +B  90c and DS  5 cm then find the radius of the circle. [CBSE 2008, ’13] 16. In the given figure, O is the centre of the circle and TP is the tangent to the circle from an external point T. If +PBT  30c, prove that [HOTS] [CBSE 2015] BA : AT  2 : 1.

ANSWERS (EXERCISE 8A)

1. 21 cm

2. 7 cm

3. 12 cm

4. AD  7 cm, BE  5 cm, CF  3 cm 9. AD  3 cm 11. 10.9 cm 13. TP  4 cm

7. 28 cm

12. AB  10 cm, AC  12 cm

15. 11 cm HINTS TO SOME SELECTED QUESTIONS

4. x  y  12, y  z  8, z  x  10. Solving, we get x  7, y  5, z  3. 

8. BC  8 cm

AD  x  7 cm, BE  y  5 cm, CF  z  3 cm.

5. OA = AP and OB = BP 

+OAP  90c and +OBP  90c



+OAP  +OBP  180c



quad. AOBP is cyclic

[a a quad. is cyclic if the sum of a pair of opposite angles is 180].

Circles

489

7. PA  PB, CA  CE, DB  DE [£ tangents from an external point are equal]. Perimeter of 3PCD  PC  CD  PD  PC  CE  DE  PD  PC  CA  DB  PD  PA  PB  2PA [a PB  PA]  (2 #14) cm  28 cm. 8. We know that the tangents to a circle from an exterior point are equal. 

AP  AR  7 cm; BQ  BP  AB  AP  (10  7) cm  3 cm; CQ  CR  5 cm.

And so, BC  BQ  CQ  (3  5) cm  8 cm. 9. When a qudrilateral ABCD is drawn to circumscribe a circle then AB  CD  AD  BC. 

AD  AB  CD  BC  (6  4  7) cm  3 cm.

10. We know that the tangents to a circle from an external point are equal. 

AQ  AR, BP  BR, CP  CQ.

Now, AB  AC & AR  BR  AQ  CQ 

AR  BP  AR  CP & BP  CP



P bisects the base BC.

1 13. R is the midpoint of PQ, i.e., PR  QR  a # 4.8k cm  2.4 cm. 2 Also, TO = PQ. In right 3PRO, we have PO 2  PR 2  RO 2 & RO  PO 2  PR 2 Let TR  x and TP  y.

 3 2  2.4 2  3.24  1.8.

In right 3PTR, we have TP 2  TR 2  PR 2 & y 2  x 2  (2.4) 2.

… (i)

In right 3OTP, we have TO 2  TP 2  PO 2 & (TR  RO) 2  TP 2  PO 2 

(x  1.8) 2  y 2  3 2 & x 2  3.6x  3.24  y 2  9.

Solving (i) and (ii), we get x  3.2, y  4. 

TP  4 cm.

14. Let APB and CQD be two parallel tangents to a circle with centre O. Join OP and OQ. Draw RO  AB. Now, +APO  +ROP  180c [co-interior angles] 

+ROP  90c

[a +APO  90c].

… (ii)

490

Secondary School Mathematics for Class 10 Similarly, +CQO  +ROQ  180c [co-interior angles] 

+ROQ  90c

[a +CQO  90c].



+POQ  +ROP  +ROQ  180c.

Hence, POQ is a straight line passing through O. 15. POQB is a square, since +PBQ  +BQO  +BPO  90c, OP  OQ  r. 

r  OQ  QB.

We know that tangents to a circle from an exterior point are equal. 

DS  DR, AR  AQ.

Now, AR  AD  DR  AD  DS  (23  5) cm  18 cm and r  QB  AB  AQ  AB  AR  (29  18) cm  11 cm. 16. +APB  90c [angle in a semicircle] 

+PAB  90c  +PBA  90c  30c  60c [a +PBA  +PBT  30c]. +PAT  +PAB  180c [linear pair]



+PAT  180c  +PAB  180c  60c  120c. +APT  +PBA  30c [angles in alternate segments].

In 3PAT, we have +APT  +PAT  +PTA  180c 

+PTA  180c  (+APT  +PAT)  180c  (30c  120c)  30c.

Now, +APT  +PTA  30c & AT  AP

… (i) [sides opposite equal angles are equal]

In right 3APB, we have AP AP cos (+PAB)  & cos 60c  & BA  2AP. BA BA From (i) and (ii), we get BA : AT  2AP : AP  2 : 1.

f

EXERCISE 8B

Very-Short-Answer Questions 1. In the adjoining figure, a circle touches all the four sides of a quadrilateral ABCD whose sides are AB  6 cm, BC  9 cm and CD  8 cm. Find the length of side AD. [CBSE 2011] 2. In the given figure, PA and PB are two tangents to the circle with centre O. If +APB  50c then what is the measure of [CBSE 2015] +OAB.

… (ii)

Circles

491

3. In the given figure, O is the centre of a circle. PT and PQ are tangents to the circle from an external point P. If +TPQ  70c, find +TRQ. [CBSE 2015]

4. In the given figure, common tangents AB and CD to the two circles with centres O1 and O2 intersect at E. Prove that [CBSE 2014] AB  CD. 5. If PT is a tangent to a circle with centre O and PQ is a chord of the circle such that +QPT  70c, then find the measure of +POQ.

Short-Answer Questions 6. In the given figure, a triangle ABC is drawn to circumscribe a circle of radius 2 cm such that the segments BD and DC into which BC is divided by the point of contact D, are of lengths 4 cm and 3 cm respectively. If the area of 3ABC  21 cm 2 then find the lengths of sides AB and AC. [CBSE 2011] 7. Two concentric circles are of radii 5 cm and 3 cm. Find the length of the chord of the larger circle (in cm) which touches the smaller circle. [CBSE 2012, ’14]

8. Prove that the perpendicular at the point of contact of the tangent to a circle passes through the centre. 9. In the given figure, two tangents RQ and RP are drawn from an external point R to the circle with centre O. If +PRQ  120c, then prove that [CBSE 2015] OR  PR  RQ.

492

Secondary School Mathematics for Class 10

10. In the given figure, a circle inscribed in a triangle ABC touches the sides AB, BC and CA at points D, E and F respectively. If AB  14 cm, BC  8 cm and CA  12 cm. Find the lengths AD, BE and CF. [CBSE 2013]

11. In the given figure, O is the centre of the circle. PA and PB are tangents. Show that AOBP is a cyclic quadrilateral. [CBSE 2014]

12. In two concentric circles, a chord of length 8 cm of the larger circle touches the smaller circle. If the radius of the larger circle is 5 cm then [CBSE 2013C] find the radius of the smaller circle. 13. In the given figure, PQ is a chord of a circle with centre O and PT is a tangent. If +QPT  60c, find +PRQ. [HOTS] [CBSE 2015]

14. In the given figure, PA and PB are two tangents to the circle with centre O. If +APB  60c then find the measure of +OAB.

15. If the angle between two tangents drawn from an external point P to a circle of radius a and centre O, is 60 then find the length of OP. [CBSE 2017]

ANSWERS (EXERCISE 8B)

1. AD  5 cm

2. 25

6. AB  7.5 cm, AC  6.5 cm 10. AD  9 cm, BE  5 cm, CF  3 cm 14. 30

3. 55

5. 140c

7. 8 cm 12. 3 cm

13. 120

15. a 3 HINTS TO SOME SELECTED QUESTIONS

1. When a quadrilateral ABCD circumscribes a circle then AB  CD  AD  BC. 

AD  AB  CD  BC  (6  8  9) cm  5 cm.

Circles 2. +APB  2+OAB & +OAB 

50c  1 #+APB  25c. 2 2

493 [See Solved Example 13.]

3. Join OT and OQ. Then, +TOQ  +TPQ  180c & +TOQ  180c  70c  110c. [See Theorem 8.] Now, +TRQ 

1 1 #+TOQ  #110c  55c 2 2 [a the angle subtended by an arc of a circle at the centre is double the angle subtended by it at any point on the remaining part of the circle].

4. We know that the tangents to a circle from an external point are equal. 

AE  CE, BE  DE



AE  BE  CE  DE & AB  CD.

5. Mark a point R in the alternate segment. Join RP and RQ. Then, +PRQ  +QPT  70c [angles in the alternate segments]. 

+POQ  2+PRQ  2 #70c  140c. [a angle subtended by an arc on the centre is double the angle subtended at any point on the remaining part of the circle.]

6. Mark the points of contact E and F of tangents AC and AB respectively. We know that the tangents to a circle from an external point are equal. 

AE  AF  x cm (say); BD  BF  4 cm; CD  CE  3 cm.

Now, ar (3ABC)   

1 #(perimeter of 3ABC)# r 2

1 #{(x  4)  (4  3)  (3  x)}# 2 & x  3.5. 2  AB (x  4) cm  7.5 cm; AC  (x  3) cm  6.5 cm.

21 

7. Let AB be the chord, P be the point where it touches the smaller circle and O be the common centre. AB is tangent to smaller circle and so, OP = AB. 

P is the midpoint of AB, i.e., AP  PB [a a perpendicular from the centre on any chord bisects the chord].

Now, OP  3 cm, OA  5 cm & AP  OA 2  OP 2  5 2  3 2  4. 

AB  2AP  8 cm.

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Secondary School Mathematics for Class 10

9. Join OP and OQ. 1 #+PRQ  60c. 2 [a two tangents from an external point are

+PRQ  120c & +PRO  +QRO 

equally inclined to the line segment joining the centre to that point. And so, +PRO  +QRO 

1 +PRQ.] 2

In right 3OPR, we have cos +PRO  Similarly, RQ  

PR PR 1 PR 1 & cos 60c  &  & PR  #OR. 2 OR 2 OR OR

1 #OR. 2

PR  RQ  OR.

10. x  y  14, y  z  8, z  x  12. Solving, x  9, y  5, z  3. 

AD  x  9 cm, BE  y  5 cm, CF  z  3 cm.

12. Let AB be the chord, P be the point where it touches the smaller circle and O be the common centre. Then, AB is the tangent to smaller circle at P. 

OP = AB.

And so, P is the midpoint of AB, i.e., AP  PB 

1 1 AB  a # 8k cm  4 cm 2 2 [a perpendicular from the centre on any chord bisects the chord].

Now, OA  5 cm, AP  4 cm & OP  OA 2  AP 2  5 2  4 2  3. 

radius of smaller circle  OP  3 cm.

13. Mark a point M in the alternate segment. Join MP and MQ. Then, +PMQ  +QPT  60c [angles in the alternate segments]. Now, +PMQ  +PRQ  180c [a PMQR is a cyclic quadrilateral] 

+PRQ  180c  +PMQ  180c  60c  120c.

Circles

495

14. PA  PB [a tangents from an external point are equal] 1 1  +PAB  +PBA  #(180c  +APB)  #(180c  60c)  60c. 2 2 Now, +PAO  90c [a radius through the point of contact is perpendicular to the tangent] 

+OAB  +PAB  90c & +OAB  90c  +PAB  90c  60c  30c.

MULTIPLE-CHOICE QUESTIONS (MCQ) Choose the correct answer in each of the following questions: 1. The number of tangents that can be drawn from an external point to a circle is [CBSE 2011, ’12] (a) 1

(b) 2

(c) 3

(d) 4

2. In the given figure, RQ is a tangent to the circle with centre O. If SQ  6 cm and QR  4 cm, then OR is equal to [CBSE 2014] (a) 2.5 cm

(b) 3 cm

(c) 5 cm

(d) 8 cm

3. In a circle of radius 7 cm, tangent PT is drawn from a point P such that PT  24 cm. If O is the centre of the circle, then length OP  ? (a) 30 cm

(b) 28 cm

(c) 25 cm

(d) 18 cm

4. Which of the following pairs of lines in a circle cannot be parallel? [CBSE 2011]

(a) Two chords

(b) A chord and a tangent

(c) Two tangents

(d) Two diameters

5. The chord of a circle of radius 10 cm subtends a right angle at its centre. The length of the chord (in cm) is [CBSE 2014] (a)

5 2

(b) 5 2

(c) 10 2

6. In the given figure, PT is a tangent to the circle with centre O. If OT  6 cm and OP  10 cm, then the length of tangent PT is (a) 8 cm

(b) 10 cm

(c) 12 cm

(d) 16 cm

(d) 10 3

496

Secondary School Mathematics for Class 10

7. In the given figure, point P is 26 cm away from the centre O of a circle and the length PT of the tangent drawn from P to the circle is 24 cm. Then, the radius of the circle is [CBSE 2011, ’12] (a) 10 cm

(b) 12 cm

(c) 13 cm

(d) 15 cm

8. PQ is a tangent to a circle with centre O at the point P. If 3OPQ is an isosceles triangle, then +OQP is equal to [CBSE 2014] (a) 30

(b) 45

(c) 60

(d) 90

9. In the given figure, AB and AC are tangents to the circle with centre O such that +BAC  40c. Then, +BOC is equal to [CBSE 2011, ’14] (a) 80

(b) 100

(c) 120

(d) 140

10. If a chord AB subtends an angle of 60 at the centre of a circle, then the angle between the tangents to the circle drawn from A and B is [CBSE 2013C]

(a) 30

(b) 60

(c) 90

11. In the given figure, O is the centre of two concentric circles of radii 6 cm and 10 cm. AB is a chord of outer circle which touches the inner circle. The length of chord AB is (a) 8 cm

(b) 14 cm

(c) 16 cm

(d)

136 cm

12. In the given figure, AB and AC are tangents to a circle with centre O and radius 8 cm. If OA  17 cm, then the length of AC (in cm) is [CBSE 2012]

(a) 9 (c)

(b) 15 353

(d) 25

13. In the given figure, O is the centre of a circle, AOC is its diameter such that +ACB  50c. If AT is the tangent to the circle at the point A then +BAT  ? (a) 40

(b) 50

(c) 60

(d) 65

(d) 120

Circles

497

14. In the given figure, O is the centre of a circle, PQ is a chord and PT is the tangent at P. If +POQ  70c, then +TPQ is equal to [CBSE 2011] (a) 35

(b) 45

(c) 55

(d) 70

15. In the given figure, AT is a tangent to the circle with centre O such that OT  4 cm and +OTA  30c. Then, AT  ? (a) 4 cm

(b) 2 cm

(c) 2 3 cm

(d) 4 3 cm

16. If PA and PB are two tangents to a circle with centre O such that +AOB  110c then +APB is equal to [CBSE 2011, ’14] (a) 55

(b) 60

(c) 70

(d) 90

17. In the given figure, the length of BC is [CBSE 2012, ’14]

(a) 7 cm (b) 10 cm (c) 14 cm (d) 15 cm 18. In the given figure, if +AOD  135c then [CBSE 2013C] +BOC is equal to (a) 25 (c) 52.5



(b) 45 (d) 62.5

19. In the given figure, O is the centre of a circle and PT is the tangent to the circle. If PQ is a chord such that +QPT  50c then +POQ  ? (a) 100

(b) 90

(c) 80

(d) 75

20. In the given figure, PA and PB are two tangents to the circle with centre O. If +APB  60c then [CBSE 2011] +OAB is (a) 15

(b) 30

(c) 60

(d) 90

498

Secondary School Mathematics for Class 10

21. If two tangents inclined at an angle of 60 are drawn to a circle of radius 3 cm then the length of each tangent is (a) 3 cm

(b)

3 3 cm 2

(c) 3 3 cm

(d) 6 cm

22. In the given figure, PQ and PR are tangents to a circle with centre A. If +QPA  27c then [CBSE 2012] +QAR equals (a) 63

(b) 117

(c) 126

(d) 153

23. In the given figure, PA and PB are two tangents drawn from an external point P to a circle with centre C and radius 4 cm. If PA = PB, then the length of each tangent is [CBSE 2013] (a) 3 cm

(b) 4 cm

(c) 5 cm

(d) 6 cm

24. If PA and PB are two tangents to a circle with centre O such that +APB  80c. Then, +AOP  ? (a) 40

(b) 50

(c) 60

(d) 70

25. In the given figure, O is the centre of the circle. AB is the tangent to the circle at the point P. If +APQ  58c then the measure of +PQB is [CBSE 2014] (a) 32 (b) 58 (c) 122

(d) 132

26. In the given figure, O is the centre of the circle. AB is the tangent to the circle at the point P. If +PAO  30c then +CPB  +ACP is equal to (a) 60

(b) 90

(c) 120

(d) 150

27. In the given figure, PQ is a tangent to a circle with centre O. A is the point of contact. If +PAB  67c, then the measure of +AQB is (a) 73

(b) 64

(c) 53

(d) 44



Circles

499

28. In the given figure, two circles touch each other at C and AB is a tangent to both the circles. The measure of [HOTS] [CBSE 2013C] +ACB is (a) 45

(b) 60

(c) 90

(d) 120

29. O is the centre of a circle of radius 5 cm. At a distance of 13 cm from O, a point P is taken. From this point, two tangents PQ and PR are drawn to the circle. Then, the area of quad. PQOR is (a) 60 cm2

(b) 32.5 cm2

(c) 65 cm2

(d) 30 cm2

30. In the given figure, PQR is a tangent to the circle at Q, whose centre is O and AB is a chord parallel to PR such that +BQR  70c. Then, +AQB  ? (a) 20

(b) 35

(c) 40

(d) 45

31. The length of the tangent from an external point P to a circle of radius 5 cm is 10 cm. The distance of the point from the centre of the circle is (a) 8 cm (b) 104 cm (c) 12 cm (d) 125 cm [CBSE 2013C]

32. In the given figure, O is the centre of a circle, BOA is its diameter and the tangent at the point P meets BA extended at T. If +PBO  30c then +PTA  ? (a) 60

(b) 30

(c) 15

(d) 45

33. In the given figure, a circle touches the side DF of 3EDF at H and touches ED and EF produced at K and M respectively. If EK  9 cm then the perimeter of 3EDF is [CBSE 2012] (a) 9 cm

(b) 12 cm

(c) 13.5 cm

(d) 18 cm

34. To draw a pair of tangents to a circle, which are inclined to each other at an angle of 45, we have to draw tangents at the end points of those

500

Secondary School Mathematics for Class 10

two radii, the angle between which is (a) 105

(b) 135

[CBSE 2011]

(c) 140

(d) 145

35. In the given figure, O is the centre of a circle; PQL and PRM are the tangents at the points Q and R respectively and S is a point on the circle such that +SQL  50c and +SRM  60c. Then, +QSR  ? (a) 40

(b) 50

(c) 60

(d) 70

36. In the given figure, a triangle PQR is drawn to circumscribe a circle of radius 6 cm such that the segments QT and TR into which QR is divided by the point of contact T, are of lengths 12 cm and 9 cm respectively. If the area of 3PQR  189 cm 2 then the length of side PQ is [CBSE 2011] (a) 17.5 cm

(b) 20 cm

(c) 22.5 cm

37. In the given figure, QR is a common tangent to the given circles, touching externally at the point T. The tangent at T meets QR at P. If PT  3.8 cm then the length of QR is [CBSE 2014] (a) 1.9 cm

(b) 3.8 cm

(c) 5.7 cm

(d) 7.6 cm

38. In the given figure, quad. ABCD is circumscribed, touching the circle at P, Q, R and S. If AP  5 cm, BC  7 cm and CS  3 cm. Then, the length AB  ? (a) 9 cm

(b) 10 cm

(c) 12 cm

(d) 8 cm

39. In the given figure, quad. ABCD is circumscribed, touching the circle at P, Q, R and S. If AP  6 cm, BP  5 cm, CQ  3 cm and DR  4 cm then perimeter of quad. ABCD is (a) 18 cm

(b) 27 cm

(c) 36 cm

(d) 32 cm

(d) 25 cm

Circles

501

40. In the given figure, O is the centre of a circle, AB is a chord and AT is the tangent at A. If +AOB  100c then +BAT is equal to [CBSE 2011] (a) 40

(b) 50

(c) 90

(d) 100

41. In a right triangle ABC, right-angled at B, BC  12 cm and AB  5 cm. The radius of the circle inscribed in the triangle is [CBSE 2014] (a) 1 cm

(b) 2 cm

(c) 3 cm

(d) 4 cm

42. In the given figure, a circle is inscribed in a quadrilateral ABCD touching its sides AB, BC, CD and AD at P, Q, R and S respectively. If the radius of the circle is 10 cm, BC  38 cm, PB  27 cm and AD = CD then the length of CD is [CBSE 2013] (a) 11 cm

(b) 15 cm

(c) 20 cm

(d) 21 cm

43. In the given figure, 3ABC is right-angled at B such that BC  6 cm and AB  8 cm. A circle with centre O has been inscribed inside the triangle. OP = AB, OQ = BC and OR = AC. If OP  OQ  OR  x cm then x  ? (a) 2 cm

(b) 2.5 cm

(c) 3 cm

(d) 3.5 cm

44. Quadrilateral ABCD is circumscribed to a circle. If AB  6 cm, BC  7 cm and CD  4 cm then the length of AD is [CBSE 2012] (a) 3 cm

(b) 4 cm

(c) 6 cm

(d) 7 cm

45. In the given figure, PA and PB are tangents to the given circle such that PA  5 cm and +APB  60c. The length of chord AB is (a) 5 2 cm

(b) 5 cm

(c) 5 3 cm

(d) 7.5 cm

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Secondary School Mathematics for Class 10

46. In the given figure, DE and DF are tangents from an external point D to a circle with centre A. If DE  5 cm and DE = DF then the radius of the circle is [CBSE 2013] (a) 3 cm

(b) 4 cm

(c) 5 cm

(d) 6 cm

47. In the given figure, three circles with centres A, B, C respectively touch each other externally. If AB  5 cm, BC  7 cm and CA  6 cm then the radius of the circle with centre A is (a) 1.5 cm

(b) 2 cm

(c) 2.5 cm

(d) 3 cm

48. In the given figure, AP, AQ and BC are tangents to the circle. If AB  5 cm, AC  6 cm and BC  4 cm then the length of AP is [CBSE 2012] (a) 15 cm (b) 10 cm (c) 9 cm

(d) 7.5 cm

49. In the given figure, O is the centre of two concentric circles of radii 5 cm and 3 cm. From an external point P tangents PA and PB are drawn to these circles. If PA  12 cm then PB is equal to (a) 5 2 cm

(b) 3 5 cm

(c) 4 10 cm

(d) 5 10 cm

True/False Type

50. Which of the following statements is not true? (a) If a point P lies inside a circle, no tangent can be drawn to the circle, passing through P. (b) If a point P lies on the circle, then one and only one tangent can be drawn to the circle at P. (c) If a point P lies outside the circle, then only two tangents can be drawn to the circle from P. (d) A circle can have more than two parallel tangents, parallel to a given line.

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51. Which of the following statements is not true? (a) A tangent to a circle intersects the circle exactly at one point. (b) The point common to the circle and its tangent is called the point of contact. (c) The tangent at any point of a circle is perpendicular to the radius of the circle through the point of contact. (d) A straight line can meet a circle at one point only. 52. Which of the following statements is not true? (a) A line which intersects a circle in two points, is called a secant of the circle. (b) A line intersecting a circle at one point only, is called a tangent to the circle. (c) The point at which a line touches the circle, is called the point of contact. (d) A tangent to the circle can be drawn from a point inside the circle. Assertion-and-Reason Type

Each question consists of two statements, namely, Assertion (A) and Reason (R). For selecting the correct answer, use the following code: (a) Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A). (b) Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A). (c) Assertion (A) is true and Reason (R) is false. (d) Assertion (A) is false and Reason (R) is true. 53.

Assertion (A)

Reason (R)

At a point P of a circle with centre The tangent at any point of a circle is O and radius 12 cm, a tangent PQ perpendicular to the radius through of length 16 cm is drawn. Then, the point of contact.

OQ  20 cm.

The correct answer is (a)/(b)/(c)/(d). 54.

Assertion (A)

Reason (R)

If two tangents are drawn to a circle A parallelogram circumscribing a from an external point then they circle is a rhombus. subtend equal angles at the centre.

The correct answer is (a)/(b)/(c)/(d).

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Secondary School Mathematics for Class 10

Assertion (A)

55.

Reason (R)

In the given figure, a quad. ABCD is In two concentric circles, the chord drawn to circumscribe a given circle, of the larger circle, which touches the smaller circle, is bisected at the as shown. point of contact. Then, AB  BC  AD  DC.

The correct answer is (a)/(b)/(c)/(d). ANSWERS (MCQ)

1. 10. 19. 28. 37. 46. 55.

