43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Physics [16 ed.] 8194767733, 9788194767732

Table of contents :
Starting Pages
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Chapter -13
Chapter 14
Chapter 15
Chapter 16
Chapter 17
Chapter 18
Chapter 19
Chapter 20
Chapter 21
Chapter 22
Chapter 23
Chapter 24
Chapter 25
Chapter 26
Chapter 27
Chapter 28
Chapter 1-Sol
Chapter 2-Sol
Chapter 3-Sol
Chapter 4-Sol
Chapter 5-Sol
Chapter 6-Sol
Chapter 7-Sol
Chapter 8-Sol
Chapter 9-Sol
Chapter 10-Sol
Chapter 11-Sol
Chapter 11-Sol 2
Chapter 12-Sol
Chapter 13_Sol
Chapter 14_Sol
Chapter 15_Sol
Chapter 16_Sol
Chapter 17_Sol
Chapter 18_Sol
Chapter 19_Sol
Chapter 20_Sol
Chapter 21_Sol
Chapter 22_Sol
Chapter 23_Sol
Chapter 24_Sol
Chapter 25_Sol
Chapter 26_Sol
Chapter 27_Sol
Chapter 28_Sol

Citation preview

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Disha Publication feels proud to release its thoroughly Revised & Updated Edition of the Book “43 Years IIT-JEE Advanced + JEE Main Chapter-wise

& Topic-wise Solved Paper MATHEMATICS”. This is the 16th Edition which speaks volumes about the Quality & Hard Work that has gone in building this book.

It is an integrated book, which contains Chapter-wise & Topic-wise collection of past JEE Advanced (including 1978 - 2012 IIT-JEE & 2013 2020 JEE Advanced) questions from 1978 to 2020 and past JEE Main from 2013 – 2020 (including all Online & Offline Papers). •

The unique feature of this new edition is the division of questions into

28 chapters as per NCERT. With this new feature this book has become the 1st to adopt NCERT Chapterisation among JEE Advanced Solved Papers. •

Each chapter divides the questions into 2 - 4 topics which are further

divided into 10 categories of questions – MCQ 1 option correct

2.

Integer Value Answer

3.

Numeric Answer

4.

Fill in the Blanks

5.

True/False

6.

MCQ more than 1 option correct answer

7.

Multiple Matching

8.

Passage Based

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1.

9.

Assertion-Reason

10. Subjective Questions.

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•

All the Screening and Mains papers of IIT-JEE have been incorporated

in the book. All online papers of JEE Main including the 16 papers of 2020

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& 16 papers of 2019 phase I & II have been incorporated in the book.

IIT-JEE/ JEE Advanced Papers

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• 1978 – 2012 IIT JEE Screening & Main • 2013 – 2020 JEE Advanced Papers 1 & 2

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•

JEE Main (Online + Offline) Papers

JEE Main 2013 – 5 (4 + 1) papers JEE Main 2014 – 4 (3 + 1) papers JEE Main 2015 – 3 (2 + 1) papers JEE Main 2016 – 3 (2 + 1) papers JEE Main 2017 – 3 (2 + 1) papers JEE Main 2018 – 4 (3 + 1) papers JEE Main 2019 – 16 Papers JEE Main 2020 – 16 Papers

Detailed solution of each and every question has been provided for

100% conceptual clarity of the student. Well elaborated detailed solutions with user friendly language provided at the end of the book.•

Solutions

have been given with enough diagrams, proper reasoning to bring

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conceptual clarity. •

The students are advised to attempt questions of a

topic immediately after they complete a topic in their class/ school/ home. The book contains around 4000 Milestone Problems in Mathematics.

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If utilised properly this book can take your JEE preparation to the next level. Although all efforts are made to ensure quality but some errors might have crept in. We would invite our readers to send us these errors so that we can incorporate the corrections in the upcoming editions. All the Best

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Disha Experts

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1.

Sets

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Class – XI

Topic 1 : Sets, Types of Sets, Subsets, Power Set, Cardinal Number of Sets, Operations on Sets Topic 2 : Venn Diagrams, Algebraic Operations on Sets, De Morgan’s Law, Number of Elements in Different Sets 2.

Relations and Functions

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Topic 1 : Relations and Functions, Domain, Codomain and Range Topic 2 : Types of Functions, Algebraic Operations on Functions 3.

Trigonometric Functions

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Topic 1 : Trigonometric Ratios, Domain and Range of Trigonometric Functions, Trigonometric Ratios of Allied Angles Topic 2 : Trigonometric Identities, Greatest and Least Value of Trigonometric Expressions Topic 3 : Solutions of Trigonometric Equations

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4.

Principle of Mathematical Induction

5.

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Topic 1 : Problems Based on Sum of Series, Problems Based on Inequality and Divisibility Complex Numbers and Quadratic Equations

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Topic 1 : Integral Powers of Iota, Algebraic Operations of Complex Numbers, Conjugate, Modulus and Argument or Amplitude of a Complex Number Topic 2 : Rotational Theorem, Square Root of a Complex Number, Cube Roots of Unity, Geometry of Complex Numbers, De-moiver’s Theorem, Powers of Complex Numbers Topic 3 : Solutions of Quadratic Equations, Sum and Product of Roots, Nature of Roots, Relation Between Roots and Co-efficients, Formation of an Equation with Given Roots Topic 4 : Condition for Common Roots, Maximum and Minimum value of Quadratic Equation, Quadratic Expression in two Variables, Solution of Quadratic Inequalities

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6.

Linear Inequalities

Topic 1 : Solution of Linear Inequality and System of Linear Inequalities

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7.

Permutations and Combinations

8.

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Topic 1 : Factorials and Permutations Topic : 2 Combinations and Dearrangement Theorem Binomial Theorem

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Topic 1 : Binomial Theorem for a Positive Integral Index ‘x’, Expansion of Binomial, General Term, Coefficient of any Power of ‘x’ Topic 2 : Middle Term, Greatest Term, Independent Term, Particular Term from end in Binomial Expansion, Greatest Binomial Coefficients Topic 3 : Properties of Binomial Coefficients, Number of Terms in the Expansion of (x+y+z)n, Binomial Theorem for any Index, Multinomial Theorem, Infinite Series 9.

Sequences and Series

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Topic 1 : Arithmetic Progression Topic 2 : Geometric Progression Topic 3 : Harmonic Progression, Relation Between A. M., G. M. and H.M. of two Positive Numbers Topic 4 : Arithmetic-Geometric Sequence (A.G.S.), Some Special Sequences

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10. Straight Lines and Pair of Straight Lines

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Topic 1 : Distance Formula, Section Formula, Locus, Slope of a Straight Line Topic 2 : Various Forms of Equation of a Line Topic - 3 : Distance Between two Lines, Angle Between two Lines and Bisector of the Angle Between the two Lines Topic 4 : Pair of Straight Lines 11. Conic Sections Topic 1 : Circles Topic 2 : Parabola Topic 3 : Ellipse Topic 4 : Hyperbola

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12. Limits and Derivatives

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Topic 1 : Limit of a Function, Sandwitch Theorem. Topic 2 : Limits Using L-hospital’s Rule, Evaluation of Limits of the form 1∞, Limits by Expansion Method Topic 3 : Derivatives of Polynomial & Trigonometric Functions, Derivative of Sum, Difference, Product & Quotient of two functions 13. Mathematical Reasoning

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Topic 1 : Statement, Truth Value of a Statement, Logical Connectives, Truth Table, Logical Equivalance, Tautology & Contradiction, Duality Topic 2 : Converse, Inverse & Contrapositive of the Conditional Statement, Negative of a Compound Statement, Algebra of Statement 14. Statistics

Topic 1 : Arithmetic Mean, Geometric Mean, Harmonic Mean, Median & Mode Topic 2 : Quartile, Measures of Dispersion, Quartile Deviation, Mean Deviation, Variance & Standard Deviation, Coefficient of Variation 15. Probability

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Topic 1: Random Experiment, Sample Space, Events, Probability of an Event, Mutually Exclusive & Exhaustive Events, Equally Likely Events Topic 2 : Odds Against & Odds in Favour of an Event, Addition Theorem, Boole’s Inequality, Demorgan’s Law

Class – XII

16. Relations and Functions

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Topic 1 : Types of Relations, Inverse of a Relation, Mappings, Mapping of Functions, Kinds of

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Mapping of Functions

Topic 2 : Composite Functions & Relations, Inverse of a Function, Binary Operations

17. Inverse Trigonometric Functions

Topic 1 : Trigonometric Functions & Their Inverses,

Domain & Range of Inverse Trigonometric Functions,

Principal

Value

of

Inverse

Trigonometric Functions

Topic 2 : Properties of Inverse Trigonometric

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Functions, Infinite Series of Inverse Trigonometric Functions

18. Matrices

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Topic 1 : Order of Matrices, Types of Matrices,

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Addition & Subtraction of Matrices, Scalar Multiplication of Matrices, Multiplication of Matrices

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Topic 2 : Transpose of Matrices, Symmetric & Skew Symmetric Matrices, Inverse of a Matrix by

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Elementary Row Operations 19. Determinants

Topic 1 : Minor & Co-factor of an Element of a Determinant, Value of a Determinant

Topic 2 : Properties of Determinants, Area of a Triangle

Topic 3 : Adjoint of a Matrix, Inverse of a Matrix, Some Special Cases of Matrix, Rank of a Matrix

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Topic 4 : Solution of System of Linear Equations 20. Continuity and Differentiability

Topic 1 : Continuity

Topic 2 : Differentiability

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Topic

3

:

Chain

Rule

of

Differentiation,

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Differentiation of Explicit & Implicit Functions, Parametric & Composite Functions, Logarithmic

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& Exponential Functions, Inverse Functions, Differentiation by Trigonometric Substitution

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Topic 4 : Differentiation of Infinite Series, Successive

Differentiation, nth Derivative of Some Standard Functions, Leibnitz’s Theorem, Rolle’s Theorem, Lagrange’s Mean Value Theorem 21. Applications of Derivatives

Topic 1 : Rate of Change of Quantities

Topic 2 : Increasing & Decreasing Functions Topic 3 : Tangents & Normals

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Topic 4 : Approximations, Maxima & Minima 22. Integrals

Topic 1 : Standard Integrals, Integration Substitution, Integration by Parts

by

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Topic 2 : Integration of the Forms: ∫ex(f(x) + f’(x))dx, ∫ekx(df(x) + f’(x))dx, Integration by Partial Fractions, Integration of Different Expressions of ex

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Topic 3 : Evaluation of Definite Integral by Substitution, Properties of Definite Integrals

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Topic 4 : Reduction Formulae for Definite Integration, Gamma & Beta Function, Walli’s Formula, Summation of Series by Integration 23. Applications of Integrals

Topic 1 : Curve & X-axis Between two Ordinates, Area of the Region Bounded by a Curve & Y-axis Between two Abscissa Topic 2 : Different Cases of Area Bounded Between the Curves

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24. Differential Equations

Topic 1 : Ordinary Differential Equations, Order & Degree of Differential Equations

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Topic 2 : General & Particular Solution of Differential Equation Topic 3 : Linear Differential Equation of First Order

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25. Vector Algebra

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Topic 1 : Algebra of Vectors, Linear Dependence & Independence of Vectors, Vector Inequality

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Topic 2 : Scalar or Dot Product of two Vectors

Topic 3 : Vector or Cross Product of two vectors, Scalar & Vector Triple Product 26. Three Dimensional Geometry

Topic 1 : Direction Ratios & Direction cosines of a Line, Angle between two lines in terms of dc’s and dr’s, Projection of a Point on a Line

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Topic 2 : Equation of a Straight Line in Cartesian and Vector Form, Angle Between two Lines, Distance Between two Parallel Lines

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Topic 3 : Equation of a Plane in Different Forms, Equation of a Plane Passing Through the Intersection of two Given Planes, Projection of a Line on a Plane 27. Probability

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Topic 1 : Multiplication Theorem on Probability, Independent events, Conditional Probability,

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Baye’s Theorem

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Topic 2 : Random Variables, Probability Distribution, Bernoulli Trails, Binomial Distribution, Poisson Distribution 28. Properties of Triangles

Topic 1 : Properties of Triangle, Solutions of Triangles, Inscribed & Circumscribed Circles, Regular Polygons

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Topic 2 : Heights & Distances

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Hints & Solutions Class – XI

Sets Relations and Functions Trigonometric Functions Principle of Mathematical Induction Complex Numbers and Quadratic Equations linear Inequalities Permutations and Combinations Binomial Theorem Sequences and Series Straight Lines and Pair of Straight Lines Conic Sections Limits and Derivatives Mathematical Reasoning Statistics Probability

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1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.

Class – XII

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Relations and Functions Inverse Trigonometric Functions Matrices Determinants Continuity and Differentiability Applications of Derivatives Integrals Applications of Integrals Differential Equations Vector Algebra Three Dimensional Geometry Probability Properties of Triangles

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16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28.

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1.

Let S = {1, 2, 3, ... , 100}. The number of non-empty subsets A of S such that the product of elements in A is even is : [Main Jan. 12, 2019 (I)]

(a) (b)

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(c) 250 – 1 (d) 250 + 1 2. Let

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3.

. Then S : [Main 2018]

contains exactly one element. contains exactly two elements. contains exactly four elements. is an empty set.

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(a) (b) (c) (d)

and

If f(x) +

, x ≠ 0 and

S = {x ∈ R : f(x) = f(–x)}; then S: [Main 2016]

(a) contains exactly two elements.

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(b) contains more than two elements. (c) is an empty set. (d) contains exactly one element. cos θ} and 4. Let P = {θ : sin θ – cos θ = sin θ} be two sets. Then Q = {θ : sin θ + cos θ =

[2011]

Q and Q – P ≠ φ P P Q Q P P= Q Let S={1, 2, 3, 4} . The total number of unordered pairs of disjoint subsets of S is equal to [2010] (a) 25 (b) 34(c) 42 (d) 41

(a) (b) (c) (d) 5.

Set A has m elements and set B has n elements. If the total number of subsets of A is 112 more than the total number of subsets of B, then the value of m⋅n is ______. [Main Sep. 06, 2020 (I)]

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6.

7.

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(a) (b) (c) (d)

Let S = {1, 2, 3, ..., 9}. For k = 1, 2, ..., 5, let Nk be the number of subsets of S, each containing five elements out of which exactly k are odd. Then N1 + N2 + N3 + N4 + N5 = [Adv. 2017] 210 252 125 126

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(a) (b) (c) (d) 2.

(a) (b) (c) (d)

Let

where each Xi contains 10 elements and each Yi

contains 5 elements. If each element of the set T is an element of exactly 20 of sets Xi’s and exactly 6 of sets Yi’s, then n is equal to [Main Sep. 04, 2020 (II)] 15 50 45 30 Let Z be the set of integers. If A = {x∈Z : 2(x + 2) ( – 5x + 6) = 1} and B = {x Z : –3 < 2x – 1< 9}, then the number of subsets of the set A × B, is : [Main Jan. 12, 2019 (II)] 15 2 218 212 210 In a class of 140 students numbered 1 to 140, all even numbered students opted Mathematics course, those whose number is divisible by 3 opted Physics course and those whose number is divisible by 5

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3.

A survey shows that 73% of the persons working in an office like coffee, whereas 65% like tea. If x denotes the percentage of them, who like both coffee and tea, then x cannot be : [Main Sep. 05, 2020 (I)] 63 36 54 38 A survey shows that 63% of the people in a city read newspaper A whereas 76% read newspaper B. If x% of the people read both the newspapers, then a possible value of x can be : [Main Sep. 04, 2020 (I)] 29 37 65 55

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1.

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(a) (b) (c) (d) 4.

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(a) (b) (c) (d) 5.

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(a) (b) (c) (d) 6.

opted Chemistry course. Then the number of students who did not opt for any of the three courses is: [Main Jan. 10, 2019 (II)] 102 42 1 38 . Then which of the Let A, B and C be sets such that following statements is not true ? [Main April 12, 2019 (II)]

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(a) (b) If , then (c) (d) If , then 7. In a certain town, 25% of the families own a phone and 15% own a car; 65% families own neither a phone nor a car and 2,000 families own both a car and a phone. Consider the following three statements : [Main Online April 10, 2015] (A) 5% families own both a car and a phone (B) 35% families own either a car or a phone (C) 40,000 families live in the town Then, (a) Only (A) and (C) are correct. (b) Only (B) and (C) are correct. (c) All (A), (B) and (C) are correct. (d) Only (A) and (B) are correct. 8. If X and Y are two sets, then X ∩ (X ∪Y)c equals. [1979] (a) X (b) Y (c) φ (d) None of these. 9.

Let X = {n N: l

n

50}. If

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10. (a) (b) (c) (d) 11.

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A = {n X: n is a multiple of 2} and B = {n X: n is a multiple of 7}, then the number of elements in the smallest subset of X containing both A and B is  [Main Jan. 7, 2020 (II)]

In a college of 300 students every student reads 5 newspapers and every newspaper is read by 60 students. The number of newpapers is [1998 - 2 Marks] at least 30 at most 20 exactly 25 none of these Suppose A1, A2, ........ A30 are thirty sets each with five elements and B1, B2, ....... Bn are n sets each with three elements. Let

. Assume that each element of S belongs to exactly

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ten of the Ai’ s and to exactly nine of the Bj’ s. Find n. [1981 - 2 Marks] 12. (i) Set A has 3 elements, and set B has 6 elements. What can be the minimum number of elements in theset A ∪ B? [1980] (ii) P, Q, R are subsets of a set A. Is the following equality true ? R × (Pc ∪ Qc)c = (R × P) ∩ (R × Q)? (iii) For any two subset X and Y of a set A define X Y = (Xc ∩ Y) ∪ (X ∩Yc) Then for any three subsets X, Y and Z of the set A, is the following equality true. (X Y) Z = X (Y Z)? 13. An investigator interviewed 100 students to determine their preferences for the three drinks : milk (M), coffee (C) and tea (T). He reported the following : 10 students had all the three drinks M, C and T; 20 had M and C; 30 had C and T; 25 had M and T; 12 had M only; 5

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had C only; and 8 had T only. Using a Venn diagram find how many did not take any of the three drinks. [1978]

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1.

Let R1 and R2 be two relations defined as follows : and

, where Q is the set of all rational

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numbers. Then :

[Main Sep. 03, 2020 (II)]

(a) Neither R1 nor R2 is transitive.

(b) R2 is transitive but R1 is not transitive. (c) R1 is transitive but R2 is not transitive. (d) R1 and R2 are both transitive. The

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2.

domain

of

the

function

is

Then a is equal to : [Main Sep. 02, 2020 (I)]

(a)

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(b)

(d) 3.

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(c)

If R = {(x, y) :

is a relation on the set of

integers Z, then the domain of R–1 is :

[Main Sep. 02, 2020 (I)]

(a) {–2, –1, 1, 2}

(b) {0, 1} (c) {–2, –1, 0, 1, 2} (d) {–1, 0, 1} 4. The domain

of

the

definition

of

the

function

is:

[Main April. 09, 2019 (II)]

(–1, 0) ∪ (1, 2) ∪ (3, ∞) (–2, –1) ∪ (–1, 0) ∪ (2, ∞) (–1, 0) ∪ (1, 2) ∪ (2, ∞) (1, 2) ∪ (2, ∞)

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(a) (b) (c) (d) 5.

Let f : R

R be defined by

Then the range of

[Main Jan. 11, 2019 (I)]

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f is :

(a)

(b) R – [–1, 1]

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(c)

6.

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(d) (–1, 1) – {0} is

Range of the function

[2003S]

(a) (b) (c) (d)

(1, ) (1,11/7] (1, 7/3] (1, 7/5]

7.

Let the function

Then the value of

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is ____

Let [t] denote the greatest integer 2[x + 2] – 7 = 0 has :

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1.

be defined by

[Adv. 2020]

Then the equation in x, [x]2 + [Main Sep. 04, 2020 (I)]

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(a) exactly two solutions (b) exactly four integral solutions (c) no integral solution

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(d) infinitely many solutions 2.

Let f (x) be a quadratic polynomial such that f (–1) + f (2) = 0. If one of the roots of f (x) = 0 is 3, then its other root lies in :

(a) (–1, 0) (b) (1, 3) (c) (–3, –1) (d) (0, 1) 3.

Let f (1, 3)

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[Main Sep. 02, 2020 (II)]

R be a function defined by f (x) =

denotes the greatest integer

, where [x]

x. Then the range of f is:

[Main Jan. 8, 2020 (II)]

(a) (b)

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(c)

(d)

If f(x) = loge

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, |x| < 1, then f

is equal to : [Main April 8, 2019 (I)]

(a) 2f(x)

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(b) 2f(x2)

(c) (f(x)) 2 (d) –2f(x)

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5.

Let f(x) = ax (a > 0) be written as f(x) = f1(x) + f2(x), where f1(x) is an even function and f2(x) is an odd function. Then f1(x + y) + f1(x – y) equals :

(a) 2f1(x) f1(y)

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[Main April. 08, 2019 (II)] (b) 2f1(x + y) f1(x – y) (c) 2f1(x)f2(y)

(d) 2f1(x + y) f2(x – y) 6.

Let

, where [n] denotes the greatest integer less

than or equal to n. Then

is equal to:

[Main Online April 19, 2014]

(a) 56 (b) 689 (c) 1287

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(d) 1399 The value of

is [2012]

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7.

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For any

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1.

the expression

3(sinq – cosq)4 + 6(sinq + cosq)2 + 4sin6q equals:

[Main Jan. 9, 2019 (I)]

13 – 4cos2q + 6sin2qcos2q 13 – 4cos6q 13 – 4cos2q + 6cos4q 13 – 4cos4q + 2sin2qcos2q

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(a) (b) (c) (d) 2.

Let

where

and

Then

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equals

[Main 2014]

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(a)

(b)

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(c)

3.

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(d) If 2cos θ + sin θ = 1

, then 7 cos q + 6 sin q is equal to:

[Main Online April 11, 2014]

(a) (b) 2 (c) (d) 4.

The expression

can be written as :

[Main 2013]

sinA cosA + 1 secA cosecA + 1 tanA + cotA secA + cosecA

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(a) (b) (c) (d) 5.

Given both θ and φ are acute angles and sin θ =

, cos φ =

, then

the value of θ + φ belongs to

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[2004S]

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(a)

(b) (c)

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(d) 6.

If ω is an imaginary cube root of unity then the value of

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is

[1994]

(a) (b) (c) (d) 7.

If tanθ = –

, then sinθ is

[1979]

but not

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(a) – (b) – (c)

or

but not –

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(d) None of these

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8.

(a) (b) (c) (d)

Which of the following number(s) is/are rational?

sin 15° cos 15° sin 15° cos 15° sin 15° cos 75°

[1998 - 2 Marks]

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In this questions there are entries in columns 1 and 2. Each entry in column 1 is related to exactly one entry in column 2. Write the correct letter from column 2 against the entry number in column 1 in your answer book. is

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9.

[1992 - 2 Marks]

Column I (A) positive (B) negative

Column II

(p)

(q)

(r)

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(s)

Let O be the origin, and

of the sides

be three unit vectors in the directions

respectively, of a triangle PQR.

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[Adv. 2017] = 10. (a) sin (P + Q) (b) sin 2R (c) sin (P + R) (d) sin (Q + R) 11. If the triangle PQR varies, then the minimum value ofcos(P + Q) + cos (Q +R) + cos (R + P) is (a)

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(b)

(d)

12.

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(c)

Find the range of values of t for which 2 sin t= .

,t∈

[2005 - 2 Marks]

1.

If

and

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then :

[Main Sep. 05, 2020 (II)]

(a)

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(b)

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(c)

(d)

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The set of all possible values of θ in the interval (0, π) for which the lie on the same side of the line x + y = points (1, 2) and 1 is : [Main Sep. 02, 2020 (II)]

(a) (b) (c) (d) 3.

The value of

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2.

is

[Main Jan. 9, 2020 (I)]

(a)

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(b) (c)

(d)

The angle of elevation of the top of a vertical tower standing on a horizontal plane is observed to be 45o from a point A on the plane. Let B be the point 30 m vertically above the point A. If the angle of elevation of the top of the tower from B be 30o, then the distance (in m) of the foot of the tower from the point A is: [Main April 12, 2019 (II)]

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4.

(a)

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(b) (c)

5.

The value of

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(d) [Main April 9, 2019 (II)]

cos210° – cos10° cos50° + cos250° is : (a)

+ cos20°

(c)

(1 + cos20°)

(d) 3/2 6.

If cos (α + β) = equal to :

(b) 3/4

, sin(α – β) =

and 0 < α, β
t2 > t3 > t4 (b) t4 > t3 > t1 > t2

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(c) t3 > t1 > t2 > t4 (d) t2 > t3 > t1 > t4 14. The values of θ ∈ (0, 2π) for which 2 sin2θ – 5 sinθ + 2 > 0, are [2006 - 3M, –1] (a) (b) (c) (d) 15.

If α + β = π/2 and β + γ = α, then tan α equals

[2001S]

2(tanβ + tanγ) tanβ + tanγ tanβ + 2tanγ 2tanβ + tanγ The maximum value of (cos α1).(cos α2)…(cos αn), under the restrictions

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0 ≤ α1, α2, …, αn ≤

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1/2n/2 1/2n 1/2n 1 Let f (θ) = sinθ(sinθ + sin3θ). Then f (θ) is

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(a) (b) (c) (d) 17.

and (cot α1).(cot α2) … (cot αn) = 1 is

only when for all real for all real only when

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18.

3

+6

+4

= [1995S]

(a) 11

12

(c)

13

(d)

14

then (sec2x – tan2x) equals

Let

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19.

(b)

[1994]

(a) (b) (c) (d) 20. (a)

Given

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(b)

then for all real values of θ

(c)

(d)

If α + β + γ = 2π, then

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(a) tan

+ tan

(b) tan

tan

(c) tan

+ tan

+ tan + tan + tan

= tan tan

[1979] tan

+ tan

= – tan

tan tan

=1

tan tan

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(d) None of these 22.

The maximum value of the expression

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is

[2010]

23.

If

, then tan (

+ 2 ) is equal to _____.

[Main Jan. 8, 2020 (II)]

24.

be the function defined by

Let

are such that

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If

25.

If K is _________ .

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If

and

then the value of

is _____

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_________

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[1991 - 2 Marks] 28.

is an identity in x, where C0, C1,

Suppose

. then the value of n is _________ [1981 - 2 Marks]

29.

If

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, then tan 2A = tan B.

[1983 - 1 Mark]

30.

Let f(x) = x sin πx, x > 0. Then for all natural numbers n, f ′(x) vanishes at [Adv. 2013]

(a) A unique point in the interval

(b) A unique point in the interval

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(c) A unique point in the interval (n, n + 1) (d) Two points in the interval (n, n + 1) 31. Let θ, ϕ ∈ [0, 2π] be such that 2 cosθ (1 – sin ϕ) = sin2θ

and

, then ϕ cannot satisfy [2012]

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(d) If

, then

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32.

[2009]

(a) (b) (c) (d) 33.

For a positive integer n, let fn(θ)

[1999 - 3 Marks]

. Then

=

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(d)

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34.

The minimum value of the expression are real numbers satisfying

(a) positive (b) zero (c) negative

, where is [1995]

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(a) (b) (c) (–1, 2) 36.

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(d) –3 35. Let 2sin2x + 3sinx – 2 > 0 and x2 – x – 2 < 0 (x is measured in radians). Then x lies in the interval [1994]

(d)

The expression 3

–

is equal to

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0 1 3 sin 4 + cos 6 none of these

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(a) (b) (c) (d) (e)

37.

is equal to [1984 - 3 Marks]

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(d)

In any triangle ABC, prove that

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38.

[2000 - 3 Marks]

.

39. Prove that

(n – k) cos

, where n

3 is an integer.

[1997 - 5 Marks]

40.

do not lie between

Prove that the values of the function

and 3 for any real x.

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[1997 - 5 Marks] 41. Find the smallest positive number p for which the equation cos(p sin . x) = sin(pcos x) has a solution x [1995 - 5 Marks] 42. Determine the smallest positive value of x (in degrees) for which tan(x + 100°) = tan (x + 50°) tan(x) tan (x – 50°). [1993 - 5 Marks]

, wherever defined never lies between

43. Show that the value of and 3.

If exp {(sin x + sin x + sin x + 4

6

...............

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2

,0 3 satisfying the equation

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27.

[2011]

Two parallel chords of a circle of radius 2 are at a distance apart . If the chords subtend at the center , angles of

and

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where k > 0, then the value of [k] is [2010]

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[Note : [k] denotes the largest integer less than or equal to k ]

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29.

The number of values of θ in the interval, for n = 0,

such that

and tan θ = cot 5θ as well as sin 2θ = cos 4θ is

[2010] The number of all possible values of θ where 0 < θ < π, for which the system of equations

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(y + z) cos 3θ = (xyz) sin 3θ

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(xyz) sin 3θ = (y + 2z) cos 3θ + y sin 3θ have a solution (x0, y0, z0) with y0 z0 0, is

[2010]

30.

The number of distinct solutions of the equation, log1/2|sin x| = 2 – log1/2|cos x| in the interval [0, 2π], is _____. [Main Jan. 9, 2020 (I)] 31. Let a, b, c be three non-zero real numbers such that the equation : , has two distinct real roots α and

β with

. Then, the value of

is _______.

[Adv. 2018]

The real roots of the equation cos7 x + sin4 x=1 in the interval (–π, π) are ..., ..., and _________ . [1997 - 2 Marks] 33. General value of θ satisfying the equation tan2 θ +sec 2 θ = 1 is_________ [1996 - 1 Mark] 34. The sides of a triangle inscribed in a given circle subtend angles at the centre. The minimum valueof the arithmetic mean

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of cos

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and cos

The set of all x in the interval [0, 0, is _________ .

is equal to _________

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[1987 - 2 Marks]

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36.

The solution set of the system of equations where x and y are real, is _________ .

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37.

There exists a value of θ between 0 and 2π that satisfies the equation . [1984 - 1 Mark]

38.

Let α and β be non–zero real numbers such that2(cosβ – cosα) + cosα cosβ = 1. Then which of the following is/are true? [Adv. 2017]

(a) (b)

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39. The number of points in (–∞ ∞), for which x2 – x sin x – cos x = 0, is 6 4 2 0

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40.

For

the solution (s) of

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is (are) (a) (b) (c) (d)

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[2009]

The number of values of x in the interval [0, 5π] satisfying the equation 3 sin2 x – 7 sin x + 2 = 0 is [1998 - 2 Marks]

(a) (b) (c) (d) 42.

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41.

= 0 are

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(a) 7 /24 (b) 5 /24 (c) 11 /24 (d) /24. 43. The number of all possible triplets (a1, a2, a3) such that a1 + a2 cos(2x) + a3sin2(x) = 0 for all x is [1987 - 2 Marks] (a) zero

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44.

one three infinite none

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(b) (c) (d) (e)

Let f (x) = sin (π cos x) and g (x) = cos (2π sin x) be two functions defined for x > 0. Define the following sets whose elements are written in the increasing order.

Column - I contains the sets X, Y, Z and W. Column - II contains some information regarding these sets. [Adv. 2019] Column I Column II (I) X (II) Y

(q) an arithmetic progression (r) NOT an arithmetic progression

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(p)

(IV)

W

(s)

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(d) (IV), (q), (t) 45. Let f (x) = sin (π cos x) and g (x) = cos (2π sin x) be two functions defined for x > 0. Define the following sets whose elements are written in the increasing order.

Column - I contains the sets X, Y, Z and W. Column - II contains some information regarding these sets. [Adv. 2019] Column I Column II (I) X (II) Y (III)Z (IV)

(p)

(q) an arithmetic progression (r) NOT an arithmetic progression

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(s)

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(t)

(u)

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Which of the following is the only CORRECT combination? (a) (I), (q), (u) (b) (I), (p), (r) (c) (II), (r), (s) (d) (II), (q), (t) 46.

Find all values of θ in the interval (1 – tan θ) (1 + tan θ) sec2 θ +

satisfying the equation = 0.

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[1996 - 2 Marks] 47. Find the values of x∈(– π, + π) which satisfy the equation

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[1984 - 2 Marks] 48. Find all the solution of

[1983 - 2 Marks]

49. (a) Draw the graph of y = (b) If cos (α + β) =

(sinx + cosx) from x = –

, sin (α − β) =

to x =

, and α, β lies between 0 and

.

,

find tan2α.

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Consider the statement: “P(n) : n2 – n + 41 is prime.” Then which one of the following is true? [Main Jan. 10, 2019 (II)] Both P(3) and P(5) are true. P(3) is false but P(5) is true. Both P(3) and P(5) are false. P(5) is false but P(3) is true.

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2. Use mathematical induction to show that (25)n+1 – 24n + 5735 is divisible by (24)2 for all n = 1, 2, ........ [2002 - 5 Marks] 2 3. Let a, b, c be positive real numbers such that b − 4ac > 0 and let α1 = c. Prove by induction that αn + 1 =

is well - defined and

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αn + 1
0 [1985 - 5 Marks] 18. If p be a natural number then prove that pn + 1 + (p + 1)2n –1 is divisible by p2 + p + 1 for every positive integer n. [1984 - 4 Marks] 19. Use mathematical induction to prove : If n is any odd positive integer, then n(n2 – 1) is divisible by 24. [1983 - 2 Marks] 20. Prove that 72n + (23n – 3)(3n – 1) is divisible by 25 for any natural number n. [1982 - 5 Marks]

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If + icos

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(b)

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(d)

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If the four complex numbers z, ,

and

represent the vertices of a square of side 4 units in the Argand plane, then |z| is equal to : [Main Sep. 05, 2020 (I)]

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3. (a) (b) (c) (d) 4.

