Secondary School Mathematics for Class 9 (Secondary School Mathematics for Class 9) [Paperback ed.] 817709727X, 9788177097276

Table of contents :
Cover......Page 1
Title......Page 2
Preface......Page 4
Contents......Page 12
Number Systems......Page 14
Polynomials......Page 81
Factorisation of Polynomials......Page 110
Linear Equations in Two Variables......Page 155
Coordinate Geometry......Page 180
Introduction to Euclid’s Geometry......Page 192
Lines and Angles......Page 208
Triangles......Page 251
Congruence of Triangles and Inequalities in a Triangle......Page 276
Quadrilaterals......Page 319
Areas of Parallelograms and Triangles......Page 376
Circles......Page 420
Geometrical Constructions......Page 514
Volume and Surface Area of Solids......Page 557
Presentation of Data in Tabular Form......Page 629
Bar Graph, Histogram and Frequency Polygon......Page 645
Mean, Median and Mode of Ungrouped Data......Page 675
Probability......Page 703

Citation preview

Secondary School Mathematics FOR CLASS 9 (In accordance with the latest CBSE syllabus)

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

Preface We feel proud to present this edition of the book. It is based on the new CBSE syllabus. Apart from the CBSE course, this book can be used as a textbook for the courses of those state boards which follow the CBSE syllabus. The book has been thoroughly revised and reset. It now has a large number of questions from the NCERT textbook with full explanatory solutions. A large number of multiple-choice questions (MCQ) on all topics have also been included. This book will also be of immense help to the students who wish to sit for the board examination. Though revised and reset, it continues to have the qualities which made it so popular among teachers and students in the past. We have emphasized the theoretical as well as the numerical aspects of the mathematics course. The underlying concepts have been gradually and systematically developed. The exposition is simple, yet rigorous. The language is precise and simple. In each chapter, all the results and concepts of a particular topic have been put together. These are followed by a large number of solved examples. Quite a large number of problems have been given as exercises. We are thankful to the editorial and production staff of Bharati Bhawan for doing such a good job. We also wish to thank all teachers and students who sent suggestions for the improvement of the book. We hope that we shall continue to receive such invaluable feedback. —Authors

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Mathematics Syllabus For Class 9

Unit I: Number Systems 1. Real Numbers

(18 Periods)

Review of representation of natural numbers, integers, rational numbers on the number line. Representation of terminating/nonterminating recurring decimals on the number line through successive magnification. Rational numbers as recurring/terminating decimals. Operations on real numbers. Examples of nonrecurring/nonterminating decimals. Existence of nonrational numbers (irrational numbers) such as 2 , 3 and their representation on the number line. Explaining that every real number is represented by a unique point on the number line and conversely, every point on the number line represents a unique real number. Definition of nth root of a real number. Existence of x for a given positive real number x and its representation on the number line with geometric proof. Rationalisation (with precise meaning) of real numbers of the type (and 1 1 their combinations) and , where x and y are natural ab x x y numbers, a and b are integers. Recall of laws of exponents with integral powers. Rational exponents with positive real bases (to be done by particular cases, allowing learner to arrive at the general laws).

Unit II: Algebra 1. Polynomials

(23 Periods)

Definition of a polynomial in one variable, its coefficients, with examples and counter-examples, its terms, zero polynomial. Degree of a polynomial. Constant, linear, quadratic, cubic polynomials; monomials, binomials, trinomials. Factors and multiples. Zeros of a polynomial. Motivate and state the Remainder Theorem with examples. Statement and proof of the Factor Theorem. Factorisation of ax 2  bx  c, a ! 0, where a, b and c are real numbers, and of cubic polynomials using the Factor Theorem. (v)

Recall of algebraic expressions and identities. Further verification of identities of the type (x  y  z) 2  x 2  y 2  z 2  2xy  2yz  2zx, (x ! y) 3  x 3 ! y 3 ! 3xy (x ! y), x 3 ! y 3  (x ! y) (x 2 " xy  y 2) x 3  y 3  z 3  3xyz  (x  y  z) (x 2  y 2  z 2  xy  yz  zx), and their use in factorization of polynomials. 2. Linear Equations in Two Variables

(14 Periods)

Recall of linear equations in one variable. Introduction to the equation in two variables. Focus on linear equations of the type ax  by  c  0. Prove that a linear equation in two variables has infinitely many solutions, and justify their being written as ordered pairs of real numbers, plotting them and showing that they lie on a line. Graph of linear equations in two variables. Examples, problems from real life, including problems on Ratio and Proportion, and with algebraic and graphical solutions being done simultaneously.

Unit III: Coordinate Geometry 1. Coordinate Geometry

(6 Periods)

The Cartesian plane, coordinates of a point, names and terms associated with the coordinate plane, notations, plotting points in the plane.