(b) (d) (a) (c) (d) (c) (d)

2. 11. 20. 29. 38. 47.

(c) (c) (b) (a) (a) (b)

3. 12. 21. 30. 39. 48.

(c) (b) (c) (c) (c) (d)

4. 13. 22. 31. 40. 49.

(d) (b) (c) (d) (b) (c)

5. 14. 23. 32. 41. 50.

(c) (a) (b) (b) (b) (d)

6. 15. 24. 33. 42. 51.

(a) (c) (b) (d) (d) (d)

7. 16. 25. 34. 43. 52.

(a) (c) (a) (b) (a) (d)

8. 17. 26. 35. 44. 53.

(b) (b) (b) (d) (a) (a)

9. 18. 27. 36. 45. 54.

(d) (b) (d) (c) (b) (b)

HINTS TO SOME SELECTED QUESTIONS 4. Every diameter passes through the centre and so no two diameters of a circle can be parallel. 5. AB  OA 2  OB 2  10 2  10 2  10 2 cm.

8. We have OP = PQ, i.e., +OPQ  90c. 3OPQ is isosceles & OP  PQ & +OQP  +POQ  45c [a in a triangle, angles opposite equal sides are equal]. 9. ABOC is a cyclic quadrilateral 

[see Solved Example 18].

+BAC  +BOC  180c & +BOC  180c  +BAC  180c  40c  140c.

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10. Let P be the point of intersection of tangents at A and B. Then, OAPB is a cyclic quadrilateral. [see Solved Example 18] 

+AOB  +APB  180c & +APB  180c  60c  120c.

13. +BAT  +ACB  50c [angles in alternate segments]. 14. Mark a point R in the alternate segment. Join PR and QR. 1 1 Then, +PRQ  +POQ  #70c  35c. 2 2 Now, +TPQ  +PRQ  35c [angles in alternate segments]. 15. In right 3OAT (having right angle at A), we have cos (+OTA) 

3 AT AT & cos 30c  & AT  d # 4n cm  2 3 cm. 2 4 OT

17. We have AF  AE, BD  BF, CD  CE. [a tangents from an external point are equal] 

CE  AC  AE  AC  AF  (11  4) cm  7 cm; BD  BF  3 cm; CD  CE  7 cm.

And so, BC  BD  CD  (3  7) cm  10 cm. 18. +AOD  +BOC  180c [see Solved Example 29] 

+BOC  180c  +AOD  180c  135c  45c.

19. Mark a point R in the alternate segment. Join PR and QR. Then, +PRQ  +QPT  50c [angles in alternate segments]. Now, +POQ  2+PRQ  100c. [a angle at the centre is double the angle on the circle] 1 +APB  30c. [See Solved Example 13.] 2 21. Let PQ and PR be tangents to a circle with centre O from

20. +APB  2+OAB & +OAB 

a point P such that +QPR  60c. 1 +QPR  30c 2 [a tangents from an external point

Then, +QPO  +RPO 

are equally inclined to the line joining the point and the centre]. In right 3OPQ, we have tan (+OPQ) 

OQ 3 1  3 & tan 30c  & & PQ  3 3 cm. PQ PQ 3 PQ

Now, PR  PQ  3 3 cm.

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Secondary School Mathematics for Class 10

22. +QPR  2+QPA  2 # 27c  54c [a AP bisects +QPR]. Now, QPRA is a cyclic quadrilateral &

+QAR  +QPR  180c

&

+QAR  180c  54c  126c.

[see Solved Example 18]

23. Join AC and BC. Then, ACBP is a square. 

PA  PB  AC  radius of the circle = 4 cm.

25. +QPR  90c [angle in a semicircle] and +QRP  +APQ  58c [angles in alternate segments]. In 3PQR, we have +PQR  +QRP  +QPR  180c & +PQR  32c. 

+PQB  +PQR  32c.

26. +DPC  90c [angle in a semicircle] and +DPA  +DCP [angles in alternate segments]. Now, +CPB  +DPA  +DPC  180c &

+CPB  +DPA  90c & +CPB  +DCP  90c

&

+CPB  +ACP  90c.

27. +BAC  90c [angle in a semicircle]. +PAB  +BAC  +CAQ  180c & +CAQ  180c  (90c  67c)  23c. +ACB  +PAB  67c [angles in alternate segments] +ACB  +ACQ  180c [linear pair] &

+ACQ  180c  67c  113c.

Now, +CAQ  +ACQ  +AQC  180c [in 3ACQ] &

+AQC  180c  (23c  113c)  44c & +AQB  44c.

28. Draw a tangent to the circles at point C. Let it meet AB at P. Then, PA  PC and PB  PC. PA  PC & +PAC  +PCA PB  PC & +PBC  +PCB 

+PAC  +PBC  +PCA  +PCB  +ACB

&

+PAC  +PBC  +ACB  2+ACB

&

180c  2+ACB & +ACB  90c.

29. Clearly, OQ  OR  5 cm, +OQP  +ORP  90c and OP  13 cm. 

PQ 2  OP 2  OQ 2  (13) 2  (5) 2  169  25  144 & PQ  144  12 cm.



ar (3OQP) 

1 1 # PQ #OQ  a #12 # 5k cm 2  30 cm 2 . 2 2

Similarly, ar (3ORP)  30 cm 2 . 

ar(quad. PQOR)  (30  30) cm 2  60 cm 2 .

30. +QAB  +BQR  70c [angles in alternate segments] +ABQ  +BQR  70c [alternate int. angles].

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In 3AQB, we have +AQB  180c  (+QAB  +ABQ)  180c  (70c  70c)  40c. 32. +APB  90c [angle in a semicircle]. 

+PAB  90c  +PBA  90c  30c  60c. +PAT  +PAB  180c [linear pair]



+PAT  180c  +PAB  180c  60c  120c. +APT  +PBA  30c [angles in alternate segments].

Now, +APT  +PAT  +PTA  180c [in 3PAT] &

+PTA  180c  (30c  120c)  30c.

34. Let PA and PB be the desired tangents to a circle with centre O from an exterior point P. Then, +APB  45c. BOAP must be a cyclic quadrilateral. 

+AOB  +APB  180c & +AOB  180c  45c  135c.

So, the angle between the two radii must be 135. 35. Since PQL is a tangent and OQ is a radius, so +OQL  90c. 

+OQS  90c  50c  40c.

Now, OQ  OS & +OSQ  +OQS  40c. Similarly, +ORS  90c  60c  30c. And, OR  OS & +OSR  +ORS  30c. 

+QSR  +OSQ  +OSR  40c  30c  70c.

36. ar (3PQR) 

1 #(perimeter of 3PQR)# r 2

&

1 #{(x  12)  (12  9)  (9  x)}# 6 2 x  10.5



PQ  (x  12) cm  (10.5  12) cm  22.5 cm.



189 

37. PQ  PT  3.8 cm, PR  PT  3.8 cm, QR  PQ  PR  (3.8  3.8) cm  7.6 cm. 38. Since the lengths of tangents drawn from an external point to a circle are equal, we have AQ  AP  5 cm. CR  CS  3 cm and BR  (BC  CR)  (7  3) cm  4 cm. BQ  BR  4 cm. 

AB  (BR  BQ)  (5  4) cm  9 cm.

40. Mark a point C in the alternate segment. Join AC and BC. 1 1 +ACB  +AOB  #100c  50c 2 2 [a angle at the centre is double the angle on the circle]. Now, +BAT  +ACB  50c [angles in alternate segments].

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Secondary School Mathematics for Class 10

41. In right 3ABC, we have AC  ar (3ABC)  

AB 2  BC 2  5 2  12 2 cm  13 cm. 1 #(perimeter of 3ABC)# r 2

1 1 # 5 #12  #(5  12  13)# r & r  2 cm. 2 2 1 :a ar (3ABC)  # BC # ABD 2

42. BQ  PB  27 cm; CQ  BC  BQ  (38  27) cm  11 cm; CR  CQ  11 cm. Join OR. Then, SDRO is a square. 

DR  SO  radius  10 cm.

And so, CD  DR  CR  (10  11) cm  21 cm. 43. AC 2  AB 2  BC 2  8 2  6 2  64  36  100 

AC  100 cm  10 cm.

CR  CQ  (BC  BQ)  (6  x) cm. AR  AP  (AB  BP)  (8  x) cm. AC  (AR  CR)  [(8  x)  (6  x)] cm  (14  2x) cm &

(14  2x)  10 & 2x  4 & x  2 cm.

44. AB  CD  AD  BC [see Solved Example 26] &

AD  AB  CD  BC  (6  4  7) cm  3 cm.

45. PA  PB & +PBA  +PAB  xc (say) Then, xc  xc  60c  180 & x  60c. 

3APB is equilateral and so, AB  PA  5 cm.

46. Join AE and AF. Then, AFDE is a square. 

radius of circle  AE  DE  5 cm.

47. Let the radii of the three circles be x, y, z respectively. Then, x  y  5, y  z  7 and z  x  6 & 2 (x  y  z)  18 & x  y  z  9. 

x  (x  y  z)  (y  z)  (9  7)  2 cm.

49. Join OB. Then, OA  5 cm, OB  3 cm. OP  OA 2  PA 2  5 2  12 2  13 cm. PB  OP 2  OB 2  13 2  3 2  160 cm  4 10 cm. 53. OQ  OP 2  PQ 2  12 2  16 2  20 cm. 

A is true. Also, R is true and is a correct explanation of A.

Hence, the correct answer is (a). 54. A and R are both true (see Theorem 4(i) and Solved Example 28) but R is not a correct explanation of A. Hence, the correct answer is (b).

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55. A is false. Correct relation is AB  CD  AD  BC. R is true (see Theorem 5). Hence, the correct answer is (d).

TEST YOURSELF MCQ 1. In the given figure, O is the centre of a circle, PQ is a chord and the tangent PT at P makes an angle of 50 with PQ. Then, +POQ  ? (a) 130

(b) 100

(c) 90

(d) 75

2. If the angle between two radii of a circle is 130 then the angle between the tangents at the ends of the radii is (a) 65

(b) 40

(c) 50

3. If tangents PA and PB from a point P to a circle with centre O are drawn so that +APB  80c then +POA  ? (a) 40

(b) 50

(c) 80

(d) 60

4. In the given figure, AD and AE are the tangents to a circle with centre O and BC touches the circle at F. If AE  5 cm then perimeter of 3ABC is (a) 15 cm

(b) 10 cm

(c) 22.5 cm

(d) 20 cm

Short-Answer Questions 5. In the given figure, a quadrilateral ABCD is drawn to circumscribe a circle such that its sides AB, BC, CD and AD touch the circle at P, Q, R and S respectively. If AB  x cm, BC  7 cm, CR  3 cm and AS  5 cm, find x.

(d) 90

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Secondary School Mathematics for Class 10

6. In the given figure, PA and PB are the tangents to a circle with centre O. Show that the points A, O, B, P are concyclic.

7. In the given figure, PA and PB are two tangents from an external point P to a circle with centre O. If +PBA  65c, find +OAB and +APB.

8. Two tangent segments BC and BD are drawn to a circle with centre O such that+CBD  120c. Prove that OB  2BC.

9. Fill in the blanks. (i) A line intersecting a circle in two distinct points is called a ...... . (ii) A circle can have ...... parallel tangents at the most. (iii) The common point of a tangent to a circle and the circle is called the …… . (iv) A circle can have …… tangents. 10. Prove that the lengths of two tangents drawn from an external point to a circle are equal. 11. Prove that the tangents drawn at the ends of the diameter of a circle are parallel. 12. In the given figure, if AB  AC, prove that BE  CE.

13. If two tangents are drawn to a circle from an external point, show that they subtend equal angles at the centre. 14. Prove that the tangents drawn at the ends of a chord of a circle make equal angles with the chord.

Circles

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15. Prove that the parallelogram circumscribing a circle, is a rhombus. 16. Two concentric circles are of radii 5 cm and 3 cm respectively. Find the length of the chord of the larger circle which touches the smaller circle. Long-Answer Questions 17. A quadrilateral is drawn to circumscribe a circle. Prove that the sums of opposite sides are equal. 18. Prove that the opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle. 19. Prove that the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line segments joining the points of contact at the centre. 20. PQ is a chord of length 16 cm of a circle of radius 10 cm. The tangents at P and Q intersect at a point T as shown in the figure. Find the length of TP.

ANSWERS (TEST YOURSELF)

1. (b)

2. (c)

3. (b)

4. (b)

5. 9 cm

7. +OAB  25c, +APB  50c 9. (i) secant 16. 8 cm

(ii) two

(iii) point of contact

20. 10.7 cm (approx.)



(iv) infinitely many

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Secondary School Mathematics for Class 10

Constructions

9 DIVISION OF A LINE SEGMENT IN A GIVEN RATIO EXAMPLE 1

Divide a line segment 6 cm long in the ratio 4 : 3. Prove your assertion.

STEPS OF CONSTRUCTION

Step 1. Draw a line segment AB  6 cm. Step 2. Draw a ray AX, making an acute angle +BAX. Step 3. Along AX, mark (4  3)  7 points A1, A2, A3, A4, A5, A6 and A7

such that AA1  A1 A2  A2 A3  A3 A4  A4 A5  A5 A6  A6 A7 . Step 4. Join A7 B. Step 5. From A4, draw A4 C parallel to A7 B (by making an angle equal

to +AA7 B), meeting AB at C. Then, C is the point on AB, which divides it in the ratio 4 : 3.

Thus, AC : CB  4 : 3. PROOF

Let AA1  A1 A2  …  A6 A7  x. In 3ABA7, we have A4 C  A7 B. AC  AA4  4x  4  [by Thales’ theorem]. CB A4 A7 3x 3 Hence, AC : CB  4 : 3. 512

Constructions

513

ALTERNATIVE METHOD STEPS OF CONSTRUCTION

Step 1. Draw a line segment AB  6 cm. Step 2. Draw a ray AX, making an acute +BAX. Step 3. Draw a ray BY parallel to AX by making +ABY  +BAX. Step 4. Locate the points A1, A2, A3, A4 on AX and B1, B2, B3 on BY

such that AA1  A1 A2  A2 A3  A3 A4  BB1  B1 B2  B2 B3 . Step 5. Join A4 B3, intersecting AB at a point C.

Then, AC : CB  4 : 3.

PROOF

Here, AA4  BB3 .  +CAA4  +CBB3

(alt. int. O).

Also, +ACA4  +BCB3

(vert. opp. O).



3CAA4 +3CBB3 [by AA-similarity]. AC  AA4  4 · And so, CB BB3 3 Hence, AC : CB  4 : 3.

TO CONSTRUCT A TRIANGLE SIMILAR TO A GIVEN TRIANGLE AS PER GIVEN SCALE FACTOR EXAMPLE 2

Construct a triangle with sides 5 cm, 5.5 cm and 6.5 cm. Then 3 construct another triangle whose sides are times the corresponding 5 sides of the first triangle. [CBSE 2014]

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Secondary School Mathematics for Class 10

STEPS OF CONSTRUCTION

Step 1. Draw a line segment BC  5.5 cm. Step 2. With B as centre and radius equal

to 5 cm, draw an arc. Step 3. With C as centre and radius equal

to 6.5 cm, draw another arc cutting the previous arc at A. Step 4. Join AB and AC.

Thus, 3ABC with given lengths of sides is obtained. Step 5. Below BC, make an acute angle +CBX. Step 6. Along BX, mark off five points B1, B2, B3, B4 and B5 such that

BB1  B1 B2  B2 B3  B3 B4  B4 B5 . Step 7. Join B5 C. Step 8. From B3, draw B3 D  B5 C (by making an angle+BB3 D  +BB5 C),

meeting BC at D. Step 9. From D, draw DE  CA (by making an angle +BDE  +BCA),

meeting AB at E. Then, 3EBD is the required triangle, each of whose sides is of the corresponding side of 3ABC. PROOF

3 5

Since DE  CA, we have 3ABC +3EBD. 

EXAMPLE 3

EB  DE  BD  3 · AB CA BC 5 Draw a triangle ABC in which AB  5 cm, BC  6 cm and 5 +ABC  60c. Then, construct a triangle whose sides are times the 7 corresponding sides of 3ABC. [CBSE 2011, ’15]

STEPS OF CONSTRUCTION

Step 1. Draw a line segment BC  6 cm. Step 2. At B, construct +CBX  60c. Step 3. With B as centre and radius 5 cm, draw an arc cutting ray BX

at A. Step 4. Join AC.

Thus, 3ABC is obtained.

Constructions

515

Step 5. Below BC, make an acute angle +CBY. Step 6. Along BY, mark off seven points B1, B2, B3, B4, B5, B6 and B7

such that BB1  B1 B2  B2 B3  B3 B4  B4 B5  B5 B6  B6 B7 . Step 7. Join B7 C. Step 8. From B5, draw B5 D  B7 C, meeting BC at D. Step 9. From D, draw DE  CA, meeting AB at E.

Then, 3EBD is the required triangle, each of whose sides is of the corresponding side of 3ABC. PROOF

5 7

Since DE  CA, we have 3ABC +3EBD. 

EXAMPLE 4

EB  DE  BD  5 · AB CA BC 7 Draw a triangle ABC with side BC  7 cm, +B  45c, and 4 +A  105c. Then, construct a triangle whose sides are times the 3 corresponding sides of 3ABC.

STEPS OF CONSTRUCTION

Step 1. Draw a line segment BC  7 cm. Step 2. At B, construct +CBX  45c and at C, construct

+BCY  180c  (+B  +A)  180c  (45c  105c)  30c. Suppose BX and CY intersect at A. 3ABC is thus obtained.

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Secondary School Mathematics for Class 10

Step 3. Below BC, make an acute angle +CBZ. Step 4. Along BZ, mark off four points B1, B2, B3 and B4 such that

BB1  B1 B2  B2 B3  B3 B4 . Step 5. Join B3 C. Step 6. From B4, draw B4 D  B3 C, meeting BC (produced) at D. Step 7. From D, draw DE  CA, meeting ray BX at E.

Then, 3EBD is the required triangle, each of whose sides is times the corresponding side of 3ABC. PROOF

4 3

Since CA  DE, we have 3ABC +3EBD. 

EXAMPLE 5

EB  DE  BD  4 · AB CA BC 3 Construct an isosceles triangle whose base is 6 cm and altitude 3 4 cm. Then, construct another triangle whose sides are times the 4 corresponding sides of the first triangle. [CBSE 2010, ’15]

STEPS OF CONSTRUCTION

Step 1. Draw a line segment BC  6 cm. Step 2. Draw the perpendicular bisector XY of BC, cutting BC at D. Step 3. With D as centre and radius 4 cm, draw an arc cutting XY

at A. Step 4. Join AB and AC.

Thus, isosceles 3ABC having base 6 cm and altitude 4 cm is obtained.

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517

Step 5. Below BC, make an acute angle +CBZ. Step 6. Along BZ, mark off four points B1, B2, B3 and B4 such that

BB1  B1 B2  B2 B3  B3 B4 . Step 7. Join B4 C. Step 8. From B3, draw B3 E  B4 C, meeting BC at E. Step 9. From E, draw EF  CA, meeting BA at F.

Then, 3FBE is the required triangle, each of whose sides is times the corresponding side of 3ABC. PROOF

3 4

Since EF  CA, we have 3FBE +3ABC. 

EXAMPLE 6

FB  EF  BE  3 · AB CA BC 4 Construct a 3ABC in which AB  4 cm, +B  60c and altitude CL  3 cm. Construct a 3ADE similar to 3ABC such that each side of 3 3ADE is times that of the corresponding side of 3ABC. 2

STEPS OF CONSTRUCTION

Step 1. Draw a line segment AB  4 cm. Step 2. Construct +ABP  60c. Step 3. Draw a line GH  AB at a distance of 3 cm from AB, intersecting

BP at C.

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Secondary School Mathematics for Class 10 Step 4. Join CA.

Thus, 3ABC is obtained. Step 5. Extend AB to D such that AD 

3 3 AB  a # 4k cm  6 cm. 2 2

Step 6. Draw DE  BC, cutting AC produced at E.

Then, 3ADE is the required triangle similar to 3ABC such 3 that each side of 3ADE is times the corresponding side of 2 3ABC. PROOF

Since DE  BC, we have 3ADE +3ABC. 

f

AD  DE  AE  3 · AB BC AC 2

EXERCISE 9A

1. Draw a line segment AB of length 7 cm. Using ruler and compasses, AP  3 · find a point P on AB such that [CBSE 2011] AB 5 2. (i) Draw a line segment of length 8 cm and divide it internally in the ratio 4 : 5. [CBSE 2017] (ii) Draw a line segment of length 7.6 cm and divide it in the ratio 5 : 8. Measure the two parts. 3. Construct a3PQR, in which PQ  6 cm, QR  7 cm and PR  8 cm. Then, 4 construct another triangle whose sides are times the corresponding 5 [CBSE 2013, ’14] sides of 3PQR. 4. Construct a triangle with sides 5 cm, 6 cm and 7 cm and then another 7 triangle whose sides are of the corresponding sides of the first 5 triangle. [CBSE 2008C]

Constructions

519

5. Construct a 3ABC with BC  7 cm, +B  60c and AB  6 cm. Construct 3 another triangle whose sides are times the corresponding sides of 4 [CBSE 2008, ’09, ’15] 3ABC. 6. Construct a3ABC in which AB  6 cm, +A  30c and+B  60c. Construct another 3ABlCl similar to 3ABC with base ABl  8 cm. [CBSE 2015] 7. Construct a 3ABC in which BC  8 cm, +B  45c and+C  60c. Construct 3 another triangle similar to 3ABC such that its sides are of the 5 corresponding sides of 3ABC. [CBSE 2010, ’12, ’14] 8. To construct a triangle similar to 3ABC in which BC  4.5 cm, +B  45c 3 and +C  60c, using a scale factor of , BC will be divided in the ratio 7 [CBSE 2012]

(a) 3 : 4

(b) 4 : 7

(c) 3 : 10

(d) 3 : 7

9. Construct an isosceles triangle whose base is 8 cm and altitude 4 cm 1 and then another triangle whose sides are 1 times the corresponding 2 sides of the isosceles triangle. 10. Draw a right triangle in which the sides (other than hypotenuse) are of lengths 4 cm and 3 cm. Then, construct another triangle whose sides are 5 times the corresponding sides of the given triangle. [CBSE 2011] 3 HINTS TO SOME SELECTED QUESTIONS 1. Divide the line segment AB in the ratio 3 : 2. 6. Construct 3ABC. Produce AB to Bl such that ABl  8 cm. Draw BlCl  BC such that BlCl meets AC produced at Cl.

CONSTRUCTION OF TANGENTS TO A CIRCLE CONSTRUCTION OF A TANGENT TO A CIRCLE AT A GIVEN POINT ON THE CIRCLE EXAMPLE 1

Draw a circle of radius 2.5 cm. Take a point P on it. Construct a tangent at the point P.

STEPS OF CONSTRUCTION

Step 1. Draw a circle of radius 2.5 cm taking a point O as its centre. Step 2. Mark a point P on this circle.

520

Secondary School Mathematics for Class 10 Step 3. Join OP. Step 4. Construct +OPT  90c. Step 5. Produce TP to Tl.

Then TlPT is the required tangent. CONSTRUCTION OF A TANGENT TO A CIRCLE AT A POINT ON IT WITHOUT USING THE CENTRE EXAMPLE 2

Draw a circle of radius 5 cm. Take a point P on it. Without using the [CBSE 2002] centre of the circle construct a tangent at the point P.

STEPS OF CONSTRUCTION

Step 1. Draw a circle of radius 5 cm. Step 2. Mark a point P on it. Step 3. Draw any chord PQ. Step 4. Take a point R on the major arc QP. Step 5. Join PR and RQ. Step 6. Make +QPT  +PRQ. Step 7. Produce TP to Tl, as shown in the figure.

Then, TlPT is the required tangent at P. CONSTRUCTION OF TANGENTS TO A CIRCLE FROM A POINT OUTSIDE IT WHEN THE CENTRE OF THE CIRCLE IS KNOWN EXAMPLE 3

Draw a circle of radius 2 cm with centre O and take a point P outside the circle such that OP  6.5 cm. From P, draw two tangents to the circle.

STEPS OF CONSTRUCTION

Step 1. Draw a circle with O as centre

and radius 2 cm. Step 2. Mark a point P outside the circle

such that OP  6.5 cm. Step 3. Join OP and bisect it at M. Step 4. Draw a circle with M as centre

and radius equal to MP, to intersect the given circle at the points T and Tl.