4 4 2 2 The value of

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is : [Main Sep. 05, 2020 (II)]

– 215 215 i – 215 i 65 If z1, z2 are complex numbers such that Re(z1) = |z1 – 1|, Re(z2) = |z2 – 1|

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is

equal

to

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then

(d) 5.

Let z be a complex number such that

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Then the value of |z + 3i| is : [Main Jan. 9, 2020 (I)]

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(b) (c)

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(d) If z be a complex number satisfying |Re(z)| + |Im(z)| = 4, then |z| cannot be: [Main Jan. 9, 2020 (II)]

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(a) (b) (c) (d) 7.

The equation

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(b) the line through the origin with slope 1. (c) a circle of radius 1.

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If a > 0 and z =

, has magnitude

, then

is equal to :

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(b)

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(c)

(d)

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9.

Let z Î C be such that |z| < 1. If w =

, then :

(a) (b) (c) (d)

5 Re (w) > 4 4 Im (w) > 5 5 Re (w) > 1 5 Im (w) < 1

10.

If value of

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[Main April 09, 2019 (II)]

is a purely imaginary number and |z| = 2, then a

is :

[Main Jan. 12, 2019 (I)]

(a) 2 (b) 1 (c) (d) 11.

Let z be a complex number such that

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Then | z | is equal to :

[Main Jan. 11, 2019 (II)]

(a)

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(b)

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(c)

(d)

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12.

Let z1 and z2 be any two non-zero complex numbers such that 3 | z1 | = 4 | z2 |. If

then:

(a) Re(z) = 0 (b) | z | = (c) | z | = (d) Im(z) = 0 13.

Let A=

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.

Then the sum of the elements in A is: [Main Jan. 9 2019 (I)] (a) (b) π

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(c) (d)

14.

The set of all α ∈ R, for which w =

is a purely

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(a) {0} (b) an empty set (c)

(d) equal to R

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15.

A value of for which

is purely imaginary, is:

(a) (b) (c) (d) 16.

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[Main 2016]

If z is a non-real complex number, then the minimum value of is :

[Main Online April 11, 2015]

–1 –4 –2 –5 If z is a complex number such that

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(a) (b) (c) (d) 17.

then the minimum value of

[Main 2014]

w

(a) is strictly greater than

but less than

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(b) is strictly greater than (c) is equal to (d) lie in the interval (1, 2)

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18.

If z is a complex number of unit modulus and argument θ, then arg equals:

[Main 2013]

–θ

(b)

(c) θ (d) π – θ 19.

If

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(a) –θ

is purely real where w = α + iβ, β ≠ 0 and z ≠ 1, then the set

of the values of z is

[2006 - 3M, –1]

(a) (b) (c) (d) 20.

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(a) (b) (c) (d) 21.

{z : |z| = 1} {z : z = } {z : z ≠ 1} {z : |z| = 1, z ≠ 1} For all complex numbers z1, z2 satisfying |z1|=12 and| z2-3-4i | = 5, the minimum value of |z1-z2| is [2002S] 0 2 7 17 If z1, z2 and z3 are complex numbers such that [2000S]

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(a) (b) (c) (d) 22.

equal to 1 less than 1 greater than 3 equal to 3 If arg(z) < 0, then arg (-z) - arg(z) =

is

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[2000S]

(c) (d) 23.

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(a) (b)

For positive integers n1, n2 the value of the expression , where

i=

is a real number if and only if

[1996 - 1 Marks]

(a) n1 = n2 +1 (b) n1 = n2 –1 (c) n1 = n2 (d) n1 > 0, n2 > 0 24. Let z and 1 or i i or – i 1 or – 1 i or – 1 Let z and

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(a) (b) (c) (d) 25.

be two complex numbers such that | z | 1,| | 1 and | z + i | = | z – i | = 2 then z equals [1995S]

be two non zero complex numbers such that| z | = | | and Arg z + Arg = π, then z equals [1995S]

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(a) (b) – (c) (d) – 26. The smallest positive integer n for which [1980] is

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27.

n=8 n = 16 n = 12 none of these

If

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(a) (b) (c) (d)

, then the greatest common

divisor of the least values of m and n is _________. [Main Sep. 03, 2020 (I)] 28.

For any integer k, let αk = cos

value of the expression

+ i sin

, where i =

is

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29. If z is any complex number satisfying |z – 3 – 2i| minimum value of |2z – 6 + 5i| is

[Adv. 2015] 2, then the [2011]

If the expression

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30.

. The

w

is real, then the set of all possible values of x is ............

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31.

For complex number , if and .

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, we have

and , we write . Then for all complex numbers z with

[1981 - 2 Marks]

32.

Let S be the set of all complex numbers z satisfying Then which of the following statements is/are TRUE?

[Adv. 2020]

for all

(a) (b)

for all

(c)

for all

(d) The set S has exactly four elements 33. Let s, t, r be non-zero complex numbers and L be the setof solutions z of the equation sz + t + r = 0, where = x + iy

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= x – iy. Then, which of the following statement(s) is (are) TRUE? [Adv. 2018] If L has exactly one element, then |s| ≠ |t| If |s| = |t|, then L has infinitely many elements The number of elements in L ∩ {z : |z – 1 + i| = 5} is at most 2 If L has more than one element, then L has infinitely many elements For a non-zero complex number z, let arg(z) denote the principal . Then, which of the following argument with statement (s) is (are) FALSE? [Adv. 2018]

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(a) (b) (c) (d) 34.

(a)

, where

(b) The function , defined by , is continuous at all points of , where

for all

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(c) For any two non-zero complex numbers z1 and z2,

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35.

Let a, b, x and y be real numbers such that a – b = 1 andy ≠ 0. If the complex number z = x + iy satisfies

= y, then which of the

following is(are) possible value(s) of x ?

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(a) (b) (c)

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(d) 36.

Let z1 and z2 be two distinct complex numbers and letz = (1 – t) z1 + tz2 for some real number t with 0 < t < 1. If Arg (w) denotes the principal argument of a non-zero complex number w, then [2010]

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(a) (b) Arg (z – z1) = Arg (z – z2) (c)

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(d) Arg (z – z1) = Arg (z2 – z1) 37.

If

, then

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[1998 - 2 Marks] (b) x = 1, y = 3

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(a) x = 3, y = 2 (c) x = 0, y = 3 (d) x = 0, y = 0

38. The value of the sum

(in + in+1), where i =

, equals

[1998 - 2 Marks]

(a) i

(b) i – 1

39.

The value of

(c) –i

(d)

0

is

[1987 - 2 Marks]

(a) (b) (c) (d) (e) 40.

–1 0 –i i None If

and

are two nonzero complex numbers such that then Arg Arg is equal to

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(a)

(b)

(c) 0 (d)

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(e) 41.

Let z1 and z2 be complex numbers such that

and

=

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. If z1 has positive real part and z2 has negative imaginary part, then

(a) zero

may be [1986 - 2 Marks]

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real and positive real and negative purely imaginary none of these. If and are complex numbers such that |z1| = |z2|=1 and Re(z1 )=0, then the pair of complex numbers and satisfies –

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(b) (c) (d) (e) 42.

[1985 - 2 Marks]

(a) (b) (c) (d) none of these 43.

Let

; k = 1, 2, ...., 9.

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List-I P. For each zk there exists as zj such1. that zk. zj = 1 Q. There exists a such2.

List-II True

False

that z1.z = zk has no solution z in the set of complex numbers

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R.

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S.

P Q R S (a) 1 2 4 3 (c) 1 2 3 4

equals

equals

3.

1

4.

2

P Q R S (b) 2 1 3 4 (d) 2 1 4 3

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PASSAGE-1

and

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[Adv. 2013]

44. Area of S = (a) (b) (c) (d)

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(b)

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(c)

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PASSAGE-2 Let A, B, C be three sets of complex numbers as defined below

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46. The number of elements in the set is (a) 0 (b) 1 (c) 2 (d) 47. Let z be any point in . Then, |z + 1 – i|2 + |z – 5 – i|2 lies between (a) 25 and 29 (b) 30 and 34 (c) 35 and 39 (d) 40 and 44 and let w be any point satisfying |w – 2 48. Let z be any point – i| < 3. Then, |z| – |w| + 3 lies between (a) –6 and 3 (b) –3 and 6 (c) –6 and 6 (d) –3 and 9 49.

Let z satisfy| z | = 1 and z = 1– . Statement 1 : z is a real number.

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(d) Statement 1 is true; Statement 2 is true; Statement 2 is not a correct explanation for Statement 1. If z1 and z2 are two complex numbers such taht |z1| < 1< |z2| then

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50.

.

prove that

51.

Let Z1 = 10 + 6i and Z2 = 4 + 6i. If Z is any complex number such

that the argument of 52.

is

, then prove that | Z – 7 – 9i| =

. [1990 - 4 Marks] Show that the area of the triangle on the Argand diagram formed by the complex numbers z, iz and z + iz is

53.

[2003 - 2 Marks]

.

[1986 - 2½ Marks] Find the real values of x and y for which the following equation is satisfied

[1980]

, prove that (x2 + y2)2 =

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54. If x + iy = Express

in the form x + iy. [1978]

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. [1979]

Let z = x + iy be a non-zero complex number such that z2 = i |z|2, where , then z lies on the: i= [Main Sep. 06, 2020 (II)] (a) line, y = –x

w

1.

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(b) imaginary axis (c) line, y = x (d) real axis 2. If a and b are real numbers such that

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where

then a + b is equal to :

[Main Sep. 04, 2020 (II)]

(a) (b) (c) (d)

9 24 33 57

3.

The value of

is :

[Main Sep. 02, 2020 (I)]

(a)

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(d)

The imaginary part of

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can be : [Main Sep. 02, 2020 (II)]

(a)

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(b) (c) 6 (d)

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5.

Let

=

. If a = (1 + )

and b =

then a and

b are the roots of the quadratic equation: (a) (b) (c) (d) 6.

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[Main Jan. 8, 2020 (II)] x + 101x + 100 = 0 x2 – 102 x + 101 = 0 x2 – 101x + 100 = 0 x2 + 102x +101 = 0 2

If Re

= 1, where z = x + iy, then the point (x, y) lies on a:

[Main Jan. 7, 2020 (I)]

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(a) circle whose centre is at

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(b) straight line whose slope is

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(c) straight line whose slope is

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8.

, then (1 + iz + z5 + iz8)9 is equal to: [Main April 08, 2019 (II)]

0 1 (– 1 + 2i)9 –1

w

(a) (b) (c) (d)

If

Let

where x and y are real numbers

then y – x equals :

(a) 91

[Main Jan. 11, 2019 (I)]

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(b) – 85 (c) 85 (d) – 91 . If

Let S be the set of all complex numbers z satisfying

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9.

the complex number z0 is such that

is the maximum of the set

, then the principal argument of

is

[Adv. 2019]

(a) (b) (c) (d)

The least positive integer n for which

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10.

[Main Online April 16, 2018]

2 6 5 3 A complex number z is said to be unimodular if |z| = 1. Suppose z1

w

(a) (b) (c) (d) 11.

= 1, is

w

and z2 are complex numbers such that

is unimodular and z2

is not unimodular. Then the point z1 lies on a: [Main 2015]

(a) circle of radius 2.

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(b) circle of radius (c) straight line parallel to x-axis (d) straight line parallel to y-axis. Let complex numbers α and

lie on circles (x – x0)2

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12.

+ (y – y0)2 = r2 and (x – x0)2 + (y – y0)2 = 4r2. respectively. If z0 = x0 + iy0 satisfies the equation

[Adv. 2013]

(a) (b) (c) (d)

Let z be a complex number such that the imaginary part of z is nonzero and a = z2 + z + 1 is real. Then a cannot take the value [2012]

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13.

(a) –1

(c)

(d)

Let z = x + iy be a complex number where x and y are integers. Then the area of the rectangle whose vertices are the roots of the equation : is

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14.

(b)

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(a) (b) (c) (d)

[2009]

48 32 40 80

15. Let z = cos θ + i sin θ. Then the value of

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at θ = 2° is [2009]

(b) (c) (d) 16.

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(a)

A particle P starts from the point z0 = 1 + 2i, where . It moves horizontally away from origin by 5 units and then vertically away from origin by 3 units to reach a point z1. From z1 the particle moves units in the direction of the vector and then it moves through an angle

in anticlockwise direction on a circle with centre

at origin, to reach a point z2. The point z2 is given by 6 + 7i –7 + 6i 7 + 6i –6 + 7i

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(a) (b) (c) (d)

17.

If | z | = 1 and z ≠ ± 1, then all the values of

[2008]

lie on [2007 -3 marks]

a line not passing through the origin |z|= the x-axis the y-axis A man walks a distance of 3 units from the origin towards the northeast (N 45° E) direction. From there, he walks a distance of 4 units towards the north-west (N 45° W) direction to reach a point P. Then the position of P in the Argand plane is [2007 -3 marks] (a) 3eiπ/4 + 4i

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(a) (b) (c) (d) 18.

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(b) (3 – 4i)eiπ/4 (c) (4 + 3i)eiπ/4 (d) (3 + 4i)eiπ/4 19. a, b, c are integers, not all simultaneously equal and ω is cube root of unity (ω ≠ 1), then minimum value of|a + bω + cω2| is [2005S] (a) 0 (b) 1 (c) (d)

The locus of z which lies in shaded region (excluding the boundaries) is best represented by [2005S]

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20.

z : |z + 1| > 2 and |arg (z+1)| < π/4 z : |z – 1| > 2 and |arg (z–1)| < π/4 z : |z + 1| < 2 and |arg (z+1)| < π/2 z : |z – 1| < 2 and |arg (z+1)| < π/2 If ω (≠ 1) be a cube root of unity and (1 + ω2)n = (1 + ω4)n, then the least positive value of n is [2004S] 2 3 5 6

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(a) (b) (c) (d) 21.

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(a) (b) (c) (d)

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22.

If

, then Re( ) is [2003S] (b)

(c) 23.

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(a) 0

(d)

Let

, then the value of the det.

is

[2002 - 2 Marks]

(a) (b) (c)

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(d) 24.

The complex numbers z1, z2 and z3 satisfying

are

the vertices of a triangle which is [2001S]

of area zero right-angled isosceles equilateral obtuse-angled isosceles Let z1 and z2 be nth roots of unity which subtend a right angle at the origin. Then n must be of the form [2001S] (a) 4k + 1 (b) 4k +2

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(a) (b) (c) (d) 25.

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(c) 4k + 3 (d) 4k , then 4 + 5

If i =

is equal to

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26.

[1999 - 2 Marks]

(a) (b) – (c) (d) 27.

If

( 1) is a cube root of unity and

=A+B

then A and B

are respectively

[1995S]

0, 1 1, 1 1, 0 – 1, 1 If a, b, c and u, v, w are complex numbers representing the vertices of two triangles such that c = (1 – r) a + rb and w = (1 – r)u + rv, where r is a complex number, then the two triangles [1985 - 2 Marks] have the same area are similar are congruent none of these The points z1, z2, z3 z4 in the complex plane are the vertices of a parallelogram taken in order if and only if [1983 - 1 Mark] z1 + z4 = z2 + z3 z1 + z3 = z2 + z4 z1 + z2 = z3 + z4 None of these

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(a) (b) (c) (d) 28.

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(a) (b) (c) (d) 29.

(a) (b) (c) (d)

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30.

If and complex plane,

, then

implies that, in the [1983 - 1 Mark]

z lies on the imaginary axis z lies on the real axis z lies on the unit circle None of these

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(a) (b) (c) (d)

represents the

31. The inequality region given by

[1982 - 2 Marks]

(a) (b) (c) (d)

Re(z) ≥ 0 Re(z) < 0 Re(z) > 0 none of these

32. If

, then

[1982 - 2 Marks]

Re(z) = 0 Im(z) = 0 Re(z) > 0, Im (z) > 0 Re(z) > 0, Im (z) < 0

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(a) (b) (c) (d)

which satisfy

33. The complex numbers

w

the equation

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(a) (b) (c) (d) 34.

lie on [1981 - 2 Marks]

the x-axis the straight line y = 5 a circle passing through the origin none of these If the cube roots of unity are 1, ω, ω2, then the roots of the equation (x – 1)3 + 8 = 0 are

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[1979] (a) – 1, 1 + 2ω, 1 + 2ω (c) – 1, – 1, – 1

2

For a complex number z, let Re(z) denote the real part of z. Let S be where the set of all complex numbers z satisfying

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35.

(b) – 1, 1 – 2ω, 1 – 2ω (d) None of these

2

Then the minimum possible value of

where

with Re(z1) > 0 and Re(z2) < 0, is _____ 36.

Let

[Adv. 2020] be a cube root of unity. Then the minimum of the set

equals _____

[Adv. 2019]

37.

Let that

, and a, b, c, x, y, z be non-zero complex numbers such [2011]

w .je

a+b+c=x a + bω + cω2 = y a + bω2 + cω = z

Then the value of

is

w

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38. The value of the expression 1•(2–ω)(2–ω2)+2•(3–ω)(3–ω2)+....+(n–1).(n–ω)(n–ω2), imaginary cube root of unity, is..... 39.

where

ω is

an

[1996 - 2 Marks] Suppose Z1, Z2, Z3 are the vertices of an equilateral triangle inscribed then Z2 = ........, Z3 = ............ in the circle |Z| = 2. If Z1 = 1 + [1994 - 2 Marks]

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ABCD is a rhombus. Its diagonals AC and BD intersect at the point M and satisfy BD = 2AC. If the points D and M represent the complex numbers 1 + i and 2 - i respectively, then A represents the complex number .........or.......... [1993 - 2 Marks] =a 41. If a and b are the numbers between 0 and 1 such that the points + i, z2 = 1 + bi and z3 = 0 form an equilateral triangle, then a = .......and b = ........... [1989 - 2 Marks] 42. For any two complex numbers z1, z2 and any real number a and b.

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40.

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[1988 - 2 Marks] | az1 – bz2 |2 + | bz1 + az2 |2 = ............. The cube roots of unity when represented on Argand diagram form the vertices of an equilateral triangle. [1988 - 1 Mark] 44. If three complex numbers are in A.P. then they lie on a circle in the complex plane. [1985 - 1 Mark] 45. If the complex numbers, Z1, Z2 and Z3 represent the vertices of an equilateral triangle such that | Z1| = | Z2 | = | Z3 | then Z1 + Z2 + Z3 = 0. [1984 - 1 Mark]

46.

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43.

Let a, b

and a2 + b2

, where

Suppose and z

0.

. If z = x+ iy

S, then (x, y) lies on

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[JEE Adv. 2016]

(a) the circle with radius a > 0, b

and centre

for

0

(b) the circle with radius

and centre

fora < 0, b

0

w

(c) the x-axis for a 0, b = 0 (d) the y-axis for a = 0, b 0

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47. Let

and P = {wn : n = 1, 2, 3, ...}. Further H1 = and

where c is the set of all

complex numbers. If

and O represents the

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origin, then ∠z1Oz2 =

[JEE Adv. 2013]

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(a) (b) (c) (d) 48.

If ω is an imaginary cube root of unity, then (1 + ω – ω2)7 equals [1998 - 2 Marks] 2 (a) 128ω (b) –128ω (c) 128ω (d) –128ω2 49.

Match the statements in Column I with those in Column II.

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[2010] [Note : Here z takes values in the complex plane and Im z and Re z denote , respectively, the imaginary part and the real part of z.] Column I Column II (A) The set of points z satisfying (p) an ellipse with eccentricity

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|z – i| z | | = |z + i | z || is contained in or equal to points z satisfying Im z = 0 (B) The set of points z satisfying points z satisfying |Im z | < 1 |z + 4 | + |z – 4 | = 10 is contained in or equal to (C) If | w | = 2, then the set of points points z satisfying | Re z | < 2 z=w–

(q) the set of (r) the set of

(s) the set of

is contained in or equal to

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(D) If | w | = 1, then the set of points points z satisfying | z | < 3 z=w+ z

the set of

is contained in or equal to.

0 is a complex number

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50.

(t)

[1992 - 2 Marks] Column II =0 (p)

Column I (A) Re z = 0

(q)

(B) Arg z =

=0

(r

) 51.

=

If one the vertices of the square circumscribing the circle |z – 1| = is . Find the other vertices of the square.

[2005 - 4 Marks]

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52. Find the centre and radius of circle given by

where, z = x + iy, α = α1 + iα2, β = β1 + iβ2 53.

Prove that

[2004 - 2 Marks] there exists no complex number z such that where |ar| < 2.[2003 - 2 Marks]

, be a root of the equation zp+q – zp – zq + Let a complex number 1 = 0, where p, q are distinct primes. Show that either 1 + α + α2 + .... + αp - 1 = 0 or 1 + α + α2 + ... + αq – 1 = 0, but not both together. [2002 - 5 Marks] 2 55. For complex numbers z and w, prove that |z| w–|w|2 z = z –w if and only if z = w or z = 1. [1999 - 10 Marks]

w

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54.

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Let z1 and z2 be roots of the equation z2 + pz + q = 0, where the coefficients p and q may be complex numbers. Let A and B represent z1 and z2 in the complex plane. If ∠AOB = α 0 and OA = OB, where O is the origin, prove that

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56.

p2 = 4q cos2 57.

.

Find all non-zero complex numbers Z satisfying

[1997 - 5 Marks]

= iZ2.

[1996 - 2 Marks]

show that

58.

If

59.

If iz + z – z + i = 0, then show that | z | = 1.

[1995 - 5 Marks]

3

2

w

w

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[1995 - 5 Marks] 60. If 1, a1, a2 ......, an – 1 are the n roots of unity, then show that (1 – a1)(1 – a2) (1 – a3) ....(1 – an – 1) = n [1984 - 2 Marks] 61. Prove that the complex numbers z1, z2 and the origin form an equilateral triangle only ifz12 + z22 –z1z2 = 0. [1983 - 3 Marks] 62. Let the complex number z1, z2 and z3 be the vertices of an equilateral triangle. Let z0 be the circumcentre of the triangle. Then prove that z12 + z22 + z32 = 3z02. [1981 - 4 Marks] 63. If x = a + b, y = aγ + bβ and z = aβ + bγ where γ and β are the complex cube roots of unity, show that xyz = a3 + b3. [1978]

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If α and β be two roots of the equation x2 – 64x + 256 = 0. Then the value of

(a) (b) (c) (d) 2.

(a) (b) (c) (d) 3.

2 3 1 4

is: [Main Sep. 06, 2020 (I)]

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1.

If α and β are the roots of the equation 2x(2x + 1) = 1, then β is equal to: [Main Sep. 06, 2020 (II)] 2α(α + 1) –2α(α + 1) 2α(α – 1) 2α2 The product of the roots of the equation 9x2 – 18| x | + 5 = 0, is : [Main Sep. 05, 2020 (I)]

(a) (c)

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(b) (d) 4.

If α and β are the roots of the equation,

w

of

the the value

is equal to : [Main Sep. 05, 2020 (II)]

w

(a)

(b)

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(c)

5.

(a) (b) (c) (d) 6.

Let

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(d) and k > 0. If the curve represented by Re(u)

+ Im(u) = 1 intersects the y-axis at the points P and Q where PQ = 5, then the value of k is : [Main Sep. 04, 2020 (I)] 3/2 1/2 4 2 be in R. If α and β are roots of the equation, Let and α and γ are the roots of the equation, then

is equal to :

[Main Sep. 04, 2020 (II)]

27 18 9 36

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(a) (b) (c) (d) 7.

If α and β are the roots of the equation x2 + px + 2 = 0 and

and

is equal to : [Main Sep. 03, 2020 (I)]

w

w

are the roots of the equation 2x2 + 2qx + 1 = 0, then

(a)

(b)

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(c) (d) The set of all real values of λ for which the quadratic equations, always have exactly one root in the interval (0, 1) is :

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8.

[Main Sep. 03, 2020 (II)]

(a) (b) (c) (d) 9.

(0, 2) (2, 4] (1, 3] (–3, –1) Let α and β be the roots of the equation,

If

then :

[Main Sep. 02, 2020 (I)]

(a) (b) (c)

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(d) 10.

The number of real roots of the equation, e4x + e3x – 4e2x + ex + 1 = 0 is: [Main Jan. 9, 2020 (I)]

1 3 2 4 If the equation, x2 + bx + 45 = 0 (b R) has conjugate complex roots , then: and they satisfy |z + l| = [Main Jan. 8, 2020 (I)] (a) b2 – b = 30 (b) b2 + b = 72 (c) b2–b = 42

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(a) (b) (c) (d) 11.

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(d) b2 + b = 12 12. Let and be the roots of the equation x2 – x – l = 0. If pk = ( )k + ( )k, k l, then which one of the following statements is not true ? [Main Jan. 7, 2020 (II)] (a) p3 = p5 – p4 (b) P5 = 11 (c) (p1 + p2 + p3 + p4 + p5) = 26 (d) p5 = p2 p3 . tan 13. Let and be two real roots of the equation (k +1) tan2x – x = (1 – k), where k( –1) and are real numbers. If tan2( + ) = 50, then a value of is: [Main Jan. 7, 2020 (I)] (a) 10 (b) 10 (c) 5 (d) 5 14. Suppose a, b denote the distinct real roots of the quadratic polynomial and suppose c, d denote the distinct complex roots Then the value of

[Adv. 2020]

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of the quadratic polynomial

0 8000 8080 16000 If a and b are the roots of the quadratic equation, x2 + x sin q – 2sinq =

w

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is (a) (b) (c) (d) 15.

0

, then

is equal to : [Main April 10, 2019 (I)]

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(a)

(c) 16.

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(b) (d)

The number of real roots of the equation

is:

[Main April 10, 2019 (II)]

(a) 3 (b) 2 (c) 4 (d) 1 17. Let p, q ∈ R. If 2 – x2 + px + q = 0, then:

is a root of the quadratic equation,

[Main April 9, 2019 (I)]

p – 4q + 12 = 0 q2 – 4p – 16 = 0 q2 + 4p + 14 = 0 p2 – 4q – 12 = 0

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(a) (b) (c) (d)

2

18.

The sum of the solutions of the equation

+2=

0, (x > 0) is equal to:

9 12 4 10 If α and β be the roots of the equation x2 – 2x + 2 = 0, then the least

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(a) (b) (c) (d) 19.

[Main April 8, 2019 (I)]

value of n for which

= 1 is : [Main April 8, 2019 (I)]

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2 5 4 3 If

be the ratio of the roots of the quadratic equation in x, 3m2x2 +

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(a) (b) (c) (d) 20.

m(m – 4)x + 2 = 0, then the least value of m for which

is :

[Main Jan. 12, 2019 (I)]

(a) (b) (c)

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(d) 21. Consider the quadratic equation (c – 5)x2 – 2cx + (c – 4) = 0, c ¹ 5. Let S be the set of all integral values of c for which one root of the equation lies in the interval (0, 2) and its other root lies in the interval (2, 3). Then the number of elements in S is: [Main Jan. 10, 2019 (I)] (a) 18 (b) 12 (c) 10 (d) 11 22. Let a and b be two roots of the equation x2 + 2x + 2 = 0, then a15 + b15 is equal to: [Main Jan. 9, 2019 (I)] (a) – 256 (b) 512 (c) – 512 (d) 256 23. The number of all possible positive integral values of α for which the roots of the quadratic equation, 6x2 – 11x + α = 0 are rational numbers is: [Main Jan. 09, 2019 (II)] (a) 3

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(b) 2 (c) 4 (d) 5 24. If an angle A of a ∆ ABC satisfies 5 cos A + 3 = 0, then the roots of the quadratic equation, 9x2 + 27x + 20 = 0 are. [Main Online April 16, 2018] (a) sin A, sec A (b) sec A, tan A (c) tan A, cos A (d) sec A, cot A 25. If tan A and tan B are the roots of the quadratic equation, 3x2 – 10x – 25 = 0 then the value of 3 sin2 (A + B) – 10 sin (A + B). cos (A + B) – 25 cos2 (A + B) is [Main Online April 15, 2018] (a) 25 (b) – 25 (c) – 10 (d) 10 are the distinct roots, of the equation , then 26. If

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is equal to :

[Main 2018]

w

w

(a) 0 (b) 1 (c) 2 (d) – 1 27. If, for a positive integer n, the quadratic equation, ) (x + n) = 10n x(x + 1) + (x + 1) (x + 2) + ..... + (x + has two consecutive integral solutions, then n is equal to: (a) (b) (c) (d)

11 12 9 10

[Main 2017]

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28.

The sum of all the real values of x satisfying the equation = 1 is :

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[Main Online April 9, 2017] (a) 16 (b) 14 (c) – 4 (d) – 5 29. The sum of all real values of x satisfying the equation is :

[Main 2016]

(a) (b) (c) (d) 30.

6 5 3 –4 If x is a solution of the equation,

, then

is equal to :

[Main Online April 10, 2016]

(a)

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(b) (c) (d) 2 31.

Let

. Suppose α1 and Β1are the roots of the equation x2

w

– 2x sec α + 1 = 0 and α2 and β2 are the roots of the equation x2 + 2x tan θ – 1 = 0. If α1 > β1 and α2 > β2, then α1 + β2 equals

w

(a) 2 (sec θ – tan θ) (c) –2 tanθ

[Adv. 2016] (b) 2 sec θ (d) 0

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32.

Let α and β be the roots of equation x2 – 6x – 2 = 0. If an = αn – βn, for n ≥ 1, then the value of

is equal to:

(a) (b) (c) (d) 33.

3 –3 6 –6 If

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[Main 2015]

and the equation

(where [x] denotes the greatest integer ) has no integral solution, then all possible values of a lie in the interval: [Main 2014] (a) (b) (c) (d)

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34. The sum of the roots of the equation, x2 + |2x – 3| – 4 = 0, is:

[Main Online April 12, 2014]

(a) 2 (b) – 2 (c)

w

(d) 35.

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The quadratic equation p(x) = 0 with real coefficients has purely imaginary roots. Then the equation p(p(x)) = 0 has [Adv. 2014] (a) one purely imaginary root (b) all real roots (c) two real and two purely imaginary roots

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(d) neither real nor purely imaginary roots 36.

then a

If p and q are non-zero real numbers and is :

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quadratic equation whose roots are

[Main Online April 25, 2013]

(a) (b) (c) (d) 37.

px – qx + p = 0 qx2 + px + q2 = 0 px2 + qx + p2 = 0 qx2 – px + q2 = 0 Let α and β be the roots of x2 – 6x – 2 = 0, with α > β. If 2

for n

2

1, then the value of

is

[2011]

1 2 3 4

Let (x0, y0) be the solution of the following equations

w .je

(a) (b) (c) (d) 38.

Then x0 is

[2011]

w

(a)

w

(b) (c)

(d) 6

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39.

Let p and q be real numbers such that

and

. If

α and β are nonzero complex numbers satisfying α + β = – p and α3 + and

as its roots is

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β3 = q, then a quadratic equation having

[2010]

(a) (b) (c) (d) 40.

Let α, β be the roots of the equation x2 – px + r = 0 and

be the

roots of the equation x2 – qx + r = 0. Then the value of r is [2007 -3 marks] (a)

w .je

(b) (c)

(d)

Let a, b, c be the sides of a triangle where a ≠ b ≠ c and λ ∈ R. If the roots of the equation x2 + 2(a + b + c)x + 3λ (ab + bc + ca) = 0 are real, then [2006 - 3M, –1]

w

41.

w

(a)

(b)

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(c)

42. (a) (b) (c) (d) 43.

If one root is square of the other root of the equation x2 + px + q = 0, then the relation between p and q is [2004S] p3 – q(3p – 1) + q2 = 0 p3 – q(3p + 1) + q2 = 0 p3 + q(3p – 1) + q2 = 0 p3 + q(3p+1)+q2 = 0 For the equation 3x2 + px + 3 = 0, p > 0, if one of the root is square of the other, then p is equal to [2000S] 1/3 1 3 2/3 If b > a, then the equation (x – a) (x – b) –1 = 0 has [2000S] both roots in (a, b) both roots in (–∞, a) both roots in (b, +∞) one root in (–∞, a) and the other in (b, +∞) If α and β (α < β) are the roots of the equation x2 + bx + c = 0, where c < 0 < b, then [2000S] 0 0. [1980]

[1980]

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Find all the real values of x for which y takes real values. 48. Given n4 < 10n for a fixed positive integer n 2, prove that 49.

(n + 1)4 < 10n + 1. [1980]

Find all integers x for which

[1978]

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(5x – 1) < (x + 1)2 < (7x – 3). 50. Sketch the solution set of the following system of inequalities: x2 + y2 – 2x 0; 3x – y – 12 0; y – x 0; y 0. 51. Show that the square of

[1978] is a rational number. [1978]

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52.

If (m , n) = [1978] m), show that

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where m and n are positive integers (n

(m, n + 1) = (m – 1, n + 1) + xm – n – 1 (m – 1, n). 53.

Solve for x : 4x –

=

– 22x – 1

w

w

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[1978]

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Two families with three members each and one family with four members are to be seated in a row. In how many ways can they be seated so that the same family members are not separated? [Main Sep. 06, 2020 (I)] 2! 3! 4! (3!)3 ⋅ (4!) (3!)2 ⋅ (4!) 3! (4!)3 up to 51th term) + (1! – 2! + The value of

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(a) (b) (c) (d) 2.

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1.

3! – ... up to 51th term) is equal to : [Main Sep. 03, 2020 (I)]

1 – 51(51)! 1 + (51)! 1 + (52)! 1 If the number of five digit numbers with distinct digits and 2 at the 10th place is 336 k, then k is equal to: [Main Jan. 9, 2020 (I)] (a) 4

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(b) 6 (c) 7 (d) 8 Total number of 6-digit numbers in which only and all the five digits 1, 3, 5, 7 and 9 appear, is: [Main Jan. 7, 2020 (I)]

(a) (b) 6! (c) 56 (d) 5.

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4.