Unit IV: Geometry 1. Introduction to Euclid’s Geometry

(6 Periods)

History—Geometry in India and Euclid’s geometry. Euclid’s method of formalizing observed phenomenon into rigorous mathematics with definitions, common/obvious notions, axioms/postulates, and theorems. The five postulates of Euclid. Equivalent versions of the fifth postulate. Showing the relationship between axiom and theorem, for example: (Axiom) 1. Given two distinct points, there exists one and only one line through them. (Theorem) 2. (Prove) Two distinct lines cannot have more than one point in common. 2. Lines and Angles (13 Periods) 1. (Motivate) If a ray stands on a line, then the sum of the two adjacent angles so formed is and the converse. 2. (Prove) If two lines intersect, vertically opposite angles are equal. (vi)

3. (Motivate) Results on corresponding angles, alternate angles, interior angles when a transversal intersects two parallel lines. 4. (Motivate) Lines, which are parallel to a given line, are parallel. 5. (Prove) The sum of the angles of a triangle is 180. 6. (Motivate) If a side of a triangle is produced, the exterior angle so formed is equal to the sum of the two interior opposite angles. 3. Triangles

(20 Periods)

1. (Motivate) Two triangles are congruent if any two sides and the included angle of one triangle are equal to any two sides and the included angle of the other triangle (SAS congruence). 2. (Prove) Two triangles are congruent if any two angles and the included side of one triangle are equal to any two angles and the included side of the other triangle (ASA congruence). 3. (Motivate) Two triangles are congruent if the three sides of one triangle are equal to three sides of the other triangle (SSS congruence). 4. (Motivate) Two right triangles are congruent if the hypotenuse and a side of one triangle are equal (respectively) to the hypotenuse and a side of the other triangle. 5. (Prove) The angles opposite to equal sides of a triangle are equal. 6. (Motivate) The sides opposite to equal angles of a triangle are equal. 7. (Motivate) Triangle inequalities and relation between ‘angle and facing side’ inequalities in triangles. 4. Quadrilaterals

(10 Periods)

1. (Prove) The diagonal divides a parallelogram into two congruent triangles. 2. (Motivate) In a parallelogram opposite sides are equal, and conversely. 3. (Motivate) In a parallelogram opposite angles are equal, and conversely. 4. (Motivate) A quadrilateral is a parallelogram if a pair of its opposite sides are parallel and equal. 5. (Motivate) In a parallelogram, the diagonals bisect each other, and conversely. 6. (Motivate) In a triangle, the line segment joining the midpoints of any two sides is parallel to the third side and is half of it, and (motivate) its converse. (vii)

5. Area

(7 Periods)

Review concept of area, recall area of a rectangle. 1. (Prove) Parallelograms on the same base and between the same parallels have the same area. 2. (Motivate) Triangles on the same (or equal base) base and between the same parallels are equal in area. 6. Circles

(15 Periods)

Through examples, arrive at definitions of circle and related concepts— radius, circumference, diameter, chord, arc, secant, sector segment, subtended angle. 1. (Prove) Equal chords of a circle subtend equal angles at the centre, and (motivate) its converse. 2. (Motivate) The perpendicular from the centre of a circle to a chord bisects the chord and conversely, the line drawn through the centre of a circle to bisect a chord is perpendicular to the chord. 3. (Motivate) There is one and only one circle passing through three given noncollinear points. 4. (Motivate) Equal chords of a circle (or of congruent circles) are equidistant from the centre (or their respective centres), and conversely. 5. (Prove) 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. 6. (Motivate) Angles in the same segment of a circle are equal. 7. (Motivate) If a line segment joining two points subtends equal angles at two other points lying on the same side of the line containing the segment, the four points lie on a circle. 8. (Motivate) The sum of either of the pair of the opposite angles of a cyclic quadrilateral is 180, and its converse. 7. Constructions

(10 Periods)

1. Construction of bisectors of line segments and angles of measure 60, 90, 45, etc., equilateral triangles. 2. Construction of a triangle given its base, sum/difference of the other two sides and one base angle. 3. Construction of a triangle of given perimeter, and base angles. (viii)

Unit V: Mensuration 1. Areas

(4 Periods)

Area of a triangle using Heron’s formula (without proof) and its application in finding the area of a quadrilateral. 2. Surface Areas and Volumes

(12 Periods)

Surface areas and volumes of cubes, cuboids, spheres (including hemispheres) and right circular cylinders/cones.

Unit VI: Statistics and Probability 1. Statistics

(13 Periods)

Introduction to Statistics: Collection of data, presentation of data— tabular form, ungrouped/grouped, bar graphs, histograms (with varying base lengths), frequency polygons. Mean, median and mode of ungrouped data. 2. Probability

(9 Periods)

History, repeated experiments and observed frequency approach to probability. Focus is on empirical probability. (A large amount of time to be devoted to group and to individual activities to motivate the concept; the experiments to be drawn from real-life situations, and from examples used in the chapter on statistics.)