Constructions

521

Step 5. Join PT and PTl.

Then, PT and PTl are the required tangents. EXAMPLE 4

Write the steps of construction for drawing two tangents to a circle of radius 6 cm from a point 10 cm away from its centre. [CBSE 2013C]

STEPS OF CONSTRUCTION

Step 1. Draw a circle with O as centre

and radius 6 cm. Step 2. Mark a point P outside the circle

such that OP  10 cm. Step 3. Join OP and bisect it at a point M. Step 4. Draw a circle with M as the

centre and radius equal to MP, to intersect the given circle at the points T and Tl. Step 5. Join PT and PTl.

Then, PT and PTl are the required tangents. EXAMPLE 5

Draw a line segment AB of length 7 cm. Taking A as centre, draw a circle of radius 3 cm and taking B as centre, draw another circle of radius 2 cm. Construct tangents to each circle from the centre of the other circle. [CBSE 2011, ’15]

STEPS OF CONSTRUCTION

Step 1. Draw

a line AB  7 cm.

segment

Step 2. With A as centre and radius

3 cm, draw a circle. Step 3. With B as centre and radius 2 cm, draw another circle. Step 4. Bisect the line segment AB

at M. Step 5. With M as centre and radius equal to AM, draw a circle

intersecting the first circle at Q and Ql and the second circle at P and Pl. Step 6. Join AP and APl. Also, join BQ and BQl.

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Secondary School Mathematics for Class 10

Then, BQ and BQl are the required tangents to the first circle and AP and APl are the required tangents to the second circle. CONSTRUCTION OF TANGENTS TO A CIRCLE FROM A POINT OUTSIDE IT WITHOUT USING THE CENTRE EXAMPLE 6

Draw a circle of radius 2.5 cm and take a point P outside it. Without using the centre of the circle, draw two tangents to the circle from the point P.

STEPS OF CONSTRUCTION

Step 1. Draw a circle of radius

2.5 cm and take a point p outside it. Step 2. Through P draw a secant

PAB to intersect the circle at A and B. Step 3. Produce AP to a point C such that PA  PC. Step 4. Draw a semicircle with CB as diameter. Step 5. Draw PD = CB, intersecting the semicircle at D. Step 6. With P as centre and PD as radius, draw arcs to intersect the

circle at T and Tl. Step 7. Join PT and PTl.

Then, PT and PTl are the required tangents. CONSTRUCTION OF TANGENTS TO A CIRCLE INCLINED TO EACH OTHER AT A GIVEN ANGLE EXAMPLE 7

Draw a circle of radius 3 cm. Draw a pair of tangents to this circle, which are inclined to each other at an angle of 60c.

STEPS OF CONSTRUCTION

Step 1. Draw a circle with O as centre and radius = 3 cm. Step 2. Draw any diameter AOB of this circle. Step 3. Construct +BOC  60c such that radius OC meets the circle

at C.

Constructions

523

Step 4. Draw AM = AB and CN = OC.

Let AM and CN intersect each other at P. Then, PA and PC are the desired tangents to the given circle, inclined at an angle of 60. PROOF

+AOC  (180c  60c)  120c. In quad. OAPC, we have +OAP  90c, +AOC  120c, +OCP  90c. 

EXAMPLE 8

+APC  [360c  (90c  120c  90c)]  60c. Draw a circle of radius 3.5 cm. Draw two tangents to the circle which are perpendicular to each other. [CBSE 2015]

STEPS OF CONSTRUCTION

Step 1. Draw a circle with O as centre

and radius = 3.5 cm. Step 2. Draw any diameter AOB of

this circle. Step 3. Construct +BOX  90c such

that ray OX meets the circle at C. Step 4. Draw AM = AB and CN =OC.

Let AM and CN intersect each other at a point P. Then, PA and PC are the required tangents (which are perpendicular to each other). PROOF

+AOC  (180c  +BOC)  180c  90c  90c. In quad. OAPC, we have +OAP  90c, +AOC  90c, +OCP  90c. 

+APC  [360c  (90c  90c  90c)]  90c.

CONSTRUCTION OF TANGENTS TO A CIRCLE FROM A POINT ON A LARGER CIRCLE CONCENTRIC WITH THE FIRST ONE EXAMPLE 9

Draw two concentric circles of radii 2 cm and 5 cm. Taking a point on the outer circle, construct a pair of tangents to the other. Measure

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Secondary School Mathematics for Class 10

the lengths of the two tangents. Also, verify the measurement by actual calculation. [CBSE 2013C] STEPS OF CONSTRUCTION

Step 1. Mark a point O on the paper. Step 2. With O as centre and radii

2 cm and 5 cm, draw two concentric circles. Step 3. Mark a point P on the outer

circle. Step 4. Join OP and bisect it at a

point M. Step 5. Draw a circle with M as the centre and radius equal to MP, to

intersect the inner circle in points T and Tl. Step 6. Join PT and PTl.

Then, PT and PTl are the required tangents. Measurement Upon measuring the two tangents, we find PT  PTl  4.6 cm. Calculation of the length of tangent

Join OT to form a right 3OTP [a PT is a tangent at point T]. Then, PT  OP 2  OT 2  5 2  2 2  25  4  21 . 4.58 cm. [a OP  radius of larger circle = 5 cm, OT  radius of smaller circle = 2 cm.] EXAMPLE 10

Construct a right triangle ABC with AB  6 cm, BC  8 cm and +B  90c. Draw BD, the perpendicular from B on AC. Draw the circle through B, C and D and construct the tangents from A to this circle. [CBSE 2014, ’15]

STEPS OF CONSTRUCTION

Step 1. Draw a line segment AB  6 cm. Step 2. At B, construct +ABX  90c. Step 3. With B as centre and radius 8 cm, draw an arc cutting ray BX

at C. Step 4. Join AC.

Thus, 3ABC is obtained.

Constructions

525

Step 5. From B, draw BD = AC. Step 6. Bisect BC at point O. Step 7. With O as centre and radius OB, draw a circle. This circle

passes through B, C and D. Thus, the required circle is obtained. Step 8. Join AO and bisect it at M. Step 9. With M as the centre and radius equal to AM, draw a circle

cutting the previous circle at the points T and B. Step 10. Join AT.

Then, AT and AB are the required tangents. f

EXERCISE 9B

1. Draw a circle of radius 3 cm. From a point P, 7 cm away from the centre of the circle, draw two tangents to the circle. Also, measure the lengths of the tangents. [CBSE 2010] 2. Draw two tangents to a circle of radius 3.5 cm from a point P at a distance of 6.2 cm from its centre. [CBSE 2013] 3. Draw a circle of radius 3 cm. Take two points P and Q on one of its diameters extended on both sides, each at a distance of 7 cm on opposite sides of its centre. Draw tangents to the circle from these two points P and Q. [CBSE 2017] 4. Draw a circle with centre O and radius 4 cm. Draw any diameter AB of this circle. Construct tangents to the circle at each of the two end points of the diameter AB.

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Secondary School Mathematics for Class 10

5. Draw a circle with the help of a bangle. Take any point P outside the circle. Construct the pair of tangents from the point P to the circle. HINT

See Example 6.

6. Draw a line segment AB of length 8 cm. Taking A as centre, draw a circle of radius 4 cm and taking B as centre, draw another circle of radius 3 cm. Construct tangents to each circle from the centre of the other circle. [CBSE 2014] 7. Draw a circle of radius 4.2 cm. Draw a pair of tangents to this circle inclined to each other at an angle of 45. 8. Write the steps of construction for drawing a pair of tangents to a circle of radius 3 cm, which are inclined to each other at an angle of 60. [CBSE 2011, ’12, ’14]

9. Draw a circle of radius 3 cm. Draw a tangent to the circle making an angle of 30 with a line passing through the centre. HINT

Draw a circle with centre O and radius 3 cm. Draw a radius OA and produce it to B. Make +AOP  60c. Draw PQ = OP, meeting OB at Q. Then, PQ is the desired tangent such that +OQP  30c.

10. Construct a tangent to a circle of radius 4 cm from a point on the concentric circle of radius 6 cm and measure its length. Also, verify the measurement by actual calculation. 11. Draw two concentric circles of radii 3 cm and 5 cm. Taking a point on the outer circle, construct the pair of tangents to the inner circle. [CBSE 2017]

12. Write the steps of construction to construct the tangents to a circle from an external point. [CBSE 2017]

TEST YOURSELF Long-Answer Questions 1. Draw a line segment AB of length 5.4 cm. Divide it into six equal parts. Write the steps of construction. 2. Draw a line segment AB of length 6.5 cm and divide it in the ratio 4 : 7. Measure each of the two parts.

Constructions

527

3. Construct a 3ABC in which BC  6.5 cm, AB  4.5 cm and +ABC  60c. 3 Construct a triangle similar to this triangle whose sides are of the 4 corresponding sides of 3ABC. [CBSE 2009] 4. Construct a 3ABC in which BC  5 cm, +C  60c and altitude from A equal to 3 cm. Construct a 3ADE similar to 3ABC such that each side 3 of 3ADE is times the corresponding side of 3ABC. Write the steps of 2 construction. 5. Construct an isosceles triangle whose base is 9 cm and altitude 5 cm. 3 Construct another triangle whose sides are of the corresponding 4 sides of the first isosceles triangle. [CBSE 2009C] 6. Draw a 3ABC, right-angled at B such that AB  3 cm and BC  4 cm. 7 Now, construct a triangle similar to 3ABC, each of whose sides is 5 times the corresponding side of 3ABC. 7. Draw a circle of radius 4.8 cm. Take a point P on it. Without using the centre of the circle, construct a tangent at the point P. Write the steps of construction. 8. Draw a circle of radius 3.5 cm. Draw a pair of tangents to this circle which are inclined to each other at an angle of 60. Write the steps of construction. 9. Draw a circle of radius 4 cm. Draw tangent to the circle making an angle of 60 with a line passing through the centre. 10. Draw two concentric circles of radii 4 cm and 6 cm. Construct a tangent to the smaller circle from a point on the larger circle. Measure the length of this tangent. 

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Secondary School Mathematics for Class 10

Trigonometric Ratios

10

It is that branch of mathematics which deals with the measurement of angles and the problems allied with angles.

TRIGONOMETRY

TRIGONOMETRIC RATIOS (T-RATIOS) OF AN ACUTE ANGLE OF A RIGHT TRIANGLE

Let +BAC   be an acute angle of a right-angled 3ABC. In right-angled 3ABC, let base  AB  x units, perpendicular  BC  y units and hypotenuse  AC  r units. We define the following ratios, known as Trigonometric Ratios for . perpendicular y (i) sine  hypotenuse  r , and is written as sin . base x (ii) cosine  hypotenuse  r , and is written as cos . (iii) tangent 

perpendicular y  , and is written as tan . x base

hypotenuse r (iv) cosecant  perpendicular  y , and is written as cosec . (v) secant 

hypotenuse r  , and is written as sec . x base

base x (vi) cotangent  perpendicular  y , and is written as cot . RECIPROCAL RELATION

Clearly, we have 1 (i) cosec   sin 

1 (ii) sec   cos  528

1 (iii) cot   tan 

Trigonometric Ratios

529

SUMMARY

Consider a 3 ABC in which +B  90c and +A  . Let AB  x, BC  y and AC  r. Then, y y x (i) sin   r (ii) cos   r (iii) tan   x 1 r 1 r (iv) cosec   sin   y (v) sec   cos   x 1 x (vi) cot   tan   y Also, we have 1 (i) cosec   sin 

1 (ii) sec   cos 

1 (iii) cot   tan 

T-RATIOS OF AN ANGLE ARE WELL-DEFINED THEOREM

PROOF

Show that the value of each of the trigonometric ratios of an angle does not depend on the size of the triangle. It only depends on the angle. Consider a 3ABC in which +B  90c and +A  c. Take a point P on AC and draw PQ = AB. Then, 3AQP is similar to 3ABC. AQ AP PQ    AB AC CB PQ CB    AP AC sin . AQ AB Similarly, AP  AC  cos  PQ CB and AQ  AB  tan . Similarly, if we produce AC to R and draw RS = AB produced then 3ASR is similar to 3ABC. AS  AR  RS  AB AC CB RS  CB   AR AC sin . AS AB Similarly, AR  AC  cos  RS  CB  and AS AB tan . Hence, the trigonometric ratios of an angle do not depend on the size of the triangle. They only depend on the angle.

530 REMARK

Secondary School Mathematics for Class 10

Consider a 3ABC in which +B  90c and +A  c. We know that in a right triangle, the hypotenuse is the longest side. 

BC AB sin   AC  1 and cos   AC  1.

Thus, sin   1 and cos   1. POWER OF T-RATIOS

We write

(sin )2  sin 2 ;

(sin )3  sin 3 ;

(cos )3  cos 3 ; etc.

QUOTIENT RELATION OF T-RATIOS THEOREM 1

For any acute angle , prove that sin  (i) tan   cos  ;

PROOF

cos  (ii) cot   sin  ;

(iii) tan  · cot   1.

Consider a right-angled 3ABC in which +B  90c and +A  c. Let AB  x units, BC  y units and AC  r units. Then, y (y/r) (i) tan  x  (x/r) [dividing num. and denom. by r]  sin  · cos  

sin  tan   cos  ·

x (x/r) (ii) cot  y  (y/r)

[dividing num. and denom. by r]

 cos  · sin  

cos  cot   sin  ·

sin  cos  (iii) tan  · cot   cos  · sin   1. SUMMARY

sin  (i) tan   cos 

cos  (ii) cot   sin 

(iii) tan  · cot   1

Trigonometric Ratios

531

SQUARE RELATION THEOREM 2

For any acute angle , prove that (i) sin 2   cos 2   1; (ii) 1  tan 2   sec 2 ; (iii) 1  cot 2   cosec 2 .

PROOF

Consider a right-angled 3ABC in which +B  90c and +A  c. Let AB  x units, BC  y units and AC  r units. Then, by Pythagoras‘ theorem, we have x2  y2  r2 . y2 x2 y2 x2 (i) sin 2   cos 2   a r k  ` r j  d 2  2 n r r (x 2  y 2) r 2  2  r2 r  1.    sin 2   cos 2   1. y2 y2 y2  x2 r2  2 (ii) 1  tan 2  1  a x k  1  2  x x2 x 2  ` r j  sec 2 . x  1  tan 2   sec 2 . 2 2 x2 x2 x  y  r2 (iii) 1  cot 2  1  a y k  1  2  2 y y y2

[a x 2  y 2  r 2]

[a x 2  y 2  r 2]

[a x 2  y 2  r 2]

 a r k  cosec 2 . y 2

 1  cot 2   cosec 2 . SUMMARY

(i) sin 2   cos 2   1

(ii) 1  tan 2   sec 2 

(iii) 1  cot 2   cosec 2 

SOLVED EXAMPLES EXAMPLE 1 SOLUTION

8 If sin A  17 , find other trigonometric ratios of +A. Let us draw a 3ABC in which +B  90c. BC 8 Then, sin A  AC  17 · Let BC  8k and AC  17k, where k is positive.

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Secondary School Mathematics for Class 10

By Pythagoras‘ theorem, we have 

AC 2  AB 2  BC 2 AB 2  AC 2  BC 2



AB 2  (17k) 2  (8k) 2  289k 2  64k 2  225k 2

AB  225k 2  15k. BC 8k  8 AB 15k  15  sin A  AC  ; cos A  AC  ; 17k 17 17k 17 sin A 8 17 8 tan A  cos A  a17 # 15 k  15 ; 1 17 1 17 cosec A  sin A  8 ; sec A  cos A  15 1 15 and cot A  tan A  8 · 

EXAMPLE 2 SOLUTION

9 If cos A  41 , find other trigonometric ratios of +A. Let us draw a 3ABC in which +B  90c. AB 9 Then, cos A  AC  41 · Let AB  9k and AC  41k, where k is positive. By Pythagoras‘ theorem, we have AC 2  AB 2  BC 2  BC 2  AC 2  AB 2  BC 2  (41k) 2  (9k) 2  1681k 2  81k 2  1600k 2  BC  1600k 2  40k. BC 40k  40 9  sin A  AC  ; cos A  41 (given); 41k 41 sin A 40 41 40 tan A  cos A  a 41 # 9 k  9 ; 1 41 1 41 cosec A  sin A  40 ; sec A  cos A  9 1 9 and cot A  tan A  40 ·

EXAMPLE 3

If tan A  3 , find other trigonometric ratios of +A.

SOLUTION

Let us draw a 3ABC in which +B  90c. 3 BC Then, tan A  AB  1 · Let BC  3 k and AB  k, where k is positive.

Trigonometric Ratios

By Pythagoras‘ theorem, we have AC 2  AB 2  BC 2  k 2  ( 3 k) 2  k 2  3k 2  4k 2 . 

AC  4k 2  2k.



BC sin A  AC  AB cos A  AC 

3k 3  2 ; 2k k 1 ; 2k 2

1 2 tan A  3 (given); cosec A  sin A  ; 3 1 1 1 · sec A  cos A  2 and cot A  tan A  3 EXAMPLE 4 SOLUTION

25 If sec  7 , find all trigonometric ratios of . Let us draw a 3ABC in which +B  90c. Let +A  c. AC 25 Then, sec  AB  7 · Let AC  25k and AB  7k, where k is positive. By Pythagoras‘ theorem, we have AC 2  AB 2  BC 2  BC 2  AC 2  AB 2  (25k)2  (7k)2  625k 2  49k 2  576k 2  BC  576k 2  24k. 

BC 24k  24 1 7 sin   AC  ; cos   sec   25 ; 25k 25 sin  24 25 24 1 25 tan   cos   a 25 # 7 k  7 ; cosec   sin   24 ; 25 1 7 sec  7 (given) and cot   tan   24 ·

EXAMPLE 5

5 cosec   4 tan  k · 3 If cos  5 , find the value of a sec   cot 

SOLUTION

Let us draw a 3ABC in which +B  90c. Let +A  c. AB 3 Then, cos  AC  5 · Let AB  3k and AC  5k, where k is positive.

533

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Secondary School Mathematics for Class 10

By Pythagoras‘ theorem, we have AC 2  AB 2  BC 2  BC 2  AC 2  AB 2  (5k) 2  (3k) 2  25k 2  9k 2  16k 2  BC  16k 2  4k. 1 5 BC 4k  4  sec   cos   3 ; tan   AB  ; 3k 3 1 3 AC 5k  5 · cot   tan   4 ; and cosec   BC  4k 4 5 4  16 k a5 # 4 4 # 3 k a 25  4 tan  3 4 5 cosec    k  a 53 53 sec   cot  a3 4 k a3 4 k 75  64  12  a11 # 12 k  11 · 29 29 12 20  9 12 EXAMPLE 6

(2cos   sin ) 12 5 ·  If sec  4 , show that 7 (cot   tan )

SOLUTION

Consider a 3ABC in which +A  and +B  90c. hypotenuse AC 5 5x    AB 4 4x (say). base AC  5x and AB  4x, where x is positive.

sec  

By Pythagoras’ theorem we have AC 2  AB 2  BC 2  BC 2  AC 2  AB 2  (5x)2  (4x)2  9x 2  BC  3x. AB 4x 4 BC 3x 3  cos   AC  5x  5 ; sin   AC  5x  5 ; AB 4x 4 BC 3x 3 cot   BC  3x  3 ; and tan   AB  4x  4 · 4 3 8 3 5 (2cos   sin ) a2 # 5  5 k a 5  5 k a 5 k     1  12 ·  7 7 43 43 (cot   tan ) k 7 a3 4 k a 3 4 k a12 k a12 EXAMPLE 7

SOLUTION

In a 3ABC it is given that +B  90c and AB : AC  1 : 2 . Find the 2tan A · value of a k 1  tan 2 A Consider a 3ABC in which +B  90c and AB : AC  1 : 2 .

Trigonometric Ratios

535

Let AB  x. Then, AC  2 x. By Pythagoras‘ theorem, we have AC 2  AB 2  BC 2 & BC 2  AC 2  AB 2  BC 2  ( 2 x)2  (x) 2  2x 2  x 2  x 2  BC  x. 

BC x tan A  AB  x  1.

So, the given expression  c

EXAMPLE 8

SOLUTION

EXAMPLE 9

SOLUTION

If 3tan  4, evaluate

2 tan A  2 #1  2  m a  k 2 1. 1 1 1  tan 2 A

3 sin   2 cos  · 3 sin   2 cos 

4 3 tan   4 & tan   3 · Given expression   3 sin  2 cos  3 sin   2 cos    3 tan  2 [dividing num. and denom. by cos ] 3 tan   2 a3 # 34  2k 6   3.  :a tan  4D 3 2 4 a3 # 3 2k If 5 cot  3, find the value of a

5 sin   3 cos  k · 4 sin   3 cos 

3 5 cot   3 & cot   5 · 5 sin   3 cos  k  (5  3 cot ) Given expression  a 4 sin   3 cos  (4  3 cot ) [dividing num. and denom. by sin ] 

a5  3 # 35 k

a4  3 # 35 k



a5  95 k

a4  95 k

EXAMPLE 10

If 7 sin 2   3 cos 2   4, show that tan 

SOLUTION

7 sin 2   3cos 2   4 

4 sin 2   3 sin 2   3 cos 2   4



4 sin 2   3(sin 2   cos 2 )  4

 a16 # 5 k  16 · 5 29 29

1 · 3

[CBSE 2008]

536

Secondary School Mathematics for Class 10



4 sin 2  3 # 1  4



1 4 sin 2   1 & sin 2   4 ·

[a sin 2   cos 2   1]

1 3 cos 2   (1  sin 2 )  a1  4 k  4 · sin 2   1 a 4 # 34 k  13 ·  tan 2   cos 2  1 · Hence, tan  3 

EXAMPLE 11

(2  2 sin )(1  sin ) 15 · If cot  8 then evaluate  (1 cos )(2  2 cos )

SOLUTION

Given expression 

(2  2 sin )(1  sin ) (1  cos )(2  2 cos )



2(1  sin )(1  sin ) 2(1  cos )(1  cos )



(1  sin 2 ) cos 2    cot 2  (1  cos 2 ) sin 2 

[CBSE 2009]

2  (cot ) 2  a15 k  225 · 8 64

225 Hence, the value of the given expression is 64 · EXAMPLE 12

SOLUTION

In 3ABC, right-angled at B, AB  5 cm and BC  12 cm. Find the values of sin A, sec A, sin C and sec C. In 3ABC, +B  90c, AB  5 cm and BC  12 cm. By Pythagoras‘ theorem, we have AC 2  (AB 2  BC 2)  {(5)2  (12)2} cm 2  (25  144) cm 2  169 cm 2 . 

AC  169 cm 2  13 cm.

For T-ratios of +A, we have base = AB = 5 cm, perpendicular = BC = 12 cm and hypotenuse = AC = 13 cm. BC 12 AC 13 sin A  AC  13 and sec A  AB  5 · For T-ratios of +C, we have



base = BC = 12 cm,

Trigonometric Ratios

537

perpendicular = AB = 5 cm and hypotenuse = AC = 13 cm.  EXAMPLE 13

SOLUTION

AB 5 AC 13 sin C  AC  13 and sec C  BC  12 ·

In a 3ABC, +B  90c, AB  5 cm and (BC  AC)  25 cm. Find the values of sin A, cos A, cosec C and sec C. Let BC  x cm. Then, AC  (25  x) cm. By Pythagoras‘ theorem, we have   

AB 2  BC 2  AC 2 5 2  x 2  (25  x) 2 25  x 2  625  50x  x 2 50x  600

 x  12.  BC  12 cm, AC  13 cm and AB  5 cm. For T-ratios of +A, we have AB 5 BC 12 sin A  AC  13 and cos A  AC  13 · For T-ratios of +C, we have base, BC = 12 cm, perpendicular, AB = 5 cm and hypotenuse, AC = 13 cm.  EXAMPLE 14

SOLUTION

AC 13 AC 13 cosec C  AB  5 and sec C  BC  12 ·

In a 3ABC, +B  90c, AB  7 cm and (AC  BC)  1 cm. Find the values of sin A, cos A, sin C and cos C. Let BC  x cm. Then, AC  (x  1) cm. By Pythagoras‘ theorem, we have AB 2  BC 2  AC 2  7 2  x 2  (x  1) 2 

49  x 2  x 2  2x  1



2x  48



x  24.



BC  24 cm, AC  25 cm and AB  7 cm.