The number of 6 digit numbers that can be formed using the digits 0, 1, 2, 5, 7 and 9 which are divisible by 11 and no digit is repeated, is: [Main April 10, 2019 (I)] (a) 72 (b)60 (c) 48 (d) 36

The number of four-digit numbers strictly greater than 4321 that can be formed using the digits 0, 1, 2, 3, 4, 5 (repetition of digits is allowed) is: [Main April 08, 2019 (II)] 288 360 306 310 Consider three boxes, each containing 10 balls labelled 1, 2,..., 10. Suppose one ball is randomly drawn from each of the boxes. Denote by ni, the label of the ball drawn from the ith box, (i = 1, 2, 3). Then, the number of ways in which the balls can be chosen such that n1 < n2 < n3 is : [Main Jan. 12, 2019 (I)]

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(a) (b) (c) (d) 9.

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120 82 240 164 The number of numbers between 2,000 and 5,000 that can be formed with the digits 0, 1, 2, 3, 4, (repetition of digits is not allowed) and are multiple of 3 is? [Main Online April 16, 2018] 30 48 24 36 The number of ways in which 5 boys and 3 girls can be seated on a round table if a particular boy B1 and a particular girl G1 never sit adjacent to each other, is : [Main Online April 9, 2017] 5 × 6! 6 × 6! 7! 5 × 7! If all the words (with or without meaning) having five letters, formed using the letters of the word SMALL and arranged as in a dictionary; then the position of the word SMALL is : [Main 2016] nd 52 58th 46th 59th If the four letter words (need not be meaningful) are to be formed using the letters from the word “MEDITERRANEAN” such that the first letter is R and the fourth letter is E, then the total number of all such words is : [Main Online April 9, 2016] 110 59

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(a) (b) (c) (d) 8.

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(a) (b)

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(c)

(c) 216 (d) 192

The number of ways of selecting 15 teams from 15 men and 15 women, such that each team consists of a man and a woman, is: [Main Online April 10, 2015] 1120 1880 1960 1240 The sum of the digits in the unit’s place of all the 4-digit numbers formed by using the numbers 3, 4, 5 and 6, without repetition, is: [Main Online April 9, 2014] 432 108 36 18 Six cards and six envelopes are numbered 1, 2, 3, 4, 5, 6 and cards are to be placed in envelopes so that each envelope contains exactly one

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(d) 56 12. The number of points, having both co-ordinates as integers, that lie in the interior of the triangle with vertices (0, 0), (0, 41) and (41, 0) is : [Main 2015] (a) 820 (b) 780 (c) 901 (d) 861 13. The number of integers greater than 6,000 that can be formed, using the digits 3, 5, 6, 7 and 8, without repetition, is : [Main 2015] (a) 120 (b) 72

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(a) (b) (c) (d) 18.

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(a) (b) (c) (d) 17.

card and no card is placed in the envelope bearing the same number and moreover the card numbered 1 is always placed in envelope numbered 2. Then the number of ways it can be done is [Adv. 2014] 264 265 53 67 5 - digit numbers are to be formed using 2, 3, 5, 7, 9 without repeating the digits. If p be the number of such numbers that exceed 20000 and q be the number of those that lie between 30000 and 90000, then p : q is : [Main Online April 25, 2013] 6:5 3:2 4:3 5:3 The number of seven digit i ntegers, with sum of the digits equal to 10 and formed by using the digits 1, 2 and 3 only, is [2009] 55 66 77 88 The letters of the word COCHIN are permuted and all the permutations are arranged in an alphabetical order as in an English dictionary. The number of words that appear before the word COCHIN is [2007 -3 marks] 360 192 96 48 If the LCM of p, q is r2t4s2, where r, s, t are prime numbers and p, q are the positive integers then the number of ordered pair (p, q) is [2006 - 3M, –1]

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(a) (b) (c) (d) 22.

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252 254 225 224 The number of arrangements of the letters of the word BANANA in which the two N’s do not appear adjacently is [2002S] 40 60 80 100 How many different nine digit numbers can be formed from the number 223355888 by rearranging its digits so that the odd digits occupy even positions ? [2000S] 16 36 60 180 A five-digit numbers divisible by 3 is to be formed using the numerals 0, 1, 2, 3, 4 and 5, without repetition. The total number of ways this can be done is [1989 - 2 Marks] 216 240 600 3125 Ten different letters of an alphabet are given. Words with five letters are formed from these given letters. Then the number of words which have at least one letter repeated are [1982 - 2 Marks] 69760 30240 99748 none of these

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(a) (b) (c) (d) 21.

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Five persons A, B, C, D and E are seated in a circular arrangement. If each of them is given a hat of one of the three colours red, blue and green, then the number of ways of distributing the hats such that the persons seated in adjacent seats get different coloured hats is ___. [Adv. 2019] 26. The number of 5 digit numbers which are divisible by 4, with digits from the set {1, 2, 3, 4, 5} and the repetition of digits is allowed, is _________. [Adv. 2018] 27. Let n1 < n2 < n3 < n4 < n5 be positive integers such that n1 + n2 + n3 + n4 + n5 = 20. Then the number of such distinct arrangements (n1, n2, n3, n4, n5) is [Adv. 2014] If the letters of the word ‘MOTHER’ be permuted and all the words so formed (with or without meaning) be listed as in a dictionary, then the position of the word ‘MOTHER’ is ________. [Main Sep. 02, 2020 (I)] 29. In a hotel, four rooms are available. Six persons are to be accommodated in these four rooms in such a way that each of these rooms contains at least one person and at most two persons. Then the number of all possible ways in which this can be done is [Adv. 2020]

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28.

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25.

There are four balls of different colours and four boxes of colours, same as those of the balls. The number of ways in which the balls, one each in a box, could be placed such that a ball does not go to a box of its own colour is _________. [1988 - 2 Marks] 31. In a certain test, ai students gave wrong answers to atleast i questions, where i = 1, 2, …, k. No student gave more than k wrong answers. The total number of wrong answers given is ................. [1982 - 2 Marks]

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(a) (b) (c) (d)

An n-digit number is a positive number with exactly n digits. Nine hundred distinct n- digit numbers are to be formed using only the three digits 2, 5 and 7. The smallest value of n for which this is possible is [1998 - 2 Marks] 6 7 8 9

In a high school, a committee has to be formed from a group of 6 boys M1, M2, M3, M4, M5, M6 and 5 girls G1, G2, G3, G4, G5. (i) Let α1 be the total number of ways in which the committee can be formed such that the committee has 5 members, having exactly 3 boys and 2 girls. (ii) Let α2 be the total number of ways in which the committee can be formed such that the committee has at least 2 members, and having an equal number of boys and girls. (iii) Let α3 be the total number of ways in which the committee can be formed such that the committee has 5 members, at least 2 of them being girls. (iv) Let α4 be the total number of ways in which the committee can be formed such that the committee has 4 members, having at least 2 girls such that both M1 and G1 are NOT in the committee together. LIST - I LIST - II P. The value of α1 is 1. 136 Q. The value of α2 is 2. 189 R. The value of α3 is 3. 192 S. The value of α4 is 4. 200 5. 381 6. 461 [Adv. 2018]

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The correct option is: (a) P → 4; Q → 6; R → 2; S → 1 (b) P → 1; Q → 4; R → 2; S → 3 (c) P → 4; Q → 6; R → 5; S → 2 (d) P → 4; Q → 2; R → 3; S → 1 34. Consider all possible permutations of the letters of the word ENDEANOEL. Match the Statements / Expressions in Column I with the Statements / Expressions in Column II and indicate your answer by darkening the appropriate bubbles in the 4 × 4 matrix given in the ORS. [2008] Column I Column II (A) The number of permutations containing the word ENDEA is (p) 5! (B) The number of permutations in which the letter E occurs in (q) 2 × 5! the first and the last positions is (C) The number of permutations in which none of the letters (r) 7 × 5! D, L, N occurs in the last five positions is (D) The number of permutations in which the letters A, E, O (s) 21 × 5! occur only in odd positions is

35.

If total number of runs scored in n matches is

(2n + 1 – n – 2)

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where n > 1, and the runs scored in the kth match are given by k. 2n + 1– k , where 1 < k < n.Find n. [2005 - 2 Marks]

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36.

Prove by permutation or otherwise

is an integer

(n ∈ I+). [2004 - 2 Marks]

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37.

m men and n women are to be seated in a row so that no two women sit together. If m > n, then show that the number of ways in which they can be seated is

1.

There are 3 sections in a question paper and each section contains 5 questions. A candidate has to answer a total of 5 questions, choosing at least one question from each section. Then the number of ways, in which the candidate can choose the questions, is : [Main Sep. 05, 2020 (II)] 3000 1500 2255 2250 Let n > 2 be an integer. Suppose that there are n Metro stations in a city located along a circular path. Each pair of stations is connected by a straight track only. Further, each pair of nearest stations is connected by blue line, whereas all remaining pairs of stations are connected by red line. If the number of red lines is 99 times the number of blue lines, then the value of n is : [Main Sep. 02, 2020 (II)] 201 200 101 199 If a, b and c are the greatest values of 19Cp, 20Cq and 21Cr respectively, then: [Main Jan. 8, 2020 (I)]

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(a) (b) (c) (d) 2.

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(a)

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(b)

(d) 4.

(a) (b) (c) (d) 5.

The number of ordered pairs (r, k) for which 6.35Cr = (k2 – 3).36Cr + 1, where k is an integer, is: [Main Jan. 7, 2020 (II)] 3 2 6 4

The number of ways of choosing 10 objects out of 31 objects of which 10 are identical and the remaining 21 are distinct is: [Main April 12, 2019 (I)] 220 – 1 221 220 220+1

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(c)

6.

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Suppose that 20 pillars of the same height have been erected along the boundary of a circular stadium. If the top of each pillar has been connected by beams with the top of all its non-adjacent pillars, then the total number of beams is : [Main April 10, 2019 (II)] 170 180 210 190 A committee of 11 members is to be formed from 8 males and 5 females. If m is the number of ways the committee is formed with at

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(a) (b) (c) (d) 9.

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(a) (b) (c) (d) 8.

least 6 males and n is the number of ways the committee is formed with at least 3 females, then: [Main April 9, 2019 (I)] m + n = 68 m = n = 78 n=m–8 m = n = 68 There are m men and two women participating in a chess tournament. Each participant plays two games with every other participant. If the number of games played by the men between themselves exceeds the number of games played between the men and the women by 84, then the value of m is [Main Jan. 12, 2019 (II)] 12 11 9 7 Consider a class of 5 girls and 7 boys. The number of different teams consisting of 2 girls and 3 boys that can be formed from this class, if there are two specific boys A and B, who refuse to be the members of the same team, is: [Main Jan. 9, 2019 (I)] 500 200 300 350 The number of four letter words that can be formed using the letters of the word BARRACK is [Main Online April 15, 2018] 144 120 264 270 From 6 different novels and 3 different dictionaries, 4 novels and 1 dictionary are to be selected and arranged in a row on a shelf so that

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13. The value of

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the dictionary is always in the middle. The number of such arrangements is : [Main 2018] less than 500 at least 500 but less than 750 at least 750 but less than 1000 at least 1000 A man X has 7 friends, 4 of them are ladies and 3 are men. His wife Y also has 7 friends, 3 of them are ladies and 4 are men. Assume X and Y have no common friends. Then the total number of ways in which X and Y together can throw a party inviting 3 ladies and 3 men, so that 3 friends of each of X and Y are in this party, is : [Main 2017] 484 485 468 469 is equal to :

[Main Online April 9, 2016]

1240 560 1085 680 A debate club consists of 6 girls and 4 boys. A team of 4 members is to be selected from this club including the selection of a captain (from among these 4 memoers) for the team. If the team has to include at most one boy, then the number of ways of selecting the team is [Adv. 2016] (a) 380 (b) 320 (c) 260 (d) 95 15. Let A and B be two sets containing four and two elements respectively. Then the number of subsets of the set A × B, each having

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at least three elements is : [Main 2015]

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275 510 219 256 If in a regular polygon the number of diagonals is 54, then the number of sides of this polygon is [Main Online April 11, 2015] (a) 12 (b) 6 (c) 10 (d) 9 17. Let A and B two sets containing 2 elements and 4 elements respectively. The number of subsets of A × B having 3 or more elements is [Main 2013] (a) 256 (b) 220 (c) 219 (d) 211 18. Let Tn be the number of all possible triangles formed by joining vertices of an n-sided regular polygon. If Tn+1 – Tn = 10, then the value of n is : [Main 2013] (a) 7 (b) 5 (c) 10 (d) 8 19. On the sides AB, BC, CA of a ∆ABC, 3, 4, 5 distinct points (excluding vertices A, B, C) are respectively chosen. The number of triangles that can be constructed using these chosen points as vertices are : [Main Online April 23, 2013] (a) 210 (b) 205

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(a) (m + n – 1)2 4m+n–1 m2n2 m(m + 1)n(n + 1) Let Tn denote the number of triangles which can be formed using the vertices of a regular polygon of n sides. If Tn + 1 − Tn = 21, then n equals [2001S] 5 7 6 4 Eight chairs are numbered 1 to 8. Two women and three men wish to occupy one chair each. First the women choose the chairs from amongst the chairs marked 1 to 4 ; and then the men select the chairs from amongst the remaining. The number of possible arrangements is [1982 - 2 Marks]

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(b) (c) (d) 22.

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(c) 215 (d) 220 20. The total number of ways in which 5 balls of different colours can be distributed among 3 persons so that each person gets at least one ball is [2012] (a) 75 (b) 150 (c) 210 (d) 243 21. A rectangle with sides of length (2m – 1) and (2n – 1) units is divided into squares of unit length by drawing parallel lines as shown in the diagram, then the number of rectangles possible with odd side lengths is [2005S]

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(a)

(b)

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(c) (d) none of these The value of the expression

is equal to

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24.

[1982 - 2 Marks]

(a) C5 (b) 52C5 (c) 52C4 (d) none of these 25. nCr–1 = 36, nCr = 84 and nCr + 1 = 126, then r is : 47

(a) (b) (c) (d) 26.

[1979]

1 2 3 None of these. Let

denote the number of elements in a set X. Let S = {1, 2, 3, 4,

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5, 6} be a sample space, where each element is equally likely to occur. If A and B are independent events associated with S, then the number , equals ____. of ordered pairs (A, B) such that

27.

[Adv. 2019] Words of length 10 are formed using the letters A, B, C, D, E, F, G, H, I, J. Let x be the number of such words where no letter is repeated; and let y be the number of such words where exactly one letter is

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repeated twice and no other letter is repeated. Then,

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= [Adv. 2017] Let n be the number of ways in which 5 boys and 5 girls can stand in a queue in such a way that all the girls stand consecutively in the queue. Let m be the number of ways in which 5 boys and 5 girls can

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stand in a queue in such a way that exactly four girls stand consecutively in the queue. Then the value of

is

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[Adv. 2015] 29. Let n ≥ 2 be an integer. Take n distinct points on a circle and join each pair of points by a line segment. Colour the line segment joining every pair of adjacent points by blue and the rest by red. If the number of red and blue line segments are equal, then the value of n is [Adv. 2014] 30. Consider the set of eight vectors Three non-coplanar vectors can be

chosen from V in 2p ways. Then p is

[Adv. 2013]

The number of words (with or without meaning) that can be formed from all the letters of the word “LETTER” in which vowels never come together is ______. [Main Sep. 06, 2020 (II)] 32. The number of words, with or without meaning, that can be formed by taking 4 letters at a time from the letters of the word ‘SYLLABUS’ such that two letters are distinct and two letters are alike, is ______. [Main Sep. 05, 2020 (I)] 33. A test consists of 6 multiple choice questions, each having 4 alternative answers of which only one is correct. The number of ways, in which a candidate answers all six questions such that exactly four of the answers are correct, is __________. [Main Sep. 04, 2020 (II)] 34. The total number of 3-digit numbers, whose sum of digits is 10, is _________. [Main Sep. 03, 2020 (II)] 25 35. If Cr ≡ Cr and C0 + 5⋅C1 + 9⋅C2 + ... + (101)⋅C25 = 225⋅k, then k is equal to ________. [Main Jan. 9, 2020 (II)]

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An urn contains 5 red marbles, 4 black marbles and 3 white marbles. Then the number of ways in which 4 marbles can be drawn so that at the most three of them are red is _________. [Main Jan. 8, 2020 (I)] 37. The number of 4 letter words (with or without meaning) that can be formed from the eleven letters of the word ‘EXAMINATION’ is _________. [Main Jan. 8, 2020 (II)] 38. An engineer is required to visit a factory for exactly four days during the first 15 days of every month and it is mandatory that no two visits take place on consecutive days. Then the number of all possible ways in which such visits to the factory can be made by the engineer during1-15 June 2021 is _____ [Adv. 2020]

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36.

Total number of ways in which six ‘+’ and four ‘–’ signs can be arranged in a line such that no two ‘–’ signs occur together is ................. [1988 - 2 Marks] 40. The side AB, BC and CA of a triangle ABC have 3, 4 and 5 interior points respectively on them. The number of triangles that can be constructed using these interior points as vertices is ................. [1984 - 2 Marks]

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39.

The product of any r consecutive natural numbers is always divisible by r!. [1985 - 1 Mark]

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41.

For nonnegative integers s and r, let

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42.

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For positive integers m and n, let

where for any nonnegative integer p,

Then which of the following statements is/are TRUE?

[Adv. 2020]

(a) (b) (c) (d)

g(m, n) = g(n, m) for all positive integers m, n g(m, n + 1) = g(m + 1, n) for all positive integers m, n g(2m, 2n) = 2g(m, n) for all positive integers m, n g(2m, 2n) = (g(m, n))2 for all positive integers m, n

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PASSAGE Let an denote the number of all n-digit positive integers formed by the digits 0, 1 or both such that no consecutive digits in them are 0. Let bn = the number of such n-digit integers ending with digit 1 and cn = the number of such n–digit integers ending with digit 0. [2012 ] 43. The value of b6 is (a) 7 (b) 8 (c) 9 (d) 11 44. Which of the following is correct? (a) a17 = a16 + a15 (b) c17 ≠ c16 + c15 (c) b17 ≠ b16 + c16 (d) a17 = c17 + b16

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(a) (b) 46.

47.

48.

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49.

A committee of 12 is to be formed from 9 women and 8 men. In how many ways this can be done if at least five women have to be included in a committee? In how many of these committees [1994 - 4 Marks] The women are in majority? The men are in majority? Eighteen guests have to be seated, half on each side of a long table. Four particular guests desire to sit on one particular side and three others on the other side. Determine the number of ways in which the sitting arrangements can be made. [1991 - 4 Marks] A box contains two white balls, three black balls and four red balls. In how many ways can three balls be drawn from the box if at least one black ball is to be included in the draw? [1986 - 2½ Marks] 7 relatives of a man comprises 4 ladies and 3 gentlemen; his wife has also 7 relatives; 3 of them are ladies and 4 gentlemen. In how many ways can they invite a dinner party of 3 ladies and 3 gentlemen so that there are 3 of man’s relatives and 3 of the wife’s relatives? [1985 - 5 Marks] Five balls of different colours are to be placed in there boxes of different size. Each box can hold all five. In how many different ways can we place the balls so that no box remains empty ? [1981 - 4 Marks] Six X’s have to be placed in the squares of figure below in such a way that each row contains at least one X. In how many different ways can this be done.

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45.

[1978]

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1.

If {p} denotes the fractional part of the number p, then equal to :

, is

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(a)

(b)

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(c)

(d)

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2.

If α and β be the coefficients of x4 and x2 respectively in the expansion of

, then: [Main Jan. 8, 2020 (II)]

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+ = 60  + = –30  – = 60  – = –132 The smallest natural number n, such that the coefficient of x in the expansion of

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(a) (b) (c) (d) 3.

is nC23 , is :

[Main April 10, 2019 (II)]

(a) (b) (c) (d) 4.

38 58 23 35

If the fourth term in the Binomial expansion of

(x > 0)

is 20 × 87, then a value of x is:

[Main April 9, 2019 (I)]

8 82 8 8–2 The sum of the co-efficients of all even degree terms in x in the

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3

expansion of

,(x > 1) is equal to :

[Main April 8, 2019 (I)]

29 32 26 24 Let (x + 10)50 + (x – 10)50 = a0 + a1x + a2x2 + .... + a50x50, for all x R;

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(a) (b) (c) (d) 6.

then

is equal to :

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(a) (b) (c) (d) 7.

12.50 12.00 12.25 12.75

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[Main Jan. 11, 2019 (II)]

If the third term in the binomial expansion of

equals

2560, then a possible value of x is:

[Main Jan. 10, 2019 (I)]

(a) (b) (c) (d) 8.

If the fractional part of the number

is equal to:

is

, then k

[Main Jan. 9, 2019 (I)]

6 8 4 14 The coefficient of x2 in the expansion of the product(2 – x2). ((1 + 2x + 3x2)6 + (1 – 4x2)6) is [Main Online April 16, 2018] 106 107 155 108 The sum of the co-efficients of all odd degree terms in the expansion of [Main 2018] is :

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(a) (b) (c) (d) 10.

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0 1 2 –1 The

coefficient

of

x–5

in

the

binomial

expansion

of

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(a) (b) (c) (d) 11.

where x ≠ 0, 1, is :

(a) 1

(b) 4

(c) – 4

[Main Online April 9, 2017] (d) – 1

12. If the coefficients of x–2 and x–4 in the expansion of

> 0), are m and n respectively, then

, (x

is equal to :

[Main Online April 10, 2016]

(a) 27 (b) 182

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13.

If

the

coefficents

of x3 and x4 in the expansion of in powers of x are both zero, then (a, b) is

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equal to:

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(b)

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(c)

14.

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(d) If

and

the set of natural numbers, then

where N is

is equal to:

[Main 2014]

(a) X (b) Y (c) N (d) Y – X 15. The number of terms in the expansion of (1 + x)101 (1 + x2 – x)100 in powers of x is:

[Main Online April 9, 2014]

302 301 202 101 The sum of the rational terms in the binomial expansion of

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(a) (b) (c) (d) 16.

is :

25 32 9 41

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[Main Online April 23, 2013]

17. For r = 0, 1, …, 10, let Ar, Brand Cr denote, respectively, the coefficient of xr in the expansions of (1 + x)10 , [2010]

(1 + x)20 and (1 + x)30. Then

Ar(B10Br – C10Ar) is equal to

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(a) B10 – C10 (b) A10(B210C10A10) (c) 0 (d) C10–B10 18. Coefficient of t24 in (1 +t2)12 (1+t12) (1 + t24) is

[2003S]

(a) C6 + 3 (b) C6 + 1 (c) C6 (d) C6 + 2 19. In the binomial expansion of (a − b)n, n ≥ 5, the sum of the 5th and 6th terms is zero. Then a/b equals [2001S] (a) (n − 5)/6 (b) (n − 4)/5 (c) 5/(n −4) (d) 6/(n − 5) 12

20.

12

12

The coefficient of x4 in

12

is

[1983 - 1 Mark]

(b)

w .je

(a) (c)

w

w

(d) none of these 21. Given positive integers r > 1, n >2 and that the coefficient of (3r)th and (r + 2)th terms in the binomial expansion of (1+ x)2n are equal . Then [1983 - 1 Mark] (a) n = 2r (c) n = 2r + 1 (c) n = 3r (d) none of these 22.

Let

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X = (10C1)2 + 2(10C2)2 + 3(10C3)2 + ... + 10(10C10)2, where 10Cr, r ∈ {1, 2, ..., 10} denote binomial coefficients. Then, the value X is _____ .

eb oo ks .in

of

[Adv. 2018] 23. Let m be the smallest positive integer such that the coefficient of x2 in the expansion of (1 + x)2 + (1 + x)3 + ... + (1 + x)49 + (1 + mx)50 is (3n + 1) 51C3 for some positive integer n. Then the value of n is [Adv. 2016] 24. The coefficients of three consecutive terms of (1 + x)n+5 are in the ratio 5 : 10 : 14. Then n = [Adv. 2013] 25.

The natural number m, for which the coefficient of x in the binomial expansion of

[Main Sep. 05, 2020 (I)] The coefficient of x in the expansion of (1 + x + x2 + x3)6 in powers of x, is ____________. [Main Sep. 05, 2020 (II)] 4

w .je

26.

is 1540, is ______.

27.

Then

is equal to ___________. [Main Sep. 04, 2020 (I)]

The sum of the rational terms in the expansion of (

w

28.

Let

is

................

w

[1997 - 2 Marks] 29. Let n be positive integer. If the coefficients of 2nd, 3rd, and 4th terms in the expansion of (1 + x)n are in A.P., then the value of n is .............. [1994 - 2 Marks] 50 50 50 30. The larger of 99 + 100 and 101 is ................

www.jeebooks.in

[1982 - 2 Marks]

equals

eb oo ks .in

31. If

[1998 - 2 Marks]

(a) (n – 1)an (b) nan nan

(c)

(d) None of the above

1.

If the constant term in the binomial expansion of

is 405,

then |k| equals:

[Main Sep. 06, 2020 (II)]

9 1 3 2

w .je

(a) (b) (c) (d) 2.

w

If for some positive integer n, the coefficients of three (1 + x) n + 5 are in the ratio 5 : 10 : 14, then the largest coefficient in this expansion is : [Main Sep. 04, 2020 (II)] 462 330 792 252 is If the number of integral terms in the expansion of

w

(a) (b) (c) (d) 3.

exactly 33, then the least value of n is :

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(a) (b) (c) (d) 4.

264 128 256 248

eb oo ks .in

[Main Sep. 03, 2020 (I)]

If the term independent of x in the expansion of

is k,

then 18k is equal to :

[Main Sep. 03, 2020 (II)]

(a) (b) (c) (d) 5.

5 9 7 11 Let α > 0, β > 0 be such that

If the maximum value of the

term independent of x in the binomial expansion of

is

10k, then k is equal to :

[Main Sep. 02, 2020 (I)]

336 352 84 176

w .je

(a) (b) (c) (d)

In the expansion of

if l1 is the least value of the

term independent of x when

and l2 is the least value of the

term independent of x when

then the ratio l2 : l1 is equal

w

6.

w

to :

(a) (b) (c) (d)

1:8 16 : 1 8:1 1 : 16

[Main Jan. 9, 2020 (II)]

www.jeebooks.in

7.

The total number is irrational terms in the binomial expansion of

eb oo ks .in

is : [Main Jan. 12, 2019 (II)]

(a) (b) (c) (d) 8.

55 49 48 54 A ratio of the 5th term from the begining to the 5th term from the end is:

in the binomial expansion of

[Main Jan. 12, 2019 (I)]

(a) (b)

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(c)

(d)

The term independent of x in expansion of

w

9.

w

(a) (b) (c) (d)

is [Main 2013]

4 120 210 310

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10.

For a positive integer n,

is expanded in increasing powers of

11.

Prove that

eb oo ks .in

x. If three consecutive coefficients in this expansion are in the ratio, 2 : 5 : 12, then n is equal to __________. [Main Sep. 02, 2020 (II)]

where

and n is an even

positive integer. 12.

Let R =

[1993 - 5 Marks] and f = R – [R], where [ ] denotes the

greatest integer function. Prove that

.

w .je

[1988 - 5 Marks]

1.

is equal to: [Main Sep. 04, 2020 (I)]

C7 – 30C7 50 C7 – 30C7 50 C6 – 30C6 51 C7 + 30C7 If 20C1 + (22) 20C2 +(32) 20C3+ ………. + (202) 20C20 = A(2b), then the ordered pair (A, b) is equal to : [Main April 12, 2019 (II)] (a) (420, 19) (b) (420, 18) 51

w

w

(a) (b) (c) (d) 2.

The value of

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w .je

eb oo ks .in

(c) (380, 18) (d) (380, 19) 3. If the coefficients of x2 and x3 are both zero, in the expansion of the expression (1 + ax + bx2) (1–3x)15 in powers of x, then the ordered pair (a, b) is equal to: [Main April 10, 2019 (I)] (a) (28, 861) (b) (–54, 315) (c) (28, 315) (d) (–21, 714) 4. The sum of the series 20 2· C0 + 5·20C1 + 8·20C2 + 11·20C3 + … + 62·20C20 is equal to : [Main April 8, 2019 (I)] (a) 226 (b) 225 (c) 223 (d) 224 5. The value of r for which is maximum, is : 15 20 11 10

w

(a) (b) (c) (d)

w

6.

[Main Jan. 11, 2019 (I)]

If

then K is equal to: [Main Jan. 10, 2019 (II)]

(a) (25)2 (b) 225 – 1 (c) 224

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(d) 225 7.

The coefficient of t4 in the expansion of

eb oo ks .in

[Main Jan. 09, 2019 (II)]

(a) 14 (b) 15 (c) 10 (d) 12 8. The value of (21C1 – 10C1) + (21C2 – 10C2) + (21C3 – 10C3) + (21C4 – 10C4) + .... + (21C10 – 10C10) is :

[Main 2017]

(a) (b) (c) (d)

– 210 – 211 – 210 – 29

If the number of terms in the expansion of

,

w .je

9.

220 221 221 220

w

w

x ≠ 0, is 28, then the sum of the coefficients of all the terms in this expansion, is : [Main 2016] (a) 243 (b) 729 (c) 64 (d) 2187 10. The sum of coefficients of integral power of x in the binomial expansion

is : [Main 2015]

(a)

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(b)

(d) 11.

eb oo ks .in

(c)

Coefficient of x11 in the expansion of

[Adv. 2014]

(1 + x ) (1 + x ) (1 + x ) is (a) 1051 (b) 1106 (c) 1113 (d) 1120 12. The value of 2 4

3 7

4 12

is

where

w .je

(a)

[2005S]

(b)

w

(c)

w

(d)

13.

The sum

is maximum

when m is [2002S]

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5 10 15 20

14.

For

eb oo ks .in

(a) (b) (c) (d)

[2000S]

(a) (b) (c) (d) 15.

6 9 12 24

w .je

(a) (b) (c) (d)

If in the expansion of (1 + x)m (1 – x)n, the coefficients of x and x2 are 3 and – 6 respectively, then m is [1999 - 2 Marks]

16.

The expression

+

is a polynomial

w

of degree

w

(a) (b) (c) (d)

[1992 - 2 Marks]

5 6 7 8

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The coefficient of x4 in the expansion of (1 + x + x2)10 is _______. [Main Jan. 9, 2020 (I)] 18. If the sum of the coefficients of all even powers of x in the product (1 + x + x2 + ... + x2n) (1 – x + x2 – x3 + ... + x2n) is 61, then n is equal to . [Main Jan. 7, 2020 (I)] 19. Suppose

eb oo ks .in

17.

holds for some positive integer n. The

equals _____

[Adv. 2019]

The sum of the coefficients of the plynomial (1 + x – 3x2)2163 is ................ [1982 - 2 Marks]

w .je

20.

21.

If Cr stands for nCr, then the sum of the series

w

where n is an even positive integer, is equal to [1986 - 2 Marks]

w

(a) 0 (b) (c)

(d)

(e) none of these.

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22.

Prove that

eb oo ks .in

[2003 - 2 Marks]

.

. Prove that

23. For any positive integer m, n (with n ≥ m), let

. Hence or otherwise, prove

that

.

[2000 - 6 Marks]

w .je

24. Let n be a positive integer and (1 + x + x2)n = a0 + a1 x + ............+ a2n x2n Show that a02 – a12 + a22 .............+ a2n2 = an

25. If

[1994 - 5 Marks]

=

and ak = 1 for allk

n, then

show that

w

[1992 - 6 Marks]

w

26.

27.

Prove that = 0, .

n > 2, where

[1989 - 5 Marks] Given sn = 1 + q + q + ...... + q ; 2

n

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eb oo ks .in

Prove that

[1984 - 4 Marks] 28. If (1 + x) = C0 + C1x + C2x + ...... + Cnx then show that the sum of the products of the Ci’s taken two at a time, represented by n

2

n

is equal to

[1983 - 3 Marks]

29. Given that C1 + 2C2x + 3C3x2 + ........... + 2n C2nx2n – 1 = 2n (1 + x)2n – 1 where Cr =

r = 0, 1, 2, .................., 2n

Prove that C12 – 2C22 + 3C32 – ...................... – 2nC2n2 = (– 1)nn Cn.

w

w

w .je

[1979]

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The common difference of the A.P. b1, b2, ..., bm is 2 more than the common difference of A.P. a1, a2, ..., an. If a40 = – 159, a100 = – 399 and b100 = a70, then b1 is equal to: [Main Sep. 06, 2020 (II)] 81 – 127 – 81 127 If 32sin2α–1, 14 and 34–2sin2α are the first three terms of an A.P. for some α, then the sixth term of this A.P is: [Main Sep. 05, 2020 (I)] 66 81 65 78 If the sum of the first 20 terms of the series is 460, then x is equal to :

w .je

(a) (b) (c) (d) 2.

eb oo ks .in

1.

w

w

(a) (b) (c) (d) 3.

(a) 7

[Main Sep. 05, 2020 (II)] 2

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be a given A.P. whose common difference is an If and and then the ordered pair is equal to :

eb oo ks .in

(b) 71/2 (c) e2 (d) 746/21 4. Let integer

[Main Sep. 04, 2020 (II)]

(a) (b) (c) (d) 5.

(2490, 249) (2480, 249) (2480, 248) (2490, 248) If the first term of an A.P. is 3 and the sum of its first 25 terms is equal to the sum of its next 15 terms, then the common difference of this A.P. is : [Main Sep. 03, 2020 (I)]

(a)

w .je

(b) (c)

(d)

In the sum of the series

w

6.

upto nth term is 488 and then nth term is

w

negative, then : [Main Sep. 03, 2020 (II)]

(a) n = 60 (b) nth term is –4 (c) n = 41

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(d) nth term is If the sum of first 11 terms of an A.P., is 0 then the sum of the A.P., is ka1, where k is equal to :

eb oo ks .in

7.