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Weightage MATHEMATICS CLASS 9 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 

8

Real Numbers 17

2. Algebra  

Polynomials Linear Equations in Two Variables

3. Coordinate Geometry 

4

Coordinate Geometry 28

4. Geometry       

Introduction to Euclid’s Geometry Lines and Angles Triangles Quadrilaterals Area Circles Constructions

5. Mensuration  

Marks

13

Area of a Triangle using Heron’s Formula Surface Areas and Volumes

6. Statistics and Probability 

Statistics



Probability Total

10

80  (x)

Contents Number Systems

1. Number Systems

1

Algebra

2. Polynomials

68

3. Factorisation of Polynomials

97

4. Linear Equations in Two Variables

142

Coordinate Geometry

5. Coordinate Geometry

167

Geometry

6. Introduction to Euclid’s Geometry

179

7. Lines and Angles

195

8. Triangles

238

9. Congruence of Triangles and Inequalities in a Triangle

263

10. Quadrilaterals

306

11. Areas of Parallelograms and Triangles

363

12. Circles

407

13. Geometrical Constructions

501

Mensuration

14. Areas of Triangles and Quadrilaterals

516

15. Volume and Surface Area of Solids

544

Statistics and Probability

16. Presentation of Data in Tabular Form

616

17. Bar Graph, Histogram and Frequency Polygon

632

18. Mean, Median and Mode of Ungrouped Data

662

19. Probability

690 (xi)

Number Systems

1 INTRODUCTION

We have learnt about various types of numbers in our earlier classes. Let us review them and learn more about numbers. NATURAL NUMBERS

Counting numbers are called natural numbers.

The collection of natural numbers is denoted by N and is written as N  {1, 2, 3, 4, 5, 6, …} . REMARKS

(i) The least natural number is 1. (ii) There are infinitely many natural numbers.

All natural numbers together with 0 form the collection W of all whole numbers, written as W  {0, 1, 2, 3, 4, 5, …} .

WHOLE NUMBERS

REMARKS

(i) (ii) (iii) (iv)

The least whole number is 0. There are infinitely many whole numbers. Every natural number is a whole number. All whole numbers are not natural numbers, as 0 is a whole number which is not a natural number.

INTEGERS All natural numbers, 0 and negatives of natural numbers form the collection of all integers. It is represented by Z after the German word ‘zahlen’ meaning ‘to count’. Thus, we write Z  {…,  5,  4,  3,  2, 1, 0, 1, 2, 3, 4, 5, …} .

REMARKS

(i) (ii) (iii) (iv)

0 is neither negative nor positive. There are infinitely many integers. Every natural number is an integer. Every whole number is an integer.

REPRESENTATION OF INTEGERS ON NUMBER LINE

A number line is a visual representation of numbers on a graduated straight line. To represent integers on the number line, draw a line XY which extends endlessly in both the directions, as indicated by the arrowheads in the diagram below.

1

2

Secondary School Mathematics for Class 9

Take any point O on this line. Let this point represent the integer 0 (zero). Now, taking a fixed length, called unit length, set off equal distances to the right as well as to the left of O. On the right-hand side of O, the points at distances of 1 unit, 2 units, 3 units, 4 units, 5 units, etc., from O denote respectively the positive integers 1, 2, 3, 4, 5, etc. Similarly, on the left-hand side of O, the points at distances of 1 unit, 2 units, 3 units, 4 units, 5 units, etc., from O denote respectively the negative integers –1, –2, –3, –4, –5, etc. Since the line can be extended endlessly on both sides of O, it follows that we can represent each and every integer by some point on this line. For instance, starting from O and moving to its right, after 836 units, we get a point which represents the integer 836. Similarly, starting from O and moving to its left, a point after 750 units, represents the integer ‘–750’. Thus, each and every integer can be represented by some point on the number line. p The numbers of the form q , where p and q are integers and q ! 0, are known as rational numbers. The collection of rational numbers is RATIONAL NUMBERS

denoted by Q and is written as p Q  ( q : p, q are integers, q ! 02 · ‘Rational’ comes from the word ‘ratio’ and Q comes from the word ‘quotient’. 1 3 11 2001 , etc., are all rational numbers. Thus, , , ,  4 2 79 2002 REMARKS

(i) There are infinitely many rational numbers. (ii) There is no least or greatest rational number. 0 · 1 (iv) Every natural number is a rational number since we can 3 1 2 write, 1  , 2  , 3  , etc. 1 1 1

(iii) 0 is a rational number, since we can write, 0 

(v) Every integer is a rational a be written as , e.g.,  31  1

number since an integer a can  31 , 0 79 · 0  and 79  1 1 1

Hence, rational numbers include natural numbers, whole numbers and integers.