538

Secondary School Mathematics for Class 10

For T-ratios of +A, we have BC 24 AB 7 sin A  AC  25 and cos A  AC  25 · For T-ratios of +C, we have AB 7 BC 24 sin C  AC  25 and cos C  AC  25 ·

EXAMPLE 15

In a 3ABC, +C  90c and tan A 

1 · Find the values of: 3

(i) (sin A $ cos B  cos A $ sin B) (ii) (cos A · cos B  sin A · sin B) [CBSE 2008] SOLUTION

Consider a 3ABC in which +C  90c and tan A  Then, tan A 

1 BC  1 · & AC 3 3

1 · 3

Let BC  x. Then, AC  3 x. By Pythagoras‘ theorem, we have AB 2  AC 2  BC 2  ( 3 x)2  x 2  (3x 2  x 2)  4x 2 

AB  4x 2  2x.

For T-ratios of +A, we have base  AC  3 x, perpendicular  BC  x and hypotenuse  AB  2x. 3x 3 BC x 1 AC sin A  AB  2x  2 and cos A  AB  2x  2 · For T-ratios of +B, we have 

base  BC  x, perpendicular  AC  3 x and hypotenuse  AB  2x. 

3x 3 AC BC x 1 sin B  AB  2x  2 and cos B  AB  2x  2 ·

3 3 1 1 (i) (sin A · cos B  cos A · sin B)  c 2 # 2  2 # 2 m  a1  3 k  1. 4 4  (sin A · cos B  cos A · sin B)  1.

Trigonometric Ratios

539

3 3 1 1 (ii) (cos A cos B  sin A sin B)  c 2 # 2  2 # 2 m  0.  (cos A cos B  sin A sin B)  0. EXAMPLE 16

SOLUTION

If +A and +B are acute angles such that cos A  cos B then prove that +A  +B. Let 3ACD and 3BEF be two right triangles given in such a way that cos A  cos B. Then, cos A  cos B AC  BE  AD BF AC AD  BE  BF  k (say)  AC  k (BE) and AD  k (BF) . 

CD  EF 

AD 2  AC 2 BF 2  BE 2 k BF 2  BE 2 k BF 2  BE 2

… (i) [using Pythagoras’ theorem] [using (i)].

AC AD CD Thus, we have: BE  BF  EF ·  3ACD + 3BEF and hence, +A  +B. f

EXERCISE 10

3 1. If sin  2 , find the value of all T-ratios of . 7 2. If cos  25 , find the values of all T-ratios of . 15 3. If tan  8 , find the values of all T-ratios of . 4. If cot  2, find the values of all T-ratios of . 5. If cosec  10 , find the values of all T-ratios of . a2  b2 , find the values of all T-ratios of . a2  b2 7. If 15 cot A  8, find the values of sin A and sec A. 6. If sin 

9 8. If sin A  41 , find the values of cos A and tan A.

540

Secondary School Mathematics for Class 10

9. If cos  0.6, show that (5sin   3tan )  0. sin   10. If cosec  2, show that acot    2. 1 cos  k (cosec 2   sec 2 ) 3 1  · 11. If tan  , show that (cosec 2   sec 2 ) 4 7 (1  sin   cos ) 3 20  · 12. If tan  21 , show that  (1 sin   cos ) 7 (sin   2 cos ) 12 5 ·  13. If sec  4 , show that 7 (tan   cot ) sec   cosec   1 · 3 14. If cot  4 , show that sec   cosec  7 cosec 2   cot 2   7 · 3 sec 2   1 a· b a If sin  , show that (sec   tan )  b ba (sin   cot ) 3 3 ·  If cos  5 , show that 160 2 tan  7 4 If tan  3 , show that (sin   cos )  5 · (a sin   b cos ) (a 2  b 2) a ·  If tan  , show that b (a sin   b cos ) (a 2  b 2) (4 cos   sin ) 4  · If 3 tan  4, show that (2 cos   sin ) 5 (4 sin   3 cos ) 1  · If 3 cot  2, show that (2 sin   6 cos ) 3 (1  tan 2 )  (cos 2   sin 2 ) . If 3 cot  4, show that (1  tan 2 ) (3  4 sin 2 ) (3  tan 2 ) 17 ·  If sec  8 , verify that 2  (4 cos  3) (1  3 tan 2 )

3 15. If sin  4 , show that 16. 17. 18. 19. 20. 21. 22. 23.

24. In the adjoining figure, +B  90c, +BAC  c, BC  CD  4 cm and AD  10 cm. Find (i) sin  and (ii) cos . HINT



AB 2  (AD 2  BD 2)  36 cm 2 AB  6 cm.

AC 2  (AB 2  BC 2)  52 cm 2 

AC  2 13 cm.

Thus, AB  6 cm and AC  2 13 cm.

25. In a 3ABC, +B  90c, AB  24 cm and BC  7 cm. Find (i) sin A

(ii) cos A

(iii) sin C

(iv) cos C.

Trigonometric Ratios

541

26. In a 3ABC, +C  90c, +ABC  c, BC  21 units and AB  29 units. 41 Show that (cos 2   sin 2 )  841 · 27. In a 3ABC, +B  90c, AB  12 cm and BC  5 cm. Find (i) cos A

(ii) cosec A

(iii) cos C

(iv) cosec C.

1 28. If sin  2 , prove that (3 cos   4 cos 3 )  0. 1 · 29. In a 3ABC, +B  90c and tan A  Prove that 3 (i) sin A · cos C  cos A · sin C  1

(ii) cos A · cos C  sin A · sin C  0

30. If +A and +B are acute angles such that sin A  sin B then prove that +A  +B. 31. If +A and +B are acute angles such that tan A  tan B then prove that +A  +B. 32. In a right 3ABC, right-angled at B, if tan A  1 then verify that 2 sin A · cos A  1. 33. In the figure of 3PQR, +P  c and +R  c. Find (i) ( x  1 ) cot  (ii) ( x 3  x 2 ) tan  (iii) cos  34. If x  cosec A  cos A and y  cosec A  cos A then prove that 2 2 2 x  ym   c  m c 1 0. x y 2 35. If x  cot A  cos A and y  cot A  cos A, prove that xy 2 xy 2 c m c m  1. xy 2 ANSWERS (EXERCISE 10)

3

1

1 2 , cosec   , sec   2 3 3 24 7 24 7 25 25 2. sin   25 , cos   25 , tan   7 , cot   24 , cosec   24 , sec   7 15 8 15 8 17 17 3. sin   17 , cos   17 , tan   8 , cot   15 , cosec   15 , sec   8 1. sin   2 , cos   2 , tan   3 , cot  

4. sin  

5 1 2 1 , cos   , tan   2 , cot   2, cosec   5 , sec   2 5 5

542

Secondary School Mathematics for Class 10

5. sin  

10 1 3 1 , cos   , tan   3 , cot   3, cosec   10 , sec   3 10 10

a2  b2 2ab a2  b2 2 2 , cos   2 2 , tan   2ab , a b a b 2  b2 a a2  b2 2ab cosec   2 2 , sec   , cot   2 2 2ab a b a b 40 9 15 17 7. sin A  17 , sec A  8 8. cos A  41 , tan A  40 6. sin  

24. (i) sin 

7

25. (i) 25

12

27. (i) 13 33. (i)

x 2

2 13 13 24 (ii) 25 13 (ii) 5 (ii)

x2 2

3 13 (ii) cos  13 24 7 (iii) 25 (iv) 25 5 13 (iii) 13 (iv) 12 2 x1 (iii) (x  2)

HINTS TO SOME SELECTED QUESTIONS 6. In right 3ABC, +B  90c and +BAC  . a 2  b 2  BC · a 2  b 2 AC 2 Let BC  (a  b 2) and AC  (a 2  b 2) . Then, Given: sin 

AB 2  (AC 2  BC 2)  (a 2  b 2) 2  (a 2  b 2) 2  4a 2 b 2 

AB  2ab.

Now, we can find all the T-ratios of . 8 15 BC 7. cot A  15 & tan A  8  AB ·  AC 2  AB 2  BC 2  (8) 2  (15) 2  64  225  289  

AC  289  17. BC  15 AC  17 · sin A  and sec A  8 AC 17 AB

9 BC 8. sin A  41  AC ·  AB 2  AC 2  BC 2  (41) 2  (9) 2  1681  81  1600  

AB  1600  40. AB 40 BC 9 cos A  AC  41 and tan A  AB  40 ·

Trigonometric Ratios 6 3 AB 9. cos  10  5  AC · BC 2  AC 2  AB 2  (5) 2  (3) 2  25  9  16 & BC  4. 

4 4 sin   5 and tan   3 ·

1 BC 10. cosec   2 & sin   2  AC · AB 2  AC 2  BC 2  (2) 2  (1) 2  4  1  3 & AB  3 . 3 1 sin   2 , cos   2 and cot   3 . (2  3 ) 1 3  2. # Given expression  ) 3  (2  3 ) (2  3 ) 

11. tan 

1  BC · 7 AB



AC 2  AB 2  BC 2  ( 7 ) 2  (1) 2  (7  1)  8 & AC  8  2 2 .



AC 2 2 cosec   BC  1  2 2 & cosec 2   8

AC 2 2 and sec   AB  & sec 2   78 · 7 20 BC 12. tan  21  AB · 

AC 2  AB 2  BC 2  (21) 2  (20) 2  441  400  841



AC  841  29.

5 4 AB 13. sec   4 & cos   5  AC · 

BC 2  AC 2  AB 2  (5) 2  (4) 2  25  16  9 & BC  3.



3 4 3 4 sin   5 , cos   5 , tan   4 , cot   3 ·

16. sin 

a  BC · b AC



AB 2  AC 2  BC 2  b 2  a 2



AB  b 2  a 2 . b a sec   and tan   b2  a2 b2  a2 (b  a) (b  a)  (sec   tan )  (b  a) · (b  a) b2  a2

 



b  a· b  a (b  a) · (b  a)



ba· ba

543

544

Secondary School Mathematics for Class 10

4 BC 18. tan  3  AB ·  AC 2  AB 2  BC 2  9  16  25 & AC  5. 4 3 7  (sin   cos )  a 5  5 k  5 ·

19.

(a sin   b cos ) (a tan   b)  [dividing num. and denom. by cos ] (a sin   b cos ) (a tan   b) a `a # b bj (a 2  b 2)   2 2 · a `a # b  bj (a  b )

(4 cos   sin ) (4  tan )  [dividing num. and denom. by cos ] (2 cos   sin ) (2  tan ) a4  34 k 8 4    · a2  34 k 10 5 (4 sin   3 cos ) (4  3 cot )  21. [dividing num. and denom. by sin ] (2 sin   6 cos ) (2  6 cot ) a4  3 # 23 k 2 1    · a2  6 # 23 k 6 3 20.

4 AB 22. cot  3  BC ·  AC 2  AB 2  BC 2  (4) 2  (3) 2  16  9  25 & AC  5. 

3 4 3 sin   5 , cos   5 and tan   4 ·

17 8 AB 23. sec   8 & cos   17  AC · 

BC 2  AC 2  AB 2  (17) 2  (8) 2  (289  64)  225

BC  225  15. 15 8 15  sin   17 , cos   17 and tan   8 · 33 Then, LHS  RHS  611 · 

25. Given, 3ABC in which +B  90c, BC  7 cm and AB  24 cm. 

AC 2  (24) 2  (7) 2  576  49  625 & AC  25 cm.

BC 7 AB 24 AB 24 (i) sin A  AC  25 · (ii) cos A  AC  25 · (iii) sin C  AC  25 · BC 7 (iv) cos C  AC  25 · 3 1 3 28. cos 2   (1  sin 2 )  a1  4 k  4 & cos   2 · 

3 3 3 3 3 3 3 (3 cos   4 cos 3 )  c3 # 2  4 # 8 m  c 2  2 m  0.

Trigonometric Ratios

545

30. Let 3ACD and 3BEF be two right triangles such that sin A  sin B. Then, sin A  sin B CD EF  AD  BF CD AD  EF  BF  k (say)  CD  k (EF) and AD  k (BF) . 

AC  BE

2

2

AD CD BF 2  EF 2

… (i)

[using Pythagoras’ theorem]

k BF 2  EF 2  k [using (i)]. BF 2  EF 2 CD  AD  AC · EF BF BE 3ACD + 3BEF and hence +A  +B. 

 

31. Let 3ACD and 3BEF be two right triangles such that tan A  tan B. Then, tan A  tan B CD EF  AC  BE CD AC  EF  BE and +C  +E  90c.  3ACD + 3BEF and hence, +A  +B. BC 32. tan A  1 & AB  1 & BC  AB  k (say).  AC 2  (AB 2  BC 2)  (k 2  k 2)  2k 2 & AC  2 k. k  1 2k 2 AB k  1 · and cos A  AC  2k 2 1 1    2 sin A cos A c2 # # m 1. 2 2 

BC sin A  AC 

33. PQ 2  (PR 2  QR 2)  (x  2) 2  x 2  4 (x  1) PQ  2 x  1 . QR  (i) cot   PQ 2 QR  (ii) tan   PQ 2



x x & ( x  1) cot   · 2 x1 x x2 x2  & ( x 3  x 2 ) tan   · 3 2 2 x1 2 x x PQ 2 x  1 ·  (iii) cos  PR (x  2) (x  y) 2 · 34. Adding, we get, cosec A  & sin A  2 (x  y) xy · Subtracting, we get, cos A  2 2 2 2 x  ym    sin 2 A  cos 2 A  1 & c  m  c 1 0. x y 2



546

Secondary School Mathematics for Class 10

T-Ratios of Some Particular Angles

11

TRIGONOMETRIC RATIOS OF 45, 60 AND 30 (GEOMETRICALLY) TRIGONOMETRIC RATIOS OF 45

Let 3ABC be a right-angled triangle in which +B  90c and +A  45c. Then, clearly, +C  45c. +A  +C & AB  BC. Let AB  BC  a units. Then, AC  AB 2  BC 2  a 2  a 2  2a 2  2 a units. Base AB  a; perpendicular BC  a and hypotenuse AC  2 a. 

BC sin 45c  AC 

a  1 ; 2a 2

AB cos 45c  AC 

BC a tan 45c  AB  a  1; 1  2; sec 45c  cos 45c

a  1 ; 2a 2

1  2; sin 45c 1  1. cot 45c  tan 45c

cosec 45c 

TRIGONOMETRIC RATIOS OF 60 AND 30

Consider an equilateral 3ABC with each side equal to 2a. Then, each angle of 3ABC is 60. From A, draw AD = BC. Then, clearly, BD  DC  a. Also, +ADB  90c.  +BAD  30c. From right-angled 3ADB, we have AD  AB 2  BD 2  (2a) 2  a 2  4a 2  a 2  3a 2  3 a. T-RATIOS OF 60

In right-angled 3ADB, we have base BD  a, perpendicular AD  3 a and hypotenuse AB  2a. 546

T-Ratios of Some Particular Angles



3a 3 AD sin 60c  AB  2a  2 ; AD tan 60c  BD  sec 60c 

547

BD a 1 cos 60c  AB  2a  2 ;

3a  3; a

1  2 ; sin 60c 3 1  1 · cot 60c  tan 60c 3

cosec 60c 

1  2; cos 60c

T-RATIOS OF 30

In right-angled 3ADB, we have base AD  3 a, perpendicular BD  a and hypotenuse AB  2a. 

3a 3 AD cos 30c  AB  2a  2 ; 1  2; cosec 30c  sin 30c

BD a 1 sin 30c  AB  2a  2 ; BD a  1 tan 30c  AD  ; 3a 3 1  2 ; sec 30c  cos 30c 3

cot 30c 

1  3. tan 30c

AXIOMS FOR T-RATIOS OF 0

We define: (i) sin 0c  0

(ii) cos 0c  1

(iii) tan 0c  0

(iv) sec 0c  1

cosec 0c and cot 0c are not defined.

NOTE

AXIOMS FOR T-RATIOS OF 90

We define: (i) sin 90c  1

(ii) cos 90c  0

(iii) cosec 90c  1

(iv) cot 90c  0

tan 90c and sec 90c are not defined.

NOTE

TABLE FOR T-RATIOS OF 0, 30, 45, 60, 90



sin 

cos 

tan 

0

0

1

0

30

1 2

1 3

2

45

1 2

3 2 1 2

1

2

60

3 2

1 2

90

1

0

3 not defined

cosec  not defined

sec  1

2 3

cot  not defined

3

2

1

2 3

2

1 3

1

not defined

0

548

Secondary School Mathematics for Class 10

SOLVED EXAMPLES EXAMPLE 1

Evaluate: (i) sin 60c · cos 30c  cos 60c · sin 30c (ii) tan 30c · cosec 60c  tan 60c · sec 30c

SOLUTION

On substituting the values of various T-ratios, we get (i) sin 60c · cos 30c  cos 60c · sin 30c c

3 3 1  # 1 m  a3  1 k  2  1 · 2 # 2 2 2 4 4 4 2 (ii) tan 30c · cosec 60c  tan 60c · sec 30c  c 1 # 2  3 # 2 m  a 2  2k  2 2 · 3 3 3 3 3 EXAMPLE 2

Evaluate: 1 1 (i) sin 2 30c · cos 2 45c  4 tan 2 30c  2 sin 2 90c  8 cot 2 60c (ii)

SOLUTION

tan 2 60c  4 sin 2 45c  3 sec 2 30c  5 cos 2 90c cosec 30c  sec 60c  cot 2 30c

On substituting the values of various T-ratios, we get 1 1 (i) sin 2 30c · cos 2 45c  4 tan 2 30c  2 sin 2 90c  8 cot 2 60c

EXAMPLE 3

2 2 2 2  a1 k · c 1 m  4 # c 1 m  1 # (1) 2  1 # c 1 m 2 2 8 2 3 3 1 1 1 1 1 1 a # 4#  #1 # k 2 3 2 8 3 4 1 4 1 1 48 a    k  8 3 2 24 24 2. tan 2 60c  4 sin 2 45c  3 sec 2 30c  5 cos 2 90c (ii) cosec 30c  sec 60c  cot 2 30c 1 2 2 2 ( 3 ) 2  4 # c m  3 # c m  5 # 02 2 3  2  2  ( 3) 2 JK N 1 4 KK3  4 # 2  3 # 3  5 # 0OOO 3  2  4  0 K k  9. Oa 1 43 L P Show that

(i) cos 60c · cos 30c  sin 60c · sin 30c  cos 90c (ii) cos 60c  1  2 sin 2 30c  2 cos 2 30c  1 tan 60c  tan 30c  (iii)  tan 30c 1 tan 60c · tan 30c

T-Ratios of Some Particular Angles SOLUTION

549

We have (i) cos 60c cos 30c  sin 60c sin 30c 3 3 1m  c 3  3 m    c1 # 0. 2 2 2 #2 4 4 Also, cos 90c  0.  cos 60c cos 30c  sin 60c sin 30c  cos 90c. 1 (ii) cos 60c  2 ; 1 2 1 1 1 1  2 sin 2 30c  :1  2 # a 2 k D  a1  2 # 4 k  a1  2 k  2 ; 2

3 3 3 1 2 cos 2 30c  1  mode > median 1 (c) mean = mode = median (d) mode  (mean + median) 2 24. Look at the cumulative frequency distribution table given below: Monthly income

Number of families

More than ` 10000 More than ` 14000 More than ` 18000 More than ` 20000 More than ` 25000 More than ` 30000

100 85 69 50 37 15

Number of families having income range 20000 to 25000 is (a) 19 (b) 16 (c) 13 (d) 22 25. The median of first 8 prime numbers is (a) 7 (b) 9 (c) 11

(d) 13

26. The mean of 20 numbers is zero. Of them, at the most, how many may be greater than zero? (a) 0 (b) 1 (c) 10 (d) 19

Mean, Median, Mode of Grouped Data, Cumulative Frequency Graph and Ogive 879

27. If the median of the data 4, 7, x  1, x  3, 16, 25, written in ascending order, is 13 then x is equal to (a) 13

(b) 14

(c) 15

(d) 16

28. The mean of 2, 7, 6 and x is 15 and the mean of 18, 1, 6, x and y is 10. What is the value of y? (a) 5

(b) 10

(c) 20

(d) 30

Matching of columns

29. Match the following columns: Column I

Column II

(a) The most frequent value in a data is known as …… .

(p) standard deviation

(b) Which of the following cannot be determined graphically out of mean, mode and median?

(q) median

(c) An ogive is used to determine …… .

(r) mean

(d) Out of mean, mode, median and standard deviation, which is not a measure of central tendency?

(s) mode

The correct answer is: (a) –……,

(b) –……,

(c) –……,

(d) –…… .

Assertion-and-Reason Type

Each question consists of two statements, namely, Assertion (A) and Reason (R). For selecting the correct answer, use the following code: (a) Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A). (b) Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A). (c) Assertion (A) is true and Reason (R) is false. (d) Assertion (A) is false and Reason (R) is true. 30.

Assertion (A)

Reason (R)

If the median and mode of a Mean, median and mode of a frequency distribution are 150 and frequency distribution are related as: 154 respectively, then its mean is 148. mode = 3 median – 2 mean.

The correct answer is: (a)/(b)/(c)/(d).

880

Secondary School Mathematics for Class 10

Assertion (A)

31.

Reason (R)

Consider the following frequency The value of the variable which distribution: occurs most often is the mode. Class interval

3– 6– 9– 12– 15– 18– 6 9 12 15 18 21

Frequency 2

5

21 23 10 12

The mode of the above data is 12.4.

The correct answer is: (a)/(b)/(c)/(d). ANSWERS (MCQ)

1. 10. 19. 28.

5. (d) 6. (b) 7. (b) 8. (b) 9. (b) (d) 2. (a) 3. (a) 4. (c) 11. 12. 13. 14. (c) (b) (b) (c) (b) 15. (c) 16. (b) 17. (a) 18. (c) (b) 20. (c) 21. (c) 22. (b) 23. (c) 24. (c) 25. (b) 26. (d) 27. (c) 30. (a) 31. (b) (c) 29. (a)–(s), (b)–(r), (c)–(q), (d)–(p) HINTS TO SOME SELECTED QUESTIONS

1. Range is not a measure of central tendency. 2. Mean cannot be determined graphically. 3. Only mean is affected by extreme values, while both median and mode remain unaffected. 4. Mode of a frequency distribution can be obtained graphically from a histogram. 6. The cumulative frequency table is useful in determining the median. 8. Clearly, we have f i (xi  x )  0. 9. We have ui 

(xi  A) · h

10. di’s are deviations from A of midpoints of the classes. 11. In computing the mean of the grouped data, we assume that the frequencies are centred at the class marks of the classes. 13. Clearly, the abscissa of the point of intersection of both the ogives gives the median. 14. The class having maximum frequency is the modal class. So, the modal class is 150–155. Its lower limit is 150. N 30 and the cumulative frequency just more than 30 is 37. Its class 2 is 160–165, whose upper limit is 165.

Also, N  60 &

Required sum = (150 + 165) = 315.

Mean, Median, Mode of Grouped Data, Cumulative Frequency Graph and Ogive 881 15. The class 30–40 has maximum frequency. So, the modal class is 30–40. 18. Mode = (3 median) – (2 mean)  (3 # 9  2 # 8.9)  (27  17.8)  9.2. 19. We have:

Class interval

35–45

45–55

55–65

65–75

Frequency

8

12

20

10

Cumulative frequency

8

20

40

50

N 25, which lies in class interval 55–65. 2 b_ ]Z] N ]] a  cf kbbb 2 b  55  (65  55) #(25  20)  57.5. ] Median  l  []h # `b 20 f bb ]] bb ]] a \ 20. The maximum frequency is 25 and the modal class is 22–26. Here, N  50 &



xk  22, f k  25, f k  1  16, f k  1  19 and h  4.



(f k  f k  1) 3 mode  xk  h · ) (2f k  f k  1  f k  1)  (22  4 #

(25  16) 2  a22  4 # 9 k  a22  12 k  (22  2.4)  24.4. 5 15 (50  16  19)

21. 3 # median = (mode + 2 mean)  (16  2 # 28)  72 

median 

72  24. 3

22. Mode = (3 median) – (2 mean) 

2 mean = (3 median) – (mode)  (3 # 26)  29  49



mean 

49  24.5. 2

23. We must have, mean = mode = median. 25. First 8 prime numbers are 2, 3, 5, 7, 11, 13, 17, 19. Required median = mean of fourth and fifth observations   7 11  9. 2 26. Mean of 20 numbers = 0. 

sum of 20 numbers  0 # 20  0.