[Main Sep. 02, 2020 (II)]

(a) (b) (c) (d) 8.

If the 10th term of an A.P. is

and its 20th term is

then the sum

of its first 200 terms is:

[Main Jan. 8, 2020 (II)]

w .je

(a) 50 (b)

(c) 100 (d)

Let f : R R be such that for all x R, (21+x + 21–x), f (x) and (3x + 3–x) are in A.P., then the minimum value of f (x) is: [Main Jan. 8, 2020 (I)]

w

9.

w

(a) (b) (c) (d)

2 3 0 4

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10.

Five numbers are in A.P., whose sum is 25 and product is 2520. If one of these five numbers is

, then the greatest number amongst them

is: (a) 27 (b) 7 (c)

eb oo ks .in

[Main Jan. 7, 2020 (I)]

w .je

(d) 16 11. Let Sn denote the sum of the first n terms of an A.P. If S4 = 16 and S6 = –48, then S10 is equal to: [Main April 12, 2019 (I)] (a) –260 (b) –410 (c) –320 (d) –380 12. If a1, a2, a3, ..... an are in A.P. and a1 + a4 + a7 + ….+ a16 = 114, then a1 + a6 + a11 + a16 is equal to: [Main April 10, 2019 (I)] (a) 98 (b) 76 (c) 38 (d) 64 13. Let the sum of the first n terms of a non-constant A.P., a1, a2, a3, ………….. be 50n +

A, where A is a constant. If d is the

w

common difference of this A.P., then the ordered pair (d, a50) is equal to: [Main April 09, 2019 (I)] (50, 50 + 46A) (50, 50 + 45A) (A, 50 + 45A) (A, 50 + 46A) If the sum and product of the first three terms in an A.P. are 33 and 1155, respectively, then a value of its 11th term is:

w

(a) (b) (c) (d) 14.

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[Main April 09, 2019 (II)]

(a) (b) (c) (d) 16. (a) (b) (c) (d) 17.

w .je

(a) (b) (c) (d) 18.

–35 25 –36 –25 The sum of all natural numbers ‘n’ such that 100 < n < 200 and H.C.F. (91, n) > 1 is : [Main April 08, 2019 (I)] 3203 3303 3221 3121 If nC4, nC5 and nC6 are in A.P., then n can be : [Main Jan. 12, 2019 (II)] 9 14 11 12 If 19th term of a non-zero A.P. is zero, then its (49th term) : (29th term) is : [Main Jan. 11, 2019 (II)] 4:1 1:3 3:1 2:1 The sum of all two digit positive numbers which when divided by 7 yield 2 or 5 as remainder is: [Main Jan. 10, 2019 (I)] 1256 1465 1365 1356

eb oo ks .in

(a) (b) (c) (d) 15.

w

w

(a) (b) (c) (d)

19. Let

(xi ≠ 0 for i = 1, 2, ...., n) be in A.P. such that x1 =

4 and x21 = 20. If n is the least positive integer for which xn > 50, then

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is equal to. (a) 3 (b) (c) (d) 20.

Let

eb oo ks .in

[Main Online April 16, 2018]

be in A.P. such that

If

and

, then m is equal to :

[Main 2018]

w .je

(a) 68 (b) 34 (c) 33 (d) 66 21. For any three positive real numbers a, b and c, 9(25a2 + b2) + 25(c2 – 3ac) = 15b(3a + c). Then :

[Main 2017]

a, b and c are in G.P. b, c and a are in G.P. b, c and a are in A.P. a, b and c are in A.P. If three positive numbers a, b and c are in A.P. such that abc = 8, then the minimum possible value of b is : [Main Online April 9, 2017] (a) 2

w

w

(a) (b) (c) (d) 22.

(b)

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(c)

eb oo ks .in

(d) 4 23. Let a1, a2, a3, ...., an, be in A.P. If a3 + a7 + a11 + a15 = 72, then the sum of its first 17 terms is equal to : [Main Online April 10, 2016] (a) 306 (b) 204 (c) 153 (d) 612 24. Let bi > 1 for i = 1, 2, ..., 101. Suppose loge b1, logeb2, ...., loge b101 are in Arithmetic Progression (A.P.) with the common difference loge 2. Suppose a1, a2, ...., a101 are in A.P. such that a1= b1 and a51= b51. If t= b1+b2 + .... + b51 and s= a1+a2+ .... + a53, then [Adv. 2016] (a) s > t and a101> b101 (b) s > t and a101 < b101 (c) s < t and a101 > b101 (d) s < t and a101 < b101 25. Let α and β be the roots of equation px2 + qx + r = 0, p ≠ 0.

w .je

If p, q, r are in A.P and is:

= 4, then the value of | α – β| [Main 2014]

(a)

w

(b)

w

(c)

(d)

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(a) (b) (c) (d) 27.

(a) (b) (c) (d) 28.

The sum of the first 20 terms common between the series 3 + 7 + 11 + 15 + ......... and 1 + 6 + 11 + 16 + ......, is [Main Online April 11, 2014] 4000 4020 4200 4220 Given sum of the first n terms of an A.P. is 2n + 3n2. Another A.P. is formed with the same first term and double of the common difference, the sum of n terms of the new A.P. is : [Main Online April 22, 2013] 2 n + 4n 6n2 – n n2 + 4n 3n + 2n2 In the sum of first n terms of an A.P. is cn2, then the sum of squares of these n terms is [2009]

w .je

(a)

eb oo ks .in

26.

(b) (c)

w

(d)

If the sum of the first 2n terms of the A.P. 2, 5, 8, …, is equal to the sum of the first n terms of the A.P. 57, 59, 61, …, then n equals [2001S] (a) 10 (b) 12 (c) 11

w

29.

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(d) 13

31.

32.

33.

w .je

34.

Let AP(a;d) denote the set of all the terms of an infinite arithmetic progression with first term a and common difference d > 0. If AP (1;3) AP (2;5) AP (3;7) = AP (a;d) then a + d equals ____. [Adv. 2019] Let X be the set consisting of the first 2018 terms of the arithmetic progression 1, 6, 11, ..., and Y be the set consisting of the first 2018 terms of the arithmetic progression 9, 16, 23, .... . Then, the number of is ____. elements in the set [Adv. 2018] The sides of a right angled triangle are in arithmetic progression. If the triangle has area 24, then what is the length of its smallest side? [Adv. 2018] Suppose that all the terms of an arithmetic progression (A.P.) are natural numbers. If the ratio of the sum of the first seven terms to the sum of the first eleven terms is 6 : 11 and the seventh term lies in between 130 and 140, then the common difference of this A.P. is [Adv. 2015] A pack contains n cards numbered from 1 to n. Two consecutive numbered cards are removed from the pack and the sum of the numbers on the remaining cards is 1224. If the smaller of the numbers on the removed cards is k, then k – 20 = [Adv. 2013] Let a1, a2, a3 .....a100 be an arithmetic progression witha1 = 3 and Sp =

eb oo ks .in

30.

35.

, let

If

w

. For any integer n with

w

does not depend on n, then a2 is

36.

[2011] Let Sk, k = 1, 2, ….. , 100, denote the sum of the infinite geometric series whose first term is

and the common ratio is

. Then the

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value of

+

is [2010]

eb oo ks .in

37. Let a1,a2,a3........, a11 be real numbers satisfying a1=15, 27–2a2> 0 and ak=2ak–1–ak–2 for k = 3, 4,..........11. then the value of

if

is equal to

[2010]

38.

The number of terms common to the two A.P.’s 3, 7, 11, ..., 407 and 2, 9, 16, ..., 709 is _____. [Main Jan. 9, 2020 (II)]

Let p and q be roots of the equation x2 – 2x + A = 0 and let r and s be the roots of the equation x2 – 18x + B = 0. Ifp < q < r < s are in arithmetic progression, then A = .............. and B = .............. [1997 - 2 Marks] 40. The sum of integers from 1 to 100 that are divisible by 2 or 5 is .............. [1984 - 2 Marks]

w .je

39.

41.

Then Sn can take value(s) [Adv. 2013]

1056 1088 1120 1332 Let Tr be the rth term of an A.P., for r = 1, 2, 3, .... If for some positive integers m, n we have

w

w

(a) (b) (c) (d) 42.

Let

Tm =

and Tn =

, then Tmn equals

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[1998 - 2 Marks]

(b) (c) 1 (d) 0

eb oo ks .in

(a)

Let Vr denote the sum of first r terms of an arithmetic progression (A.P.) whose first term is r and the common difference is (2r – 1). Let Tr = Vr + 1 – Vr – 2 and Qr = Tr + 1 – Tr for r = 1, 2, ... 43. The sum V1 + V2 + ... + Vn is [2007 -4 marks] (a) (b)

w .je

(c) (d)

Tr is always an odd number an even number a prime number a composite number Which one of the following is a correct statement ? Q1, Q2, Q3, ... are in A.P. with common difference 5 Q1, Q2, Q3, ... are in A.P. with common difference 6 Q1, Q2, Q3, ... are in A.P. with common difference 11 Q1= Q2 = Q3 = ....

w

w

44. (a) (b) (c) (d) 45. (a) (b) (c) (d)

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The fourth power of the common difference of an arithmatic progression with integer entries is added to the product of any four consecutive terms of it. Prove that the resulting sum is the square of an integer. [2000 - 4 Marks] 47. The real numbers x1, x2, x3 satisfying the equation are in AP. Find the intervals in which β and γ lie.

eb oo ks .in

46.

[1996 - 3 Marks]

48.

If

,

, and

are in arithmetic

progression, determine the value of x.

w .je

49.

[1990 - 4 Marks] The interior angles of a polygon are in arithmetic progression. The smallest angle is 120°, and the common difference is 5°, Find the number of sides of the polygon. [1980]

1.

If f (x + y) = f (x) f (y) and

of all natural numbers, then the value of

, x, y ∈ N, where N is the set is : [Main Sep. 06, 2020 (I)]

w

(a)

w

(b) (c)

(d)

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Let a, b, c, d and p be any non zero distinct real numbers such that (a2 + b2 + c2)p2 – 2 (ab + bc + cd)p + (b2 + c2 + d2) = 0. Then : [Main Sep. 06, 2020 (I)] (a) a, c, p are in A.P. (b) a, c, p are in G.P. (c) a, b, c, d are in G.P.

eb oo ks .in

2.

(d) a, b, c, d are in A.P. 3. If 210 + 29⋅31 + 28⋅32 + . . . . + 2×39 + 310 = S – 211 then S is equal to: [Main Sep. 05, 2020 (I)] (a) 311 – 212 (b) 311 (c)

(d) 2⋅311 4. If the sum of the second, third and fourth terms of a positive term G.P. is 3 and the sum of its sixth, seventh and eighth terms is 243, then the sum of the first 50 terms of this G.P. is: [Main Sep. 05, 2020 (II)]

w .je

(a) (b) (c)

w

(d)

w

5.

Let α and β be the roots of

and γ and δ be the roots of

If α, β, γ, δ form a geometric progression. Then ratio

(2q + p) : (2q – p) is : [Main Sep. 04, 2020 (I)]

(a) 3 : 1

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(b) (c) (d) 7.

If series

eb oo ks .in

(b) 9 : 7 (c) 5 : 3 (d) 33 : 31 6. The sum of the first three terms of a G.P. is S and their product is 27. Then all such S lie in : [Main Sep. 02, 2020 (I)] (a)

and

then the sum to infinity of the following

[Main Sep. 02, 2020 (I)] is :

(a)

w .je

(b) (c)

(d)

Let S be the sum of the first 9 terms of the series :

w

w

8.

(a) – 5

where

and

If

then k is equal to : [Main Sep. 02, 2020 (II)]

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(b) 1 (c) – 3 (d) 3 The product

is equal to:

eb oo ks .in

9.

[Main Jan. 9, 2020 (I)]

(a) (b)

(c) 1 (d) 2 10. Let an be the nth term of a G.P. of positive terms. and

If

is equal to :

[Main Jan. 9, 2020 (II)]

300 225 175 150

11.

If

w .je

(a) (b) (c) (d)

tan2n θ and

for 0 < θ
An, for all n ≥ p is [Main Online April 15, 2018]

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5 7 11 9 If the 2nd, 5th and 9th terms of a non-constant A.P. are in G.P., then the common ratio of this G.P. is : [Main 2016] (a) 1 (b)

(c) (d) 25.

eb oo ks .in

(a) (b) (c) (d) 24.

Let z = 1 + ai be a complex number, a > 0, such that z3 is areal number. Then the sum 1 + z + z2 + .... + z11 is equal to: [Main Online April 10, 2016]

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26.

[Main 2015]

4 lmn2 4 l 2 m 2 n2 4 l2 mn 4 lm2n The sum of the 3rd and the 4th terms of a G.P. is 60 and the product of its first three terms is 1000. If the first term of this G.P. is positive, then its 7th term is :

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If m is the A.M. of two distinct real numbers l and n(l, n > 1) and G1, G2 and G3 are three geometric means between l and n, then equals.

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[Main Online April 11, 2015] 7290 640 2430 320 Three positive numbers form an increasing G. P. If the middle term in this G.P. is doubled, the new numbers are in A.P. then the common ratio of the G.P. is: [Main 2014]

(a) (b) (c) (d) 29.

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(a) (b) (c) (d) 28.

The least positive integer n such that , is:

[Main Online April 12, 2014]

4 5 6 7

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(a) (b) (c) (d) 30.

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In the quadratic equation ax2 + bx + c = 0, ∆ = b2 – 4ac and α + β, α2 + β2, α3 + β3, are in G.P. where α, β are the root of ax2 + bx + c = 0, then [2005S] ∆≠0 b∆ = 0 c∆ = 0 ∆=0 An infinite G.P. has first term ‘x’ and sum ‘5’, then x belongs to [2004S] x < – 10

(a)

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– 10 < x < 0 0 < x < 10 x > 10 Suppose a, b, c are in A.P. and a2, b2, c2 are in G.P.if a < b < c and a + b+c=

, then the value of a is

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(b) (c) (d) 32.

[2002S]

(a) (b) (c) 33.

Let α, β be the roots of x2 − x + p = 0 and γ, δ be the roots of x2 − 4x + q = 0. If α, β, γ, δ are in G.P., then the integral values of p and q respectively, are [2001S] –2, –32 –2, 3 –6, 3 –6, –32 Consider an infinite geometric series with first term a and common ratio r. If its sum is 4 and the second term is 3/4, then [2000S]

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(a) (b) (c) (d) 34.

(d)

(b)

(c)

(d) a = 3, r =

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35.

Sum of the first n terms of the series is equal to [1988 - 2 Marks]

(a) (b)

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(c) n + (d) 36.

–1 + 1. and

If a, b, c are in G.P., then the equations are in ––

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have a common root if

[1985 - 2 Marks]

(a) (b) (c) (d) 37.

A.P. G.P. H.P. none of these The rational number, which equals the number decimal is

with recurring

[1983 - 1 Mark]

(a) (b) (c)

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(d) none of these 38. The third term of a geometric progression is 4. The product of the first five terms is [1982 - 2 Marks] 3 (a) 4 (b) 45 (c) 44 (d) none of these

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Let a1, a2, a3, ... be a sequence of positive integers in arithmetic progression with common difference 2. Also, let b1, b2, b3, ... be a sequence of positive integers in geometric progression with common

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ratio 2. If a1 = b1 = c, then the number of all possible values of c, for which the equality

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holds for some positive integer n, is _____ [Adv. 2020]

40.

Suppose that a function f : R→R satisfies f (x + y) = f(x)f(y) for all x, and f (a) = 3. If

, then n is equal to ______.

[Main Sep. 06, 2020 (II)]

41.

is equal to

The value of _________ .

[Main Sep. 03, 2020 (I)]

is an integer. If a, b, c are

42. Let a, b, c be positive integers such that

in geometric progression and the arithmetic mean of a, b, c is b + 2,

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then the value of

43.

is [Adv. 2020]

Let n be an odd integer. If sin nθ =

br sinr θ, for every value of θ,

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then

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(a) (b) (c) (d) 44.

[1998 - 2 Marks]

b0 = 1, b1 = 3 b0 = 0, b1 = n b0 = –1, b1 = n b0 = 0, b1 = n2 – 3n + 3 if For

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then: (a) (b) (c) (d)

45.

If

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[1993 - 2 Marks]

and

bn = 1 – an, then find the least natural number n0 such that .

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46.

[2006 - 6M] Let a, b, c, d be real numbers in G.P. If u, v, w, satisfy the system of equations [1999 - 10 Marks]

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then show that the roots of the equation

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and 20x2 + 10 (a - d)2 x – 9 = 0 are reciprocals of each other. ............... , are the sums of infinite geometric series 47. If whose first terms are 1, 2, 3, ..............., n and whose common ratios are , ...............

respectively, then find the values of

+ ............... +

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1.

eb oo ks .in

[1991 - 4 Marks] 48. Find three numbers a, b, c, between 2 and 18 such that (i) their sum is 25 (ii) the numbers 2, a, b sare consecutive terms of an A.P. and (iii) the numbers b, c, 18 are consecutive terms of a G.P. [1983 - 2 Marks] 49. Does there exist a geometric progression containing 27, 8 and 12 as three of its terms ? If it exits, how many such progressions are possible ? [1982 - 3 Marks]

If the arithmetic mean of two numbers a and b, a > b > 0, is five times their geometric mean, then

is equal to :

[Main Online April 8, 2017]

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(a) (b) (c)

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Let x, y, z be positive real numbers such that x + y + z = 12 and x3y4z5 = (0.1) (600)3. Then x3 + y3 + z3 is equal to : [Main Online April 9, 2016] (a) 342 (b) 216

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(c) 258 (d) 270 3. Let a1, a2, a3, ..... be in harmonic progression with a1 = 5 and a20 = 25. The least positive integer n for which an < 0 is [2012] (a) 22 (b) 23 (c) 24 (d) 25 4. Let the positive numbers a, b, c, d be in A.P. Then abc, abd, acd, bcd are [2001S] (a) NOT in A.P./G.P./H.P. (b) in A.P. (c) in G.P. (d) in H.P. 5. The harmonic mean of the roots of the equation is [1999 - 2 Marks]

2 4 6 8 Let a1, a2, ..... a10 be in A, P, and h1, h2,....h10 be in H.P. Ifa1 = h1 = 2 and a10 = h10 = 3, then a4h7 is [1999 - 2 Marks] 2 3 5 6 7. If ln(a + c), ln (a – c), ln (a – 2b + c) are in A.P., then [1994] a, b, c are in A.P. a2, b2, c2 are in A.P. a, b, c are in G.P. a, b, c are in H.P.

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(a) (b) (c) (d) 6.

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(a) (b) (c) (d)

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8.

Let m be the minimum possible value of

where

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y1, y2, y3 are real numbers for which y1 + y2 + y3 = 9. Let M be the where x1, x2, maximum possible value of x3 are positive real numbers for which x1 + x2 + x3 = 9. Then the value is _____ [Adv. 2020] of

If m arithmetic means (A.Ms) and three geometric means (G.Ms) are inserted between 3 and 243 such that 4th A.M. is equal to 2nd G.M., then m is equal to ___________. [Main Sep. 03, 2020 (II)]

10.

Let the harmonic mean and geometric mean of two positive numbers be the ratio 4 : 5. Then the two number are in the ratio .............. [1992 - 2 Marks]

11.

A straight line through the vertex P of a triangle PQR intersects the side QR at the point S and the circumcircle of the triangle PQR at the point T. If S is not the centre of the circumcircle, then [2008]

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9.

(b)

(a)

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(d)

12.

If x > 1, y > 1, z > 1 are in G.P., then

are in [1998 - 2 Marks]

(a) A.P.

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(b) H.P. (c) G..P. (d) None of these 13. If the first and the (2n – 1)st terms of an A.P., a G.P. and an H.P. are equal and their n-th terms are a, b and c respectively, then [1988 - 2 Marks] (a) a = b = c (b) a b c (c) a + c = b (d)

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Let A1, G1, H1 denote the arithmetic, geometric and harmonic means, respectively, of two distinct positive numbers. For n ≥ 2, Let An – 1 and Hn – 1 have arithmetic, geometric and harmonic means as An, Gn, Hn respectively. [2007 -4 marks] 14. Which one of the following statements is correct ? (a) G1 > G2 > G3 > ... (b) G1 < G2 < G3 < ... (c) G1 = G2 = G3 = ... (d) G1 < G3 < G5 < ... and G2 > G4 > G6 > .... 15. Which one of the following statements is correct ? (a) A1 > A2 > A3 > ... (b) A1 < A2 < A3 < ... (c) A1 > A3 > A5 > ... and A2 < A4 < A6 < ... (d) A1 < A3 < A5 < ... and A2 > A4 > A6 > ... 16. Which one of the following statements is correct ? (a) H1 > H2 > H3 > ... (b) H1 < H2 < H3 < ... (c) H1 > H3 > H5 > ... and H2 < H4 < H6 < ... (d) H1 < H3 < H6 < ... and H2 > H4 > H6 > ...

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Suppose four distinct positive numbers a1, a2, a3, a4 are in G.P. Let b1 = a1, b2 = b1 + a2, b3 = b2 + a3 and b4 = b3 + a4. STATEMENT - 1 : The numbers b1, b2, b3, b4 are neither in A.P. nor in G.P. and STATEMENT - 2 : The numbers b1, b2, b3, b4 are in H.P. [2008] (a) STATEMENT - 1 is True, STATEMENT - 2 is True; STATEMENT - 2 is a correct explanation for STATEMENT - 1 (b) STATEMENT - 1 is True, STATEMENT - 2 is True; STATEMENT - 2 is NOT a correct explanation for STATEMENT - 1 (c) STATEMENT - 1 is True, STATEMENT - 2 is False (d) STATEMENT - 1 is False, STATEMENT - 2 is True 18.

If a, b, c are in A.P., a2, b2, c2 are in H.P., then prove that either a = b = c or a, b,

form a G.P.

[2003 - 4 Marks] Let a, b be positive real numbers. If a, A1, A2, b are in arithmetic progression, a, G1, G2, b are in geometric progression and a, H1, H2, b are in harmonic progression, show that

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19.

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17.

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[2002 - 5 Marks] 20. Let a1, a2, …, an be positive real numbers in geometric progression. For each n, let An, Gn, Hn be respectively, the arithmetic mean, geometric mean, and harmonic mean ofa1, a2, …, an. Find an expression for the geometric mean of G1, G2, …, Gn in terms of A1, A2 , …, An, H1, H2, …, Hn. [2001 - 5 Marks] 21. Let p be the first of the n arithmetic means between two numbers and q the first of n harmonic means between the same numbers. Show that q does not lie between p and

.

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[1991 - 4 Marks] If a > 0, b > 0 and c > 0, prove that

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22.

[1984 - 2 Marks] 23. The harmonic mean of two numbers is 4. Their arithmetic mean A and the geometric mean G satisfy the relation. 2A + G2 = 27 Find the two numbers. [1979]

1.

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(a) (b) (c) (d) 2.

If 1 + (1 – 22 ⋅ 1) + (1 – 42 ⋅ 3) + (1 – 62 ⋅ 5) + . . . . . . . + (1 – 202 ⋅ 19) then an ordered pair is equal to : [Main Sep. 04, 2020 (I)] (10, 97) (11, 103) (10, 103) (11, 97) be a function which satisfies Let If f (a) = 2 and , then the value of n, for which g(n) = 20, is :

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(a) (b) (c) (d) 3.

5 20 4 9 If the sum of the first 40 terms of the series, 3 + 4 + 8 + 9 + 13 + 14 + 18 +19 + ... is (102)m, then m is equal to:

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[Main Jan. 7, 2020 (II)] 20 25 5 10 For x Î R, let [x] denote the greatest integer < x, then the sum of the series

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(a) (b) (c) (d) 4.

is

[Main April 12, 2019 (I)]

(a) (b) (c) (d) 5.

–153 –133 –131 –135 The sum is :

…... upto 10th term,

[Main April 10, 2019 (I)]

680 600 660 620 The sum of the series 1 + 2 × 3 + 3 × 5 + 4 × 7 + ….. upto 11th term is: [Main April 09, 2019 (II)] 915 946 945 916

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(a) (b) (c) (d) 6.

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The sum

is equal to : [Main April 08, 2019 (II)]

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(a) 2 –

(c) 2 – (d) 2 – 8.

Let Sk =

If

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(b) 1 –

Then A is equal to

[Main Jan. 12, 2019 (I)]

283 301 303 156 The sum of the following series [Main Jan. 09, 2019 (II)]

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(a) (b) (c) (d) 9.

+... up to 15 terms, is:

7520 7510 7830 7820 The sum of the first 20 terms of the series

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is? [Main Online April 16, 2018]

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(a)

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(b) (c) (d) 11.

Let A be the sum of the first 20 terms and B be the sum of the first 40 terms of the series

If

, then λ is equal to :

[Main 2018]

248 464 496 232 Let a, b, c ∈ R. If f(x) = ax2 + bx + c is such that a + b + c = 3 and f(x

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(a) (b) (c) (d) 12.

+ y) = f(x) + f(y) + xy,

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13.

is equal to : [Main 2017]

255 330 165 190

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(a) (b) (c) (d)

x, y ∈ R, then

Let Sn =

.....

, If 100

Sn = n, then n is equal to :

(a) 199

[Main Online April 9, 2017]

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the

sum

of

the

first

ten

terms

of

the

series

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(b) 99 (c) 200 (d) 19 14. If

is

then m is equal to :

[Main 2016]

(a) (b) (c) (d) 15.

100 99 102 101 For x2016 =

, if (1 + x)2016 + x (1 + x)2015 + x2(1 + x)2014 + .... +

, then a17 is equal to :

[Main Online April 9, 2016]

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(a) (b) (c)

(d)

The sum of first 9 terms of the series. [Main 2015]

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16.

(a) 142 (b) 192

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(c) 71 (d) 96 is equal to :

The value of

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17.

[Main Online April 10, 2015]

(a) (b) (c) (d)

7770 7785 7775 7780

18.

If

then k is equal to:

[Main 2014]

(a) 100 (b) 110 (c)

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(d)

19.

The number of terms in an A.P. is even; the sum of the odd terms in it is 24 and that the even terms is 30. If the last term exceeds the first term by

[Main Online April 19, 2014]

4 8 12 16 The sum of first 20 terms of the sequence 0.7, 0.77, 0.777,....., is [Main 2013]

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(a) (b) (c) (d) 20.

, then the number of terms in the A.P. is:

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(a)

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(b) (c) (d) 21. (a) (b) (c) (d)

The sum,

is equal to __________.

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22.

The value of l2 + 32 + 52 + .......................+ 252 is : [Main Online April 25, 2013] 2925 1469 1728 1456

23.

The sum

For any odd integer n

25.

The

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sum

of

[Main Jan. 8, 2020 (II)]

is _______. [Main Jan. 8, 2020 (I)]

1, n3– (n–1)3+...+(–1)n–1 13 = .............. [1996 - 1 Mark] the first n terms of the series

when n is even. When n is odd, the sum is .............. [1988 - 2 Marks]

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Let α and β be the roots of x2 – x – 1 = 0, with α > β. For all positive integers n, define

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26.

and Then which of the following options is/are correct ?

[Adv. 2019]

(a)

(b) bn = αn + βn for all n ≥ 1 (c) a1 + a2 + a3 + ….. an = an+2 – 1 for all n ≥ 1 (d) 27.

For a positive integer n, let

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a (n) = 1 +

100 a (100) a (100) > 100 a (200) 100 a (200) > 100

28.

Find the sum of the series :

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(a) (b) (c) (d)

. Then

[1999 - 3 Marks]

[1985 - 5 Marks]

29. If n is a natural number such that and p1, p2, ....., pk are distinct primes, then show that ln n ≥ k ln2

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[1984 - 2 Marks]

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1.

A triangle ABC lying in the first quadrant has two vertices as A(1, 2) and ar sq. units, then the and B(3, 1). If abscissa of the vertex C is:

[Main Sep. 04, 2020 (I)]

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(a) (b) (c)

(d)

If the perpendicular bisector of the line segment joining the points P (1, 4) and Q (k, 3) has y-intercept equal to – 4, then a value of k is : [Main Sep. 04, 2020 (II)] (a) – 2 (b) – 4

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2.

(c)

(d)

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If a ∆ABC has vertices A(–1, 7), B(–7, 1) and C(5, –5), then its orthocentre has coordinates : [Main Sep. 03, 2020 (II)]

(a) (b) (–3, 3) (c)

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3.

(d) (3, –3) 4. A triangle has a vertex at (1, 2) and the mid points of the two sides through it are (–1, 1) and (2, 3). Then the centroid of this triangle is : [Main April 12, 2019 (II)] (a) (b)

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(c) (d) 5.

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Two vertices of a triangle are (0, 2) and (4, 3). If its orthocentre is at the origin, then its third vertex lies in which quadrant? [Main Jan. 10, 2019 (II)] third second first fourth Let the orthocentre and centroid of a triangle be A(–3, 5) and B(3, 3) respectively. If C is the circumcentre of this triangle, then the radius of the circle having line segment AC as diameter, is :

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[Main 2018] (a)

(c) (d) 7.

A square, of each side 2, lies above the x-axis and has one vertex at the origin. If one of the sides passing through the origin makes an angle 30° with the positive direction of the x-axis, then the sum of the x-coordinates of the vertices of the square is : [Main Online April 9, 2017]

(a) (b) (c) (d)

Let L be the line passing through the point P(1, 2) such that its intercepted segment between the co-ordinate axes is bisected at P. If L1 is the line perpendicular to L and passing through the point (–2, 1), then the point of intersection of L and L1 is : [Main Online April 10, 2015]

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(d)

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9.

The x-coordinate of the incentre of the triangle that has the coordinates of mid points of its sides as (0, 1) (1, 1) and(1, 0) is : [Main 2013]

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(a) (b) (c) (d) 10.

A light ray emerging from the point source placed at P( l, 3) is reflected at a point Q in the axis of x. If the reflected ray passes through the point R (6, 7), then the abscissa of Q is: [Main Online April 9, 2013]

(a) 1 (b) 3 (c) (d)

Let O(0, 0), P(3, 4), Q(6, 0) be the vertices of the triangles OPQ. The point R inside the triangle OPQ is such that the triangles OPR, PQR, OQR are of equal area. The coordinates of R are [2007 -3 marks]

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11.

(a)

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(d)

Area of the triangle formed by the line x + y = 3 and angle bisectors of the pair of straight lines x2 – y2 + 2y = 1 is [2004S] (a) 2 sq. units

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(b) 4 sq. units (c) 6 sq. units (d) 8 sq. units 13. Orthocentre of triangle with vertices (0, 0), (3, 4) and (4, 0) is (a)

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[2003S]

(b) (3, 12)

(c)

(d) (3, 9)

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14. The number of intergral points (integral point means both the coordinates should be integer) exactly in the interior of the triangle with vertices (0,0), (0,21) and (21,0), is [2003S] (a) 133 (b) 190 (c) 233 (d) 105 15. A straight line through the origin O meets the parallel lines 4x + 2y = 9 and 2x + y + 6 = 0 at points P and Q respectively. Then the point O divides the segemnt PQ in the ratio [2002S] (a) 1 : 2 (b) 3 : 4 (c) 2 : 1 (d) 4 : 3 16. Area of the parallelogram formed by the lines y = mx,y = mx + 1, y = nx and y = nx + 1 equals [2001S] 2 (a) |m + n|/(m − n) (b) 2/|m + n| (c) 1/(|m + n|) (d) 1/(|m − n|) 17. The number of integer values of m, for which the x-coordinate of the point of intersection of the lines 3x + 4y = 9 and y = mx + 1 is also an integer, is [2001S] (a) 2 (b) 0 (c) 4

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(d) 1 18.

The incentre of the triangle with vertices

, (0, 0) and (2, 0) is

(a) (b) (c) (d) 19.

If x1, x2, x3 as well as y1, y2, y3, are in G.P. with the same common ratio, then the points (x1, y1), (x2, y2) and (x3, y3). [1999 - 2 Marks] lie on a straight line lie on an ellipse lie on a circle are vertices of a triangle The orthocentre of the triangle formed by the lines xy = 0 and x + y = 1 is [1995S]

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(a) (b) (c) (d) 20.

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(b)

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21.

The locus of a variable point whose distance from (–2, 0) is 2/3 times its distance from the line x =

is [1994]

(a) (b) (c) (d) 23.

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(a) (b) (c) (d) 24.

ellipse parabola hyperbola none of these If the sum of the distances of a point from two perpendicular lines in a plane is 1, then its locus is [1992 - 2 Marks] square circle straight line two intersecting lines If P = (1, 0), Q = (–1, 0) and R = (2, 0) are three given points, then locus of the point S satisfying the relation SQ2 + SR2 = 2SP2, is [1988 - 2 Marks] a straight line parallel to x-axis a circle passing through the origin a circle with the centre at the origin a straigth line parallel to y-axis. The straight lines x + y = 0, 3x + y – 4 = 0, x + 3y – 4 = 0 form a triangle which is [1983 - 1 Mark] isosceles equilateral right angled none of these The point (4, 1) undergoes the following three transformations successively. [1980] Reflection about the line y = x. Translation through a distance 2 units along the positive direction of x-axis.

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(a) (b) (c) (d) 22.

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(i) (ii)

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(iii) Rotation through an angle p/4 about the origin in the counter clockwise direction. Then the final position of the point is given by the coordinates.

(b) (–

,7

)

(c) (d) (

26.

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)

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(a)

Let A(l, 0), B(6, 2) and C

be the vertices of a triangle ABC.