Number Systems

3

Rational numbers do not have a unique p representation in the form q , where p and q are integers and q ! 0. EQUIVALENT RATIONAL NUMBERS

Thus,

1  2  3  4   15  16   144  … … …. 2 4 6 8 30 32 288

These are known as equivalent rational numbers. SIMPLEST FORM OF A RATIONAL NUMBER

p A rational number q is said to be in its

simplest form, if p and q are integers having no common factor other than 1 (that is, p and q are co-primes) and q ! 0. 1 2 3 4 5 Thus, the simplest form of each of , , , , etc., is · 2 4 6 8 10 Similarly, the simplest form of

95 5 6 2 is and that of is · 9 3 133 7

EXAMPLE 1

Write four rational numbers equivalent to

SOLUTION

We have

4· 7

4  4#2  4#3  4# 4  4#5 7 7#2 7#3 7#4 7#5 &

4  8  12  16  20 · 7 14 21 28 35

Thus, four rational numbers equivalent to

4 are 7

20 · 8 , 12 , 16 and 25 14 21 28 REPRESENTATION OF RATIONAL NUMBERS ON REAL LINE

Draw a line XY which extends endlessly in both the directions. Take a point O on it and let it represent 0 (zero). Taking a fixed length, called unit length, mark off OA  1 unit. The midpoint B of OA denotes the rational number set off equal distances each equal to OB 

1 unit. 2

1· Starting from O, 2

4

Secondary School Mathematics for Class 9

From the point O, on its right, the points at distances equal to OB, 2OB, 1 2 3 4 3OB, 4OB, etc., denote respectively the rational numbers , , , , etc. 2 2 2 2 Similarly, from the point O, on its left, the points at distances equal to 1  2 , , OB, 2OB, 3OB, 4OB, etc., denote respectively the rational numbers 2 2 3 4 , , etc. 2 2 Thus, each rational number with 2 as its denominator can be represented by some point on the number line. Next, draw the line XY. Take a point O on it representing 0. Let OA  1 unit. Divide OA into three equal parts with OC as the first part. 1 Then, C represents the rational number · 3

From the point O, set off equal distances, each equal to OC  on both sides of O.

1 unit 3

The points at distances equal to OC, 2OC, 3OC, 4OC, etc., from the point 1 2 3 4 O on its right denote respectively the rational numbers , , , , etc. 3 3 3 3 Similarly, the points at distances equal to OC, 2OC, 3OC, 4OC, etc., 1 , from the point O on its left denote respectively the rational numbers 3 2 3 4 , , , etc. 3 3 3 Thus, each rational number with 3 as its denominator can be represented by some point on the number line (or, the real line). Proceeding in this manner, we can represent each and every rational number by some point on the line. 3 5 and (ii) 1 on real line. 8 7

EXAMPLE 2

Represent (i) 2

SOLUTION

Draw a line XY and taking a fixed length as unit length, represent integers on this line. E

X

–2

Q

D

O

A

B

–1

0

1

2

P

C 3

Y

Number Systems

5

(i) On the right of O, take OA  1 unit. Then, OB  2 units. Divide the 3rd unit BC into 8 equal parts. 3 3 BP represents of a unit. Therefore, P represents 2 · 8 8 (ii) On the left of O, take OD  1 unit. Divide the 2nd unit DE into 7 equal parts. 5 5 DQ represents of a unit. Therefore, Q represents 1 · 7 7 EXAMPLE 3 SOLUTION

8 11 and (ii)  on the number line. 5 7 Draw a line XY and taking a fixed length as unit length, represent integers on this line. Represent (i)

8 3· 1 5 5 On the right of O, take OA  1 unit. Divide the 2nd unit AB into 5 equal parts. 3 3 AP represents of a unit. Therefore, P represents 1 · 5 5 11   4 · (ii)  1 7 7 On the left of O, take OD  1 unit. Divide the 2nd unit DE into 7 equal parts. 4 Then, DQ represents of a unit. Therefore, Q represents 7  11 · 7 (i)

FINDING RATIONAL NUMBERS BETWEEN TWO GIVEN RATIONAL NUMBERS

Suppose we are required to find one rational number between two rational numbers x and y such that x  y.

METHOD 1

Then, EXAMPLE 4 SOLUTION

1  (x y) is a rational number lying between x and y. 2 3 1 1 2 Find a rational number lying between (i) and ; (ii) and  · 2 3 3 4 1 1 (i) Let x  and y  · 2 3  required rational number lying between x and y  1 (x  y)  1 b1  1 l  1 # 5  5 · 2 2 3 2 2 6 12

6

Secondary School Mathematics for Class 9

Hence, (ii) Let x  

5 1 1 is a rational number lying between and · 3 2 12 3 2 and y   · 3 4

required rational number lying between

3 2 and 3 4

 1 (x  y)  1