It is possible that 19 of these numbers may be positive and if their sum is a, the 20th number is ( a) . 27. Clearly, median of 6 observations = mean of 3rd and 4th observations

882

Secondary School Mathematics for Class 10

(x  1)  (x  3) 2x  4   x  2. 2 2  x  2  13 or x  13  2  15. 276x  k 5 or 15  x  20 or x  5. 28. We have a 4 18  1  6  x  y m  10 or 25  5  y  50 or y  20. Also, c 5 

30. Reason (R) is clearly true. Using the relation given in Reason (R), we have 2 mean = (3 median) – (mode)  (3 #150)  (154)  450  154  296. 

mean = 148, which is true.

Thus, Assertion (A) and Reason (R) are both true and Reason (R) is the correct explanation of Assertion (A). Hence, the correct answer is (a). 31. Reason (R) is clearly true. The maximum frequency is 23 and the modal class is 12–15. 

xk  12, f k  23, f k  1  21, f k  1  10 and h  3.



mode  (12  3 #



Assertion (A) is true.

(23  21) 2  a12  3 # 2 k  12.4. 15 (2 # 23  21  10)

But, Reason (R) is not a correct explanation of Assertion (A). Hence, the correct answer is (b).

TEST YOURSELF MCQ 1. Which one of the following measures is determined only after the construction of cumulative frequency distribution? (a) Mean

(b) Median

(c) Mode

(d) None of these

2. If the mean of a data is 27 and its median is 33 then the mode is (a) 30

(b) 43

(c) 45

(d) 47

3. Consider the following distribution: Class

0–5

5–10

10–15

15–20

20–25

Frequency

10

15

12

20

9

The sum of the lower limits of the median class and the modal class is (a) 15

(b) 25

(c) 30

(d) 35

Mean, Median, Mode of Grouped Data, Cumulative Frequency Graph and Ogive 883

4. Consider the following frequency distribution: Class

0–5

6–11

12–17

18–23

24–29

Frequency

13

10

15

8

11

The upper limit of the median class is (a) 16.5

(b) 18.5

(c) 18

(d) 17.5

Very-Short-Answer Questions 5. If the mean and mode of a frequency distribution be 53.4 and 55.2 respectively, find the median. 6. In the table given below, the times taken by 120 athletes to run a 100-mhurdle race are given. Class

13.8–14

Frequency

2

14–14.2 14.2–14.4 14.4–14.6 14.6–14.8 14.8–15 4

15

54

25

20

Find the number of athletes who completed the race in less than 14.6 seconds. 7. Consider the following frequency distribution: Class

0–5

6–11

12–17

18–23

24–29

Frequency

13

10

15

8

11

Find the upper limit of the median class. 8. The annual profits earned by 30 shops of a shopping complex in a locality are recorded in the table shown below: Profit (in lakhs `)

Number of shops

More than or equal to 5

30

More than or equal to 10

28

More than or equal to 15

16

More than or equal to 20

14

More than or equal to 25

10

More than or equal to 30

7

More than or equal to 35

3

If we draw the frequency distribution table for the above data, find the frequency corresponding to the class 20–25.

884

Secondary School Mathematics for Class 10

Short-Answers Questions 9. Find the mean of the following frequency distribution: Class

1–3

3–5

5–7

7–9

Frequency

9

22

27

18

10. The maximum bowling speeds (in km/hr) of 33 players at a cricket coaching centre are given below: Speed in km/hr

85–100

100–115

115–130

130–145

No. of players

10

4

7

9

Calculate the median bowling speed. 11. The arithmetic mean of the following frequency distribution is 50. Class

0–10

10–20

20–30

30–40

40–50

Frequency

16

p

30

32

14

Find the value of p. 12. Find the median of the following frequency distribution: Marks

0–10

10–20

20–30

30–40

40–50

Number of students

6

16

30

9

4

13. Following is the distribution of marks of 70 students in a periodical test: Less than Less than Less than Less than Less than 10 20 30 40 50

Marks Number of students

3

11

28

48

70

Draw a cumulative frequency curve for the above data. 14. Find the median of the following data. Class interval

0–10

10–20

20–30

30–40

40–50

Total

Frequency

8

16

36

34

6

100 [CBSE 2014]

15. For the following distribution draw a ’less than type‘ ogive and from the curve find the median. Marks obtained

Less than 20

Less than 30

Less than 40

Less than 50

Less than 60

Less than 70

Less than 80

Less than 90

Less than 100

Number of students

2

7

17

40

60

82

85

90

100

[CBSE 2014]

Mean, Median, Mode of Grouped Data, Cumulative Frequency Graph and Ogive 885

16. The median value for the following frequency distribution is 35 and the sum of all the frequencies is 170. Using the formula for median, find the missing frequencies. Class

0–10

10–20

20–30

30–40

40–50

50–60

60–70

Frequency

10

20

?

40

?

25

15

17. Find the missing frequencies f1 and f 2 in the table given below, it being given that the mean of the given frequency distribution is 50. Class

0–20

20–40

40–60

60–80

80–100

Total

Frequency

17

f1

32

f2

19

120

18. Find the mean of the following frequency distribution using stepdeviation method: Class

84–90

90–96

96–102

Frequency

15

22

20

102–108 108–114 114–120 18

20

25

Long-Answer Questions 19. Find the mean, median and mode of the following data: Class

0–10

10–20

20–30

30–40

40–50

50–60

60–70

Frequency

6

8

10

15

5

4

2

20. Draw ’less than ogive‘ and ’more than ogive‘ on a single graph paper and hence find the median of the following data: Class interval

5–10

10–15

15–20

20–25

25–30

30–35

35–40

Frequency

2

12

2

4

3

4

3 [CBSE 2014]

21. The production yield per hectare of wheat of some farms of a village are given in the following table: Production yield 40–45 45–50 50–55 55–60 60–65 65–70 70–75 75–80 80–85 (in kg/ha) Number of farms

1

9

15

18

40

26

16

14

10

Draw a less than type ogive and a more than type ogive for this data. [CBSE 2014]

886

Secondary School Mathematics for Class 10

22. The following table gives the marks obtained by 50 students in a class test: Marks

11–15 16–20 21–25 26–30 31–35 36–40 41–45 46–50

Number of students

2

3

6

7

14

12

4

2

Calculate the mean, median and mode for the above data. ANSWERS (TEST YOURSELF)

1. (b)

2. (c)

3. (b)

4. (d)

6. 75

7. 17.5

8. 4

9. 5.42

10. 117.1 km/hr 11. 28

12. 24

14. 27.22

17. f 1  28, f 2  24

18. 102.75

19. Mean = 30, median = 30.67, mode = 32.01 22. Mean = 32, median = 33, mode = 35



5. 54 16. 35, 25

Probability

INTRODUCTION

In class IX, we have studied the concept of empirical probability. Since empirical probability is based on experiments, we also call it experimental probability. Suppose we toss a coin 500 times and get a head, say, 240 times and tail 260 times. Then, we would say that in a single throw of a coin, the 240 12 , i.e., probability of getting a head is ⋅ 500 25 Again, suppose we toss a coin 1000 times and get a head, say, 530 times and tail 470 times. Then, we would say that in a single throw of a coin, the 530 53 probability of getting a head is , i.e., ⋅ 1000 100 Thus, in various experiments, we would get different probabilities for the same event. However, theoretical probability overcomes the above problem. In this chapter, by probability, we shall mean theoretical probability. PROBABILITY

Probability is a concept which numerically measures the degree of certainty of the occurrence of events. Before defining probability, we shall define certain concepts used therein. EXPERIMENT An operation which can produce some well-defined outcomes is called an experiment. RANDOM EXPERIMENT An experiment in which all possible outcomes are known, and the exact outcome cannot be predicted in advance, is called a random experiment. By a trial, we mean ‘performing a random experiment’. Examples (i) Tossing a fair coin (ii) Rolling an unbiased die (iii) Drawing a card from a pack of well-shuffled cards (iv) Picking up a ball from a bag of balls of different colours These are all examples of a random experiment. 887

888

Secondary School Mathematics for Class 10

SOME DETAILS ABOUT THESE EXPERIMENTS

I. Tossing a coin When we throw a coin, either a head (H) or a tail (T) appears on the upper face. II. Throwing a die A die is a solid cube, having 6 faces, marked 1, 2, 3, 4, 5 and 6, or having 1, 2, 3, 4, 5 and 6 dots. In throwing a die, the outcome is the number or number of dots appearing on the uppermost face. The plural of die is dice. III. Drawing a card from a well-shuffled deck of 52 cards. A deck of playing cards has in all 52 cards. (i) It has 13 cards each of four suits, namely spades, clubs, hearts and diamonds.

(a) Cards of spades and clubs are black cards. (b) Cards of hearts and diamonds are red cards.

(ii) Kings, queens and jacks (or knaves) are known as face cards. Thus, there are in all 12 face cards.

LOOKING AT ALL POSSIBLE OUTCOMES IN VARIOUS EXPERIMENTS

I. When we toss a coin, we get either a head (H) or a tail (T). Thus, all possible outcomes are H, T. II. Suppose two coins are tossed simultaneously. Then, all possible outcomes are HH, HT, TH, TT. REMARKS

HH means head on first coin and head on second coin. HT means head on first coin and tail on second coin, etc.

III. On rolling a die, the number on the upper face is the outcome. Thus, all possible outcomes are 1, 2, 3, 4, 5, 6.

Probability

889

IV. In drawing a card from a well-shuffled deck of 52 cards, total number of possible outcomes is 52. EVENT

The collection of all or some of the possible outcomes is called an event.

Examples

(i) In throwing a coin, H is the event of getting a head. (ii) Suppose we throw two coins simultaneously and let E be the event of getting at least one head. Then, E contains HT, TH, HH.

EQUALLY LIKELY EVENTS A given number of events are said to be equally likely if none of them is expected to occur in preference to the others.

For example, if we roll an unbiased die, each number is equally likely to occur. If, however, a die is so formed that a particular face occurs most often then the die is biased. In this case, the outcomes are not equally likely to happen. PROBABILITY OF OCCURRENCE OF AN EVENT

Probability of occurrence of an event E, denoted by P(E) is defined as: P(E) =

number of outcomes favourable to E ⋅ total number of possible outcomes

SURE EVENT

It is evident that in a single toss of die, we will always get a number less than 7. So, getting a number less than 7 is a sure event. 6 P(getting a number less than 7) = = 1. 6 Thus, the probability of a sure event is 1. IMPOSSIBLE EVENT

In a single toss of a die, what is the probability of getting a number 8? We know that in tossing a coin, 8 will never come up. So, getting 8 is an impossible event. P(getting 8 in a single throw of a die) =

0 = 0. 6

Thus, the probability of an impossible event is zero.

890

Secondary School Mathematics for Class 10

COMPLEMENTARY EVENT

Let E be an event and (not E) be an event which occurs only when E does not occur. We denote (not E) by E′, or E, called complement of event E. The event (not E) is called the complementary event of E. Clearly, P(E) + P(not E) = 1. ∴

P(E) = 1 − P(not E).

SUMMARY

(i) For some event E, we have 0 ≤ P(E) ≤ 1. (ii) P(E) = 0 , when E is an impossible event. (iii) P(E) = 1, when E is a sure event. (iv) P(not E) = 1 − P(E). Thus, P(E )=1 − P(E).

SOLVED EXAMPLE 1 SOLUTION

EXAMPLES

A coin is tossed once. What is the probability of getting a head? When a coin is tossed once, all possible outcomes are H and T. Total number of possible outcomes = 2. The favourable outcome is H. Number of favourable outcomes = 1. ∴

EXAMPLE 2 SOLUTION

P(getting a head) number of favourable outcomes 1 = P ( H) = = ⋅ total number of possible outcomes 2

A die is thrown once. What is the probability of getting a prime number? In a single throw of a die, all possible outcomes are 1, 2, 3, 4, 5, 6. Total number of possible outcomes = 6. Let E be the event of getting a prime number. Then, the favourable outcomes are 2, 3, 5. Number of favourable outcomes = 3. 3 1 = ⋅ 6 2 A die is thrown once. What is the probability that it shows (i) a ’3’, (ii) a ‘5’, (iii) an odd number, (iv) a number greater than 4? When a die is thrown, all possible outcomes are 1, 2, 3, 4, 5, 6. Total number of possible outcomes = 6.

∴ EXAMPLE 3 SOLUTION

P(getting a prime number) = P(E) =

Probability

891

(i) Let E1 be the event of getting a 3. Then, the number of favourable outcomes = 1. 1 P(getting a 3) = P(E1 ) = ⋅ ∴ 6 (ii) Let E2 be the event of getting a 5. Then, the number of favourable outcomes = 1. 1 P(getting a 5) = P(E2 ) = ⋅ ∴ 6 (iii) Let E3 be the event of getting an odd number. Then, the favourable outcomes are 1, 3, 5. Number of favourable outcomes = 3. 3 1 P(getting an odd number) = P(E3 ) = = ⋅ ∴ 6 2

EXAMPLE 4 SOLUTION

(iv) Let E4 be the event of getting a number greater than 4. Then, the favourable outcomes are 5, 6. Number of favourable outcomes = 2. 2 1 ∴ P(getting a number greater than 4) = P(E4 ) = = ⋅ 6 3 A die is thrown once. Find the probability of getting (i) an even prime number, (ii) a multiple of 3. [CBSE 2012] When a die is thrown, all possible outcomes are 1, 2, 3, 4, 5, 6. Total number of possible outcomes = 6. (i) Let E1 be the event of getting an even prime number. Then, the favourable outcome is 2 only. Number of favourable outcomes = 1 . 1 P(getting an even prime number) = P(E1 ) = ⋅ ∴ 6

EXAMPLE 5 SOLUTION

(ii) Let E2 be the event of getting a multiple of 3. Then, the favourable outcomes are 3 and 6. Number of favourable outcomes = 2. 2 1 P(getting a multiple of 3) = P(E2 ) = = ⋅ ∴ 6 3 Two coins are tossed simultaneously. What is the probability of getting at least one head? [CBSE 2014] When two coins are tossed simultaneously, all possible outcomes are HH, HT, TH, TT. Total number of possible outcomes = 4. Let E be the event of getting at least one head.

892

EXAMPLE 6

SOLUTION

Secondary School Mathematics for Class 10

Then, E is the event of getting 1 head or 2 heads. So, the favourable outcomes are HT, TH, HH. Number of favourable outcomes = 3. 3 ∴ P(getting at least one head) = P(E) = ⋅ 4 Three unbiased coins are tossed simultaneously. Find the probability of getting (i) exactly 2 heads, (ii) at least 2 heads, (iii) at most 2 heads. [CBSE 2015] When 3 coins are tossed simultaneously, all possible outcomes are HHH, HHT, HTH, THH, HTT, THT, TTH, TTT. Total number of possible outcomes = 8.

EXAMPLE 7

SOLUTION

(i) Let E1 be the event of getting exactly 2 heads. Then, the favourable outcomes are HHT, HTH, THH. Number of favourable outcomes = 3. 3 P( getting exactly 2 heads) = P(E1 ) = ⋅ ∴ 8 (ii) Let E2 be the event of getting at least 2 heads. Then, E2 is the event of getting 2 or 3 heads. So, the favourable outcomes are HHT, HTH, THH, HHH. Number of favourable outcomes = 4. 4 1 P(getting at least 2 heads) = P(E2 ) = = ⋅ ∴ 8 2 (iii) Let E3 be the event of getting at most 2 heads. Then, E3 is the event of getting 0 or 1 head or 2 heads. So, the favourable outcomes are TTT, HTT, THT, TTH, HHT, HTH, THH. Number of favourable outcomes = 7. 7 P(getting at most 2 heads) = P(E3 ) = ⋅ ∴ 8 Cards numbered 11 to 60 are kept in a box. If a card is drawn at random from the box, find the probability that the number on the drawn card is (i) an odd number, (ii) a perfect square number, (iii) divisible by 5, (iv) a prime number less than 20. [CBSE 2014] All possible outcomes are 11, 12, 13, ... , 60. Number of all possible outcomes = ( 60 − 10) = 50. (i) Let E1 be the event that the number on the drawn card is an odd number.

Probability

893

Then, the favourable outcomes are 11, 13, 15, ..., 59. Clearly, these numbers form an AP with a = 11 and d = 2. Let the number of these numbers be n. Then, Tn = 59 ⇒ 11 + (n − 1) × 2 = 59 ⇒ (n − 1) × 2 = 48 ⇒ (n − 1) = 24 ⇒ n = 25. So, the number of favourable outcomes = 25. 25 1 P(getting an odd number) = P(E1 ) = = ⋅ ∴ 50 2 (ii) Let E2 be the event that the number on the drawn card is a perfect square number. Then, the favourable outcomes are 16, 25, 36, 49. So, the numbr of favourable outcomes = 4. ∴ P(getting a perfect square number) = P(E2 ) =

4 2 = ⋅ 50 25

(iii) Let E3 be the event that the number on the drawn card is divisible by 5. Then, the favourable outcomes are 15, 20, 25, ... , 60. Clearly, these numbers form an AP with a = 15 and d = 5. Let the number of these terms be m. Then, Tm = 60 ⇒ 15 + (m − 1) × 5 = 60 ⇒ (m − 1) × 5 = 45 ⇒ m − 1 = 9 ⇒ m = 10. So, the number of favourable outcomes = 10. 10 1 ∴ P(getting a number divisible by 5) = P(E3 ) = = ⋅ 50 5 (iv) Let E4 be the event that the number on the drawn card is a prime number less than 20. Then, the favourable outcomes are 11, 13, 17, 19. So, the number of favourable outcomes = 4. 4 2 = ⋅ 50 25 A box contains 100 red balls, 200 yellow balls and 50 blue balls. If a ball is drawn at random from the box, then find the probability that it will be (i) a blue ball, (ii) not a yellow ball, (iii) neither yellow nor a blue ball. [CBSE 2012] ∴ P(getting a prime number less than 20) = P(E4 ) =

EXAMPLE 8

SOLUTION

Total number of all possible outcomes = total number of balls = 100 + 200 + 50 = 350. (i) Number of blue balls = 50.

894

Secondary School Mathematics for Class 10

∴

P(getting a blue ball) =

50 1 = ⋅ 350 7

(ii) Number of balls which are not yellow = 100 + 50 = 150. 150 3 P(getting a ball which is not yellow) = ∴ = ⋅ 350 7

EXAMPLE 9

SOLUTION

(iii) Number of balls which are neither yellow nor blue = 100. ∴ P(getting a ball which is neither yellow nor blue) 100 2 = = ⋅ 350 7 A bag contains 5 red balls and some blue balls. If the probability of drawing a blue ball from the bag is thrice that of a red ball, find the number of blue balls in the bag. [CBSE 2007] Let the number of blue balls in the bag be x. Then, total number of balls = (5 + x). Given, P(a blue ball) = 3 × P(a red ball) x 5 ∴ = 3× ⇒ x = 15. (5 + x) (5 + x)

EXAMPLE 10

SOLUTION

Hence, the number of blue balls in the bag is 15. A bag contains white, black and red balls only. A ball is drawn at random from the bag. If the probability of getting a white ball is 3 10 and that of a black ball is 2 5 then find the probability of getting a red ball. If the bag contains 20 black balls then find the total number of balls in the bag. [CBSE 2015] Let E be the event of getting a red ball. Then, P(getting a white ball) + P(getting a black ball) + P(E) = 1 3 2 7 ⇒ + + P(E) = 1 ⇒ + P(E) = 1 10 5 10 7 3 ⇒ P(E) = 1 − = 10 10 3 ⇒ P(getting a red ball) = ⋅ 10 Since P(getting a white ball) = P(getting a red ball), so the number of white balls is equal to the number of red balls, say x. x x ∴ P(geting a red ball) = = ⋅ x + 20 + x 2 x + 20 x 3 ∴ = ⇒ 10 x = 6 x + 60 ⇒ 4 x = 60 ⇒ x = 15. 2 x + 20 10

Probability

895

Hence, the total number of balls in the bag EXAMPLE 11

SOLUTION

= 2 x + 20 = 2 × 15 + 20 = 50. Two different dice are rolled together. Find the probability of getting (i) the sum of numbers on two dice to be 5, (ii) even number on both dice, (iii) a doublet. [CBSE 2015] When two dice are thrown simultaneously, all possible outcomes are (1, 1),

(1, 2),

(1, 3),

(1, 4),

(1, 5),

(1, 6),

(2, 1),

(2, 2),

(2, 3),

(2, 4),

(2, 5),

(2, 6),

(3, 1),

(3, 2),

(3, 3),

(3, 4),

(3, 5),

(3, 6),

(4, 1),

(4, 2),

(4, 3),

(4, 4),

(4, 5),

(4, 6),

(5, 1),

(5, 2),

(5, 3),

(5, 4),

(5, 5),

(5, 6),

(6, 1),

(6, 2),

(6, 3),

(6, 4),

(6, 5),

(6, 6).

Number of all possible outcomes = 36. (i) Let E1 be the event of getting two numbers whose sum is 5. Then, the favourable outcomes are (1, 4) (2, 3), (3, 2), (4, 1). Number of favourable outcomes = 4. 4 1 = ⋅ ∴ P( getting two numbers whose sum is 5) = P(E1 ) = 36 9 (ii) Let E2 be the event of getting a even numbers on both dice. Then, the favourable outcomes are (2, 2), (2, 4), (2, 6), (4, 2), (4, 4), (4, 6), (6, 2), (6, 4), (6, 6). Number of favourable outcomes = 9. 9 1 P( getting even number on both dice) = P(E2 ) = = ⋅ ∴ 36 4

EXAMPLE 12

(iii) Let E3 be the event of getting a doublet. Then, the favourable outcomes are (1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6). Number of favourable outcomes = 6. 6 1 P( getting a doublet) = P(E3 ) = = ⋅ ∴ 36 6 Two dice are thrown at the same time. Find the probability that the sum of the two numbers appearing on the top of the dice is more than 9. [CBSE 2009C]

896 SOLUTION

Secondary School Mathematics for Class 10

We know that when two dice are thrown the same time, then the number of all possible outcomes is 36. Let E be the event that the sum of the numbers appearing on the top of the two dice is more than 9. The favourable outcomes are (4, 6), (5, 5), (5, 6), (6, 4), (6, 5), (6, 6). Number of favourable outcomes = 6. 6 1 = ⋅ 36 6 A piggy bank contains hundred 50-p coins, fifty ` 1 coins, twenty ` 2 and ten ` 5 coins. If it is equally likely that one of the coins will fall out when the bank is turned upside down, find the probability the coin falling out will be (i) a 50-p coin, (ii) of value more than ` 1, (iii) of value less than ` 5 (iv) a ` 1 or ` 2 coin. [CBSE 2014] Total number of coins = (100 + 50 + 20 + 10) = 180.

∴ EXAMPLE 13

SOLUTION

P(getting a sum more than 9) = P(E) =

So, the number of all possible outcomes is 180. (i) Let E1 be the event of getting a 50-p coin. Then, the number of favourable outcomes = 100. 100 5 P(getting a 50-p coin) = P(E1 ) = = ⋅ ∴ 180 9 (ii) LetE2 be the event of getting a coin of value more than ` 1. Then, it can be ` 2 or ` 5 coin. Number of all such coins = 20 + 10 = 30. P(getting a coin of value more than ` 1) ∴ 30 1 = P(E2 ) = = ⋅ 180 6 (iii) Let E3 be the event of getting a coin of value less than ` 5. Then, it can be 50-p or ` 1 or ` 2 coin. Number of favourable outcomes = number of all coins of 50-p, ` 1 and ` 2 = (100 + 50 + 20 = 170. P(getting a coin of value less than ` 5) ∴ 170 17 = P(E3 ) = = ⋅ 180 18 (iv) Let E4 be the event of getting a ` 1 or ` 2 coin. Number of all such coins = 50 + 20 = 70. Number of favourable outcomes = 70.