If P is a point inside the triangle ABC such that the triangles APC, APB and BPC have equal areas, then the length of the line segment

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PQ, where Q is the point

, is



[Main Jan. 7, 2020 (I)]

The vertices of a triangle are A (–1, –7), B (5, 1) and C (1, 4). The is ................... equation of the bisector of the angle [1993 - 2 Marks] 28. The orthocentre of the triangle formed by the lines x + y = 1, 2x + 3y = 6 and 4x – y + 4 = 0 lies in quadrant number ................... . [1985 - 2 Marks] 29. Given the points A (0, 4) and B (0, –4), the equation of the locus of the point P(x, y) such that | AP – BP | = 6 is ................... [1983 - 1 Mark] 30. The area enclosed within the curve | x | + | y | = 1 is ...................

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[1981 - 2 Marks] The straight line 5x + 4y = 0 passes through the point of intersection of the straight lines x + 2y – 10 = 0 and 2x + y + 5 = 0. [1983 - 1 Mark]

32.

(a) (b) (c) (d) (e)

The diagonals of a parallelogram PQRS are along the linesx +3y = 4 and 6x – 2y = 7. Then PQRS must be a. [1998 - 2 Marks] rectangle square cyclic quadrilateral rhombus. If (P(1, 2), Q(4, 6), R(5, 7) and S(a, b) are the vertices of a parallelogram PQRS, then [1998 - 2 Marks] a = 2, b = 4 a = 3, b = 4 a = 2, b = 3 a = 3, b = 5 All points lying inside the triangle formed by the points(1, 3), (5, 0) and (–1, 2) satisfy [1986 - 2 Marks] 3x + 2y 0 2x + y – 13 0 2x – 3y – 12 0 –2x + y 0 none of these.

35.

The points

(a) (b) (c) (d) 33.

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(a) (b) (c) (d) 34.

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31.

(1, 3) and (82, 30) are vertices of

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(a) an obtuse angled triangle [1986 - 2 Marks]

(b) an acute angled triangle (c) a right angled triangle

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(d) an isosceles triangle (e) none of these.

37.

38.

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39.

The area of the triangle formed by the intersection of a line parallel to x-axis and passing through P (h, k) with the lines y = x and x + y = 2 is 4h2. Find the locus of the point P. [2005 - 2 Marks] A straight line L with negative slope passes through the point (8, 2) and cuts the positive coordinate axes at points P and Q. Find the absolute minimum value of OP + OQ, as L varies, where O is the origin. [2002 - 5 Marks] Let ABC and PQR be any two triangles in the same plane. Assume that the prependiculars from the points A, B, C to the sides QR, RP, PQ respectively are concurrent. Using vector methods or otherwise, prove that the prependiculars from P, Q, R to BC, CA, AB respectively are also concurrent. [2000 - 10 Marks] Using co-ordinate geometry, prove that the three altitudes of any triangle are concurrent. [1998 - 8 Marks] A rectangle PQRS has its side PQ parallel to the line y = mx and vertices P, Q and S on the lines y = a, x = b and x = –b, respectively. Find the locus of the vertex R. [1996 - 2 Marks] Tangent at a point P1 {other than (0, 0)} on the curve y = x3 meets the curve again at P2. The tangent at P2 meets the curve at P3, and so on. form a G.P. Also find Show that the abscissae of the ratio.

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36.

40.

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41.

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[area

[1993 - 5 Marks]

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42.

Determine all values of

for which the point

lies inside the

triangle formed by the lines 2x + 3y – 1 = 0

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[1992 - 6 Marks]

x + 2y – 3 = 0 5x – 6y – 1 = 0 43.

A line cuts the x-axis at A (7, 0) and the y-axis at B(0, – 5). A variable

line PQ is drawn perpendicular to AB cutting the x-axis in P and the y-axis in Q. If AQ and BP intersect at R, find the locus of R.

[1990 - 4 Marks]

44.

Two sides of a rhombus ABCD are parallel to the lines

and

. If the diagonals of the rhombus intersect at the point (1, 2) and the vertex A is on they-axis, find possible co-ordinates of A.

[1985 - 5 Marks]

45.

One of the diameters of the circle circumscribing the rectangle ABCD . If A and B are the points (–3, 4) and (5, 4) respectively,

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is

then find the area of rectangle.

[1985 - 3 Marks] 46. The coordinates of A, B, C are (6, 3), (–3, 5), (4, – 2) respectively, and P is any point (x, y). Show that the ratio of the area of the triangles

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∆PBC and ∆ABC is

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[1983 - 2 Marks] 47. The vertices of a triangle are [at1t2, a(t1 + t2)],[at2t3, a(t2 + t3)], [at3t1, a(t3 + t1) ]. Find the orthocentre of the triangle. [1983 - 3 Marks] 48. (a) Two vertices of a triangle are (5, –1) and (–2, 3). If the orthocentre of the triangle is the origin, find the coordinates of the third point.

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1.

Let C be the centroid of the triangle with vertices (3, –1), (1, 3) and (2, 4). Let P be the point of intersection of the lines x + 3y – 1 = 0 and 3x – y + 1 = 0. Then the line passing through the points C and P also passes through the point: [Main Jan. 9, 2020 (I)] (–9, –6) (9, 7) (7, 6) (–9, –7) Slope of a line passing through P(2, 3) and intersecting the line x + y = 7 at a distance of 4 units from P, is: [Main April 9, 2019 (I)]

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(a) (b) (c) (d) 2.

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(b) Find the equation of the line which bisects the obtuse angle between the lines x – 2y + 4 = 0 and 4x – 3y + 2 = 0. [1979] 49. The area of a triangle is 5. Two of its vertices are A (2, 1) and B (3, – 2). The third vertex C lies on y = x + 3. Find C. [1978] moves with its ends on two 50. A straight line segment of length mutually perpendicular lines. Find the locus of the point which divides the line segment in the ratio 1 : 2. [1978]

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(a)

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(b) (c)

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(d)

(a) (b) (c) (d) 4.

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(a) (b) (c) (d) 5.

A point on the straight line, 3x + 5y = 15 which is equidistant from the coordinate axes will lie only in : [Main April 8, 2019 (I)] th 4 quadrant 1st quadrant 1st and 2nd quadrants 1st, 2nd and 4th quadrants If a straight line passing through the point P(–3, 4) is such that its intercepted portion between the coordinate axes is bisected at P, then its equation is : [Main Jan. 12, 2019 (II)] 3x – 4y + 25 = 0 4x – 3y + 24 = 0 x–y+7=0 4x + 3y = 0 If in a parallelogram ABDC, the coordinates of A, B and C are respectively (1, 2), (3, 4) and (2, 5), then the equation of the diagonal AD is : [Main Jan. 11, 2019 (II)] 5x – 3y + 1 = 0 5x + 3y – 11 = 0 3x – 5y + 7 = 0 3x + 5y – 13 = 0 A point P moves on the line 2x – 3y + 4 = 0. If Q(1, 4) and R(3, –2) are fixed points, then the locus of the centroid of DPQR is a line: [Main Jan. 10, 2019 (I)]

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3.

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(a) (b) (c) (d) 6.

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(a) with slope

(b) parallel to x-axis (c) with slope

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(d) parallel to y-axis 7. A straight line through a fixed point (2, 3) intersects the coordinate axes at distinct points P and Q. If O is the origin and the rectangle OPRQ is completed, then the locus of R is : [Main 2018]

8.

In a triangle ABC, coordinates of A are (1, 2) and the equations of the medians through B and C are x + y = 5 and x = 4 respectively. Then area of ∆ ABC (in sq. units) is [Main Online April 15, 2018]

(a) (b) (c) (d) 9.

5 9 12 4 Two sides of a rhombus are along the lines, x – y + 1 = 0 and 7x – y – 5 = 0. If its diagonals intersect at (–1, –2), then which one of the following is a vertex of this rhombus? [Main 2016]

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(a)

(b)

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(c) (–3, –9) (d) (–3, –8) 10. A straight line through origin O meets the lines 3y = 10 – 4x and 8x + 6y + 5 = 0 at points A and B respectively. Then O divides the segment AB in the ratio : [Main Online April 10, 2016] (a) 2 : 3 (b) 1 : 2 (c) 4 : 1 (d) 3 : 4

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11.

The point (2, 1) is translated parallel to the line L : x – y = 4 by units. If the new points Q lies in the third quadrant, then the equation of the line passing through Q and perpendicular to L is : [Main Online April 9, 2016]

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(a) (b) (c) (d) 12.

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(a) (b) (c) (d) 13.

A straight line L through the point (3, – 2) is inclined at an angle of x + y = 1. If L also intersects the x-axis, then the 60° to the line equation of L is : [Main Online April 11, 2015] x+2–3 =0 y+ y+x–3+2 =0 x+2+3 =0 y– y–x+3+2 =0 Given three points P, Q, R with P(5, 3) and R lies on the x-axis. If equation of RQ is x – 2y = 2 and PQ is parallel to the x-axis, then the centroid of ∆PQR lies on the line [Main Online April 9, 2014] 2x + y – 9 = 0 x – 2y + 1 = 0 5x – 2y = 0 2x – 5y = 0 gets reflected upon reaching xA ray of light along x +

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(a) (b) (c) (d) 14.

axis, the equation of the reflected ray is [Main 2013]

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(a) y = x + (b)

=x–

(c) y =

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(d)

(a) (b) (c) (d) 16.

Let A (–3, 2) and B (–2, 1) be the vertices of a triangle ABC. If the centroid of this triangle lies on the line 3x + 4y + 2 = 0, then the vertex C lies on the line : [Main Online April 25, 2013] 4x + 3y + 5 = 0 3x + 4y + 3 = 0 4x + 3y + 3 = 0 3x + 4y + 5 = 0 A straight line L through the point (3, –2) is inclined at an angle 60° to If L also intersects thex-axis, then the equation of the line L is

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15.

[2011]

(a) (b) (c) (d)

Let PS be the median of the triangle with vertices P(2, 2),Q(6, –1) and R(7, 3). The equation of the line passing through (1,–1) and parallel to PS is [2000S] 2x – 9y – 7 = 0 2x – 9y – 11 = 0 2x + 9y – 11 = 0 2x + 9y + 7 = 0 The equations to a pair of opposite sides of parallelogram are x2 – 5x + 6 = 0 and y2 – 6y + 5 = 0, the equations to its diagonals are [1994] x + 4y = 13, y = 4x – 7 4x + y = 13, 4y = x – 7 4x + y = 13, y = 4x – 7 y – 4x = 13, y + 4x = 7

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17.

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(a) (b) (c) (d) 18.

(a) (b) (c) (d)

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Let ƒ: R

R be defined as

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19.

The value of λ for which ƒ” (0) exists, is ______.

[Main Sep. 06, 2020 (I)]

20.

Let L1 be a straight line passing through the origin and L2 be the straight line x +y = 1. If the intercepts made by the circle 0 on L1 and L2 are equal, then which of the following equations can represent L1?

[1999 - 3 Marks]

(b) x – y = 0 (d) x – 7y = 0

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(a) x + y = 0 (c) x + 7y = 0 21.

Let a, b, c be real numbers with a2 + b2 + c2 = 1. Show that equation

= 0

the

represents a

0) and A = (3, 2). Prove that the set of points in the first quadrant which are equidistant (with respect to the new distance) from O and A consists of the union of a line segment of finite length and an infinite

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[2001- 6 Marks] For points P = (x1, y1) and Q = (x2, y2) of theco-ordinate plane, a new distance d(P, Q) is defined byd(P, Q) = |x1 – x2| + |y1 – y2|. Let O = (0,

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straight line.

ray. Sketch this set in a labelled diagram.

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2x = 5.

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23.

[2000 - 10 Marks] Find the equation of the line passing through the point(2, 3) and making intercept of length 2 units between the lines y + 2x = 3 and y + [1991 - 4 Marks]

Straight lines 3x + 4y = 5 and 4x – 3y = 15 intersect at the point A. Points B and C are chosen on these two lines such that AB = AC. Determine the possible equations of the line BC passing through the point (1, 2). [1990 - 4 Marks] and + my + n = 0 intersect at the 25. Lines

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24.

with each other. Find the equation of a point P and make an angle line L different from L2 which passes through P and makes the same angle with L1.

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26.

[1988 - 5 Marks] Two equal sides of an isosceles triangle are given by the equations and and its third side passes through the

27.

point (1, –10). Determine the equation of the third side. [1984 - 4 Marks] A straight line L is perpendicular to the line 5x – y = 1. The area of the triangle formed by the line L and the coordinate axes is 5. Find the equation of the line L.

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[1980] 28. One side of a rectangle lies along the line 4x + 7y + 5 = 0. Two of its

1.

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vertices are (–3, 1) and (1, 1). Find the equations of the other three sides. [1978]

Let L denote the line in the xy-plane with x and y intercepts as 3 and 1 respectively. Then the image of the point (–1, – 4) in this line is: [Main Sep. 06, 2020 (II)]

(a) (b)

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(c) (d) 2.

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(a) (b) (c) (d) 3.

The locus of the mid-points of the perpendiculars drawn from points on the line, x = 2y to the line x = y is: [Main Jan. 7, 2020 (II)] 2x – 3y = 0 5x – 7y = 0 3x – 2y = 0 7x – 5y = 0 A straight line L at a distance of 4 units from the origin makes positive intercepts on the coordinate axes and the perpendicular from the origin to this line makes an angle of 60o with the line x + y = 0. Then an equation of the line L is:

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[Main April 12, 2019 (II)] (a) (c) (d) None of these 4.

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(b)

Lines are drawn parallel to the line 4x – 3y + 2 = 0, at a distance

from the origin. Then which one of the following points lies on any of these lines ? [Main April 10, 2019 (II)] (a) (b) (c)

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(d) 5.

If the two lines x + (a–1) y = 1 and 2x + a2y =1 (a ∈ R – {0,1}) are perpendicular, then the distance of their point of intersection from the origin is: [Main April 09, 2019 (II)]

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(a)

(b)

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(c)

(d)

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Suppose that the points (h, k), (1, 2) and (– 3, 4) lie on the line L1. If a line L2 passing through the points (h, k) and (4, 3) is perpendicular on L1, then equals : [Main April 08, 2019 (II)]

(a) (b) 0 (c) 3 (d) 7.

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6.

If the straight line, 2x – 3y + 17 = 0 is perpendicular to the line passing through the points (7, 17) and (15, β), then β equals : [Main Jan. 12, 2019 (I)]

(a) (b) –5 (c)

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(d) 5 8. Consider the set of all lines px + qy + r = 0 such that 3p + 2q + 4r = 0. Which one of the following statements is true? [Main Jan. 9, 2019 (I)] (a) The lines are concurrent at the point

.

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(b) Each line passes through the origin. (c) The lines are all parallel. (d) The lines are not concurrent. 9. The sides of a rhombus ABCD are parallel to the lines, x – y + 2 = 0 and 7x – y + 3 = 0. If the diagonals of the rhombus intersect at P(1, 2) and the vertex A (different from the origin) is on the y-axis, then the ordinate of A is [Main Online April 15, 2018] (a) 2

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(b)

(d) 10.

(a) (b) (c) (d) 11.

Let a, b, c and d be non-zero numbers. If the point of intersection of the lines 4ax + 2ay + c = 0 and 5bx + 2by + d =0 lies in the fourth quadrant and is equidistant from the two axes then [Main 2014] 3bc – 2ad = 0 3bc + 2ad = 0 2bc – 3ad = 0 2bc + 3ad = 0 Let PS be the median of the triangle verticesP(2, 2), Q(6, –1) and R(7, 3). The equation of the line passing through (1, –1) and parallel to PS is: [Main 2014] 4x + 7y + 3 = 0 2x – 9y – 11 = 0 4x – 7y – 11 = 0 2x + 9y + 7 = 0 If a line L is perpendicular to the line 5x – y = 1, and the area of the triangle formed by the line L and the coordinate axes is 5, then the distance of line L from the line x + 5y = 0 is: [Main Online April 19, 2014]

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(a) (b) (c) (d) 12.

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(c)

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(a)

(b)

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(c)

13.

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(d) The base of an equilateral triangle is along the line given by 3x + 4y = 9. If a vertex of the triangle is (1, 2), then the length of a side of the triangle is: [Main Online April 11, 2014]

(a) (b) (c) (d)

If the image of point P(2, 3) in a line L is Q(4, 5), then the image of point R(0, 0) in the same line is: [Main Online April 25, 2013] (2, 2) (4, 5) (3, 4) (7, 7)

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14.

(a) (b) (c) (d)

Let P = (-1, 0), Q = (0, 0) and R = (3,

) be three points. Then the

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15.

equation of the bisector of the angle PQR is

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(a)

(b)

[2002S]

(c)

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(d) Let

be fixed angle. If

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16.

,

then Q is obtained from P by

[2002S]

(a) clockwise rotation around origin through an angle α (b) anticlockwise rotation around origin through an angle α (c) reflection in the line through origin with slope tan α (d) reflection in the line through origin with slope tan (α 2) 17. Line L has intercepts a and b on the coordinate axes. When the axes are rotated through a given angle, keeping the origin fixed, the same line L has intercepts p and q, then [1990 - 2 Marks] (a) (b)

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(c) (d)

18.

[1979]

Collinear Vertices of a parallelogram Vertices of a rectangle None of these

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(a) (b) (c) (d)

The points (–a, – b), (0, 0), (a, b) and (a2, ab) are :

19.

For a point P in the plane, let d1(P) and d2(P) be the distance of the point P from the lines x – y = 0 and x + y = 0 respectively. The area of the region R consisting of all points P lying in the first quadrant of the is plane and satisfying

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[Adv. 2014]

If the line, 2x – y + 3 = 0 is at a distance

and

from the lines

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20.

4x – 2y + α = 0 and 6x – 3y + β = 0, respectively, then the sum of all possible value of α and β is ______. [Main Sep. 05, 2020 (I)]

Let the algebraic sum of the perpendicular distances from the points (2, 0), (0, 2) and (1, 1) to a variable straight line be zero; then the line passes through a fixed point whose cordinates are ................... . [1991 - 2 Marks] 22. If a, b and c are in A.P., then the straight line ax + by + c = 0 will always pass through a fixed point whose coordinates are ................... [1984 - 2 Marks] 23. The set of lines ax+by+c = 0, where 3a + 2b + 4c = 0 is concurrent at the point ................... [1982 - 2 Marks]

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21.

The straight line 5x + 4y = 0 passes through the point of intersection of the straight lines x + 2y – 10 = 0 and 2x + y + 5 = 0. [1983 - 1 Mark]

25.

For a > b > c > 0, the distance between (1, 1) and the point of intersection of the lines ax + by + c = 0 and bx + ay + c = 0 is less . Then than

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24.

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(a) (b) (c) (d)

[Adv. 2013]

a+b–c>0 a–b+c0 a+b–c 0. [Adv. 2013] 47. Length of chord PQ is (a) 7a (b) 5a (c) 2a (d) 3a 48. If chord PQ subtends an angle θ at the vertex of y2 = 4ax, then tan θ = (a)

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(b)

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(c) (d)

PASSAGE-3 Consider the circle x + y = 9 and the parabola y2 = 8x. They intersect at P and Q in the first and the fourth quadrants, respectively. Tangents to the curcle at P and Q intersect the x-axis at R and tangents to the parabola at P and Q intersect the x-axis at S. [2007 -4 marks] 49. The ratio of the areas of the triangles PQS and PQR is (a) (b) 1 : 2 (c) 1 : 4 (d) 1 : 8 50. The radius of the circumcircle of the triangle PRS is (a) 5 (b) (c) (d) 51. The radius of the incircle of the triangle PQR is (a) 4 (b) 3 2

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2

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(c)

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(d) 2 52.

Given : A circle, 2x2 + 2y2 = 5 and a parabola, y2 = Statement-1 : An equation of a common tangent to these curves is y . =

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Statement-2 : If the line, y = mx +

(m ≠ 0) is their common

tangent, then m satisfies m4 – 3m2 + 2 = 0.

(b) (c) (d) 53.

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(a)

[Main 2013] Statement-1 is true; Statement-2 is true; Statement-2 is a correct explanation for Statement-1. Statement-1 is true; Statement-2 is true; Statement-2 is not a correct explanation for Statement-1. Statement-1 is true; Statement-2 is false. Statement-1 is false; Statement-2 is true. Statement-1: The line x – 2y = 2 meets the parabola, y2 + 2x = 0 only at the point (– 2, – 2).

is tangent to the

Statement-2: The line

parabola, y2 = – 2x at the point

.

[Main Online April 22, 2013]

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(a) Statement-1 is true; Statement-2 is false. (b) Statement-1 is true; Statement-2 is true; Statement-2 is a correct explanation for statement-1. (c) Statement-1 is false; Statement-2 is true. (d) Statement-1 a true; Statement-2 is true; Statement-2 is not a correct explanation for statement-1. 54. STATEMENT-1 : The curve

is symmetric with

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respect to the line x = 1. because STATEMENT-2 : A parabola is symmetric about its axis [2007 -3 marks] (a) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1 (b) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1

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(c) Statement-1 is True, Statement-2 is False (d) Statement-1 is False, Statement-2 is True.

56.

57.

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58.

Tangent is drawn to parabola y2 – 2y – 4x + 5 = 0 at a point P which cuts the directrix at the point Q. A point R is such that it divides QP externally in the ratio 1/2 : 1. Find the locus of point R. [2004 - 4 Marks] Normals are drawn from the point P with slopes m1, m2, m3 to the parabola y2 = 4x. If locus of P with m1 m2 = α is a part of the parabola itself then find α. [2003 - 4 Marks] 2 Let C1 and C2 be respectively, the parabolas x = y – 1 andy2 = x – 1. Let P be any point on C1 and Q be any point on C2. Let P1 and Q1 be the reflections of P and Q, respectively, with respect to the line y = x. . Prove that P1 lies on C2, Q1 lies on C1 and Hence or otherwise determine points P0 and Q0 on the parabolas C1 for all pairs of points (P,Q) and C2 respectively such that with P on C1 and Q on C2. [2000 - 10 Marks] From a point A common tangents are drawn to the circlex2 + y2 = a2/2 and parabola y2 = 4ax. Find the area of the quadrilateral formed by the common tangents, the chord of contact of the circle and the chord of contact of the parabola. [1996 - 2 Marks] Points A, B and C lie on the parabola y2 = 4ax. The tangents to the parabola at A, B and C, taken in pairs, intersect at points P, Q and R. Determine the ratio of the areas of the triangles ABC and PQR. [1996 - 3 Marks] Show that the locus of a point that divides a chord of slope 2 of the parabola y2 = 4x internally in the ratio 1: 2 is a parabola. Find the vertex of this parabola. [1995 - 5 Marks]

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55.

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59.

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60.

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Through the vertex O of parabola y2 = 4x, chords OP and OQ are drawn at right angles to one another. Show that for all positions of P, PQ cuts the axis of the parabola at a fixed point. Also find the locus of the middle point of PQ. [1994 - 4 Marks] 62. Three normals are drawn from the point (c, 0) to the curvey2 = x. Show that c must be greater than 1/2. One normal is always the xaxis. Find c for which the other two normals are perpendicular to each other. [1991 - 4 Marks] 63. A is a point on the parabola y2 = 4ax. The normal at A cuts the parabola again at point B. If AB subtends a right angle at the vertex of the parabola. find the slope of AB. [1982 - 5 Marks] 64. Suppose that the normals drawn at three different points on the parabola y2 = 4x pass through the point (h, k). Show that h > 2. [1981 - 4 Marks]

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61.

1.

Which of the following points lies on the locus of the foot of perpendicular drawn upon any tangent to the ellipse,

=1

from any of its foci? [Main Sep. 06, 2020 (I)]

(a)

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(b)

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(c) (d) (1, 2) 2. If the normal at an end of a latus rectum of an ellipsepasses through an extermity of the minor axis, then the eccentricity e of the ellipse satisfies: [Main Sep. 06, 2020 (II)]

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3.

(a) (b) (c) (d) 4. (a) (b) (c) (d)

If the co-ordinates of two points A and B are

and

respectively and P is any point on the conic, 9x2 + 16y2 = 144, then PA + PB is equal to : [Main Sep. 05, 2020 (I)] 16 8 6 9 If the point P on the curve, 4x2 + 5y2 = 20 is farthest from the point Q(0, – 4), then PQ2 is equals to : [Main Sep. 05, 2020 (I)] 36 48 21 29 Let

be a given ellipse, length of whose latus

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5.

e4 + 2e2 – 1 = 0 e2 + e – 1 = 0 e4 + e2 – 1 = 0 e2 + 2e – 1 = 0

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(a) (b) (c) (d)

rectum is 10. If its eccentricity is the maximum value of the function, then a2 + b2 is equal to:

145 116 126 135 Let x = 4 be a directrix to an ellipse whose centre is at the origin and

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(a) (b) (c) (d) 6.

[Main Sep. 04, 2020 (I)]

its eccentricity is

. If P(1, β), β > 0 is a point on this ellipse, then

the equation of the normal to it at P is : [Main Sep. 04, 2020 (II)]

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4x – 3y = 2 8x – 2y = 5 7x – 4y = 1 4x – 2y = 1 has the same A hyperbola having the transverse axis of length foci as that of the ellipse 3x2 + 4y2 = 12, then this hyperbola does not pass through which of the following points? [Main Sep. 03, 2020 (I)]

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(a) (b) (c) (d) 7.

(a) (b) (c) (d)

Area (in sq. units) of the region outside

and inside the

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8.

ellipse

is : [Main Sep. 02, 2020 (I)]

(a) (b)

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(c) (d)

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9.

If e1 and e2 are the eccentricities of the ellipse, hyperbola,

and the

respectively and (e1, e2) is a point on the

ellipse, 15x2 + 3y2 = k, then k is equal to [Main Jan. 9, 2020 (I)]

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16 17 15 14 The length of the minor axis (along y-axis) of an ellipse in the

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(a) (b) (c) (d) 10.

standard form is

If this ellipse touches the line, x + 6y = 8; then

its eccentricity is: [Main Jan. 9, 2020 (II)] (a) (b) (c) (d) 11.

Let the line y = mx and the ellipse 2x2 + y2 = 1 intersect at a point P in the first quadrant. If the normal to this ellipse at P meets the coand (0,

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ordinate axes at

), then

is equal to:

[Main Jan. 8, 2020 (I)]

(a)

w

(b) (c)

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(d)

12.

If the distance between the foci of an ellipse is 6 and the distance between its directrices is 12, then the length of its latus rectum is:

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[Main Jan. 7, 2020 (I)]

(c) (d) 13.

eb oo ks .in

(a) (b)

If 3x + 4y = 12

is a tangent to the ellipse

= 1 for some a

R, then the distance between the foci of the ellipse is: [Main Jan. 7, 2020 (II)] (a) (b) 4 (c) (d) 14.

If the line x – 2y = 12 is tangent to the ellipse

, then the length of the latus rectum of the ellipse is :

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point

at the

[Main April 10, 2019 (I)]

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(a) 9 (b) (c) 5 (d) 15. If the tangent to the parabola y2 = x at a point (α, β), (β > 0) is also a tangent to the ellipse, x2 + 2y2 = 1, then α is equal to: [Main April 09, 2019 (II)] (a) (b) (c) (d)

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(a) (b) (c) (d)

In an ellipse, with centre at the origin, if the difference of the lengths ), of major axis and minor axis is 10 and one of the foci is at (0, 5 then the length of its latus rectum is: [Main April 08, 2019 (II)] 10 5 8 6

17.

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16.

If the tangent at a point on the ellipse

meets the

coordinate axes at A and B, and O is the origin, then the minimum area (in sq. units) of the triangle OAB is : [Main Online April 9, 2016] (a) (b) (c) 9 (d)

The area (in sq. units) of the quadrilateral formed by the tangents at

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18.

the end points of the latera recta to the ellipse

is [Main 2015]

(a)

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(b) 27 (c)

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(d) 18 19. The locus of the foot of perpendicular drawn from the centre of the ellipse x2 + 3y2 = 6 on any tangent to it is [Main 2014]

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(a)

(c) (d) 20.

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(b)

The equation of the circle passing through the foci of the ellipse = 1, and having centre at (0, 3) is

[Main 2013]

(a) (b) (c) (d) 21.

x + y – 6y – 7 = 0 x2 + y2 – 6y + 7 = 0 x2 + y2 – 6y – 5 = 0 x2 + y2 – 6y + 5 = 0 2

2

The ellipse

is inscribed in a rectangle R whose sides

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are parallel to the coordinate axes. Another ellipse E2 passing through the point (0, 4) circumscribes the rectangle R. The eccentricity of the ellipse E2 is [2012] (b)

(a)

(d)

The normal at a point P on the ellipse x2 + 4y2 = 16 meets the x - axis at Q. If M is the mid point of the line segment PQ, then the locus of M intersects the latus rectums of the given ellipse at the points [2009]

w

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22.

(c)

(a)

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(b)

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(c) (d) 23.

The line passing through the extremity A of the major axis and extremity B of the minor axis of the ellipse x2 + 9y2 = 9 meets its auxiliary circle at the point M. Then the area of the triangle with vertices at A, M and the origin O is [2009] (a) (b)

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(c) (d)

24.

The minimum area of triangle formed by the tangent to the & coordinate axes is [2005S]

(b)

sq. units

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(a) ab sq. units

sq. units

(c)

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(d)

If tangents are drawn to the ellipse x2 + 2y2 = 2, then the locus of the mid-point of the intercept made by the tangents between the coordinate axes is [2004S]

(a) (b) (c) (d) 26.

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25.

sq. units

The area of the quadrilateral formed by the tangents at the end points

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of latus rectum to the ellipse

is

[2003S]

(a) 27/4 sq. units (b) 9 sq. units (c) 27/2 sq. units (d) 27 sq. units 27. The radius of the circle passing through the foci of the ellipse

w

, and having its centre at (0, 3) is [1995S]

w

(a) 4 (b) 3 (c)

(d)

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28.

Let E be the ellipse

and C be the circlex2 + y2 = 9. Let P

and Q be the points (1, 2) and (2, 1) respectively. Then (a) (b) (c) (d)

29.

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[1994] Q lies inside C but outside E Q lies outside both C and E P lies inside both C and E P lies inside C but outside E

Suppose that the foci of the ellipse

= 1 are (f1, 0) and (f2, 0)

where f1 > 0 and f2 < 0. Let P1 and P2 be two parabolas with a common vertex at (0, 0) and with foci at (f1, 0) and (2f2, 0), respectively. Let T1 be a tangent to P1 which passes through (2f2, 0) and T2 be a tangent to P2 which passes through (f1, 0). If m1 is the slope of T1 and m2 is the is

slope of T2, then the value of

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[Adv. 2015] 30. A vertical line passing through the point (h, 0) intersects the ellipse

at the points P and Q. Let the tangents to the ellipse at P

and Q meet at the point R. If ∆(h) = area of the triangle PQR, ∆1

w

and ∆2

w

(a) (b) (c) (d)

then [Adv. 2013]

g(x) is continuous but not differentiable at a g(x) is differentiable on R g(x) is continuous but not differentiable at b g(x) is continuous and differentiable at either (a) or (b) but not both

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Consider two straight lines, each of which is tangent to both the circle x2 + y2 =

and the parabola y2 = 4x. Let these lines intersect at the

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31.

point Q. Consider the ellipse whose center is at the origin O (0, 0) and whose semi-major axis is OQ. If the length of the minor axis of this , then which of the following statement(s) is (are) ellipse is TRUE? [Adv. 2018] (a) For the ellipse, the eccentricity is

and the length of the latus

rectum is 1

(b) For the ellipse, the eccentricity is is

and the length of the latus rectum

(c) The area of the region bounded by the ellipse between the lines x = and x = 1 is

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(d) The area of the region bounded by the ellipse between the lines x = and x = 1 is

Let E1 and E2 be two ellipses whose centers are at the origin. The major axes of E1 and E2 lie along the x-axis and the y-axis, respectively. Let S be the circle x2 + (y – 1)2 = 2. The straight line x + y = 3 touches the curves S, E1 and E2 at P, Q and R respectively.

w

32.

Suppose that PQ = PR =

w

(π – 2)

. If e1 and e2 are the eccentricities of E1

and E2, respectively, then the correct expression(s) is (are)

[Adv. 2015]

(a)

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(b) e1e2 =

(d) e1e2 = 33.

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(c)

In a triangle ABC with fixed base BC, the vertex A moves such that .

w .je

If a, b and c denote the lengths of the sides of the triangle opposite to the angles A, B and C, respectively, then [2009] (a) b + c = 4a (b) b + c = 2a (c) locus of point A is an ellipse (d) locus of point A is a pair of straight lines 34. Let P(x1, y1) and Q(x2, y2), y1 < 0, y2 < 0, be the end points of the latus rectum of the ellipse x2 + 4y2 = 4. The equations of parabolas with latus rectum PQ are [2008] (a) (b) (c)

(d)

On the ellipse

w

35.

, the points at which the tangents are

parallel to the line 8x = 9y are

w

[1999 - 3 Marks]

(a)

(b)

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(c)

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(d) 36.

If P = (x, y), F1 = (3, 0), F2 = (–3, 0) and 16x2 + 25y2 = 400, then PF1 +PF2 equals [1998 - 2 Marks]

(a) (b) (c) (d) 37.

8 6 10 12 The number of values of c such that the straight liney = 4x + c touches the curve (x2/4) + y2 = 1 is [1998 - 2 Marks] 0 1 2 infinite.

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(a) (b) (c) (d)

PASSAGE-1 Let F1(x1, 0) and F2(x2, 0) for x1 < 0 and x2 > 0, be the foci of the ellipse . Suppose a parabola having vertex at the origin and focus at F2

w

w

intersects the ellipse at point M in the first quadrant and at point N in the fourth quadrant. 38. The orthocentre of the triangle F1MN is [Adv. 2016] (a)

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(b)

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(c) (d) 39.

(a) (b) (c) (d)

If the tangents to the ellipse at M and N meet at R and the normal to the parabola at M meets the x-axis at Q, then the ratio of area of the triangle MQR to area of the quadrilateral MF1NF2 is [Adv. 2016] 3:4 4:5 5:8 2:3 PASSAGE-2

Tangents are drawn from the point P(3, 4) to the ellipse

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touching the ellipse at points A and B.

[2010]

40. The coordinates of A and B are (a) (3, 0) and (0, 2) (b)

w

(c)

w

(d) (3,0) and 41.

The orthocenter of the triangle PAB is

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(a)

(b)

(c)

(d)

The equation of the locus of the point whose distances from the point P and the line AB are equal, is (a) 9x2 + y2 –6xy –54x –62y + 241 = 0

eb oo ks .in

42.

(b) x2 + 9y2 +6xy –54x +62y –241 = 0 (c) 9x2 +9y2 –6xy –54x –62y–241 = 0

(d) x2 + y2 – 2xy + 27x + 31y – 120 = 0 43.

Find the equation of the common tangent in 1st quadrant to the circle x2 + y2 = 16 and the ellipse

. Also find the length of the

intercept of the tangent between the coordinate axes.

[2005 - 4 Marks] 44. Prove that, in an ellipse, the perpendicular from a focus upon any tangent and the line joining the centre of the ellipse to the point of contact meet on the corresponding directrix. [2002 - 5 Marks] Let P be a point on the ellipse

w .je

45.

= 1, 0 < b < a. Let the line

w

w

parallel to y−axis passing through P meet the circlex2 + y2 = a2 at the point Q such that P and Q are on the same side of x−axis. For two positive real numbers r and s, find the locus of the point R on PQ such that PR : RQ = r : s as P varies over the ellipse. [2001 - 4 Marks] 46. Let ABC be an equilateral triangle inscribed in the circle x2 + y2 = a2. Suppose perpendiculars from A, B, C to the major axis of the ellipse meets the ellipse respectively, at P, Q, R. so that

P, Q, R lie on the same side of the major axis as A, B, C respectively. Prove that the normals to the ellipse drawn at the points P, Q and R are concurrent.

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[2000 - 7 Marks] ,

47. Find the co-ordinates of all the points P on the ellipse

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for which the area of the triangle PON is maximum, where O denotes the origin and N, the foot of the perpendicular from O to the tangent at P. [1999 - 10 Marks] 2 2 2 48. Consider the family of circles x + y = r , 2 < r < 5. If in the first quadrant, the common tangent to a circle of this family and the ellipse 4x2 + 25y2 = 100 meets the co-ordinate axes at A and B, then find the equation of the locus of the mid-point of AB. [1999 - 10 Marks] 2 2 49. A tangent to the ellipse x + 4y = 4 meets the ellipsex2 + 2y2 = 6 at P and Q. Prove that the tangents at P and Q of the ellipse x2 + 2y2 = 6 are at right angles. [1997 - 5 Marks] 50. Let ‘d’ be the perpendicular distance from the centre of the ellipse to the tangent drawn at a point P on the ellipse. If F1 and

are

the

two

foci

w

w .je

F2

w

1.

of

the

ellipse,

then

show

that

. [1995 - 5 Marks]

If the line y = mx + c is a common tangent to the hyperbola and the circle x2 + y2 = 36, then which one of the

following is true? [Main Sep. 05, 2020 (II)]

(a) c2 = 369

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(b) 5m = 4 (c) 4c2 = 369 (d) 8m + 5 = 0 Let P(3, 3) be a point on the hyperbola,

If the normal to

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2.

it at P intersects the x-axis at (9, 0) and e is its eccentricity, then the ordered pair (a2, e2) is equal to : [Main Sep. 04, 2020 (I)] (a) (b) (c) (d) (9, 3) 3.

Let e1 and e2 be the eccentricities of the ellipse,

respectively satisfying e1e2 = 1. If α

w .je

and the hyperbola,

and β are the distances between the foci of the ellipse and the foci of the hyperbola respectively, then the ordered pair (α, β) is equal to : [Main Sep. 03, 2020 (II)] (a) (8, 12)

w

(b) (c)

w

(d) (8, 10) 4. A line parallel to the straight line 2x – y = 0 is tangent to the hyperbola

at the point (x1, y1). Then

is equal to :

www.jeebooks.in

(a) (b) (c) (d) 5.

6 8 10 5 For some

eb oo ks .in

[Main Sep. 02, 2020 (I)]

, if the eccentricity of the hyperbola,

is

times the eccentricity of the ellipse,

then the length of the latus rectum of the ellipse, is

:

[Main Sep. 02, 2020 (II)]

(a) (b) (c) (d)

If a hyperbola passes through the point P(10,16) and it has vertices at ( 6,0), then the equation of the normal to it at P is: [Main Jan. 8, 2020 (II)] 3x + 4y = 94 2x + 5y = 100 x + 2y = 42 x + 3y = 58 The equation of a common tangent to the curves, y2 = 16x and xy = – 4, is : [Main April 12, 2019 (II)] x–y+4=0 x+y+4=0 x – 2y + 16 = 0 2x – y + 2 = 0

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6.

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(a) (b) (c) (d) 7.

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(a) (b) (c) (d)

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8.

If the line y = mx +

is normal to the hyperbola

= 1,

then a value of m is :

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[Main April 09, 2019 (I)] (a) (b) (c) (d) 9.

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(a) (b) (c) (d) 10.

If the eccentricity of the standard hyperbola passing through the point (4, 6) is 2, then the equation of the tangent to the hyperbola at (4, 6) is : [Main April. 08, 2019 (II)] x – 2y + 8 = 0 2x – 3y + 10 = 0 2x – y – 2 = 0 3x – 2y = 0 If the vertices of a hyperbola be at (–2, 0) and (2, 0) and one of its foci be at (–3, 0), then which one of the following points does not lie on this hyperbola? [Main Jan. 12, 2019 (I)]

w

(a)

(b)

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(c)

(d) 11.

If a hyperbola has length of its conjugate axis equal to 5 and the distance between its foci is 13, then the eccentricity of the hyperbola

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is : [Main Jan. 11, 2019 (II)] (a)

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(b) 2 (c) (d) 12.

(a) (b) (c) (d) 13.

The equation of a tangent to the hyperbola 4x2 – 5y2 = 20 parallel to the line x – y = 2 is: [Main Jan 10, 2019 (I)] x–y+1=0 x–y+7=0 x–y+9=0 x–y–3=0 at the points P and Tangents are drawn to the hyperbola Q. If these tangents intersect at the point T(0, 3) then the area (in sq. units) of ∆PTQ is : [Main 2018]

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(a)

(b) (c)

The eccentricity of the hyperbola whose length of the latus rectum is equal to 8 and the length of its conjugate axis is equal to half of the distance between its foci, is : [Main 2016]

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(d) 14.

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(a)

(b)

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(c)

15.

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(d) Let P(6, 3) be a point on the hyperbola

. If the normal at

the point P intersects the x-axis at (9, 0), then the eccentricity of the hyperbola is [2011] (a) (b) (c) (d) 16.

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The locus of the orthocentre of the triangle formed by the lines (1+ p) x – py + p (1+ p) = 0, (1+ q) x – qy + q (1+ q) = 0, and y = 0, where p q, is a hyperbola a parabola an ellipse a straight line Consider a branch of the hyperbola

w

(a) (b) (c) (d) 17.

[2009]

w

with vertex at the point A. Let B be one of the end points of its latus rectum. If C is the focus of the hyperbola nearest to the point A, then the area of the triangle ABC is [2008] (a)

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(b)

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(c) (d) 18. (a) (b) (c) (d) 19.

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(a) (b) (c) (d) 20.

Let a and b be non-zero real numbers. Then, the equation (ax2 + by2 + c) (x2 – 5xy + 6y2) = 0 represents [2008] four straight lines, when c = 0 and a, b are of the same sign. two straight lines and a circle, when a = b, and c is of sign opposite to that of a two straight lines and a hyperbola, when a and b are of the same sign and c is of sign opposite to that of a a circle and an ellipse, when a and b are of the same sign and c is of sign opposite to that of a A hyperbola, having the transverse axis of length 2 sin θ, is confocal with the ellipse 3x2 + 4y2 = 12. Then its equation is [2007 - 3 marks] 2 2 2 2 x cosec θ – y sec θ = 1 x2sec2θ – y2cosec2θ = 1 x2sin2θ – y2cos2θ = 1 x2cos2θ – y2sin2θ = 1 y = 2 touches the hyperbola x2 – 2y2 = 4, then the If the line 2x + point of contact is

[2004S]

w

(a) (– 2, √6) (b) (– 5, 2√6)

w

(c)

(d)

www.jeebooks.in

21.

For hyperbola

which of the following remains

constant with change in ‘α’ (a) (b) (c) (d) 22. (a) (b) (c) (d) 23.

abscissae of vertices abscissae of foci eccentricity directrix The equation of the common tangent to the curves y2 = 8x and xy = –1 is [2002S] 3y = 9x + 2 y = 2x + 1 2y = x + 8 y=x+2 The curve described parametrically by x = t2 + t + 1,y = t2 – t + 1 represents [1999 - 2 Marks] a pair of straight lines an ellipse a parabola a hyperbola If x = 9 is the chord of contact of the hyperbola x2 – y2 = 9, then the equation of the corresponding pair of tangents is [1999 - 2 Marks] 9x2 – 8y2 + 18x – 9 = 0 9x2 – 8y2 – 18x + 9 = 0 9x2 – 8y2 – 18x – 9 = 0 9x2 – 8y2 + 18x + 9 = 0 Let P (a sec , b tanθ) and Q (a sec φ, b tan φ), whereθ + φ = π / 2, be

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(a) (b) (c) (d) 24.

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[2003S]

w

w

(a) (b) (c) (d) 25.

two points on the hyperbola

= 1. If (h, k) is the point of

intersection of the normals at P and Q, then k is equal to

www.jeebooks.in

[1999 - 2 Marks]

eb oo ks .in

(a) (b) (c) (d) 26.

The equation 2x2 + 3y2 – 8x – 18y + 35 = k represents

[1994]

no locus if k > 0 an ellipse if k < 0 a point if k = 0 a hyperbola if k > 0 Each of the four inequalties given below defines a region in the xy plane. One of these four regions does not have the following property. For any two points (x1, y1) and (x2, y2) in the region, the point

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(a) (b) (c) (d) 27.

is also in the region. The inequality defining this

region is

[1981 - 2 Marks]

(a)

w

(b) Max

w

(c)

(d)

28.

The equation

represents

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(a) (b) (c) (d)

29.

an ellipse a hyperbola a circle none of these

eb oo ks .in

[1981 - 2 Marks]

The line 2x + y = 1 is tangent to the hyperbola

. If this

line passes through the point of intersection of the nearest directrix and the x-axis, then the eccentricity of the hyperbola is [2010]

30.

An ellipse has eccentricity

and one focus at the point P

. Its

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one directrix is the common tangent, nearer to the point P, to the circle x2 + y2 =1 and the hyperbolax2 – y2 =1. The equation of the ellipse, in the standard form, is............ [1996 - 2 Marks]

31.

Let a and b be positive real numbers such that a > 1 and b < a. Let P be a point in the first quadrant that lies on the hyperbola

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Suppose the tangent to the hyperbola at P passes through the point (1, 0), and suppose the normal to the hyperbola at P cuts off equal intercepts on the coordinate axes. Let ∆ denote the area of the triangle formed by the tangent at P, the normal at P and the x-axis. If e denotes the eccentricity of the hyperbola, then which of the following statements is/are TRUE? [Adv. 2020]

(a)

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(b) (c)

32.

eb oo ks .in

(d) If 2x – y + 1 = 0 is a tangent to the hyperbola

= 1, then

which of the following cannot be sides of a right angled triangle? [Adv. 2017] (a) a, 4, 1 (b) a, 4, 2 (c) 2a, 8, 1 (d) 2a, 4, 1 33. Consider the hyperbola H : x2 – y2 = 1 and a circle S with center N(x2, 0). Suppose that H and S touch each other at a point P(x1, y1) with x1 > 1 and y1 > 0. The common tangent to H and S at P intersects the x-axis at point M. If (l, m) is the centroid of the triangle PMN, then the correct expression(s) is(are) [Adv. 2015] (a)

=1–

for x1 > 1

for x1 > 1

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(b)

(c)

=1+

for y1 > 0

(d)

Tangents are drawn to the hyperbola

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34.

for x1 > 1

parallel to the

straight line 2x – y = 1. The points of contact of the tangents on the hyperbola are [2012]

(a)

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(b)

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(c) (d) 35.

Let the eccentricity of the hyperbola

be reciprocal to that

of the ellipse x2 + 4y2 = 4. If the hyperbola passes through a focus of the ellipse, then [2011] (a) the equation of the hyperbola is

(b) a focus of the hyperbola is (2, 0)

(c) the eccentricity of the hyperbola is

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(d) the equation of the hyperbola is x2 – 3y2 = 3 36. An ellipse intersects the hyperbola 2x2 – 2y2 = 1 orthogonally. The eccentricity of the ellipse is reciprocal of that of the hyperbola. If the axes of the ellipse are along the coordinate axes, then [2009] (a) equation of ellipse is x2 + 2y2 = 2 (b) the foci of ellipse are

w

(c) equation of ellipse is x2 + 2y2 = 4 (d) the foci of ellipse are

w

37.

Let a hyperbola passes through the focus of the ellipse

.

The transverse and conjugate axes of this hyperbola coincide with the major and minor axes of the given ellipse, also the product of eccentricities of given ellipse and hyperbola is 1, then [2006 - 5M, –1]

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(a) the equation of hyperbola is

eb oo ks .in

(b) the equation of hyperbola is (c) focus of hyperbola is (5, 0) (d) vertex of hyperbola is

(Qs. 38-41) : By appropriately matching the information given in the three columns of the following table. Column 1, 2, and 3 contain conics, equations of tangents to the conics and points of contact, respectively. Column 1 Column 2 (I) x2 + y2 = a2 (i) my = m2x + a

Column 3

(P)

(II) x2 + a2y2 =(ii) y = mx + a(Q) a2 (iii)y

=

mx

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(III)y2 = 4ax

(IV)x2 – a2y2 =(iv) y a2

38.

Let

=

mx

+(R)

+(S)

, where a > b > 0, be a hyperbola in the xy-plane

w

w

whose conjugate axis LM subtends an angle of 60° at one of its . vertices N. Let the area of the triangle LMN be [Adv. 2018]

List I

List II P. The length of the conjugate1. 8

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axis of H is Q. The eccentricity of H is

2.

eb oo ks .in

R. The distance between the3. foci of H is

S. The length of the latus4. 4 rectum of H is The correct option is: (a) P → 4; Q → 2; R → 1; S → 3 (b) P → 4; Q → 3; R → 1; S → 2 (c) P → 4; Q → 1; R → 3; S → 2 (d) P → 3; Q → 4; R → 2; S → 1 39.

The tangent to a suitable conic (Column 1) at

is found to be

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w

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= 4, then which of the following options is the only correct combination? [Adv. 2018] (a) (IV) (iii) (S) (b) (IV) (iv) (S) (c) (II) (iii) (R) (d) (II) (iv) (R) 40. If a tangent to a suitable conic (column 1) is found to be y = x + 8 and its point of contact is (8, 16), then which of the following options is the only correct combination? [Adv. 2018] (a) (I) (ii) (Q) (b) (II) (iv) (R) (c) (III) (i) (P) (d) (III) (ii) (Q) , if a tangent is drawn to a suitable conic (Column 1) at 41. For the point of contact (– 1, 1), then which of the following options is the

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only correct combination for obtaining its equation? Adv. 2017] (I) (i) (P) (I) (ii) (Q) (II) (ii) (Q) (III) (i) (P) Match the conics in Column I with the statements/expressions in Column II. [2009]

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(a) (b) (c) (d) 42.

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Column I Column II (A)Circle (p) The locus of the point (h,k) for which the line hx + ky =1 touches the circle x2 + y2 =4 (B)Parabola (q) Points z in the complex plane satisfying | z + 2| – | z – 2 |= ±3 (C)Ellipse (r) Points of the conic have parametric representation

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Re

Match the statements in Column I with the properties in Column II and indicate your answer by darkening the appropriate bubbles in the 4 × 4 matrix given in the ORS. [2007 -6 marks]

Column I (A)Two intersecting circles

Column II (p)have a common

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(B)Two mutually external circles (C)Two circles, one strictly inside the other (D)Two branches of a hyperbola

tangent (q)have a common normal (r) do not have a common tangent (s) do not have a common normal

PASSAGE

The circle x2 + y2 – 8x = 0 and hyperbola points A and B.

intersect at the [2010]

Equation of the circle with AB as its diameter is x + y2 – 12x + 24 = 0 x2 + y2 + 12x + 24 = 0 x2 + y2 + 24x – 12 = 0 x2 + y2 – 24x – 12 = 0 Equation of a common tangent with positive slope to the circle as well as to the hyperbola is 2

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(b)

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47.

Tangents are drawn from any point on the hyperbola

to

the circle x2 + y2 = 9. Find the locus ofmid-point of the chord of contact. [2005 - 4 Marks] The angle between a pair of tangents drawn from a point P to the parabola y2 =4ax is 45°. Show that the locus of the point P is a

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hyperbola.

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[1998 - 8 Marks]

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1.

If α is the positive root of the equation, p(x) = x2 – x – 2 = 0, then is equal to:

[Main Sep. 05, 2020 (I)]

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(a)

(b) (c)

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(d)

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[Main Sep. 05, 2020 (II)]

(a) is equal to

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(b) is equal to 1 (c) is equal to 0 (d) does not exist 3. Let [t] denote the greatest integer

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If for some

then L is equal to :

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(a) 1 (b) 2 (c) (d) 0 4.

Let f(x) =

value at a

and

and

g(x)

, xÎ R. If f(x) attains maximum

attains

minimum

value

at

b, then

is equal to :

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(b) –3/2

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(a) 1/2 (c) –1/2 (d) 3/2 5.

, then a + b is equal to : [Main April 10, 2019 (II)]

–4 5 –7 1

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(a) (b) (c) (d)

If

6.

equals : [Main April 8, 2019 (I)]

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(a) 4 (b)

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(c) 2 (d) 4

is:

7.

[Main Jan. 12, 2019 (I)]

(a) 4 (b) (c) (d) 8

is equal to :

8.

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(a) 0

(b) 2 (c) 4

(d) 1

For each t Î R, let [t] be the greatest integer less than or equal to t. Then, [Main Jan. 10, 2019 (I)]

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(a) equals 1

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(b) equals 0 (c) equals – 1

10.

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(d) does not exist

[Main Jan. 9, 2019 (I)]

(a) exists and equals

(b) exists and equals (c) exists and equals

(d) does not exist 11. For each x∈R, let [x] be greatest integer less than or equal to x. Then [Main Jan. 09, 2019 (II)] is equal to:

– sin 1 1 sin 1 0

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(a) (b) (c) (d)

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equals. [Main Online April 15, 2018]

(a) 1

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(d) equals :

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[Main 2017]

(a) (b) (c) (d) 14.

is equal to :

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(a) 2

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(b) (c) 4 (d) 3 15.

is equal to : [Main Online April 10, 2015]

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(a) 2 (b) 3

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(c)

(d)

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16.

is equal to:

(a) (b) (c) (d) 1

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[Main 2014]

is equal to

17.

[Main 2013]

(a) (b) (c) 1 (d) 2 18. Let

α(a)

and

β(a)

be

the

roots

of

the

equation

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where a > –1. Then

and

and 1

(b)

and – 1

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(a)

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and 2

(d)

and 3

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19.

If

then

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(a) (b) (c) (d)

a = 1, b = 4 a = 1, b = – 4 a = 2, b = –3 a = 2, b = 3

20. If

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[2012]

where n is nonzero real number,

then a is equal to

[2003S]

(a) 0 (b) (c) n (d) 21.

equals

[2001S]

−π π π/2 1

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(a) (b) (c) (d)

22.

is equal to [1984 - 2 Marks]

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(a) 0

(b) –

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(d) none of these

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23.

If

then

has the value

(a) (b) (c) (d) none of these 24.

If f (x) =

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[1983 - 1 Mark]

then

f (x) is

[1979]

(a) (b) (c) (d)

The value of the limit

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0 ∞ 1 none of these

is _____

Let m and n be two positive integers greater than 1. If

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[Adv. 2020]

=–

then the value of

is [Adv. 2015]

27. The largest value of non-negative integer a for which

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is

28.

If

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[Adv. 2014]

then the value

of k is __________.

[Main Sep. 03, 2020 (I)]

is equal to

29. 30.

.

[Main Jan. 7, 2020 (I)] Let e denote the base of the natural logarithm. The value of the real number a for which the right hand limit

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is equal to a nonzero real number, is _____ [Adv. 2020]

If f (9) = 9,

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(9) = 4, then

equals............ [1988 - 2 Marks]

= ................ [1987 - 2 Marks]

33. If f(x)

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and g(x) g[f(x)] is .......................

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then

[1986 - 2 Marks]

34.

If

exists then both

and

exist.

[1981 - 2 Marks]

35.

Let f(x) =

for x ≠ 1. Then

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(a) (b)

does not exist =0

(c)

does not exist

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(d)

36.

For a ∈

(the set of all real numbers), a ≠ –1, [Adv. 2013]

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(d)

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37. [1998 - 2 Marks]

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(a) exists and it equals (b) exists and it equals – (c) does not exist because x –1 0 (d) does not exist because the left hand limit is not equal to the right hand limit. 38. The value of

[1991 - 2 Marks]

(a) (b) (c) (d) 39.

1 –1 0 none of these Let

be a function. We say that f has

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PROPERTY 1: If 2: If

exists and is finite, and PROPERTY

exists and is finite

Then which of the following options is/are correct? has PROPERTY 1

(b)

has PROPERTY 2

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(d)

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has PROPERTY 1 has PROPERTY 2

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and

f(0)

=

0.

Using

this

find

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40.

[2004 - 2 Marks]

41. Use the formula

to find

[1982 - 2 Marks]

42. Evaluate :

[1980]

43. f (x) is the integral of

, x ≠ 0, find

f ‘(x)

[1979]

44.

Evaluate

, (a ≠ 0)

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[1978]

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1.

is equal to : [Main Sep. 03, 2020 (II)]

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(a)

(b)

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(d)

2.

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(c)

is equal to :

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(a) (b) (c) (d)

e 2 1 e2

3.

is equal to:

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(a)

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(b)

(c) e2 (d) e 4.

is equal to: [Main Jan. 8, 2020 (II)]

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(a) 0

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(b) (c)

(d)

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5.

If a and b are the roots of the equation 375x2–25x–2=0, then is equal to :

(a) (b) (c) (d) 6.

For each Then

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[Main April 12, 2019 (I)]

, let [t] be the greatest integer less than or equal to t.

[Main 2018]

is equal to 15. is equal to 120. does not exist (in R). is equal to 0.

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(a) (b) (c) (d)

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equals.

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(b) (c)

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then log p is equal to :

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8.

[Main 2016]

(a) (b) (c) 2 (d) 1 9.

If value of θ is

=

and

then the

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(a)

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(b) (c)

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(d)

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equals

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(a)

(c) (d) 4 f (2) 11.

The value of

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(b)

, where x > 0 is

[2006 - 3M, –1]

(a) (b) (c) (d) 12.

0 –1 1 2 If f(x) is differentiable and strictly increasing function, then the value of

[2004S]

1 0 –1 2

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(a) (b) (c) (d)

,given that f ‘ (2) = 6 and

13.

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f ‘(1) = 4

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(a) (b) (c) (d)

[2003S]

does not exist is equal to – 3/2 is equal to 3/2 is equal to 3

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14.

Let

be such that f (1) = 3 and

= 6. Then

equals

(a) (b) (c) (d)

1 e1/2 e2 e3

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[2002S]

15. The integer n for which number is

is a finite non-zero

[2002S]

(a) (b) (c) (d)

Let α, β ∈

be such that

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16.

1 2 3 4

17.

If

= 1. Then6 (α + β) equals.

[Adv. 2016]

then the value of n is

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equal to ________.

18.

[Main Sep. 02, 2020 (I)]

= ... [1996 - 1 Mark]

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19.

= .......................

20.

Find

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[1990 - 2 Marks]

[1993 - 2 Marks]

1.

Let f (x) be a polynomial of degree 4 having extreme values at x = 1 and x = 2. If

= 3 then f (– 1) is equal to

[Main Online April 15, 2018]

(a)

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(b) (c)

(d) 2.

Let f(x) be a polynomial of degree four having extreme values at x =

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1 and x = 2. If

then f(2) is equal to : [Main 2015]

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(a) 0 (b) 4 (c) – 8

(d) – 4

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3.

Let f (1) = –2 and f ′ (x) (6) lies in the interval :

4.2 for 1

. The possible value of f

(a) [15, 19) (b) (– , 12) (c) [12, 15) (d) [19, )

Find the derivative of sin (x2 + 1) with respect to x from first principle. [1978]

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1.

The negation of the Boolean expression :

is equivalent to

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(a) (b) (c) (d) 2.

is equivalent to: [Main Sep. 05, 2020 (I)]

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(a) (b) (c) (d) 3.

The negation of the Boolean expression

Given the following two statements : is a tautology. is a fallacy. Then :

(a) both (S1) and (S2) are correct (b) only (S1) is correct

[Main Sep. 04, 2020 (I)]

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(c) only (S2) is correct (d) both (S1) and (S2) are not correct 4. The proposition

is equivalent to : [Main Sep. 03, 2020 (I)]

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(a) q (b) (c) (d) 5. Let p, q, r be three statements such that the truth value of is F. Then the truth values of p, q, r are respectively : [Main Sep. 03, 2020 (II)] (a) T, F, T (b) T, T, T (c) F, T, F (d) T, T, F 6. If p → (p ∧ ~q) is false, then the truth values of p and q are respectively: [Main Jan. 9, 2020 (II)] (a) F, F (b) T, F (c) T, T (d) F, T 7. Which one of the following is a tautology? [Main Jan. 8, 2020 (I)] (a) (p ∧ (p q)) q (b) q (p ∧ (p q)) (c) p ∧ (p q) (d) p (p ∧ q) 8. Which of the following statements is a tautology? [Main Jan. 8, 2020 (II)] (a) p ( q) p q (b) (p q) p q (c) (p q) p q

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9.

(d) (p q) p q The logical statement

is equivalent to: [Main Jan. 7, 2020 (I)]

p q ~p ~q If the truth value of the statement p → (~q r) is false (F), then the truth values of the statements p, q, r are respectively. [Main April 12, 2019 (I)] (a) T, T, F (b) T, F, F (c) T, F, T (d ) F, T, T is equivalent to : 11. The Boolean expression ~

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(a) (b) (c) (d) 10.

[Main April 12, 2019 (II)]

(a) p ∧ q (b) (c) p ∨ q (d) 12. If p ⇒(q

F, T, T T, F, F T, T, F F, F, F Which one of the following statements is not a tautology? [Main April 08, 2019 (II)] (p q) → (p (~ q)) (p ∧ q) → (~ p) q p → (p q) (p ∧ q) → p The Boolean expression is equivalent to :

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(a) (b) (c) (d) 13.

r) is false, then the truth values of p, q, r are respectively: [Main April 09, 2019 (II)]

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(a) (b) (c) (d) 14.

[Main Jan. 12, 2019 (I)]

(a)

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(b) (c )

If q is false and p ^ q r is true, then which one of the following statements is a tautology? [Main Jan. 11, 2019 (I)]

(a) (b) (c) (d) 16.

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15.

(d)

If the Boolean expression , where

is equivalent to

then the ordered pair

is:

[Main Jan. 09, 2019 (I)]

(a) (b) (c) (d)

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17. The logical statement [~ (~ p ∨ q) ∨ (p ∧ r)] ∧ (~ p ∧ r) is equivalent to:

(~ p ∧ ~ q) ∧ r ~p∨r (p ∧ r) ∧ ~ q (p ∧ ~ q) ∨ r The Boolean expression is equivalent to :

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(a) (b) (c) (d) 18.

(a) (b) (c) (d)

[Main Jan. 09, 2019 (II)]

[Main 2018]

p q ~q ~p

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(a) (b) (c) (d) 20. (a) (b) (c) (d) 21.

If p → (∼ p ∨ ∼ q) is false, then the truth values of p and q are respectively. [Main Online April 16, 2018] T, F F, F F, T T, T If (p∧ ~ q) ∧ (p ∧ r) → ~ p ∨ q is false, then the truth values of p, q and r are respectively [Main Online April 15, 2018] F, T, F T, F, T F, F, F T, T, T Which of the following is a tautology? [Main 2017]

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19.

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(a) (b) (c) (d) 22. The following statement (p → q) → [(~p → q) → q] is : a fallacy a tautology equivalent to ~ p → q equivalent to p → ~q The proposition (~p)

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(a) (b) (c) (d) 23.

[Main 2017]

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The Boolean Expression

(a) (b) (c) (d) 25.

The negation of

is equivalent to: [Main 2016]

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24.

is equivalent to :

[Main 2015]

(a) (b) (c) (d) 26.

The statement

is:

[Main 2014]

a tautology a fallacy eqivalent to equivalent to Let p, q, r denote arbitrary statements. Then the logically equivalent is: of the statement

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(a) (b) (c) (d) 27.

[Main Online April 12, 2014]

(a)

(b) (c)

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(d)

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28.

Consider Statement-1 : (p ^ ~ q) ^ (~ p ^ q) is a fallacy. Statement-2 : (p → q) ↔ (~ q → ~ p) is a tautology.

[Main 2013] (a) Statement-1 is true; Statement-2 is true; Statement-2 is a correct explanation for Statement-1.

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(b) Statement-1 is true; Statement-2 is true; Statement-2 is not a correct explanation for Statement-1. (c) Statement-1 is true; Statement-2 is false. (d) Statement-1 is false; Statement-2 is true. . If r 29. Let p and q be any two logical statements andr : p has a truth value F, then the truth values of p and q are respectively : [Main Online April 25, 2013] (a) F, F (b) T, T (c) T, F (d) F, T 30. For integers m and n, both greater than 1, consider the following three statements : P : m divides n Q : m divides n2 R : m is prime, then [Main Online April 23, 2013] (a) (b) (c) (d)

Consider the statement: “For an integer n, if n3 – 1 is even, then n is odd.” The contrapositive statement of this statement is: [Main Sep. 06, 2020 (II)] 3 For an integer n, if n is even, then n – 1 is odd. For an intetger n, if n3 – 1 is not even, then n is not odd. For an integer n, if n is even, then n3 – 1 is even. For an integer n, if n is odd, then n3 – 1 is even. is : The statement [Main Sep. 05, 2020 (II)]

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(a) (b) (c) (d) 2.

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(a) equivalent to (b) a contradiction (c) equivalent to (d) a tautology 3. Contrapositive of the statement : ‘If a function f is differentiable at a, then it is also continuous at a’, is : [Main Sep. 04, 2020 (II)] (a) If a function f is continuous at a, then it is not differentiable at a. (b) If a function f is not continuous at a, then it is not differentiable at a. (c) If a function f is not continuous at a, then it is differentiable at a (d) If a function f is continuous at a, then it is differentiable at a. 4. The contrapositive of the statement “If I reach the station in time, then I will catch the train” is: [Main Sep. 02, 2020 (I)] (a) If I do not reach the station in time, then I will catch the train. (b) If I do not reach the station in time, then I will not catch the train. (c) If I will catch the train, then I reach the station in time. (d) If I will not catch the train, then I do not reach the station in time. 5. Negation of the statement: [Main Jan. 9, 2020 (I)] is an integer of 5 is irrational is: is not an integer or 5 is not irrational (a) is not an integer and 5 is not irrational (b) is irrational or 5 is an integer. (c) is an integer and 5 is irrational (d) Let A, B, C and D be four non-empty sets. The contrapositive statement of “If A B and B D, then A C ” is: [Main Jan. 7, 2020 (II)] C, then A B and B D If A If A C, then B A or D B C, then A B and B D If A C, then A B or B D If A

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The negation of the Boolean expression ~ s (~ r ∧ s) is equivalent to : [Main April 10, 2019 (II)] (a) ~ s ∧ ~ r (b) r (c) s r (d) s ∧ r 8. For any two statements p and q, the negation of the expression p (~ p ∧ q) is: [Main April 9, 2019 (I)] (a) ~ p ∧ ~ q (b) p ∧ q (c) p ↔ q (d) ~ p ~ q 9. Contrapositive of the statement “If two numbers are not equal, then their squares are not equal”. is : [Main Jan. 11, 2019 (II)] (a) If the squares of two numbers are not equal, then the numbers are equal. (b) If the squares of two numbers are equal, then the numbers are not equal. (c) If the squares of two numbers are equal, then the numbers are equal. (d) If the squares of two numbers are not equal, then the numbers are not equal. 10. Consider the following two statements. Statement p: The value of sin 120° can be divided by taking θ = 240° in the equation 2

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7.

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sin

Statement q:

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The angles A, B, C and D of any quadrilateral ABCD satisfy the equation

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Then the truth values of p and q are respectively. [Main Online April 15, 2018] (a) F, T (b) T, T (c) F, F (d) T, F 11. Contrapositive of the statement ‘If two numbers are not equal, then their squares are not equal’, is : [Main Online April 9, 2017] (a) If the squares of two numbers are equal, then the numbers are equal. (b) If the squares of two numbers are equal, then the numbers are not equal. (c) If the squares of two numbers are not equal, then the numbers are not equal. (d) If the squares of two numbers are not equal, then the numbers are equal. 12. The contrapositive of the following statement, “If the side of a square doubles, then its area increases four times”, is : [Main Online April 10, 2016] (a) If the area of a square increases four times, then its side is not doubled. (b) If the area of a square increases four times, then its side is doubled. (c) If the area of a square does not increases four times, then its side is not doubled. (d) If the side of a square is not doubled, then its area does not increase four times. 13. The contrapositive of the statement “If it is raining, then I will not come”, is : [Main Online April 10, 2015] (a) If I will not come, then it is raining. (b) If I will not come, then it is not raining. (c) If I will come, then it is raining. (d) If I will come, then it is not raining.