Probability

70 7 = ⋅ 180 18 A game consists of tossing a one-rupee coin three times and noting its outcome each time. Hanif wins if all the tosses give the same result, i.e., three heads or three tails and loses otherwise. Calculate the probability that Hanif will lose the game. [CBSE 2009C, ’11, ’17] In tossing a one-rupee coin three times, all possible outcomes are HHH, HHT, HTH, THH, HTT, THT, TTH, TTT. ∴

EXAMPLE 14

SOLUTION

897

P(getting a ` 1 or ` 2 coin) = P(E4 ) =

Total number of all possible outcomes = 8. Let E be the event of getting 3 heads or 3 tails. Then, E consists of HHH, TTT. Number of favourable outcomes of E = 2. P(that Hanif wins the game) = P(getting 3 heads or 3 tails) 2 1 = P(E) = = ⋅ 8 4 1 3 P(that Hanif loses the game) = 1 − P(E) = ⎛⎜ 1 − ⎞⎟ = ⋅ ⎝ ⎠ 4 4 FACTS ABOUT PLAYING CARDS

1. A deck of playing cards has 52 cards. 2. There are 4 suits, namely (i) spades, (ii) clubs, (iii) hearts and (iv) diamonds. There are 13 cards of each suit. I. Spades and clubs are black cards. II. Hearts and diamonds are red cards. 3. There are 12 face cards, namely 4 kings, 4 queens and 4 jacks. EXAMPLE 15

SOLUTION

One card is drawn at random from a well-shuffled deck of 52 cards. Find the probability that the card drawn is (i) a king, (ii) a red eight, (iii) a spade, (iv) a red card, (v) the six of the clubs and (vi) a face card. Total number of all possible outcomes = 52. (i) There are 4 kings in all. ∴

P(drawing a king) =

4 1 = ⋅ 52 13

(ii) There are 2 red eights in all. ∴

P(drawing a red eight) =

2 1 = ⋅ 52 26

(iii) There are 13 cards of spades. 13 1 P(drawing a spade) = ∴ = ⋅ 52 4

898

Secondary School Mathematics for Class 10

(iv) There are 26 red cards. ∴

P(drawing a red card) =

26 1 = ⋅ 52 2

(v) There is one six of the clubs. ∴

P(drawing the six of the clubs) =

1 ⋅ 52

(vi) There are 12 face cards. 12 3 = ⋅ 52 13 One card is drawn from a well-shuffled deck of 52 cards. Find the probability of getting (i) a king of red suit, (ii) a queen of black suit, (iii) a jack of hearts, (iv) a red face card. [CBSE 2010] Total number of possible outcomes = 52. ∴

EXAMPLE 16

SOLUTION

P(drawing a face card) =

(i) Number of kings of red suit = 2. ∴

P( getting a king of red suit) =

2 1 = ⋅ 52 26

(ii) Number of queens of black suit = 2. ∴

P( getting a queen of black suit) =

2 1 = ⋅ 52 26

(iii) Number of jacks of hearts = 1. ∴

P( getting a jack of hearts) =

1 ⋅ 52

(iv) Red face cards are 2 kings, 2 queens, 2 jacks. Number of red face cards = 6. 6 3 = ⋅ 52 26 A card is drawn at random from a well-shuffled deck of 52 playing cards. Find the probability that the card drawn is (i) a card of spades or an ace, (ii) a black king, (iii) neither a jack nor a king, (iv) either a king or a queen. [CBSE 2015] Total number of all possible outcomes = 52. ∴

EXAMPLE 17

SOLUTION

P( getting a red face card) =

(i) There are 13 cards of spades including one ace and there are 3 more aces. So, the number of favourable cases = 13 + 3 = 16. 16 4 P( getting a card of spades or an ace) = ∴ = ⋅ 52 13

Probability

899

(ii) There are 2 black kings. ∴

P( getting a black king) =

2 1 = ⋅ 52 26

(iii) There are 4 jacks and 4 kings. So, the number of cards which are neither jacks nor kings = {52 − ( 4 + 4)} = 44. P( getting a card which is neither a jack nor a king) ∴ 44 11 = = ⋅ 52 13

EXAMPLE 18

SOLUTION

(iv) There are 4 kings and 4 queens. So, the number of cards which are either kings or queens = 4 + 4 = 8. P( getting a card which is either a king or a queen) ∴ 8 2 = = ⋅ 52 13 One card is drawn at random from a well-shuffled deck of 52 playing cards. Find the probability that the card drawn is (i) either a red card or a king, (ii) neither a red card nor a queen. Total number of all possible outcomes = 52. (i) Let E1 be the event of getting a red card or a king. There are 26 red cards (including 2 kings) and there are 2 more kings. So, the number of favourable outcomes = 26 + 2 = 28. 28 7 P( getting a red card or a king) = P(E1 ) = = ⋅ ∴ 52 13

EXAMPLE 19

(ii) Let E1 be the event of getting a card which is neither a red card nor a queen. There are 26 red cards (including 2 queens) and there are 2 more queens. So, the number of non-favourable outcomes = 26 + 2 = 28. ∴ the number of favourable outcomes = 52 − 28 = 24. 24 6 ∴ P( getting neither a red card nor a queen) = = ⋅ 52 13 From a pack of 52 playing cards jacks, queens, kings and aces of red colour are removed. From the remaining, a card is drawn at random. Find the probability that the card drawn is (i) a black queen (ii) a red card (iii) a ten [CBSE 2006C] (iv) a picture card (jacks, queens and kings are picture cards).

900 SOLUTION

Secondary School Mathematics for Class 10

Number of cards removed = 2 + 2 + 2 + 2 = 8. Total number of remaining cards = 52 − 8 = 44. Now, there are 2 jacks, 2 queens, 2 kings and 2 aces of black colour only. (i) Number of black queens = 2. ∴

P(getting a black queen) =

2 1 = ⋅ 44 22

(ii) Remaining number of red cards = 26 − 8 = 18. 18 9 P( getting a red card) = ∴ = ⋅ 44 22 (iii) Number of tens = 4. ∴

EXAMPLE 20

SOLUTION

P( getting a ten) =

4 1 = ⋅ 44 11

(iv) We know that jacks, queens and kings are picture cards. Out of 12 picture cards, it is given that 6 have been removed. So, the remaining number of picture cards = 12 − 6 = 6. 6 3 P( getting a picture card) = ∴ = ⋅ 44 22 All the black face cards are removed from a pack of 52 playing cards. The remaining cards are well shuffled and then a card is drawn at random. Find the probability of getting a (i) face card, (ii) red card, (iii) black card, (iv) king. [CBSE 2014] Out of 52 playing cards; 2 black jacks, 2 black queens and 2 black kings have been removed. Total number of remaining cards = (52 − 6) = 46. (i) Now, there are 6 face cards in the remaining cards. 6 3 P( getting a face card) = ∴ = ⋅ 46 23 (ii) There are 26 red cards. ∴

P( getting a red card) =

26 13 = ⋅ 46 23

(iii) Out of 46 cards, number of black cards = 26 − 6 = 20. 20 10 P( getting a black card) = = ⋅ ∴ 46 23 (iv) Now, these 46 cards have 2 kings. 2 1 P( getting a king) = ∴ = ⋅ 46 23

Probability EXAMPLE 21

SOLUTION

901

Red queens and black jacks are removed from a pack of 52 playing cards. A card is drawn at random from the remaining cards, after reshuffling them. Find the probability that the drawn card is (i) a king, (ii) of red colour, (iii) a face card, (iv) a queen. [CBSE 2014] After removing 2 red queens and 2 black jacks, the number of remaining cards = 52 − ( 2 + 2) = 48. (i) Out of 48 cards, there are 4 kings. 4 1 P( getting a king) = ∴ = ⋅ 48 12 (ii) Number of cards of red colour = 26 − 2 = 24. Total number of cards = 48. ∴

P( getting a card of red colour) =

24 1 = ⋅ 48 2

(iii) Number of face cards = 12 − ( 2 + 2) = 8. Total number of cards = 48. ∴

P( getting a face card) =

8 1 = ⋅ 48 6

(iv) Number of queens in 48 cards = 4 − 2 = 2. 2 1 P( getting a queen) = ∴ = ⋅ 48 24

EXERCISE 19A Very-Short-Answer Questions 1. Fill in the blanks: (i) The probability of an impossible event is ...... . (ii) The probability of a sure event is ...... . (iii) For any event E, P(E) + P(not E) = ...... . (iv) The probability of a possible but not a sure event lies between ...... and ...... . (v) The sum of probabilities of all the outcomes of an experiment is ...... . 2. A coin is tossed once. What is the probability of getting a tail? 3. Two coins are tossed simultaneously. Find the probability of getting (i) exactly 1 head (ii) at most 1 head (iii) at least 1 head. 4. A die is thrown once. Find the probability of getting (i) an even number (ii) a number less than 5 (iii) a number greater than 2 (iv) a number between 3 and 6 (v) a number other than 3 (vi) the number 5.

902

Secondary School Mathematics for Class 10

Short-Answer Questions 5. A letter of English alphabet is chosen at random. Determine the probability that the chosen letter is a consonant. [CBSE 2015] 6. A child has a die whose 6 faces show the letters given below: A

B

C

A

D

A

The die is thrown once. What is the probability of getting (i) A, (ii) D? 7. It is known that a box of 200 electric bulbs contains 16 defective bulbs. One bulb is taken out at random from the box. What is the probability that the bulb drawn is (i) defective, (ii) nondefective? 8. If the probability of winning a game is 0.7, what is the probability of losing it? 9. There are 35 students in a class of whom 20 are boys and 15 are girls. From these students one is chosen at random. What is the probability that the chosen student is a (i) boy, (ii) girl? 10. In a lottery there are 10 prizes and 25 blanks. What is the probability of getting a prize? 11. 250 lottery tickets were sold and there are 5 prizes on these tickets. If Kunal has purchased one lottery ticket, what is the probability that he wins a prize? 12. 17 cards numbered 1, 2, 3, 4, …, 17 are put in a box and mixed thoroughly. A card is drawn at random from the box. Find the probability that the card drawn bears (i) an odd number (ii) a number divisible by 5. [CBSE 2012] 13. A game of chance consists of spinning an arrow, which comes to rest pointing at one of the numbers 1, 2, 3, 4, 5, 6, 7, 8 and these are equally likely outcomes. Find the probability that the arrow will point at any factor of 8. [CBSE 2015] 14. In a family of 3 children, find the probability of having at least one boy. [CBSE 2014]

15. A bag contains 4 white balls, 5 red balls, 2 black balls and 4 green balls. A ball is drawn at random from the bag. Find the probability that it is (i) black, (ii) not green, (iii) red or white, (iv) neither red nor green. [CBSE 2012]

16. A card is drawn at random from a well-shuffled pack of 52 cards. Find the probability of getting (i) a red king, (ii) a queen or a jack. [CBSE 2012]

Probability

903

17. A card is drawn at random from a well-shuffled pack of 52 cards. Find the probability that the drawn card is neither a king nor a queen. [CBSE 2013]

18. A card is drawn from a well-shuffled pack of 52 cards. Find the [CBSE 2013C] probability of getting (i) a red face card (ii) a black king. 19. Two different dice are tossed together. Find the probability that (i) the number on each die is even, (ii) the sum of the numbers appearing on [CBSE 2014] the two dice is 5. 20. Two different dice are rolled simultaneously. Find the probability that [CBSE 2014] the sum of the numbers on the two dice is 10. 21. Two different dice are thrown together. Find the probability that (i) the sum of the numbers appeared is less than 7. [CBSE 2011] (ii) the product of the numbers appeared is less than 18.

[CBSE 2017]

22. Two dice are rolled together. Find the probability of getting such [CBSE 2011] numbers on two dice whose product is a perfect square. 23. Two dice are rolled together. Find the probability of getting such [CBSE 2013] numbers on the two dice whose product is 12. 24. Cards marked with numbers 5 to 50 are placed in a box and mixed thoroughly. A card is drawn from the box at random. Find the probability that the number on the taken out card is (i) a prime number [CBSE 2008] less than 10 (ii) a number which is a perfect square. 25. A game of chance consists of spinning an arrow which is equally likely to come to rest pointing to one of the numbers 1, 2, 3, …, 12 as shown in the figure. What is the probability that it will point to (i) 6? (iii) a prime number?

(ii) an even number? (iv) a number which is a multiple of 5?

26. 12 defective pens are accidently mixed with 132 good ones. It is not possible to just look at pen and tell whether or not it is defective. One pen is taken out at random from this lot. Find the probability that the pen taken out is good one. 27. A lot consists of 144 ballpoint pens of which 20 are defective and others good. Tanvy will buy a pen if it is good, but will not buy it if it is defective. The shopkeeper draws one pen at random and gives it to her. What is the probability that (i) she will buy it, (ii) she will not buy it? 28. A box contains 90 discs which are numbered from 1 to 90. If one disc is drawn at random from the box, find the probability that it bears (i) a

904

Secondary School Mathematics for Class 10

two-digit number, (ii) a perfect square number, (iii) a number divisible by 5. 29.

(i) A lot of 20 bulbs contain 4 defective ones. One bulb is drawn at random from the lot. What is the probability that this bulb is defective? (ii) Suppose the bulb drawn in (i) is not defective and not replaced. Now, bulb is drawn at random from the rest. What is the probability that this bulb is not defective?

30. A bag contains lemon-flavoured candies only. Hema takes out one candy without looking into the bag. What is the probability that she takes out (i) an orange-flavoured candy? (ii) a lemon-flavoured candy? 31. There are 40 students in a class of whom 25 are girls and 15 are boys. The class teacher has to select one student as a class representative. He writes the name of each student on a separate card, the cards being identical. Then she puts cards in a bag and stirs them thoroughly. She then draws one card from the bag. What is the probability that the name written on the card is the name of (i) a girl? (ii) a boy? 32. One card is drawn from a well-shuffled deck of 52 cards. Find the probability of drawing (i) an ace (iii) a ‘9’ of a black suit

(ii) a ‘4’ of spades (iv) a red king.

33. A card is drawn at random from a well-shuffled deck of 52 cards. Find [CBSE 2003] the probability of getting (i) a queen (iii) a king or an ace

(ii) a diamond (iv) a red ace.

34. One card is drawn from a well-shuffled deck of 52 cards. Find the probability of getting (i) a king of red suit (iii) a red face card (v) a jack of hearts

(ii) a face card (iv) a queen of black suit (vi) a spade.

35. A card is drawn at random from a well-shuffled deck of playing cards. Find the probability that the card drawn is [CBSE 2006] (i) a card of spades or an ace (iii) either a king or a queen

(ii) a red king (iv) neither a king nor a queen.

36. Two different dice are thrown together. Find the probability that the numbers obtained have (i) even sum

(ii) even product.

[CBSE 2017]

Probability

905

37. Two different dice are thrown together. Find the probability that the numbers obtained (i) have a sum less than 7

(ii) have a product less than 16

(iii) is a doublet of odd numbers.

[CBSE 2017]

38. The king, the jack and the 10 of spades are lost from a pack of 52 cards and a card is drawn from the remaining cards after shuffling. Find the probability of getting a (i) red card

(ii) black jack

(iii) red king

(iv) 10 of hearts.

[CBSE 2017]

39. Peter throws two different dice together and finds the product of the two numbers obtained. Rina throws a die and squares the number obtained. Who has the better chance to get the number 25? [CBSE 2017] ANSWERS (EXERCISE 19A)

1 3 3 (ii) (iii) 2 4 4 1 2 2 1 5 1 21 1 1 5. 6. (i) (ii) 4. (i) (ii) (iii) (iv) (v) (vi) 2 3 3 3 6 6 26 2 6 2 23 4 3 2 1 (ii) 8. 0.3 9. (i) (ii) 10. 11. 7. (i) 25 25 7 7 7 50 9 3 3 7 2 11 3 2 (ii) 13. 14. 15. (i) (ii) (iii) (iv) 12. (i) 17 17 8 8 15 15 5 5 1 2 11 3 1 1 1 1 16. (i) (ii) 17. 18. (i) (ii) 19. (i) (ii) 20. 26 13 13 26 26 4 9 12 5 13 2 1 1 5 (ii) 22. 23. 24. (i) (ii) 21. (i) 12 18 9 9 23 46 1 1 5 1 11 31 5 25. (i) (ii) (iii) (iv) 26. 27. (i) (ii) 12 2 12 6 12 36 36 9 1 1 1 15 (ii) (iii) 29. (i) (ii) 30. (i) 0 (ii) 1 28. (i) 10 10 5 5 19 5 3 1 1 1 1 31. (i) (ii) 32. (i) (ii) (iii) (iv) 8 8 13 52 26 26 1 1 2 1 1 3 3 1 1 1 (ii) (iii) (iv) 34. (i) (ii) (iii) (iv) (v) (vi) 33. (i) 13 4 13 26 26 13 26 26 52 4 4 1 2 11 1 3 5 25 1 (ii) (iii) (iv) 36. (i) (ii) (ii) (iii) 35. (i) 37. (i) 13 26 13 13 2 4 12 36 4 26 1 2 1 38. (i) (ii) (iii) (iv) 39. Rina 49 49 49 49 1. (i) 0 (ii) 1 (iii) 1 (iv) 0, 1 (v) 1

2.

1 2

3. (i)

906

Secondary School Mathematics for Class 10 HINTS TO SOME SELECTED QUESTIONS

5. Out of 26 letters of English alphabet, there are 11 consonants. 21 ⋅ ∴ P(getting a consonant) = 26 6. There are 6 letters in all consisting of 3As, 1B, 1C and 1D. 3 1 1 (ii) P(getting D) = ⋅ ∴ (i) P(getting A) = = ⋅ 6 2 6 7. Total number of bulbs = 200. Number of defective bulbs = 16. Number of non-defective bulbs = 200 − 16 = 184. 16 2 (i) P(getting a defective bulb) = = ⋅ 200 25 184 23 (ii) P(getting a non-defective bulb) = = ⋅ 200 25 8. Let E be the event of winning the game. Then, P( E) = 0 . 7 . Probability of losing the game = 1 − P( E) = ( 1 − 0 . 7 ) = 0. 3. 9. (i) P(choosing a boy) =

20 4 = ⋅ 35 7

(ii) P(choosing a girl) =

15 3 = ⋅ 35 7

10. Total number of tickets = 10 + 25 = 35. Number of prizes = 10. 10 2 P(getting a prize) = = ⋅ 35 7 11. P(getting a prize) =

5 1 = ⋅ 250 50

12. Total number of cards = 17. (i) Let E1 be the event of choosing an odd number. These numbers are 1, 3, 5, ... , 17. Let their number be n. Then, Tn = 17 ⇒ 1 + (n − 1) × 2 = 17 ⇒ n = 9. 9 ⋅ ∴ P( E1 ) = 17 (ii) Let E2 be the event of choosing a number divisible by 5. Numbers divisible by 5 are 5, 10, 15. Their number is 3. 3 ⋅ ∴ P( E2 ) = 17 13. All possible outcomes are 1, 2, 3, 4, 5, 6, 7, 8. Number of all possible outcomes is 8. All factors of 8 are 2, 4, 8. Number of favourable outcomes = 3. Probability that the arrow will point at any factor of 8 =

3 ⋅ 8

Probability

907

14. All possible outcomes are BBB, BBG, BGB, GBB, BGG, GBG, GGB, GGG. Number of all possible outcomes = 8. Let E be the event of having at least one boy. Then, E contains GGB, GBG, BGG, BBG, BGB, GBB, BBB. Number of cases favourable to E = 7. 7 ∴ required probability = P( E) = ⋅ 8 15. Total number of balls = 4 + 5 + 2 + 4 = 15. (i) Number of black balls = 2. 2 P(getting a black ball) = ⋅ 15 (ii) Number of balls which are not green = 4 + 5 + 2 = 11. 11 P(getting a ball which is not green) = ⋅ 15 (iii) Number of balls which are red or white = 5 + 4 = 9. 9 3 P(getting a ball which is red or white) = = ⋅ 15 5 (iv) Number of balls which are neither red nor green = 4 + 2 = 6. 6 2 P(getting a ball which is neither red nor green) = = ⋅ 15 5 16. Total number of cards = 52. (i) Number of red kings = 2. ∴ P(getting a red king) =

2 1 = ⋅ 52 26

(ii) There are 4 queens and 4 jacks. ∴ P(getting a queen or a jack) =

8 2 = ⋅ 52 13

17. Toal number of cards = 52. Total number of kings and queens = 4 + 4 = 8. Remaining number of cards = 52 − 8 = 44. ∴

P(getting a card which is neither a king nor a queen) =

44 11 = ⋅ 52 13

18. Total number of cards = 52. (i) 4 kings, 4 queens and 4 jacks are all face cards. Number of red face cards = 2 + 2 + 2 = 6. 6 3 = ⋅ ∴ P(getting a red face card) = 52 26 (ii) Number of black kings = 2. ∴

P(getting a black king) =

2 1 = ⋅ 52 26

19. When two different dice are thrown, then total number of outcomes = 36. (i) Let E1 be the event of getting an even number on each die. These numbers are (2, 2), (2, 4), (2, 6), (4, 2), (4, 4), (4, 6), (6, 2), (6, 4), (6, 6).

908

Secondary School Mathematics for Class 10 Number of favourable outcomes = 9. 9 1 = ⋅ 36 4 (ii) Let E2 be the event of getting the sum 5. Then these numbers are (1, 4), (2, 3), (4, 1), (3, 2). ∴ P(getting an even number on both dice) = P( E1 ) =

Number of favourable outcomes = 4. 4 1 = ⋅ ∴ P( E2 ) = 36 9 20. Number of all possible outcomes is 36. Let E be the event of getting the sum 10 on the two dice. Then, the favourable outcomes are (4, 6), (6, 4), (5, 5). Number of favouable outcomes = 3. 3 1 = ⋅ ∴ P( E) = 36 12 21.

(i) Number of all possible outcomes is 36. Let E be the event of getting the sum less than 7 on the two dice. Then, the favourable outcomes are (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (2, 1), (2, 2), (2, 3), (2, 4), (3, 1), (3, 2), (3, 3), (4, 1), (4, 2), (5, 1). Number of favourable outcomes = 15. 15 5 = ⋅ ∴ P( E) = 36 12

22. Number of all possible outcomes is 36. Let E be the event of getting the product of numbers on the two dice, as a perfect square. Then, the favourable outcomes are (1, 1), (1, 4), (4, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6). Number of favourable outcomes = 8. 8 2 = ⋅ ∴ P( E) = 36 9 23. Number of all possible outcomes is 36. Let E be the event of getting all those two numbers whose product is 12. Then, the favourable outcomes are (2, 6), (6, 2), (3, 4), (4, 3). Number of favourable outcomes = 4. 4 1 = ⋅ ∴ P( E) = 36 9 24. Total number of cards = 50 − 4 = 46. (i) Out of the given numbers, prime numbers less than 10 are 5 and 7. ∴ number of prime numbers less than 10 = 2. 2 1 = ⋅ ∴ P(getting a prime number less than 10) = 46 23

Probability (ii) From given numbers, the perfect square numbers are 9, 16, 25, 36, 49. Their number is 5. ∴ P(getting a perfect square number) =

5 ⋅ 46

25. Number of all possible outcomes = 12. 1 (i) P(getting a 6) = ⋅ 12 (ii) Out of the given numbers there are 6 even numbers. 6 1 ∴ P(getting an even number) = = ⋅ 12 2 (iii) Out of the given numbers, the prime numbers are 2, 3, 5, 7, 11. Their number is 5. 5 ⋅ 12 (iv) Out of the given numbers, the multiples of 5 are 5, 10. ∴ P(getting a prime number) =

Their number is 2. ∴ P(getting a multiple of 5) =

2 1 = ⋅ 12 6

26. Total number of pens = 132 + 12 = 144. Number of good pens = 132. 132 11 = ⋅ ∴ P(getting a good pen) = 144 12 27. Total number of pens = 144. Number of defective pens = 20. Number of non-defective pens = 144 − 20 = 124. 124 31 = ⋅ 144 36 20 5 (ii) P(not buying the pen) = P(getting a defective pen) = = ⋅ 144 36 (i) P(buying the pen) = P(getting a non-defective pen) =

28. Total number of discs = 90. (i) Number of discs bearing 2-digit numbers = number of numbers from 10 to 90 = ( 90 − 9 ) = 81. 81 9 P(getting a 2-digit number) = = ⋅ 90 10 (ii) Perfect square numbers are 12 , 2 2 , 3 2 , ... , 9 2 . Their number is 9. Number of discs bearing perfect square numbers = 9. 9 1 = ⋅ ∴ P(getting a perfect square number) = 90 10 (iii) Numbers divisible by 5 are 5, 10, 15, ... , 90. They are 18 in number. 18 1 = ⋅ ∴ P(getting a number divisible by 5) = 90 5 29.

(i) Total number of bulbs = 20. Number of defective bulbs = 4. Number of non-defective bulbs = 20 − 4 = 16.