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The contrapositive of the statement “if I am not feeling well, then I will go to the doctor” is [Main Online April 19, 2014] If I am feeling well, then I will not go to the doctor If I will go to the doctor, then I am feeling well If I will not go to the doctor, then I am feeling well If I will go to the doctor, then I am not feeling well.

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14.

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1.

The minimum value of

is :

[Main Sep. 04, 2020 (II)]

(a) (b)

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(c)

(d)

If for some x ∈ R, the frequency distribution of the marks obtained by 20 students in a test is :

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then the mean of the marks is :

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(a) (b) (c) (d)

[Main April 10, 2019 (I)]

3.2 3.0 2.5 2.8

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3.

The mean and the median of the following ten numbers in increasing order 10, 22, 26, 29, 34, x, 42, 67, 70, y are 42 and 35 respectively, then

is equal to:

(a) (b) (c) (d) 4.

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[Main April. 09, 2019 (II)]

9/4 7/2 8/3 7/3 The mean of a set of 30 observations is 75. If each other observation is multiplied by a non-zero number λ and then each of them is decreased by 25, their mean remains the same. The λ is equal to [Main Online April 15, 2018]

(a) (b) (c)

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The mean of the data set comprising of 16 observations is 16. If one of the observation valued 16 is deleted and three new observations valued 3, 4 and 5 are added to the data, then the mean of the resultant data, is: [Main 2015] 15.8 14.0 16.8 16.0 Let the sum of the first three terms of an A. P, be 39 and the sum of its last four terms be 178. If the first term of this A.P. is 10, then the median of the A.P. is : [Main Online April 10, 2015] 28 26.5

(a) (b)

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(c) 29.5 (d) 31 7. In a set of 2n distinct observations, each of the observations below the median of all the observations is increased by 5 and each of the remaining observations is decreased by 3. Then the mean of the new set of observations: [Main Online April 9, 2014] (a) increases by 1 (b) decreases by 1 (c) decreases by 2 (d) increases by 2 8. If the median and the range of four numbers {x, y, 2x + y, x – y}, where 0 < y < x < 2y, are 10 and 28 respectively, then the mean of the numbers is : [Main Online April 23, 2013] (a) 18 (b) 10 (c) 5 (d) 14 9. If x1, x2,..............., xn are any real numbers and n is any postive integer, then [1982 - 2 Marks] (a)

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(d) none of these

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10.

Consider the data on x taking the values 0, 2, 4, 8, ..., 2n with frequencies nC0, nC1, nC2, ..., nCn respectively. If the mean of this data , then n is equal to ______.

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is

[Main Sep. 06, 2020 (II)]

11.

A variable takes value x with frequency n+x –1Cx, x = 0, 1, 2, …n. The mode of the variable is.................. [1982 - 2 Marks]

12.

In a college of 300 students every student reads 5 newspapers and every newspaper is read by 60 students. The number of newpapers is [1998 - 2 Marks] at least 30 at most 20 exactly 25 none of these

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(a) (b) (c) (d)

1.

If

and

, (n, a > 1), then the standard

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deviation of n observations x1, x2, ..., xn is : [Main Sep. 06, 2020 (I)]

(a) a – 1

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(b) (c)

(d)

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2.

The mean and variance of 7 observations are 8 and 16, respectively. If five observations are 2, 4, 10, 12, 14, then the absolute difference of the remaining two observations is :

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[Main Sep. 05, 2020 (I)]

(a) 1 (b) 4

(c) 2 (d) 3 3. If the mean and the standard deviation of the data 3, 5, 7, a, b are 5 and 2 respectively, then a and b are the roots of the equation : [Main Sep. 05, 2020 (II)] (a) (b) (c) (d)

The mean and variance of 8 observations are 10 and 13.5, respectively. If 6 of these observations are 5, 7, 10, 12, 14, 15, then the absolute difference of the remaining two observations is : [Main Sep. 04, 2020 (I)]

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4.

9 5 3 7

For the frequency distribution :

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(a) (b) (c) (d) 5.

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where 0 < x1 < x2 < x3 < ... < x15 = 10 and

the standard deviation

(a) 4 (b) 1 (c) 6 (d) 2 6.

Let

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cannot be : [Main Sep. 03, 2020 (I)]

be ten observations of a random variable X. If

and

where

then the

standard deviation of these observations is :

[Main Sep. 03, 2020 (II)]

(a) (b)

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(c)

(d)

Let

and and If mean and variance of elements of Y are 17 and 216 respectively then a + b is equal to : [Main Sep. 02, 2020 (I)]

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7.

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(c) –27 (d) 9

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8.

Let the observations xi(1 ≤ i ≤ 10) satisfy the equations,

(a) (b) (c) (d) 9.

(a)

= 40. If µ and λ are the mean and the variance of

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10 and

=

the observations, x1 – 3, x2 – 3, ..., x10 – 3, then the ordered pair (µ, λ) is equal to: [Main Jan. 9, 2020 (I)] (3, 3) (6, 3) (6, 6) (3, 6) The mean and the standard deviation (s.d.) of 10 observations are 20 and 2 respectively. Each of these 10 observations is multiplied by p and then reduced by q, where p 0 and q 0. If the new mean and new s.d. become half of their original values, then q is equal to: [Main Jan. 8, 2020 (I)] –5

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(b) 10 (c) –20 (d) –10 10. The mean and variance of 20 observations are found to be 10 and 4, respectively. On rechecking, it was found that an observation 9 was incorrect and the correct observation was 11. Then the correct variance is: [Main Jan. 8, 2020 (II)] (a) 3.99 (b) 4.01 (c) 4.02 (d) 3.98 11.

If the data x1, x2, ……, x10 is such that the mean of first four of these is 11, the mean of the remaining six is 16 and the sum of squares of all

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of these is 2,000 ; then the standard deviation of this data is : [Main April 12, 2019 (I)] (a)

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(b) 2 (c) 4 (d) 12.

(a) (b) (c) (d) 13.

If both the mean and the standard deviation of 50 observations x1, x2, ….. , x50 are equal to 16, then the mean of (x1 – 4)2, (x2 – 4)2, …, (x50 – 4)2 is: [Main April 10, 2019 (II)] 400 380 525 480

where k > 0, If the standard deviation of the numbers –1, 0, 1, k is then k is equal to: [Main April 09, 2019 (I)]

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(a)

(b) (c)

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14.

The mean and variance of seven observations are 8 and 16, respectively. If 5 of the observations are 2, 4, 10, 12, 14, then the product of the remaining two observations is : [Main April 08, 2019 (I)]

(a) 45 (b) 49

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(c) 48 (d) 40 The mean and the variance of five observations are 4 and 5.20, respectively. If three of the observations are 3, 4 and 4; then the absolute value of the difference of the other two observations, is :

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15.

[Main Jan. 12, 2019 (II)]

(a) 7 (b) 5 (c) 1 (d) 3 16.

The outcome of each of 30 items was observed; 10 items gave an outcome

each, 10 items gave outcome

remaining 10 items gave outcome outcome data is

each and the

each. If the variance of this

then |d| equals :

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[Main Jan. 11, 2019 (I)]

(a)

(b) 2 (c)

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17.

The mean of five observations is 5 and their variance is 9.20. If three of the given five observations are 1, 3 and 8, then a ratio of other two observations is: [Main Jan. 10, 2019 (I)]

(a) 10 : 3 (b) 4 : 9

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(c) 5 : 8 (d) 6 : 7 5 students of a class have an average height 150 cm and variance 18 cm2. A new student, whose height is 156 cm, joined them. The variance (in cm2) of the height of these six students is:

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18.

[Main Jan. 9, 2019 (I)]

(a) 16 (b) 22 (c) 20 (d) 18 19.

If the mean of the data : 7, 8, 9, 7, 8, 7, λ, 8 is 8, then the variance of this data is

[Main Online April 15, 2018]

(a) (b) 2

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(c) (d) 1 20.

If

and

, then the standard deviation of

the 9 items x1, x2, ..., x9 is :

[Main 2018]

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(a) 4

(b) 2

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(c) 3

(d) 9 21. The sum of 100 observations and the sum of their squares are 400 and 2475, respectively. Later on, three observations, 3, 4 and 5, were

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found to be incorrect. If the incorrect observations are omitted, then the variance of the remaining observations is : [Main Online April 9, 2017] (b) 8.50 (c) 8.00 (d) 9.00 22.

(a) (b) (c) (d) 23.

If the standard deviation of the numbers 2, 3, a and 11 is 3.5, then which of the following is true? [Main 2016] 3a2 – 34a + 91 = 0 3a2 – 23a + 44 = 0 3a2 – 26a + 55 = 0 3a2 – 32a + 84 = 0 The mean of 5 observations is 5 and their variance is 124. If three of the observations are 1, 2 and 6; then the mean deviation from the mean of the data is : [Main Online April 10, 2016] 2.5 2.6 2.8 2.4 The variance of first 50 even natural numbers is [Main 2014] 437

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(a) (b) (c) (d) 24.

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(a) 8.25

(a)

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(d) 833 and M.D. be the mean and the mean deviation about of n 25. Let observations xi, i = 1, 2, ........, n. If each of the observations is

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increased by 5, then the new mean and the mean deviation about the new mean, respectively, are: [Main Online April 12, 2014] (a)

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(b) (c) (d) 26.

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(a) (b) (c) (d) 27.

All the students of a class performed poorly in Mathematics. The teacher decided to give grace marks of 10 to each of the students. Which of the following statistical measures will not change even after the grace marks were given ? [Main 2013] mean median mode variance In a set of 2n observations, half of them are equal to ‘a’ and the remaining half are equal to ‘ –a’. If the standard deviation of all the observations is 2 ; then the value of | a | is : [Main Online April 25, 2013] 2 (c)4

(a) (b)

Consider any set of 201 observations x1, x2, ....x200, x201. It is given that x1< x2 0, q > 0, r > 0 assumes its

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[1995]

(a) (b) (c) (d) p = q = r 20. Which of the following functions is periodic? [1983 - 1 Mark] (a) f(x) = x – [x] where [x] denotes the largest integer less than or equal to the real number x

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(b)

for

,

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(c) f(x) = x cosx (d) none of these 21. If x satisfies (a) (b) (c)

, then [1983 - 1 Mark]

or or

(d) None of these

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22. Let f(x) = | x – 1 |. Then [1983 - 1 Mark] (a) (b) (c) (d)

f(x ) = (f(x)) f(x + y) = f(x) + f(y) f(| x |) = | f(x) | None of these

23.

The entire graphs of the equation y = x2 + kx – x + 9 is strictly above the x-axis if and only if [1979] k 4 f (x) has three real roots if a < – 4 f (x) has three real roots if – 4 < a < 4 The function f(x) = 2|x| + |x + 2| – | |x + 2| – 2 |x| | has a local minimum or a local maximum at x = [Adv. 2013] (a) – 2 (b)

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(c) 2 (d)

35. Let f : (–1, 1) → IR be such that

Then the

is (are)

[Adv. 2012]

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value (s) of

for

(a)

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(b) (c)

(d)

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36.

+ cos

If f(x) =

where [x] stands for the greatest integer function, then [1991 - 2 Marks]

(a)

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(b) (c) (d)

37. Let g (x) be a function defined on [– 1, 1]. If the area of the equilateral triangle with two of its vertices at (0,0) and[x, g(x)] is

, then the function g(x) is

[1989 - 2 Marks]

(a) g(x) = + (b) g(x) = (c) g(x) = – (d) g(x) = 38. If

then

[1984 - 3 Marks]

x = f (y) f(1) = 3 y increases with x for x < 1 f is a rational function of x

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(a) (b) (c) (d)

39. Match the statements given in Column-I with the intervals/union of intervals given in Column-II. [2011]

Column-I

(A)

The set

: z is a complex

Column-II (p)

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number, |z| = 1 z ≠ ± 1}is (B)The domain of the function f (x) = sin–1(q) is

(C)

(r) [2, ∞)

If f (θ) =

, then the set is

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(s) (D)If f (x) = x (3x – 10), x ≥ 0 then f (x) is(t) increasing in 3/2

Let

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40.

1

Match of expressions/statements in Column I with expressions/statements in Column II and indicate your answer by darkening the appropriate bubbles in the 4 × 4 matrix given in the ORS. [2007 -6 marks] Column I Column II (A) If –1 < x < 1, then f(x) (p) 0 < f(x) < 1 satisfies (B) If 1 < x < 2, then f(x) (q) f(x) < 0 satisfies (C) If 3 < x < 5, then f(x) (r) f(x) > 0 satisfies (D) If x > 5, then f(x) (s) f(x) < 1 satisfies 41.

Let the function defined in column 1 have domain

and range

[1992 - 2 Marks]

(p) (q) (r) (s)

Column II onto but not one-one one- one but not onto one- one and onto neither one-one nor onto

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Column I (A) 1 + 2x (B) tan x

42. Let f(x) = Ax2 + Bx +C where A, B, C are real numbers. Prove that if f(x) is an integer whenever x is an integer, then the numbers 2A, A + B and C are all integers. Conversely, prove that if the numbers 2A, A+B and C are all integers then f(x) is an integer whenever x is an integer. [1998 - 8 Marks] 43. A function f :IR IR, where IR is the set of real numbers, is defined by

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. Find the interval of values of α for which f is onto. Is the function one-

to-one for α = 3? Justify your answer.

[1996 - 5 Marks] Let {x} and [x] denotes the fractional and integral part of a real number x respectively. Solve 4{x} = x + [x]. [1994 - 4 Marks] 45. Find the natural number ‘a’ for which

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44.

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, where the function ‘f’ satisfies the relation f(x + y) = f (x) f (y) for all natural numbers x, y and further f(1) = 2. [1992 - 6 Marks] R be such that for all x and y in R | f (x) – f(y) |

eb oo ks .in

46. Let R be the set of real numbers and f : R | x – y . Prove that f(x) is a constant.

[1988 - 2 Marks]

47.

A relation R on the set of complex numbers is defined byz1 R z2 if and only if

is real.

Show that R is an equivalence relation.

[1982 - 2 Marks] Let A and B be two sets each with a finite number of elements. Assume that there is an injective mapping from A to B and that there is an injective mapping from B to A. Prove that there is a bijective mapping from A to B. [1981 - 2 Marks] 49. Consider the following relations in the set of real numbers R. R = {(x, y); x ∈ R, y ∈ R, x2 + y2 25}

48.

R’ =

Find the domain and range of R ∩ R’. Is the relation R ∩ R’ a function?

[1979]

50. If f (x) = x9 – 6x8 – 2x7 + 12x6 + x4 – 7x3 + 6x2 + x – 3, find f (6).

[1979]

51. Draw the graph of y = | x |1/2 for – 1

x

1.

[1978]

Find the domain and range of the function f (x) =

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52.

[1978]

For a suitably chosen real constant a, let a function, f : R –{– a}→ R be defined by f (x) =

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1.

. Is the function one-to-one?

. Further suppose that for any real number x ≠ – a and f (x) ≠ – a, ( fof ) (x) = x. Then

w

is equal to: [Main Sep. 06, 2020 (II)]

(a)

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(b) (c) – 3 (d) 3 The inverse function of

is ________.

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2.

[Main Jan. 8, 2020 (I)]

(a) (b) (c) (d) 3.

If g(x) = x2 + x – 1 and (gof) (x) = 4x2 – 10x + 5, then

is equal to:

[Main Jan. 7, 2020 (I)]

(a) (b)

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(c)

(d) 4.

For x

, let f(x) =

, g(x) = tan x and h(x) =

If φ (x) = ((hof)og) (x), then φ

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is equal to :

[Main April 12, 2019 (I)]

w

(a) tan

(b) tan (c) tan

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(d) tan For x∈ R – {0, 1}, let f1 (x) =

f3 (x) =

, f2 (x) = 1 – x and

be three given functions. If a function, J(x) satisfies (f2oJof1) (x) = f3(x) then J(x) is

equal to:

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5.

[Main Jan. 09, 2019 (I)]

(a) f3 (x) (b)

f3(x)

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(c) f2 (x) (d) f1 (x) 6. Let N denote the set of all natural numbers. Define two binary relations on N as R1 = {(x, y) ∈ N × N : 2x + y = 10}and R2 = {(x, y) ∈ N × N : x + 2y = 10}. Then [Main Online April 16, 2018] (a) Both R1 and R2 are transitive relations (b) Both R1 and R2 are symmetric relations (c) Range of R2 is {1, 2, 3, 4} (d) Range of R1 is {2, 4, 8} 7. Consider the following two binary relations on the set A = {a, b, c} : R1 = {(c, a) (b, b) , (a, c), (c, c), (b, c), (a, a)} and R2 = {(a, b), (b, a), (c, c), (c, a), (a, a), (b, b), (a, c). Then [Main Online April 15, 2018] (a) R2 is symmetric but it is not transitive (b) Both R1 and R2 are transitive (c) Both R1 and R2 are not symmetric (d) R1 is not symmetric but it is transitive 8.

If g is the inverse of a function f and

then

is equal to: [Main 2014]

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(a)

(b)

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(c) 1 + x5 (d) 5x4 9. Let f (x) = x2 and g(x) = sin x for all x R. Then the set of all x satisfying (f o g o g o f) (x) = (g o g o f) (x), where (f o g) (x) = f (g(x)), is [Adv. 2011] (a)

www.jeebooks.in

(b) (c) (d) for n ≥ 2 and

eb oo ks .in

10. Let g(x) =

Then

equals.

[2007 -3 marks]

(a) (b) (c) (d) 11.

If

where f”(x) = –f(x) andg(x) = f ‘(x) and given that F(5) = 5,

then F(10) is equal to

[2006 - 3M, –1] 5 10 0 15 X and Y are two sets and f : X → Y. If {f(c) = y; c ⊂ X,y ⊂ Y} and {f–1(d) = x; d ⊂ Y, x ⊂ X}, then the true statement is [2005S] f(f–1(b)) = b f–1(f(a)) = a f(f–1(b)) = b, b ⊂ y f–1(f(a)) = a, a ⊂ x If f(x) = sin x + cos x, g (x) = x2 – 1, then g (f(x)) is invertible in the domain [2004S]

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(a) (b) (c) (d) 12.

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(b)

(c)

(d)

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[0, π] such that min f (x) > max g (x), then the

14. If relation between b and c, is

[2003S]

eb oo ks .in

(a) no real value of b & c (b) (c) (d)

15. Domain of definition of the function

for real valued x, is

[2003S]

(a)

(b)

(c)

(d)

16. Let f(x) =

, x ≠ −1. Then, for what value of α is f (f(x)) = x ?

[2001S]

(a) (b) (c) 1 (d) −1

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17. The domain of definition of f(x) = (a) (b) (c) (d)

is

[2001S]

R \ {−1, −2} (−2, ∞) R \ {−1, −2, −3} (−3, ∞) \ {−1, −2}

If f:[1, ∞) → [2, ∞) is given by f(x) = x +

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18.

[2001S]

)/2

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(a) (x + (b) x/(1 + x2) (c)

then f -1(x) equals

(d) 1 +

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19. Let g(x) = 1 + x − [x] and

. Then for allx, f(g(x)) is equal to [2001S]

x 1 f(x) g(x) If the function f: [1, ) f(x) = 2x (x-1), then f–1 (x) is

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(a) (b) (c) (d) 20.

[1,

) is defined by

[1999 - 2 Marks]

(a) (b) (c) (d) not defined 21. Let

. Then the set

is

[1995]

(a)

{0, 1, –1} {0, –1} empty Let f(x) = sinx and g(x) = ln | x |. If the ranges of the composition functions fog and gof are R1 and R2 respectively, then [1994 - 2 Marks]

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(b) (c) (d) 22.

(a) (b) (c)

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(d)

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23. The domain of definition of the function is [1983 - 1 Mark]

(a) (–3, –2) excluding – 2.5(b) [0, 1] excluding 0.5

(c) [–2, 1) excluding 0

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(d) none of these 24. If

then

has the value [1983 - 1 Mark]

(b) 1/2 (d) none of these

25. The value of is __________.

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(a) –1 (c) – 2

[Adv. 2018]

26. Let f : [0, 4π] → [0, π] be defined by f (x) = cos (cos x). The number of points –1

satisfying the equation is

[Adv. 2014]

27. If f(x) = sin2 x + sin2

=1, then(gof) (x) = .................

[1996 - 2 Marks]

If

, then domain of f(x) is .... and its range is .................

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28.

29. The domain of the function

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30. The values of

[1985 - 2 Marks] is given by ................. [1984 - 2 Marks] lie in the interval ................. [1983 - 1 Mark]

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31. If f(x) = (a –xn)1/n where a > 0 and n is a positive integer, then f[f(x)] = x. [1983 - 1 Mark]

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for all x ∈ R and g(x) =

32. Let

sin x for all x ∈ R. Let (fog)(x)

(a) Range of f is (b) Range of fog is (c) (d) There is an 33. Let f :

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denote f(g(x)) and (gof)(x) denote g(f(x)). Then which of the following is (are) true? [Adv. 2015]

such that

be given by

f (x) = (log(sec x + tan x))3. Then

[Adv. 2014]

(a) (b) (c) (d)

f (x) is an odd function f (x) is one-one function f (x) is an onto function f (x) is an even function

34.

Let f : (0, 1) → R be defined by f (x) = Then

, where b is a constant such that 0 < b < 1.

[2011]

(a) f is not invertible on (0, 1)

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(b) f ≠ f –1 on (0, 1) and f ′(b) =

(c) f = f –1 on (0, 1) and f ′ (b) =

(d) f –1 is differentiable (0, 1) 35. If g (f(x)) = | sin x | and f (g(x)) = (sin

)2, then

(a) f(x) = sin2 x, g(x) =

[1998 - 2 Marks]

f(x) = sin x, g(x) = | x| f(x) = x2, g(x) = sin f and g cannot be determined. If f(x) = 3x – 5, then f–1(x)

[1998 - 2 Marks]

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(b) (c) (d) 36.

(a) is given by (b) is given by (c) does not exist because f is not one-one (d) does not exist because f is not onto.

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Let f : E1 →

and E2 =

.

eb oo ks .in

37. Let E1 =

be the function defined by f (x) = loge

defined by g(x) = sin–1

and g : E2 →

be the function

.

[Adv. 2018]

P.

LIST - I The range of f is

1.

LIST - II

Q. R.

The range of g contains The domain of f contains

2. 3.

(0, 1)

S.

The domain of g is

4. 5.

(– ∞, 0) ∪ (0, ∞)

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6.

(a) (b) (c) (d)

(– ∞, 0) ∪

The correct option is:

P → 4; Q → 2; R → 1; S → 1 P → 3; Q → 3; R → 6; S → 5 P → 4; Q → 2; R → 1; S → 6 P → 4; Q → 3; R → 6; S → 5

Let f be a one-one function with domain {x, y, z} and range {1, 2, 3}. It is given that exactly one of the following statements is true and the remaining two are false determine f–1(1). [1982 - 3 Marks]

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38.

www.jeebooks.in

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If α = cos–1 equal to :

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1.

, β = tan–1

, where 0 < α, β
0 sinβ < 0 cos(α + β) > 0 cosα < 0

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9.

equals ____.

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(a) (b) (c) (d)

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10. The principal value of

is

(a) (b) (c) (d) none 11.

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[1986 - 2 Marks]

Match List I with List II and select the correct answer using the code given below the lists : [Adv. 2013] List I List II 1. P. takes value

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Q. If cosx + cosy + cosz = 0 = sinx + siny + sinz then possible value of

R.

If

2.

is

cos 2x + sinx sin 2 secx = cosx sin2x secx

3.

w

w

+

S.

cos 2x then possible value of secx is

If

4. 1

then possible value of x is Codes:

www.jeebooks.in

(a) (b) (c) (d)

12.

Prove that

Q 3 3 4 4

R 1 2 2 1

S 2 1 1 2

eb oo ks .in

P 4 4 3 3

.

[2002 - 5 Marks]

1.

is equal to :

[Main Sep. 03, 2020 (I)]

w .je

(a)

(b) (c)

w

(d)

w

2.

If

S

is

the

sum

of

the

first

10

terms

of

the

series

+...., then tan (S) is equal to: [Main Sep. 05, 2020 (I)]

www.jeebooks.in

(a)

(c) (d) 3.

The value of

eb oo ks .in

(b)

is equal to :

Main April 12, 2019 (I)]

(a) (b) (c)

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(d) 4.

, where – 1 < x < 1, –2 < y < 2,

, then for all x, y, 4x2 –4xy cosα + y2 is equal to: [Main April 10, 2019 (II)]

w

x
cos–1x, is the interval: [Main Online April 25, 2013]

(a)

(b)

(c) (0, 1)

w

(d)

w

14.

The value of cot

is [Adv. 2013]

(a)

www.jeebooks.in

(b)

(d) 15.

If sin–1

eb oo ks .in

(c)

=

+ cos–1

for 0 < |x|
0. Then the minimum value

of is:

[Main Jan. 10, 2019 (II)]

w .je

a)

b) c)

d)

w

6. If

, then the ordered pair (A, B) is equal

w

to :

(a) (b) (c) (d)

[Main 2018]

(– 4, 3) (– 4, 5) (4, 5) (– 4, – 5)

www.jeebooks.in

If S= to

,then

is equal

eb oo ks .in

7.

[Main Online April 8, 2017]

a) b) c) d) 8.

If :

= , then the determinant of the matrix(A2016 – 2A2015 – A2014) is [Main Online April 10, 2016]

–175 2014 2016 –25

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(a) (b) (c) (d)

9

if = ax –12, then ‘a’ is equal to :

[Main Online April 11, 2015]

24 –12 –24 12

w

w

(a) (b) (c) (d)

.

10.

If f(θ

= and

www.jeebooks.in

A and B are respectively the maximum and the minimum values of f(θ), then (A, B) is equal to: [Main Online April 12, 2014] b)

eb oo ks .in

(a) (3, – 1) c) d) 11.

et be a cube root of unity and S be the set of allnon-singular matrices of the f

rm

where each of a, b and c is either ω or ω2. Then the number of distinct matrices in the set S is [2011] (a) 2 (b) 6 (c) 4 (d) 8 12. Consider three poin s

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nd , wh

w

13.

a) b)

[2008]

P lies on the line segment RQ Q lies on the line segment PR R lies on the line segment QP P, Q, R are non-collinear

w

(a) (b) (c) (d)

re . Then,

et . Then the value of the determin

nt is

[2002S] c)

www.jeebooks.in

d) 14.

The

parameter,

on

which

the

value

of

the

determin

eb oo ks .in

nt does not depend upon is [1997 - 2 Marks]

(a) a (b) p (c) d

(d) x 15. Consider the set A of all determinants of order 3 with entries 0 or 1 only. Let B be the subset of A consisting of all determinants with value 1. Let C be the subset of A consisting of all determinants with value –1. Then [1981 - 2 Marks] (a) C is empty (b) B has as many elements as C c) (d) B has twice as many elements as elements as C The trace of a square matrix is defined to be the sum of its diagonal entries. If A is a 2 × 2 matrix such that the trace of A is 3 and the trace of A3 is –18, then the value of the determinant of A is _____ [Adv. 2020] 17. Let P be a matrix of order 3 × 3 such that all the entries in P are from the set {–1, 0, 1}. Then, the maximum possible value of the determinant of P is ____ . [Adv. 2018]

w

w .je

16.

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18.

Let

= , where

= , and r, s ∈ {1, 2, 3}. Let

=

and I be the identity matrix of order 2. Then the total number of ordered pairs (r, s) for whichP2 = –I is [Adv. 2016]

www.jeebooks.in

Let ω be the complex num

20.

er Then the number of distinct

complex numbers z satisfy

ng is equal to [2010]

The value of the determin

nt is ..................

eb oo ks .in

19.

[1988 - 2 Marks]

21.

Given t

at is a roo

of the other two roots are

................. and ..............

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[1983 - 2 Marks] 22. A determinant is chosen at random from the set of all determinants of order 2 with elements 0 or 1 only. The probability that the value of determinant chosen is positive is .................. [1982 - 2 Marks] The solution set of the equat

w

23.

[1981 - 2 Marks] be an identity in ,

w

24. Let

on = 0 is .................

where p, q, r, s and t are constants. Then, the value of t is .................. [1981 - 2 Marks

www.jeebooks.in

25.

The determinants

and

are not identically equ

26.

eb oo ks .in

l.

Let x ∈ R and let

and R = PQP –1 Then which of the following options is/are correct?

[Adv. 2019]

, for all x ∈ R

(a) det

(b) For x = 1, there exists a unit vector

for which

(c) There exists a real number x such that PQ = QP , then a + b = 5

w .je

(d) For x = 0,if 27.

Which

of

the

following

values

of

α

satisfy

the

equation

[Adv. 2015]

–4 9 –9 4 Let M and N be two 3 × 3 matrices such that MN = NM. Further, if M ≠ N 2 and M 2 = N 4, then [Adv. 2014] 2 2 (a) determinant of (M + MN ) is 0

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(a) (b) (c) (d) 28.

www.jeebooks.in

29.

eb oo ks .in

(b) there is 3 × 3 non-zero matrix U such that (M 2 + MN2)U is the zero matrix (c) determinant of (M 2 + MN 2) ≥ 1 (d) for a 3 × 3 matrix U, if (M 2 + MN 2)U equals the zero matrix then U is the zero matrix The determinant

is equal to zero, if

[1986 - 2 Marks]

(a) a, b, c are in A. P. (b) a, b, c are in G.. P. (c) a , b , c are in H. P. d) is a root of the equat on + bx + c = 0 (e) ( – ) is a factor of ax2

+

2bx

+

c.

30.

Consider the lines given by L1 : x + 3y – 5 = 0; L2 : 3x – ky – 1 = 0; L3 : 5x + 2y – 12 = 0

Match the Statements / Expressions in Column I with the Statements / Expressions in Column II and indicate your answer by darkening the

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appropriate bubbles in the 4 × 4 matrix given in the ORS.

[2008]

Column I Column II (A) L1, L2, L3 are concurrent, if (B) One of L1, L2, L3 is parallel

(p) k = –9

q)

w

to at least one of the other two, if

(C) L1, L2, L3 from a triangle, if

w

(D) L1, L2, L3 do not form a triangle, if

r)

(s) k = 5 PASSAGE

www.jeebooks.in

Let p be an odd prime number and Tp be the following set of2 × 2 matrices :

w .je

eb oo ks .in

[2010] 31. The number of A in Tp such that A is either symmetric or skew-symmetric or both, and det(A) divisible by p is (a) (p – 1)2 (b) 2 (p – 1) (c) (p – 1)2 + 1 (d) 2p – 1 32. The number of A in Tp such that the trace of A is not divisible by p but det (A) is divisible by p is [Note: The trace of a matrix is the sum of its diagonal entries.] (a) (p – 1) (p2 – p + 1) (b) p3 – (p – 1)2 (c) (p – 1)2 (d) (p – 1) (p2 – 2) 33. The number of A in Tpsuch that det (A) is not divisible by p is (a) 2p2 (b) p3 – 5p (c) p3 – 3p (d) p3–p2 If .

and AX = U has

w

w

34

35.

infinitely many solutions, prove thatBX = V has no unique solution. Also show that if fd 0, then BX = V has no solution. [2004 - 4 Marks] Let a > 0, d > 0. Find the value of the determinant [1996 - 5 Mark

www.jeebooks.in

eb oo ks .in

]

36. For a fixed positive integer n, if

[1992 - 4 Marks]

=

then show t

37.

I

a

at is divisible by n.

p

b q

c

r

nd

= 0. Then find the value

w .je

of

38.

[1991 - 4 Marks] Let the three digit numbers A 28, 3B9, and 62 C, where A, B, and C are integers between 0 and 9, be divisible by a fixed integer k. Show that the

w

determin

w

39.

nt is divisible by k. [1990 - 4 Marks]

Show that

www.jeebooks.in

=

eb oo ks .in

40.

[1985 - 2 Marks] Let a, b, c be positive and not all equal. Show that the value of the determin nt is negative.

[1981 - 4 Marks

s 1.

If the minimum and the maximum values of the funct defined by

on ,

w .je

are m and M respectively, then the ordered pair (m, M) is equal to : [Main Sep. 05, 2020 (I)] (a) 0, ) (b) (– 4, 0 ) (c) (– 4, 4) (d) (0, 4) 2. If a + x = b + y = c + z + 1, where a, b, c, x, y, z are non-zero distinct real

w

numbers, t

w

(a) (b) (c) (d)

en is equal to : [Main Sep. 05, 2020 (II)]

y (b – a) y (a – b) 0 y (a – c)

www.jeebooks.in

Let two points be A(l, – 1) and B(0, 2). If a point P(x , y ) be such that the area of PAB = 5 sq. units and it lies on the line, 3x + y – 4 = 0, then a value of is: [Main Jan. 8, 2020 (I)]

(a) (b) (c) (d) 4.

4 3 1 –3 Let A = [aij] and B = [bij] be two 3 3 real matrices such that bij = (3)(i + j – 2) aij, where i, j = 1, 2, 3. If the determinant of B is 81, then the determinant of A is: [Main Jan. 7, 2020 (II)] 1/3 3 1/81 1/9

(a) (b) (c) (d) 5.

eb oo ks .in

3.