909

910

Secondary School Mathematics for Class 10 4 1 = ⋅ 20 5 (ii) After removing 1 non-defective bulb, we have ∴ P(getting a defective bulb) =

remaining number of bulbs = 20 − 1 = 19. Out of these, the number of non-defective bulbs = 16 − 1 = 15. 15 ⋅ ∴ P(getting a non-defective bulb) = 19 30. Suppose there are x candies in the bag. Then, number of orange-flavoured candies in the bag = 0. And, the number of lemon-flavoured candies in the bag = x. 0 ∴ (i) P(getting an orange-flavoured candy) = = 0. x x (ii) P(getting a lemon-flavoured candy) = = 1. x 31. Total number of students in the class = 40. Number of girls = 25, number of boys = 15. 25 5 (i) P(selecting the name of a girl) = = ⋅ 40 8 15 3 (ii) P(selecting the name of a boy) = = ⋅ 40 8 32.

(i) There are 4 aces in all. ∴ P(getting an ace) =

4 1 = ⋅ 52 13

(ii) There is one ‘4’ of spades. ∴ P(getting 4 of spades) =

1 ⋅ 52

(iii) There are two nines of black suits. ∴ P(getting a ‘9’ of a black suit) =

2 1 = ⋅ 52 26

(iv) There are two red kings. ∴ P(getting a red king) = 33.

2 1 = ⋅ 52 26

(i) There are 4 queens. ∴ P(getting a queen) =

4 1 = ⋅ 52 13

(ii) There are 13 diamonds. ∴ P(getting a diamond) =

13 1 = ⋅ 52 4

(iii) There are 4 kings and 4 aces. ∴ P(getting a king or an ace) =

8 2 = ⋅ 52 13

Probability

911

(iv) There are 2 red aces. 2 1 = ⋅ 52 26 (i) There are 4 kings of red suits. ∴ P(getting a red ace) =

34.

∴ P(getting a king of a red shuit) =

2 1 = ⋅ 52 26

(ii) There are 12 face cards in all. 12 3 = ⋅ 52 13 (iii) There are 6 red face cards. ∴ P(getting a face card) =

6 3 = ⋅ 52 26 (iv) There are 2 queens of black suits. ∴ P(getting a red face card) =

∴ P(getting a queen of a black suit) =

2 1 = ⋅ 52 26

(v) There is 1 jack of hearts. ∴ P(getting a jack of hearts) =

1 ⋅ 52

(vi) There are 3 cards of spades. ∴ P(getting a spade) = 35.

13 1 = ⋅ 52 4

(i) There are 13 cards of spades including 1 ace and 3 more aces are there. ∴ P(getting a card of spades or an ace) =

13 + 3 16 4 = = ⋅ 52 52 13

(ii) There are 2 red kings. ∴ P(getting a red king) =

2 1 = ⋅ 52 26

(iii) There are 4 kings and 4 queens. ∴ P(getting either a king or a queen) =

4+ 4 8 2 = = ⋅ 52 52 13

(iv) P(neither a king nor a queen) = 1 − P(either a king or a queen) 2 11 = ⎛⎜ 1 − ⎞⎟ = ⋅ ⎝ 13 ⎠ 13

................................................................

EXERCISE 19B Long-Answer Questions 1. A box contains 25 cards numbered from 1 to 25. A card is drawn at random from the bag. Find the probability that the number on the drawn card is (i) divisible by 2 or 3, (ii) a prime number. [CBSE 2015]

912

Secondary School Mathematics for Class 10

2. A box contains cards numbered 3, 5, 7, 9,... , 35, 37. A card is drawn at random from the box. Find the probability that the number on the card is a prime number. [CBSE 2013] 3. Cards numbered 1 to 30 are put in a bag. A card is drawn at random from the bag. Find the probability that the number on the drawn card is (i) not divisible by 3, (ii) a prime number greater that 7, (iii)not a perfect square number. [CBSE 2014] 4. Cards bearing numbers 1, 3, 5,... , 35 are kept in a bag. A card is drawn at random from the bag. Find the probability of getting a card bearing (i) a prime number less than 15, (ii) a number divisible by 3 and 5. [CBSE 2010]

5. A box contains cards bearing numbers 6 to 70. If one card is drawn at random from the box, find the probability that it bears (i) a one-digit number, (ii) a number divisible by 5, (iii) an odd number less than 30, (iv) a composite number between 50 and 70. [CBSE 2015] 6. Cards marked with numbers 1, 3, 5, ..., 101 are placed in a bag and mixed thoroughly. A card is drawn at random from the bag. Find the probability that the number on the drawn card is (i) less than 19, (ii) a prime number less than 20. [CBSE 2012] 7. Tickets numbered 2, 3, 4, 5, ..., 100, 101 are placed in a box and mixed thoroughly. One ticket is drawn at random from the box. Find the probability that the number on the ticket is (i) an even number (ii) a number less than 16 (iii) a number which is a perfect square (iv) a prime number less than 40. 8.

(i) A box contains 80 discs, which are numbered from 1 to 80. If one disc is drawn at random from the box, find the probability that it bears a perfect square number. [CBSE 2011] (ii) A box contains 90 discs which are numbered 1 to 90. If one disc is drawn at random from the box, find the probability that it bears [CBSE 2017] (a) a two-digit number (b) a number divisible by 5

9. A piggy bank contains hundred 50-p coins, seventy ` 1 coin, fifty ` 2 coins and thirty ` 5 coins. If it is equally likely that one of the coins will fall out when the bank is turned upside down, what is the probability that the coin (i) will be a ` 1 coin? (ii) will not be a ` 5 coin (iii) will be [CBSE 2013C] 50-p or a ` 2 coin?

Probability

913

10. The probability of selecting a red ball at random from a jar that contains 1 only red, blue and orange balls is ⋅ The probability of selecting a blue 4 1 ball at random from the same jar is ⋅ If the jar contains 10 orange balls, 3 [CBSE 2015] find the total number of balls in the jar. 11. A bag contains 18 balls out of which x balls are red. (i) If one ball is drawn at random from the bag, what is the probability that it is not red? (ii) If two more red balls are put in the bag, the probability of drawing 9 a red ball will be times the probability of drawing a red ball in the 8 first case. Find the value of x. [CBSE 2015] 12. A jar contains 24 marbles. Some of these are green and others are blue. If a marble is drawn at random from the jar, the probability that it is green 2 is ⋅ Find the number of blue marbles in the jar. 3 13. A jar contains 54 marbles, each of which some are blue, some are green and some are white. The probability of selecting a blue marble at 1 random is and the probability of selecting a green marble at random 3 4 is ⋅ How many white marbles does the jar contain? 9 14. A carton consists of 100 shirts of which 88 are good and 8 have minor defects. Rohit, a trader, will only accept the shirts which are good. But, Kamal, an another trader, will only reject the shirts which have major defects. One shirt is drawn at random from the carton. What is the probability that it is acceptable to (i) Rohit and (ii) Kamal? 15. A group consists of 12 persons, of which 3 are extremely patient, other 6 are extremely honest and rest are extremely kind. A person from the group is selected at random. Assuming that each person is equally likely to be selected, find the probability of selecting a person who is (i) extremely patient, (ii) extremely kind or honest. Which of the above values you prefer more? [CBSE 2013] 16. A die is rolled twice. Find the probability that (i) 5 will not come up either time, (ii) 5 will come up exactly one time, (iii) 5 will come up both the times.

[CBSE 2014]

914

Secondary School Mathematics for Class 10

17. Two dice are rolled once. Find the probability of getting such numbers on two dice whose product is a perfect square. [CBSE 2011] 18. A letter is chosen at random from the letters of the word ‘ASSOCIATION’. Find the probability that the chosen letter is a (i) vowel (ii) consonant (iii) an S. 19. Five cards—the ten, jack, queen, king and ace of diamonds are well shuffled with their faces downwards. One card is then picked up at random. (a) What is the probability that the drawn card is the queen? (b) If the queen is drawn and put aside and a second card is drawn, find the probability that the second card is (i) an ace, (ii) a queen. [CBSE 2014] 20. A card is drawn at random from a well shuffled pack of 52 cards. Find the probability that the card drawn is neither a red card nor a queen. [CBSE 2011]

21. What is the probability that an ordinary year has 53 Mondays? 22. All red face cards are removed from a pack of playing cards. The remaining cards are well shuffled and then a card is drawn at random from them. Find the probability that the drawn card is (i) a red card, (ii) a face card, (iii) a card of clubs. [CBSE 2015] 23. All kings, queens and aces are removed from a pack of 52 cards. The remaining cards are well-shuffled and then a card is drawn from it. Find the probability that the drawn card is (i) a black face card, (ii) a red card. [CBSE 2012] 24. A game consists of tossing a one-rupee coin three times, and noting its outcome each time. Find the probability of getting (i) three heads, (ii) at least 2 tails. [CBSE 2015] 25. Find the probability that a leap year selected at random will contain 53 Sundays. 26. The probability of selecting a rotten apple randomly from a heap of 900 apples is 0.18. What is the number of rotten apples in the heap? [CBSE 2017]

27. A bag contains 15 white and some black balls. If the probability of drawing a black ball from the bag is thrice that of drawing a white ball find the number of black balls in the bag. [CBSE 2017] 28. Find the probability of getting the sum of two numbers, less than 3 or more than 11, when a pair of distinct dice is thrown together. [CBSE 2017]

Probability

915

ANSWERS (EXERCISE 19B)

16 9 11 2 1 5 5 1 (ii) 2. 3. (i) (ii) (iii) 4. (i) (ii) 25 25 18 3 5 6 18 9 4 1 12 3 3 7 (ii) (iii) (iv) 6. (i) (ii) 5. (i) 65 5 65 13 17 51 1 7 9 3 1 9 1 (iii) (iv) 8. (i) (ii) (a) (b) 7. (i) (ii) 2 50 100 25 10 10 5 (18 − x) 7 22 3 (ii) (iii) 10. 24 11. (i) (ii) x = 8 9. (i) 25 25 5 18 22 24 1 3 12. 8 13. 12 14. (i) (ii) 15. (i) (ii) 25 25 4 4 25 5 1 2 6 5 2 (ii) (ii) 17. 18. (i) (ii) (iii) 16. (i) 36 18 36 9 11 11 11 1 1 6 1 10 3 6 19. (a) (b) (i) (ii) 0 20. 21. 22. (i) (ii) (iii) 5 4 13 7 23 23 23 1 1 1 1 2 (ii) 24. (i) (ii) 25. 26. 162 23. (i) 20 2 8 2 7 1 27. 45 28. 18 1. (i)

HINTS TO SOME SELECTED QUESTIONS 1. Number of all possible outcomes = 25. (i) Numbers divisible by 2 or 3 are 2, 4, 6, 8,10, 12, 14, 16, 18, 20, 22, 24, 3, 9, 15, 21. Their number is 16. ∴ P(getting a number divisible by 2 or 3) =

16 ⋅ 25

(ii) Prime numbers from 1 to 25 are 2, 3, 5, 7, 11, 13, 17, 19, 23. Their number is 9. ∴ P(getting a prime number) =

9 ⋅ 25

2. Given numbers 3, 5, 7, 9, ..., 35, 37 form an AP with a = 3 and d = 2. Let Tn = 37. Then, 3 + (n − 1) × 2 = 37 ⇒ (n − 1) × 2 = 34 ⇒ n − 1 = 17 ⇒ n = 18. Out of these numbers, the prime numbers are 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37. Their number is 11. ∴

P(getting a prime number) =

11 ⋅ 18

3. Number of all possible outcomes = 30. (i) There are 10 numbers divisible by 3. Number of numbers not divisible by 3 = 30 − 10 = 20.

916

Secondary School Mathematics for Class 10 20 2 = ⋅ 30 3 (ii) Prime numbers greater than 7 are 11, 13, 17, 19, 23, 29. ∴ P(getting a number not divisible by 3) =

Number of these prime numbers = 6. 6 1 = ⋅ 30 5 (iii) Perfect square numbers are 1, 4, 9, 16, 25. These are 5 in number. ∴ P(getting a prime number greater than 7) =

Number of those numbers which are not perfect squares = 30 − 5 = 25. 25 5 = ⋅ ∴ P(getting non-perfect square numbers) = 30 6 4. The numbers 1, 3, 5, ..., 35 form an AP with a = 1 and d = 2. Let Tn = 35. Then, 1 + (n − 1) × 2 = 35 ⇒ (n − 1) × 2 = 34 ⇒ n − 1 = 17 ⇒ n = 18. ∴

number of all possible outcomes = 18. (i) Out of the given numbers, the prime numbers less than 15 are 3, 5, 7, 11, 13. Their number is 5. 5 ∴ P(getting a prime number) = ⋅ 18 (ii) A number is divisible by 3 and 5 means, it must be divisible by 15. The numbers divisible by 15 are 15 and 30. Their number is 2. ∴ P(getting a number divisible by both 3 and 5) =

2 1 = ⋅ 18 9

5. Given numbers 6, 7, 8, ..., 70 form an AP with a = 6 and d = 1. Let Tn = 70. Then, 6 + (n − 1) × 1 = 70 ⇒ n − 1 = 64 ⇒ n = 65. ∴

total number of cards = 65. (i) Out of the given numbers, the one-digit numbers are 6, 7, 8, 9. Number of one-digit numbers = 4. ∴ P(getting a one-digit number) =

4 ⋅ 65

(ii) Out of the given numbers, those divisible by 5 are 10, 15, 20, 25, ..., 70. Let Tn = 70. Then, 10 + (n − 1) × 5 = 70 ⇒ (n − 1) × 5 = 60 ⇒ n − 1 = 12 ⇒ n = 13. 13 1 = ⋅ ∴ P(getting a number divisible by 5) = 65 5 (iii) Out of the given numbers, odd numbers less than 30 are 7, 9, 11, 13, ... , 29. Let Tm = 29. Then, 7 + (m − 1) × 2 = 29 ⇒ (m − 1) × 2 = 22 ⇒ m − 1 = 11 ⇒ m = 12. 12 ⋅ 65

∴ P(getting an odd number less than 30) =

(iv) Number of numbers between 50 and 70 = numbers from 51 to 69. Their number = ( 69 − 51) + 1 = 19. Prime numbers between 50 and 70 = 53 , 59 , 61, 67 .

Probability

917

Number of prime numbers = 4. Number of composite numbers = 19 − 4 = 15. 15 3 = ⋅ ∴ P(getting a composite number) = 65 13 6. Given numbers 1, 3, 5, ..., 101 form an AP with a = 1 and d = 2. Let Tn = 101. Then, 1 + (n − 1) × 2 = 101 ⇒ (n − 1) × 2 = 100 ⇒ n − 1 = 50 ⇒ n = 51. ∴

total number of cards = 51. (i) Out of the given numbers, those less than 19 are 1, 3, 5, ..., 17. Let t m = 17. Then, 1 + (m − 1) × 2 = 17 ⇒ (m − 1) × 2 = 16 ⇒ m − 1 = 8 ⇒ m = 9. 9 3 = ⋅ ∴ P(getting a number less than 19) = 51 17 (ii) Out of the given numbers, the prime numbers less than 20 are 3, 5, 7, 11, 17, 19. Their number is 7.

7 ⋅ 51 7. The tickets bear the numbers 2, 3, 4, ... , 100, 101. So, the number of tickets = 100. ∴ P(getting a prime number less than 20) =

(i) Out of the given numbers, the even numbers are 2, 4, 6, 8, ..., 100. Their number is 50. ∴ P(getting an even number) =

50 1 = ⋅ 100 2

(ii) Out of the given numbers, the number of numbers less than 16 is 14. 14 7 = ⋅ ∴ P(getting number less than 16) = 100 50 (iii) Out of the given numbers, the perfect square numbers are 2 2 , 3 2 , 4 2 , ..., 10 2 . Their number is 9. ∴ P(getting a perfect square number) =

9 ⋅ 100

(iv) Prime numbers less than 40 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37. Their number is 12. ∴ P(getting a prime number less than 40) = 10. P(getting a red ball) =

12 3 = ⋅ 100 25

1 1 , P(getting a blue ball) = ⋅ 4 3

Let P(getting an orange ball) be x. Since there are only thre types of balls in the jar, the sum of probabilities of drawing these balls must be 1. 1 1 7 7 5 ⋅ + + x = 1⇒ + x = 1 ⇒ x = ⎛⎜ 1 − ⎞⎟ = ∴ ⎝ 4 3 12 12 ⎠ 12 ∴

P(getting an orange ball) =

5 ⋅ 12

Let the total number of balls in the jar be n.

... (i)

918

Secondary School Mathematics for Class 10 Number of orange balls = 10. 10 ⋅ n

∴

P(getting an orange ball) =

⇒

10 5 = ⇒ 5n = 120 ⇒ n = 24 [using (i)]. n 12

Hence, the total number of balls in the jar is 24. 11.

(i) Total number of balls = 18. Number of red balls = x. Number of balls which are not red = 18 − x. 18 − x ∴ P(getting a ball which is not red) = ⋅ 18 (ii) Now, total number of balls = 18 + 2 = 20.

Number of red balls now = x + 2. x+ 2 x P(getting a red ball now) = and P(getting a red ball in first case) = 20 18 x+ 2 9 x = × ⇒ 144( x + 2 ) = 180 x ⇒ 20 8 18 288 = 8. ⇒ 180 x − 144 x = 288 ⇒ 36 x = 288 ⇒ x = 36 Hence, x = 8. 12. Total number of marbles in the jar = 24. Let the number of blue marbles be x. Then, the number of green marbles = 24 − x. 24 − x P(getting a green marble) = ⋅ 24 2 But, P(getting a green marble) = (given). 3 24 − x 2 = ⇒ 72 − 3 x = 48 ⇒ 3 x = 24 ⇒ x = 8. ∴ 24 3 Hence, the number of blue marbles in the jar is 8. 13. Total number of marbles in the jar = 54. 1 4 P(getting a blue marble) = and P(getting a green marble) = ⋅ 3 9 Let P(getting a white marble) be x. Since there are only three given types of marbles in the jar, we have 1 4 7 7 2 + + x=1 ⇒ + x = 1 ⇒ x = 1− = ⋅ 3 9 9 9 9 2 ∴ P(getting a white marble) = ⋅ 9 Let the number of white marbles be n. n Then, P(getting a white marble) = ⋅ 54 n 2 = ⇒ 9n = 108 ⇒ n = 12. ∴ 54 9 Hence, there are 12 white marbles in the jar.

Probability

919

14. Total number of shirts = 100. Number of good shirts = 88. Number of shirts having minor defects = 8. Number of shirts having major defects = 100 − ( 88 + 8 ) = 4. 88 22 (i) P(that the drawn shirt is acceptable to Rohit) = = ⋅ ∴ 100 25 88 + 8 96 24 (ii) P(that the drawn shirt is acceptable to Kamal) = = = ⋅ 100 100 25 15. Total number of persons in the group = 12. (i) Number of persons who are extremely patient = 3. 3 1 P(selecting a person who is extremely patient) = = ⋅ 12 4 (ii) Number of persons who are extremely honest = 6. Number of persons who are extremely kind = 12 − ( 3 + 6 ) = 3. ∴

P(selecting a person who is extremely kind or extremely honest) =

3+ 6 9 3 = = ⋅ 12 12 4

From the given three values, we prefer honesty. Honesty can get rid of rampant corruption which is a burning issue of the present society. 16. Number of all possible outcomes = 36. (i) All those cases where 5 comes up on at least one face are (1, 5), (2, 5), (3, 5), (4, 5), (5, 5), (6, 5), (5, 1), (5, 2), (5, 3), (5, 4), (5, 6). Number of such cases = 11. Number of those cases where 5 will not come up any time = 36 − 11 = 25. 25 ⋅ ∴ P(that 5 will not come up either time) = 36 (ii) All those case where 5 comes up exactly one time are (1, 5), (2, 5), (3, 5), (4, 5), (6, 5), (5, 1), (5, 2), (5, 3), (5, 4), (5, 6). Number of such cases = 10. ∴ P(that 5 will come up exactly one time) =

10 5 = ⋅ 36 18

(iii) There is only one case, namely (5, 5) when 5 comes up both the times. 1 ⋅ ∴ P(that 5 will come up both the times) = 36 17. Number of all possible outcomes = 36. Let E be the event of getting 2 numbers on the two dice whose product is a perfect square. Then, the favourable outcomes are (1, 1), (1, 4), (4, 1), (2, 2), (3, 3), (4, 4) (5, 5), (6, 6). 8 2 Their number is 8. So, P( E) = = ⋅ 36 9 18. Total number of letters in the the given word = 11. (i) Number of vowels in the given word = 6. 6 ⋅ ∴ P(selecting a vowel) = 11

920

Secondary School Mathematics for Class 10 (ii) Number of consonants in the given word = 5. 5 P(selecting a consonant) = ⋅ 11 (iii) Number of S in the given word = 2. 2 ⋅ ∴ P(getting an S) = 11

19. Total number of cards = 5. 1 (a) P(getting a queen) = ⋅ 5 (b) When the queen is set aside, remaining number of cards = 4. 1 (i) P(getting an ace now) = ⋅ 4 0 (ii) P(getting a queen now) = = 0. 4 20. There are 26 red cards containing 2 queens and 2 more queens are there. 28 7 P(getting a red card or a queen) = = ⋅ 52 13 7 6 = ⋅ ∴ P(getting neither a red card nor a queen) = 1 − 13 13 21. An ordinary year has 365 days consisting of 52 weeks 1 day. This day can be any day of the week. 1 P(of this day to be Monday) = ⋅ 7 22. There are 6 red face cards. These are removed. Remaining number of cards = 52 − 6 = 46. (i) Number of red cards now = 26 − 6 = 20. 20 10 = ⋅ ∴ P(getting a red card) = 46 23 (ii) Remaining number of face cards = 12 − 6 = 6. 6 3 = ⋅ ∴ P(getting a face card) = 46 23 (iii) There are 12 cards of clubs. 12 6 = ⋅ ∴ P(getting a card of clubs) = 46 23 23. 4 kings, 4 queens and 4 aces have been removed. Remaining number of cards = 52 − 12 = 40. (i) There are 2 black face cards in the remaining cards. These are 2 black jacks. 2 1 ∴ P(getting 2 black face cards) = = ⋅ 40 20 (ii) Remaining number of red cards = 26 − ( 2 + 2 + 2 ) = 20. 20 1 = ⋅ ∴ P(getting a red card) = 40 2 24. When a coin is tossed 3 times, all possible outcomes are HHH, HHT, HTH, THH, HTT, THT, TTH, TTT.

Probability

921

Number of all possible outcomes = 8. 1 (i) P(getting 3 heads) = ⋅ 8 (ii) At least 2 tails means 2 or 3 tails. All such cases are HTT, THT, TTH, TTT. Their number is 4. 4 1 ∴ P(getting at least 2 tails) = = ⋅ 8 2 25. A leap year has 366 days = 52 weeks and 2 days. Now, 52 weeks contain 52 Sundays. The remaining 2 days can be: (i) Sunday and Monday (iii) Tuesday and Wednesday (v) Thursday and Friday (vii) Saturday and Sunday

(ii) Monday and Tuesday (iv) Wednesday and Thursday (vi) Friday and Saturday

Out of these 7 cases, their are 2 cases favouring it to be Sunday. 2 ∴ P(a leap year having 53 Sundays) = ⋅ 7

................................................................

MULTIPLE-CHOICE QUESTIONS (MCQ) Choose the correct answer in each of the following questions: 1. If P(E) denotes the probability of an event E then (a) P(E) < 0

(b) P(E) > 1

(c) 0 ≤ P(E) ≤ 1

[CBSE 2013C]

(d) −1 ≤ P(E) ≤ 1

2. If the probability of occurrence of an event is p then the probability of non-happening of this event is [CBSE 2013C] (a) ( p − 1)

(b) (1 − p)

(c) p

3. What is the probability of an impossible event? 1 (b) 0 (c) 1 (a) 2

⎛ 1⎞ (d) ⎜ 1 − ⎟ p⎠ ⎝ (d) More than 1

4. What is the probability of a sure event? 1 (c) 1 (d) Less than 1 (a) 0 (b) 2 5. Which of the following cannot be the probability of an event? [CBSE 2011] 3 (c) 25% (d) 0.3 (a) 1.5 (b) 5 6. A number is selected at random from the numbers 1 to 30. What is the probability that the selected number is a prime number? [CBSE 2014] 2 1 1 11 (b) (c) (d) (a) 3 6 3 30

922

Secondary School Mathematics for Class 10

7. The probability that a number selected at random from the numbers [CBSE 2014] 1, 2, 3, ..., 15 is a multiple of 4, is 4 2 1 1 (a) (b) (c) (d) 15 15 5 3 8. A box contains cards numbered 6 to 50. A card is drawn at random from the box. The probability that the drawn card has a number which is a [CBSE 2013] perfect square is 1 2 4 1 (b) (c) (d) (a) 45 15 45 9 9. A box contains 90 discs, numbered from 1 to 90. If one disc is drawn at random from the box, the probability that it bears prime number less [CBSE 2013] than 23 is 7 1 4 8 (b) (c) (d) (a) 90 9 45 89 10. Cards bearing numbers 2, 3, 4, ..., 11 are kept in a bag. A card is drawn at random from the bag. The probability of getting a card with a prime [CBSE 2012] number is 1 2 3 5 (a) (b) (c) (d) 2 5 10 9 11. One ticket is drawn at random from a bag containing tickets numbered 1 to 40. The probability that the selected ticket has a number, which is a [CBSE 2013C] multiple of 7, is 1 1 1 7 (a) (b) (c) (d) 7 8 5 40 12. Which of the following cannot be the probability of an event? 1 7 (a) (b) 0.3 (c) 33% (d) 3 6 13. If the probability of winning a game is 0.4 then the probability of losing it, is 1 (c) 0.6 (d) none of these (a) 0.96 (b) 0.4 14. If an event cannot occur then its probability is 1 3 (a) 1 (b) (c) 2 4

(d) 0

15. There are 20 tickets numbered as 1, 2, 3, ..., 20 respectively. One ticket is drawn at random. What is the probability that the number on the ticket drawn is a multiple of 5? 1 1 2 3 (b) (c) (d) (a) 4 5 5 10

Probability

923

16. There are 25 tickets numbered as 1, 2, 3, 4, ..., 25 respectively. One ticket is drawn at random. What is the probability that the number on the ticket is a multiple of 3 or 5? 2 11 12 13 (b) (c) (d) (a) 5 25 25 25 17. Cards, each marked with one of the numbers 6, 7, 8, …, 15, are placed in a box and mixed thoroughly. One card is drawn at random from the box. What is the probability of getting a card with number less than 10? [CBSE 2009C]

3 (a) 5

1 (b) 3

1 (c) 2

2 (d) 5

18. A die is thrown once. The probability of getting an even number is [CBSE 2013]

1 (a) 2

1 (b) 3

1 (c) 6

5 (d) 6

19. The probability of throwing a number greater than 2 with a fair die is [CBSE 2011]

2 (a) 5

5 (b) 6

1 (c) 3

2 (d) 3

20. A die is thrown once. The probability of getting an odd number greater than 3 is [CBSE 2013C] 1 1 1 (b) (c) (d) 0 (a) 3 6 2 21. A die is thrown once. The probability of getting a prime number is [CBSE 2013]

2 (a) 3

1 (b) 3

1 (c) 2

1 (d) 6

22. Two dice are thrown together. The probability of getting the same number on both dice is [CBSE 2012] 1 1 1 1 (b) (c) (d) (a) 2 3 6 12 23. The probability of getting 2 heads, when two coins are tossed, is [CBSE 2012]

(a) 1

3 (b) 4

1 (c) 2

1 (d) 4

24. Two dice are thrown together. The probability of getting a doublet is [CBSE 2013]

1 (a) 3

1 (b) 6

1 (c) 4

2 (d) 3

924

Secondary School Mathematics for Class 10

25. Two coins are tossed simultaneously. What is the probability of getting at most one head? 1 1 2 3 (b) (c) (d) (a) 4 2 3 4 26. Three coins are tossed simultaneously. What is the probability of getting exactly two heads? 1 1 3 3 (b) (c) (d) (a) 2 4 8 4 27. In a lottery, there are 8 prizes and 16 blanks. What is the probability of getting a prize? 1 1 2 (b) (c) (d) None of these (a) 2 3 3 28. In a lottery, there are 6 prizes and 24 blanks. What is the probability of not getting a prize? 3 3 4 (a) (b) (c) (d) None of these 4 5 5 29. A box contains 3 blue, 2 white and 4 red marbles. If a marble is drawn at random from the box, what is the probability that it will not be a [CBSE 2009C] white marble? 1 4 7 2 (b) (c) (d) (a) 3 9 9 9 30. A bag contains 4 red and 6 black balls. A ball is taken out of the bag at random. What is the probability of getting a black ball? [CBSE 2008] 2 3 1 (b) (c) (d) None of these (a) 5 5 10 31. A bag contains 8 red, 2 black and 5 white balls. One ball is drawn at random. What is the probability that the ball drawn is not black? 8 2 13 1 (b) (c) (d) (a) 15 15 15 3 32. A bag contains 3 white, 4 red and 5 black balls. One ball is drawn at random. What is the probability that the ball drawn is neither black nor white? 1 1 1 3 (b) (c) (d) (a) 4 2 3 4 33. A card is drawn at random from a well-shuffled deck of 52 cards. What is the probability of getting a black king? [CBSE 2009C] (a)

1 13

(b)

1 26

(c)

2 39

(d) None of these

Probability

925

34. From a well-shuffled deck of 52 cards, one card is drawn at random. What is the probability of getting a queen? 1 1 4 (b) (c) (d) None of these (a) 13 26 39 35. One card is drawn at random from a well-shuffled deck of 52 cards. What is the probability of getting a face card? 1 3 3 4 (b) (c) (d) (a) 26 26 13 13 36. One card is drawn at random from a well-shuffled deck of 52 cards. What is the probability of getting a black face card? 1 3 3 3 (b) (c) (d) (a) 26 26 13 14 37. One card is drawn at random from a well-shuffled deck of 52 cards. What is the probability of getting a 6? 3 1 1 (b) (c) (d) None of these (a) 26 52 13 ANSWERS (MCQ)

1. (c)

2. (b)

3. (b)

4. (c)

5. (a)

6. (c)

7. (c)

8. (d)

9. (c) 10. (a)

11. (b) 12. (d) 13. (c) 14. (d) 15. (b) 16. (c) 17. (d) 18. (a) 19. (d) 20. (b) 21. (c) 22. (c) 23. (d) 24. (b) 25. (d) 26. (c) 27. (b) 28. (c) 29. (c) 30. (b) 31. (c) 32. (c) 33. (b) 34. (a) 35. (c) 36. (b) 37. (c) HINTS TO SOME SELECTED QUESTIONS 2. P(occurrent of an event) = p ⇒ P(non-occurence of this event) = ( 1 − p ). 5. The probability of an event cannot be greater than 1. 6. Prime numbers from 1 to 30 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. Their number is 10. 10 1 = ⋅ 30 3 7. Total number of given numbers = 15. ∴

P(getting a prime number) =

From given numbers, the multiples of 4 are 4, 8, 12. They are 3 in number. ∴

P(getting a multiple of 4) =

3 1 = ⋅ 15 5

8. Given numbers are 6, 7, 8, 9, ..., 50. Number of these numbers = 50 − 5 = 45.

926

Secondary School Mathematics for Class 10 Perfect square numbers from these are 3 2 , 4 2 , 5 2 , 6 2 , 7 2 . Their number is 5.

∴

P(getting a perfect square number) =

5 1 = ⋅ 45 9

9. Total number of discs = 90. Prime number less than 23 are 2, 3, 5, 7, 11, 13, 17, 19. Their number is 8. P(getting a prime number less than 23) =

8 4 = ⋅ 90 45

10. Total number of cards = 10. Prime numbers from given numbers are 2, 3, 5, 7, 11. Their number is 5. ∴

P(getting a prime number) =

5 1 = ⋅ 10 2

11. Total number of tickets = 40. Tickets bearing the numbers as multiple of 7 bear the numbers 7, 14, 21, 28, 35. Their number is 5. 5 1 = ⋅ 40 8 13. P(losing the game) = 1 − P(winning the game) = ( 1 − 0.4 ) = 0.6. ∴

P(getting a multiple of 7) =

14. If an event cannot occur then its probability is 0. 15. Total number of tickets = 20. Multiples of 5 are 5, 10, 15, 20. Their number is 4. ∴

P(getting a multiple of 5) =

4 1 = ⋅ 20 5

16. Total number of tickets = 25. Multiples of 3 or 5 are 3, 6, 9, 12, 15, 18, 21, 24, 5, 10, 20, 25. Number of these numbers = 12. ∴

P(getting a multiple of 3 or 5) =

12 ⋅ 25

17. Total number of cards = 15 − 5 = 10. Number of cards with number less than 10 = 4. ∴

P(getting a card with number less than 10) =

4 2 = ⋅ 10 5

18. Number of all possible outcomes = 6. Even numbers are 2, 4, 6. Their number is 3. 3 1 ∴ P(getting an even number) = = ⋅ 6 2 19. Number of all possible outcomes = 6. Numbers greater than 2 are 3, 4, 5, 6. Their numbr is 4. 4 2 ∴ P(getting a number greater than 2) = = ⋅ 6 3

Probability

927

20. Number of all possible outcomes = 6. Odd number greater than 3 is 5 only. Its number is 1. 1 ∴ P(getting an odd number greater than 3) = ⋅ 6 21. Number of all possible outcomes = 6. Prime numbers are 2, 3, 5. Their number is 3. 3 1 ∴ P(getting a prime number) = = ⋅ 6 2 22. Total number of all possible outcomes = 36. Getting same number on both dice means getting (1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6). Their number is 6. ∴

P(getting the same number on both dice) =

6 1 = ⋅ 36 6

23. All possible outcomes are HH, HT, TH, TT. Their number is 4. Getting 2 heads, means getting HH. Its number is 1. 1 ∴ P(getting 2 heads) = ⋅ 4 24. Number of all possible outcomes = 36. The doublets are (1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6). Their number is 6. ∴

P(getting a doublet) =

6 1 = ⋅ 36 6

25. When two coins are tossed simultaneously, all possible outcomes are HH, HT, TH, TT. Their number is 4. All favourable outcomes are TT, TH, HT. Their number is 3. 3 ∴ P(getting at most 1 head) = ⋅ 4 26. When 3 coins are tossed, all possible outcomes are HHH, HHT, HTH, THH, HTT, THT, TTH, TTT. Their number is 8. Favourable outcomes are HHT, HTH, THH. Their number is 3. 3 ∴ P(getting exactly 2 heads) = ⋅ 8 27. Total number of lottery tickets = 8 + 16 = 24. Number of prizes = 8. ∴

P(getting a prize) =

8 1 = ⋅ 24 3

28. Total number of tickets = 6 + 24 = 30. Number of blanks = 24. ∴

P(not getting a prize) =

24 4 = ⋅ 30 5

928

Secondary School Mathematics for Class 10

29. Total number of marbles = 3 + 2 + 4 = 9. Number of non-white marbles = 3 + 4 = 7. 7 ∴ P(getting a non-white marble) = ⋅ 9 30. Total number of balls in the bag = 4 + 6 = 10. Number of black balls = 6. ∴

P(getting a black ball) =

6 3 = ⋅ 10 5

31. Total number of balls in the bag = 8 + 2 + 5 = 15. Number of non-black balls = 8 + 5 = 13. 13 P(getting a non-black ball) = ⋅ 15 32. Total number of balls in the bag = 3 + 4 + 5 = 12. Number of non-black and non-white balls = 4. ∴

P(getting a ball which is neither black nor white) =

33. Number of all possible outcomes = 52. Number of black kings = 2. ∴

P(getting a black king) =

2 1 = ⋅ 52 26

34. Number of all possible outcomes = 52. Number of queens = 4. ∴

P(getting a queen) =

4 1 = ⋅ 52 13

35. Total number of cards = 52. Number of face cards = 12 (4 kings + 4 queens + 4 jacks). 12 3 P(getting a face card) = = ⋅ 52 13

∴

36. Total number of cards = 52. Number of black face cards = 6 (2 kings +2 queens + 2 jacks). 6 3 = ⋅ ∴ P(getting a face card) = 52 26 37. Total number of cards = 52. Number of 6 s = 4. ∴

P(getting a 6) =

4 1 = ⋅ 52 13

_

4 1 = ⋅ 12 3

Sample Paper I

929

SAMPLE PAPER I Time: 3 hours

80 Marks

GENERAL INSTRUCTIONS

1. All questions are compulsory. 2. The paper consists of 30 questions divided into four sections: A, B, C and D. 3. (i) Section A consists of 6 Very-Short-Answer (VSA) questions of 1 mark each. (ii) Section B consists of 6 Short-Answer (SA) questions of 2 marks each. (iii) Section C consists of 10 Short-Answer (SA) questions of 3 marks each. (iv) Section D consists of 8 Long-Answer (LA) questions of 4 marks each. 4. Use of calculator is not permitted.

SECTION A 1. What is the distance between the points A(sin   cos , 0) and B(0, sin   cos ) ? 2. The angle of depression of a car standing on the ground from the top of a 66 m tower, is 30. Find the distance of the car from the base of the tower. 3. A rational number in its decimal expansion is 554.6023. What can you say about the prime factors of q, when this number is expressed in the p form q ? 4. If sin 

1 then find the value of 2 cot 2  2. 3

5. In a family of 3 children, find the probability of having at least one boy. 929

930

Secondary School Mathematics for Class 10

6. If  and  are the zeros of the quadratic polynomial f (x)  5x 2  7x  1, 1 1 find the value of   · 

SECTION B 7. The length of the minute hand of a clock is 5 cm. Find the area swept by the minute hand during the time from 4:05 a.m. to 4:40 a.m. 8. Prove that three times the square of any side of an equilateral triangle is equal to four times the square of the altitude. 9. In the given figure, O is the centre of the circle. PA and PB are tangent segments. Show that the quadrilateral AOBP is cyclic.

A O

B

P

10. Find the lengths of the medians of a 3ABC whose vertices are A(7, –3), B(5, 3) and C(3, –1). 11. If p, q are prime positive integers, prove that p  q is an irrational number. 12. Determine the value of k so that the following system of linear equations has no solution: (3k  1) x  3y  2  0, (k 2  1) x  (k  2) y  5  0.

SECTION C 13. Show that the square of any positive integer cannot be of the form 5q  2 or 5q  3 for any integer q. 14. Prove that

tan A  cot A   1 tan A  cot A  1  sec A · cosec A. 1  cot A 1  tan A

15. A straight highway leads to the foot of a tower. A man standing at the top of the tower observes a car at angle of depression of 30°, which is approaching the foot of the tower with a uniform speed. Six seconds later, the angle of depression of the car is found to be 60°. Find the further time taken by the car to reach the foot of the tower.

Sample Paper I

931

16. If the points A(1, –2), B(2, 3), C(–3, 2) and D(–4, –3) are the vertices of a parallelogram ABCD then taking AB as the base find the height of the parallelogram. 17. A plane left 30 minutes later than the scheduled time and in order to reach its destination 1500 km away in time, it has to increase its speed by 250 km/hr from its usual speed. Find its usual speed. 18. Obtain all the zeros of the polynomial f (x)  3x 4  6x 3  2x 2  10x  5 if 5 5 · two of its zeros are and  3 3 19. Find the mean, median and mode of the following data: Classes

0–20

20–40

40–60

60–80

Frequency

6

8

10

12

80–100 100–120 120–140 6

5

3

20. Two customers are visiting a particular shop in the same week (Monday to Saturday). Each is equally likely to visit the shop on any one day as on another. What is the probability that both will visit the shop on (i) the same day?

(ii) different days?

(iii) consecutive days? 21. PQ is a chord of length 8 cm of a circle of radius 5 cm. The tangents at P and Q intersect at a point T. Find the length TP. 22. Solve the following system of equations in x and y: (a  b) x  (a  b) y  a 2  2ab  b 2, (a  b)(x  y)  a 2  b 2.

SECTION D 23. An open metal bucket is in the shape of a frustum of a cone, mounted on a hollow cylindrical base made of the same metallic sheet. The diameters of the two circular ends of the bucket are 45 cm and 25 cm, the total vertical height of the bucket is 40 cm and that of the cylindrical base is 6 cm. Find the area of the metallic sheet used to make the bucket. Also, find the volume of water the bucket can hold, in litres. 24. It can take 12 hours to fill a swimming pool using two pipes. If the pipe of larger diameter is used for 4 hours and the pipe of smaller diameter for 9 hours, only half of the pool can be filled. How long would it take for each pipe to fill the pool separately?

932

Secondary School Mathematics for Class 10

25. If the angle of elevation of a cloud from a point h metres above a lake be  and the angle of depression of its reflection in the lake be , prove that 2h sec  · the distance of the cloud from the point of observation is tan   tan  26. The following table gives the yield per hectare of wheat of 100 farms of a village: Yield in kg/hectare

50–55

55–60

60–65

65–70

70–75

75–80

Number of farms

2

8

12

24

38

16

Change the above distribution to ‘more than type’ distribution and draw its ogive. 27. Sides of a triangular field are 15 m, 16 m and 17 m. With the three corners of the field a cow, a buffalo and a horse are tied separately with ropes of length 7 m each to graze in the field. Find the area of the field which cannot be grazed by the three animals. 28. The radius of incircle of a triangle is 4 cm and the segments into which one side is divided by the point of contact are 6 cm and 8 cm. Determine the other two sides of the triangle. 29. Two poles of height a metres and b metres are p metres apart. Prove that the height of the point of intersection of the lines joining the top of each ab pole to the foot of the opposite pole is given by metres. ab 30. In the centre of a rectangular lawn of dimensions 50 m # 40 m, a rectangular pond has to be constructed so that the area of the grass surrounding the pond would be 1184 m 2. Find the length and breadth of the pond. ANSWERS

1.

2

2. 22 3 m

3. Since 554.6023 is a terminating decimal number so q must be of the

form 2 m # 5 n, where m, n are natural numbers. 5 7 4. 18 5. 6. 7 7. 45 cm 2 10. 5 units; 5 units; 10 units 6 8 24 12. k  1 15. 3 seconds 16. units 17. 750 km/hr 26 5  5   18. , , 1, 1 19. Mean = 61.66, median = 62.4, mode = 65 3 3

Sample Paper I

20. (i)

1 6

(ii)

5 6

(iii)

5 36

21.

20 cm 3

933

22. x  a  b, y  

23. 4860.9 cm 2, 33.62 litres

24. 20 hours, 30 hours

27. (24 21  77) m 2

28. 15 cm, 13 cm

30. Length = 34 m, Breadth = 24 m



2ab ab

934

Secondary School Mathematics for Class 10

SAMPLE PAPER II Time: 3 hours

80 Marks

GENERAL INSTRUCTIONS

1. All questions are compulsory. 2. The paper consists of 30 questions divided into four sections: A, B, C and D. 3. (i) Section A consists of 6 Very-Short-Answer (VSA) questions of 1 mark each. (ii) Section B consists of 6 Short-Answer (SA) questions of 2 marks each. (iii) Section C consists of 10 Short-Answer (SA) questions of 3 marks each. (iv) Section D consists of 8 Long-Answer (LA) questions of 4 marks each. 4. Use of calculator is not permitted.

SECTION A 1. Evaluate: sin 35° · sin 55° – cos 35° · cos 55°. 2. The decimal expansion of the rational number after how many places of decimals?

83 will terminate 23 # 54

3. If the point P(x, y) is equidistant from A(5, 1) and B(–1, 5) then find the relation between x and y. 4. If  and  are the zeros of the polynomial f (x)  x 2  5x  k such that     1, find the value of k. 5. The tops of two towers of heights x and y, standing on level ground, subtend angles of 30° and 60° respectively at the centre of the line joining their feet. Find x : y. 934

Sample Sample Paper Paper II I

935

6. A die is rolled twice. Find the probability that 4 will come up exactly one time.

SECTION B 7. Find the value of a and b for which the following system of equations has infinitely many solutions: x  2y  1, (a  b) x  (a  b) y  a  b  2. 8. Use Euclid’s division algorithm to find the HCF of 441, 567 and 693. 9. Find k so that the point (–4, 6) lies on the line segment joining A(k, 10) and B(3, –8). Also, find the ratio in which P divides AB. 10. Prove that the segment joining the points of contact of two parallel tangents passes through the centre. 11. The perimeter of a right triangle is 60 cm. Its hypotenuse is 25 cm. Find the area of the triangle. 12. Find the area of the shaded region in the figure given below where radii of the two concentric circles with centre O are 7 cm and 14 cm respectively and +AOC  40c.

O B 40º D

A C

SECTION C 13. Three unbiased coins are tossed simultaneously. Find the probability of getting (i) exactly two heads;

(ii) at least two heads;

(iii) at most 2 heads. 14. Prove that the angle between two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line segments joining the points of contact at the centre. 15. Solve the following system of equations in x and y: ax  by  c, bx  ay  1  c.

936

Secondary School Mathematics for Class 10

16. Use Euclid’s division lemma to show that the cube of any positive integer is either of the form 9m, 9m + 1 or 9m + 8 for some integer m. 1· 2 18. From the top of a building AB, 60 m high, the angles of depression of the top and bottom of a vertical lamp-post CD are observed to be 30° and 60° respectively. Find 17. If 1  sin 2   3 sin  · cos , prove that tan  1 or

(i) the horizontal distance between AB and CD, (ii) the height of the lamp-post. 19. Find the area of a parallelogram ABCD if three of its vertices are A(2, 4), B(2  3 , 5) and C(2, 6). 20. A train travelling at a uniform speed for 360 km would have taken 48 minutes less to travel the same distance if its speed were 5 km/hr more. Find the original speed of the train. 21. If two zeros of the polynomial f (x)  x 4  6x 3  26x 2  138x  35 are 2 ! 3 , find other zeros. 22. The following table shows the marks obtained by 100 students of a class in a school during a particular academic session. Find the modal marks. Marks Number of students

Less Less Less Less Less Less Less Less than 10 than 20 than 30 than 40 than 50 than 60 than 70 than 80 7

21

34

46

66

77

92

100

SECTION D 23. There is a square field whose side is 44 m. A square flower bed is prepared in its centre leaving a gravel path all around the flower bed. The total cost of laying the flower bed and gravelling the path at ` 2.75 and ` 1.50 per square metre respectively, is ` 4904. Find the width of the gravel path. 24. Prove that the ratio of the areas of two similar triangles is equal to the ratio of the squares of any two corresponding sides. 25. From an external point P, a tangent PT and a line segment PAB is drawn to a circle with centre O. ON is perpendicular on the chord AB. Prove that (i) PA · PB  PN 2  AN 2 (ii) PN 2  AN 2  OP 2  OT 2 (iii) PA · PB  PT 2

Sample Sample Paper Paper III

937

26. ABCD is a trapezium with AB  DC, AB  18 cm, DC  32 cm and the distance between AB and DC is 14 cm. If arcs of equal radii 7 cm with centres A, B, C and D are drawn find the area of the remaining region, i.e., region of the trapezium excluding the circular arcs. 27. The table given below shows the frequency distribution of the scores obtained by 200 candidates in an entrance examination: Scores

200– 250

250– 300

300– 350

350– 400

400– 450

450– 500

500– 550

550– 600

No. of candidates

30

15

45

20

25

40

10

15

Draw cumulative frequency curve using ‘less than series’. 28. A window of a house is h metres above the ground. From the window, the angles of elevation and depression of the top and bottom of another house situated on the opposite side of the lane are found to be  and  respectively. Prove that the height of the house is h(1  tan  · tan ) metres. 29. 8 men and 12 boys can finish a piece of work in 10 days while 6 men and 8 boys can finish it in 14 days. Find the time taken by one man alone and that by one boy alone to finish the work. 30. From a solid cylinder whose height is 15 cm and diameter 16 cm, a conical cavity of the same height and same diameter is hollowed out. Find the total surface area of the remaining solid. [Use  3.14.] ANSWERS

1. 0

2. 4

3. 2x  3y

4. 6

5 6. 18

7. a  3, b  1

8. 63

9. k   6; 2 : 7

154 cm 2 3

13. (i)

11. 150 cm 2 15. x 

12.

3 8

(ii)

c  b ,  c  a y a  b a2  b2 a  b a2  b2

19. 2 3 sq units

20. 45 km/hr

23. 2 metres

26. 196 cm

1 2

(iii)

(ii) 40 m

21. 7, –5

22. 44.71

29. Time taken by one man alone = 140 days and

time taken by one boy alone = 280 days



7 8

18. (i) 34.64 m

2

30. 443.14 cm 2

5. 1 : 3