A value of q ∈ (0, p/3), for whi

= 0, is :

h

[Main April 12, 2019 (II)]

b)

w .je

a) 6.

c)

d)

Let the numbers 2, b, c be in an A.P. and = . If det(A)?[2, 16], then c lies in the interval : [Main April 08, 2019 (II)]

w

(a) [2, 3) (b) (2 + 23/4, 4)

w

(c) [4, 6] (d) [3, 2 + 23/4]

www.jeebooks.in

then for all, θ∈ interval :

det (A) lies in the

eb oo ks .in

7. If ;

[Main Jan. 12, 2019 (II)]

a) b) c) d) 8.

Let d∈R, and

=

w .je

q ∈ [0, 2p]. If the minimum value of det (A) is 8, then a value of d is: [Main Jan 10, 2019 (I)] (a) – 5 (b) – 7 c)

w

d)

w

9.

f

then A is: (a) invertible for all t∈R. (b) invertible only if t = π.

[Main Jan. 09, 2019 (II)]

www.jeebooks.in

(c) not invertible for any t∈R. (d) invertible only if

Let k be an integer such that triangle with vertices (k, –3k), (5, k) and (–k, 2) has area 28 sq. units. Then the orthocentre of this triangle is at the point : [Main 2017]

a) b) c) d)

f

w .je

11.

eb oo ks .in

10.

=.

then the value

of

[Main Online April 19, 2014]

depends only on a depends only on n depends both on a and n is independent of both a and n Let P = [aij] be a 3 × 3 matrix and let Q = [bij], where bij = 2i + j aijfor 1 ≤ i , j ≤ 3. If the determinant of P is 2, then the determinant of the matrix Q is [2012] (a) 210 (b) 211 (c) 212 (d) 213

w

w

(a) (b) (c) (d) 12.

www.jeebooks.in

13.

If and | A3 | = 125 then the value of α is

(a) (b) (c) (d)

±1 ±2 ±3 ±5

14.

If f(x

eb oo ks .in

[2004S]

= then

f (100) is equal to

[1999 - 2 Marks]

(a) (b) (c) (d)

0 1 100 –100

(1) is a cube root of unity, the

w .je

15. I

=

(a) 0 (b) 1 (c) i

d

Let k be a positive real number and let

w

w

16.

[1995S]

If det (adj A) + det (adj B) = 106. then [k] is equal to [Note : adj M denotes the adjoint of square matrix M and [k] denotes the largest integer less than or equal k. [2010]

www.jeebooks.in

17.

For positive numbers x, y and z, the numerical value of the determin is ..................

eb oo ks .in

nt

[1993 - 2 Marks] 18.

If

then the two triangles with vertice must be congruent. [1985 - 1 Marks] 19.

n

Which of the following is(are) not the square of a 3 × 3 matrix with real entries? [Adv. 2017]

w .je

a)

b)

w

c)

w

d)

www.jeebooks.in

20 If.

= x + iy, then

eb oo ks .in

[1998 - 2 Marks] (a) (b) (c) (d)

x = 3, y = 1 x = 1, y = 3 x = 0, y = 3 x = 0,y = 0

21.

If M is a 3 × 3 matrix, where det M = 1 and MMT = I, where ‘I’ is an identity matrix, prove that det (M – I) = 0. [2004 - 2 Marks]

22.

If matrix A =

where a, b, c are real positive numbers, abc =

1 and ATA = I, then find the value of a3 + b3 + c3. Prove

that

for

all

w .je

23.

[2003 - 2 Marks] values of

For all values of A, B, C and P, Q, R show that [1994 - 4 Mark ]

w

w

24.

[2000 - 3 Marks]

www.jeebooks.in

et

Show that,

=.

eb oo ks .in

25.

, a constant.

[1989 - 5

Marks]

26. Without

expanding

a

determinant

at

any

stage,

show

t

at , where A and B are determinants

of order 3 not involving x.

[1982 - 5 Marks]

w .je

s

1.

Let A be a 3 × 3 matrix such that adj A And

then the ordered pair, is equal to : [Main Sep. 03, 2020 (II)]

w

A).If

and B = adj(adj

a)

w

b)

(c) (3, 81)

d)

www.jeebooks.in

If the matrices

and C = 3A, t

= B = adj A

en is equal to :

eb oo ks .in

2.

[Main Jan. 9, 2020 (I)]

(a) (b) (c) (d) 3.

8 16 72 2 If

= is the inverse of a 3 × 3 matrix A, then the sum of all

values of a for which det (A) + 1 = 0, is :

[Main April 12, 2019 (I)]

(a) (b) (c) (d)

If.

........

w .je

4

0 –1 1 2

then the inverse

...,

of is:

[Main April 09, 2019 (II)]

a)

w

b)

w

c)

d)

www.jeebooks.in

Let A and B be two invertible matrices of order 3 × 3. If det (ABAT) = 8 and det (AB–1) = 8, then det (BA–1 BT) is equal to : [Main Jan. 11, 2019 (II)]

a) 6.

(b) 1 Let

c)

(d) 16

eb oo ks .in

5.

=

Where α = α(θ) and β = β(θ) are real numbers, and I is the 2 × 2 identity matrix. If a* is the minimum of the set {α(θ) : θ ∈ [0, 2 π)} and β* is the minimum of the set {β(θ) : θ ∈ [0, 2π)}. Then the value of a* + b* is [Adv. 2019] a) b) c) d)

Suppose A is any 3 × 3 non-singular matrix and (A – 3I) (A – 5I) = O, where I = I3 and O = O3. If αA + βA–1 = 4I, then α + β is equal to [Main Online April 15, 2018]

w .je

7.

(a) (b) (c) (d) 8.

w

w

(a) (b) (c) (d)

8 12 13 7 Let A be any 3 × 3 invertible matrix. Then which one of the following is not always true ? [Main Online April 8, 2017] adj (A)= |A| . A–1 adj (adj(A)) = |A|.A adj (adj(A)) = |A|2 .(adj(A))–1 adj (adj(A)) = |A|.(adj(A))–1

9

If.

and A adj A = A AT, then 5a + b is equal to: [Main 2016]

www.jeebooks.in

4 13 –1 5 If A is an 3 × 3 non-singular matrix such that AA’ = A’A andB = A–1A’, then BB’ equals: [Main 2014]

(a) B –1 11.

b)

If equal to :

eb oo ks .in

(a) (b) (c) (d) 10.

c)

(d) I

= is the adjoint of a 3 × 3 matrix A and|A| = 4, then α is

[Main 2013]

(a) (b) (c) (d)

4 11 5 0

12.

w .je

And

and A–1=,

then the value of c and d are

(–6, –11) (6, 11) (–6, 11) (6, – 1)

w

(a) (b) (c) (d)

[2005S]

Let M be a 3 × 3 invertible matrix with real entries and let I denote the 3 × 3 identity matrix. If M–1 = adj (adj M), then which of the following statements is/are ALWAYS TRUE? (a) M = I (b) det M = 1 [Adv. 2020] 2 (c) M = I

w

13.

www.jeebooks.in

(d) (adj M)2 = I Let

and Wh

eb oo ks .in

14.

re denotes the transpose of the matrix Pk. Then which of the following options is/are correct?

[Adv. 2019]

(a) (b) (c)

X is a symmetric matrix The sum of diagonal entries of X is 18 X – 30I is an invertible matrix

(d)

If then a = 30

Let

= and (adj M

w .je

15.

= where a and b are real

numbers. Which of the following options is/are correct ?

a+b=3 det (adj M 2) = 81 (adjM)–1 + adjM–1 = –M

(d)

I

w

(a) (b) (c)

[Adv. 2019]

M , then α – β + γ = 3

w

16. For 3 × 3 matrices M and N, which of the following statement(s) is (are) NOT correct? [Adv. 2013]

www.jeebooks.in

eb oo ks .in

(a) NTMN is symmetric or skew symmetric, according as M is symmetric or skew symmetric (b) MN – NM is skew symmetric for all symmetric matrices M and N (c) MN is symmetric for all symmetric matrices M and N (d) (adj M) (adj N) = adj (MN) for all invertible matrices M and N 17. Let M and N be two 3 × 3 non-singular skew- symmetricmatrices such that MN = NM. If PT denotes the transpose of P, then M2N2 (MTN)–1 (MN–1)T is equal to

[2011]

(a) (b) (c) (d)

M2 – N2 – M2 MN

PASSAGE

Let

w .je

and U1, U2 and U3 are columns of a 3 × 3 matrix U. If column

matrices U1, U2 and U3 satisfying

18.

evaluate as directed in the following questions. The value |U| is

(a) 3

(b)

w

c)

w

19.

[2006 - 5M, –2]

–3

(d)

2

The sum of the elements of the matrix U–1 is [2006 - 5M, –2]

(a) –1(b) 0 (c) 1 (d) 3

www.jeebooks.in

20.

The value of [3 2 b)

(c) 4

d)

1.

The values of λ and µ for which the system of linear equations [Main Sep. 06, 2020 (I)] x+y+z=2 x + 2y + 3z = 5 x + 3y + λz = µ has infinitely many solutions are, respectively : 6 and 8 5 and 7 5 and 8 4 and 9 et . The system of linear equation

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(a) (b) (c) (d) 2.

[2006 - 5M, –2]

eb oo ks .in

(a) 5

] U is

]

[Main Sep. 05, 2020 (I)

w

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(a) exactly one negative value of (b) exactly one positive value of (c) every value of (d) exactly two value of 3. If the system of linear equation

www.jeebooks.in

has a non-zero solution (x, y, z) for s

me t

en is equal to : [Main Sep. 05, 2020 (II)]

–3 9 3 –9 Suppose the vectors x1, x2 and x3 are the solutions of the system of linear equations, Ax = b when the vector b on the right side is equal to b1, b2 and

eb oo ks .in

(a) (b) (c) (d) 4.

b3 respectively. I

a

d

then the determinant of A is equal to :

[Main Sep. 04, 2020 (II)]

(a) 4

c)

d)

If the system of equation

w .je

5.

(b) 2

has infinitely many solutions, then :

[Main Sep. 04, 2020 (II)]

a)

b)

w

c)

w

d) 6.

Let S be the set of

ll for which the system of linear equations [Main Sep. 02, 2020 (I]

www.jeebooks.in

7.

eb oo ks .in

has no solution. Then the set S (a) contains more than two elements. (b) is an empty set. (c) is a singleton. (d) contains exactly two elements. Let A = {X = (x,

y,

z)T : PX = 0

And

where,

then the set A :

[Main Sep. 02, 2020 (II)]

(a) is a singleton (b) is an empty set (c) contains more than two elements (d) contains exactly two elements 8. The following system of linear equations 7x + 6y – 2z = 0 3x + 4y + 2z = 0 x – 2y – 6z = 0, has

[Main Jan. 9, 2020 (II)]

infinitely many solutions, (x, y, z) satisfying y = 2z. no solution. infinitely many solutions, (x, y, z) satisfying x = 2z. only the trivial solution. For which of the following ordered pairs ( , ), the system of linear equations x + 2y + 3z = 1 3x + 4y + 5z = 4x + 4y + 4z = is inconsistent? [Main Jan. 8, 2020 (I)] (a) (4, 3) (b) (4, 6) (c) (1, 0) (d) (3, 4)

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(a) (b) (c) (d) 9.

www.jeebooks.in

The system of linear equations x + 2y + 2z = 5 2 x + 3y + 5z = 8 4x + y + 6z = 10 has: [Main Jan. 8, 2020 (II)] (a) no solution when = 8 (b) a unique solution when = –8 (c) no solution when = 2 (d) infinitely many solutions when = 2 11. If the system of linear equations 2x + 2ay + az = 0 2x + 3by + bz = 0 2x + 4cy + cz = 0, where a, b, c R are non-zero and distinct; has a non-zero solution, then: [Main Jan. 7, 2020 (I)] a)

eb oo ks .in

10.

are in A.P.

(b) a, b, c are in G.P.

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(c) a + b + c = 0 (d) a, b, c are in A.P. 12. If the system of linear equations x+y+z=5 x + 2y + 2z = 6 x + 3y + lz = m, (l, m ∈R), has infinitely many solutions, then the value of l + m is : [Main April 10, 2019 (I)] (a) 12 (b) 9 (c) 7 (d) 10 13. If the system of equations 2x + 3y – z =0, x + ky – 2z = 0 and 2x – y + z = 0

w

has a non-trivial solution (x, y, z), then

a)

b)

is is equal to: [Main April 09, 2019 (II)]

c)

(d) –4

14. The set of all values of λ for which the system of linear equations x – 2y – 2z = λx

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x + 2y + z = λy –x – y = λ2 has a non-trivial solution : [Main Jan. 12, 2019 (II)] is a singleton contains exactly two elements is an empty set contains more than two elements The number of values of θ ∈ (0, π) for which the system of linear equations x + 3y + 7z = 0 – x + 4y + 7z = 0 (sin 3θ)x + (cos 2θ)y + 2z = 0 has a non-trivial solution, is: [Main Jan. 10, 2019 (II)] (a) three (b) two (c) four (d) one 16. If the system of linear equations x + ky + 3z = 0 3x + ky – 2z = 0 2x + 4y – 3z = 0

eb oo ks .in

(a) (b) (c) (d) 15.

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has a non-zero solution (x, y, z), t

en is equal to :

[Main 2018]

10 – 30 30 – 10 If S is the set of distinct values of ‘b’ for which the following system of linear equations [Main 2017] x+ y+ z=1 x + ay + z = 1 ax + by + z = 0 has no solution, then S is : (a) a singleton (b) an empty set

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(a) (b) (c) (d) 17.

www.jeebooks.in

eb oo ks .in

(c) an infinite set (d) a finite set containing two or more elements 18. The number of values of k, for which the system of equations: (k + 1) x + 8y = 4k kx + (k + 3)y = 3k – 1 has no solution, is

[Main 2013]

(a) (b) (c) (d) 19.

infinite 1 2 3 The number of 3 × 3 matrices A whose entries are either 0 or 1 and for which the system

has exactly two distinct solutions, is

[2010]

w

w

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(a) 0 (b) 29–1 (c) 168 (d) 2 20. Given 2x – y + 2z = 2, x – 2y + z = – 4, x + y + λz = 4 then the value of λ such that the given system of equation has NO solution, is [2004S] (a) 3 (b) 1 (c) 0 (d) – 3 21. If the system of equations x + ay = 0, az + y = 0 and ax + z = 0 has infinite solutions, then the value of a is [2003S] (a) –1 (b) 1 (c) 0 (d) no real values 22. The number of values of k for which the system of equations (k + 1)x + 8y = 4k; kx + (k + 3) y = 3k – 1 has infinitely many solutions is [2002S]

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(a) (b) (c) (d) 24.

0 1 2 infinite If the system of equations x – ky – z = 0, kx – y – z = 0, x + y – z = 0 has a non-zero solution, then the possible values of k are [2000S] –1, 2 1, 2 0, 1 –1, 1 Let a, b, c be the real numbers. Then following system of equations in x, y and z [1995] +

– =

,

–

+ =1,

–

+

+ = 1 has

no solution unique solution infinitely many solutions finitely many solutions

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(a) (b) (c) (d)

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(a) (b) (c) (d) 23.

25.

For a real number α, if the system

of linear equations, has infinitely many solutions, then 1 + α + a2 = [Adv. 2017] The sum of distinct values of for whcih the system of equation

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26.

2x + (3λ + 1) y + 3(λ – 1) z = 0, has non-zero solutions, is ______. 27.

If the system of equations

[Main Sep. 06, 2020 (II)] ,

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such that

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has infinitely many solutions, thena – b is equal to __________.[Main Sep. 04, 2020 (I)] 28. Let S be the set of all integer solutions, (x, y, z), of the system of equations

Then, the number of elements in the set S is

equal to ____________.

[Main Sep. 03, 2020 (II)]

29.

If the system of linear equations, x+y+z=6

x + 2y + 3z = 10 3x + 2y + z =

has more than two solutions, then

–

2

is equal to



[Main Jan. 7, 2020 (I)]

The system of equations

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30.

Will have a non-zero solution if real values of λ are given by .................. [1984 - 2 Marks

Let S be the set of all column matrices

such that b1, b2, b3¸ ∈

and

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31.

the system of equations (in real variables) –x + 2y + 5z = b1 2x – 4y + 3z = b2 x – 2y + 2z = b3

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has at least one solution. Then, which of the following system(s) (in real ∈ S?

eb oo ks .in

variables) has (have) at least one solution for each

[Adv. 2018]

(a) x + 2y + 3z = b1, 4y + 5z = b2 and x + 2y + 6z = b3 (b) x + y + 3z = b1, 5x + 2y + 6z = b2 and –2x – y – 3z = b3 (c) –x + 2y – 5z = b1, 2x – 4y + 10z = b2 and x – 2y + 5z = b3 (d) sx + 2y + 5z = b1, 2x + 3z = b2 and x + 4y – 5z = b3 32. t . Consider the system of linear equations ax + 2 = 3x – 2 = Which of the following statement(s) is (are) correct? (a) (b) (c) (d)

[Adv. 2016] If a = –3, then the system has infinitely many solutions for all values of and. I a –3, then the system has a unique solution for all values of and. I += 0, then the system has infinitely many solutions for a = –3. I +0, then the system has no solution fora = –3.

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PASSAGE

A

Let be the set of all 3× 3 symmetric matrices all of whose entries are either 0 or 1. Five of these entries are 1 and four of them are 0.

A is

The number of matrices in 12 6 9 3

34.

The number of matrices A in

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[2009]

A for which the system of linear equations

A

has a unique solution, is

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[2009] less than 4 at least 4 but less than 7 at least 7 but less than 10 at least 10

35.

The number of matrices A in

eb oo ks .in

(a) (b) (c) (d)

A for which the system of linear equations

A

is inconsistent, is

[2009]

(a) 0 (b) more than 2 (c) 2

Consider the system of equations x – 2y + 3z = –1 –x + y – 2z = k x – 3y + 4z = 1 STATEMENT - 1 : The system of equations has no solution

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36.

(d 1

STATEMENT-2 : The determinant ,

(a)

and

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[2008] STATEMENT - 1 is True, STATEMENT - 2 is True; STATEMENT - 2 is a correct explanation for STATEMENT - 1 STATEMENT - 1 is True, STATEMENT - 2 is True; STATEMENT - 2 is NOT a correct explaination for STATEMENT - 1 STATEMENT - 1 is True, STATEMENT - 2 is False STATEMENT - 1 is False, STATEMENT - 2 is True

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(b)

for

or

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37.

Let λ and α be real. Find the set of all values of λ for which the system of linear equatio s ,

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has a non-trivial solution. For λ = 1, find all values of α. 38.

Consider the system of linear equations in x, y, z

[1993 - 5 Marks] :

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x–y+z= 0 x + 4y + 3z = 0 2x + 7y + 7z = 0 Find the values of for which this system has nontrivial solutions. [1986 - 5 Marks] 39. For what value of k do the following system of equations possess a non trivial (i.e., not all zero) solution over the set of rationals Q? x + ky + 3z = 0 3x + ky – 2z = 0 2x + 3y – 4z = 0 For that value of k, find all the solutions for the system. [1979]

www.jeebooks.in

eb oo ks .in

1.

If a function f (x) defined by

be continuous for some

then the value of a is :

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and

[Main Sep. 02, 2020 (I)]

(a)

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(b) (c)

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(d)

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2.

Let [t] denote the greatest integer ≤ t and

Then the

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function, f(x) = [x2] sin(πx) is discontinuous, when x is equal to : [Main Jan. 9, 2020 (II)] (a) (b) (c) (d) 3.

If

where [x] denotes the greatest integer

function, then:

[Main April 09, 2019 (II)]

(a) fis continuous at x = 4. f(x) exists but

(b)

f(x) and

(c) Both (d)

f(x) exist but are not equal.

f(x) does not exist.

If the function

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4.

f(x) exists but

f(x) does not exist.

f(x) =

is continuous at x = 5, then the value of a – b is: [Main April 09, 2019 (II)]

w

(a)

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(b) (c)

(d)

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5.

Let f : [– 1, 3] → R be defined as

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f(x) = where [t] denotes the greatest integer less than or equal to t. Then, f is discontinuous at : [Main April 08, 2019 (II)] (a) only one point (b) only two points (c) only three points (d) four or more points 6.

Let f (x) =

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The value of k for which f is continuous at x = 2 is [Main Online April 15, 2018] (a) e–2 (b) e (c) e–1 (d) 1 7. The value of k for which the function

w

is continuous at x =

, is :

[Main Online April 9, 2017]

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(a)

(b)

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(c)

8.

eb oo ks .in

(d) Let k be a non–zero real number.

[Main Online April 11, 2015]

If f(x) =

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is a continuous function then the value of k is: (a) 4 (b) 1 (c) 3 (d) 2 9. If the function

is continuous at x = π, then k equals: [Main Online April 19, 2014]

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(a) 0 (b)

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(c) 2 (d)

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10.

Let

f

be

a

composite

function

of

x

defined

by

.

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Then the number of points x where f is discontinuous is : [Main Online April 23, 2013] (a) 4 (b) 3 (c) 2 (d) 1 11. (a) (b) (c) (d)

The function f(x) = [x]2 – [x2] (where [y] is the greatest integer less than or equal to y), is discontinuous at [1999 - 2 Marks] all integers all integers except 0 and 1 all integers except 0 all integers except 1

12.

The function f(x) = [x]

, [.] denotes the greatest integer

function, is discontinuous at

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[1995S]

(a) All x (b) All integer points (c) No x (d) x which is not an integer

Let f (x) =

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13.

14.

, for –10 < x < 10, where [t] denotes the greatest

integer function. Then the number of points of discontinuity of f is equal to ______. [Main Sep. 05, 2020 (I)]

If the function f defined on

by

www.jeebooks.in

f(x) =

is continuous, then k is equal to

eb oo ks .in

.

[Main Jan. 7, 2020 (II)]

15.

Let f(x) be a continuous function defined for 1 x 3.If f(x) takes rational values for all x and f (2) = 10, thenf(1.5) =............. [1997 - 2 Marks]

16.

Let f (x) = [x] sin

, where

denotes the greatest integer

function. The domain of f is... and the points of discontinuity of f in the domain are..... (1996 - 2 Marks) 17. Let f(x) =

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If f(x) is continuous for all x, then k = ..........

[1981 - 2 Marks]

Let [x] be the greatest integer less than or equals to x. Then, at which of the following point(s) the functionf(x) = x cos(π(x + [x])) is discontinuous? [Adv. 2017] (a) x = – 1 (b) x = 0 (c) x = 1 (d) x = 2 19. For every pair of continuous functions f, g : [0, 1] → R such that max = max , the correct statement(s) is

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(are): [Adv. 2014]

www.jeebooks.in

eb oo ks .in

(a) (f (c))2 + 3f (c) = (g(c))2 + 3g(c) for some c [0, 1] (b) (f (c))2 + f (c) = (g(c))2 + 3g(c) for some c [0, 1] (c) (f (c))2 + 3f (c) = (g(c))2 + g(c) for some c [0, 1] (d) (f (c))2 = (g(c))2 for some c ∈ [0, 1] 20. For every integer n, let an and bn be real numbers. Let function f : IR → IR be given by [2012]

for all integers n. If f is continuous, then which of the following hold(s) for all n ? (a) an–1 – bn–1= 0 (b) an – bn= 1 (c) an – bn+1= 1 (d) an–1 – bn= –1 21. The following functions are continuous on (0, ). [1991 - 2 Marks] (a) tan x

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(b)

w

(c)

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(d)

22.

If f (x) =

x – 1, then on the interval [0, π] [1989 - 2 Marks]

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(a) tan [f(x)] and 1/f(x) are both continuous (b) tan [f(x)] and 1/f(x) are both discontinuous (x) are both continuous (c) tan [f(x)] and

23.

Let

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(d) tan [f(x)] is continuous but 1/f(x) is not.

[1994 - 4 Marks]

Determine a and b such that f(x) is continuous at x = 0

24.

Let

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[1990 - 4 Marks] Determine the value of a, if possible, so that the function is continuous at x =0 25. Find the values of a and b so that the function

w

f (x) =

is continuous for

w

[1989 - 2 Marks] 26. Let f (x) be a continuous and g (x) be a discontinuous function. prove that f (x) + g (x) is a discontinuous function. [1987 - 2 Marks]

www.jeebooks.in

27.

Let

1.

(a) (b) (c) (d)

Let f : R → R be a function defined by f (x) = max{x, x2}. Let S denote the set of all points in R, where f is not differentiable. Then: [Main Sep. 06, 2020 (II)] {0, 1} {0} (an empty set) {1} If the function

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2.

eb oo ks .in

[1983 - 2 Marks] Determine the form of g(x) = f ( f(x)) and hence find the points of discontinuity of g, if any 28. Let f (x + y) = f (x) + f (y) for all x and y. If the function f(x)is continuous at x = 0, then show that f (x) is continuous at all x. [1981 - 2 Marks]

is twice differentiable,

then the ordered pair (k1, k2) is equal to: [Main Sep. 05, 2020 (I)]

(a)

w

(b) (1, 0)

w

(c)

(d) (1, 1) 3. Let f be a twice differentiable function on (1, 6). If f (2) = 8, f ‘(2) = and for all then : 5, f ‘(x) [Main Sep. 04, 2020 (I)]

www.jeebooks.in

4.

The function

eb oo ks .in

(a) (b) (c) (d)

[Main Sep. 04, 2020 (II)]

(a) continuous on R – {1} and differentiable on R – {–1, 1}. (b) both continuous and differentiable on R – {1}. (c) continuous on R – {–1} and differentiable on R – {–1, 1}. (d) both continuous and differentiable on R – {–1}.

If

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5.

is continuous at x = 0, then a + 2b is equal to: [Main Jan. 9, 2020 (I)]

(a) 1

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(b) –1 (c) 0

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(d) –2 6.

Let f and g be differentiable functions on R such that fog is the identity function. If for some a, b ∈ R, g′ (a) = 5 and g (a) = b, then f ′ (b) is equal to:

[Main Jan. 9, 2020 (II)]

www.jeebooks.in

(a) (b) 1

eb oo ks .in

(c) 5 (d) 7.

Let S be the set of all functions f : [0,1]

R, which are continuous on

[0, 1] and differentiable on (0,1). Then for every f in S, there exists a c (0,1), depending on f, such that:

[Main Jan. 8, 2020 (II)]

(a) | f (c) – f (l)| < (l – c)|f =f

(b)

(c)

(c) | f (c) + f (1)| < (1 + c) |f (d) | f (c) – f (1)| < | f 8.

(c)|

(c)|

(c)|

Let the function, f: [–7, 0]

R be continuous on [ –7, 0] and

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differentiable on (–7, 0). If f(–7) = –3 and f ′ (x) d”2, for all x (–7, 0), then for all such functions f, f ′(–1) + f(0) lies in the interval:

(a) (–

[Main Jan. 7, 2020 (I)]

, 20]

(b) [–3, 11] 

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(c) (–

(d) [–6, 20]

Let f(x) = loge (sinx), (0 < x < p) and g(x) = sin–1 (e–x), (x > 0). If a is a positive real number such that a = (fog) (a) andb = (fog) ( a), then: [Main April 10, 2019 (II)] (a) aa2 + ba + a = 0

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9.

www.jeebooks.in

eb oo ks .in

(b) aa2 – ba – a =1 (c) aa2 – ba – a = 0 (d) aa2 + ba – a = – 2a2 10. Let f : R → R be differentiable at c ∈ R and f (c) = 0. If g (x) = , then at x = c, g is :

[Main April 10, 2019 (I)]

(a) not differentiable if f ‘(c) = 0 (b) differentiable if f “(c) ≠ 0 (c) differentiable if f ‘ (c) = 0

(d) not differentiable 11. If f (1) = 1, f(1) = 3, then the derivative of f (f (f (x))) + (f (x))2 at x = 1 is :

[Main April 08, 2019 (II)]

(a) 33 (b) 12 (c) 15

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(d) 9 12. Let f be a differentiable function such that f(1) = 2 and f ′ (x) = f (x) for all x R. If h (x) = f( f (x)), then h′ (1) is equal to : [Main Jan. 12, 2019 (II)] 2 (a) 2e (b) 4e (c) 2e (d) 4e2 Let

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g(x) = |f(x)| + f(|x|). Then, in the interval (–2, 2), g is : [Main Jan. 11, 2019 (I)] (a) differentiable at all points

www.jeebooks.in

15.

Let f (x) =

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(b) not continuous (c) not differentiable at two points (d) not differentiable at one point 14. Let K be the set of all real values of x where the function f (x) = sin | x | – | x | + 2 (x – π) cos | x | is not differentiable. Then the set K is equal to : [Main Jan. 11, 2019 (II)] (a) φ (an empty set) (b) {π} (c) {0} (d) {0, π}

Let S be the set of points in the interval (– 4, 4) at which f is not differentiable. Then S:

[Main Jan 10, 2019 (I)]

(a) is an empty set (b) equals {– 2, – 1, 0, 1, 2}

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(c) equals {– 2, – 1, 1, 2} (d) equals {– 2, 2} 16.

is not differentiable at t}.

Then the set S is equal to : [Main 2018] {0} {π} {0, π} φ (an empty set) If the function.

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(a) (b) (c) (d) 17.

Let S = {

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is differentiable, then the value of k + m

(a) (b) 4 (c) 2 (d)

18.

[Main 2015]

eb oo ks .in

is :

Let f, g: R → R be two functions defined by

,

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and g(x) = x f(x) Statement I: f is a continuous function at x = 0. Statement II: g is a differentiable function at x = 0. [Main Online April 12, 2014] (a) Both statement I and II are false. (b) Both statement I and II are true. (c) Statement I is true, statement II is false. (d) Statement I is false, statement II is true. Let

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19.

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(a) (b) (c) (d)

[2012]

differentiable both at x = 0 and at x = 2 differentiable at x = 0 but not differentiable at x = 2 not differentiable at x = 0 but differentiable at x = 2 differentiable neither at x = 0 nor at x = 2

www.jeebooks.in

20.

Let

0 < x < 2, m and n are integers, , and let p be the left hand derivative of |x – 1| at x = 1. If

eb oo ks .in

g(x) = p, then

[2008]

(a) (b) (c) (d) 21.

n = 1, m = 1 n = 1, m = – 1 n = 2, m = 2 n > 2, m = n Let f (x) be differentiable on the interval (0, ∞) such thatf (1) = 1, and for each x > 0. Thenf (x) is\

[2007 - 3 marks]

(a)

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(b) (c)

(d)

If f (x) is continuous and differentiable function and f (1/n) = 0 n≥ 1and n ∈ I, then [2005S] f(x) = 0, x ∈ (0, 1] f(0) = 0, f ‘(0) = 0 f(0) = 0 = f ‘(0), x ∈ (0, 1] f(0) = 0 and f ‘(0) need not to be zero The function given by y = ||x| – 1| is differentiable for all real numbers except the points [2005S]

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(a) (b) (c) (d) 23.

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{0, 1,–1} ±1 1 –1 The domain of the derivative of the function

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(a) (b) (c) (d) 24.

[2002S]

(a) (b) (c) (d) 25.

R – {0} R – {1} R – {–1} R – {–1, 1} Which of the following functions is differentiable at x = 0?

[2001S]

cos(|x|) + |x| cos (|x|) − |x| sin (|x|) + |x| sin (|x|) − |x| Let f : R → R be a function defined by f (x) = max {x, x3}. The set of all points where f (x) is NOT differentiable is [2001S] {−1, 1} {−1, 0} {0, 1} (d){−1, 0, 1} The left-hand derivative of f(x) = [x] sin(π x) at x = k, k an integer, is [2001S] k (−1) (k − 1)π (−1)k − 1(k −1)π (−1)kkπ (−1)k − 1kπ is NOT The function f (x) =

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(a) (b) (c) (d) 26. (a) (b) (c) 27.

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(a) (b) (c) (d) 28.

differentiable at [1999 - 2 Marks]

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–1 0 1 2 Let [.] denote the greatest integer function and f(x) = [tan2 x], then: [1993 - 1 Mark] does not exist (a)

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(a) (b) (c) (d) 29.

(b) f (x) is continuous at x = 0 (c) f (x) is not differentiable at x = 0 (d) f ‘ (0) = 1 R be a differentiable function and f (1) = 4. Then the 30. Let f : R value of

is

[1990 - 2 Marks]

(a) 8 (b) 4 (c) 2 If

,

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(d) 31.

,

, then the value of

is [1983 - 1 Mark]

(a) –5 (b)

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(c) 5 (d) none of these 32. For a real number y, let [y] denotes the greatest integer less than or equal to y : Then the function

(a) discontinuous at some x

is [1981 - 2 Marks]

www.jeebooks.in

33.

eb oo ks .in

(b) continuous at all x, but the derivative f ′ (x) does not exist for some x (c) f ′ (x) exists for all x, but the second derivative f ′ (x) does not exist for some x (d) f ′ (x) exists for all x Let the functions by

and

be defined

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and where [x] denotes the greatest integer less than or equal to x. Let be the composite function defined by Suppose c is the number of points in the is NOT continuous, and suppose d is interval (–1, 1) at which is NOT the number of points in the interval (–1, 1) at which differentiable. Then the value of c + d is _____ [Adv. 2020] and be respectively given by f (x) = | x | + 1 34. Let by and g(x) = x2 + 1. Define

The number of points at which h(x) is not differentiable is 35.

Let f : [1,

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2. If

36.

)

[2,

[Adv. 2014] ) be a differentiable function such that f (1) = for all

, then the value of f (2) is [2011]

Suppose a differentiable function f (x) satisfies the identity for all real x and y. If

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then f ‘(3) is equal to ___________. [Main Sep. 04, 2020 (I)] Let S be the set of points where the function,

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37.

f(x) = |2 – |x – 3||, x R, is not differentiable. f(f(x)) is equal to

Then

.

[Main Jan. 7, 2020 (I)]

38.

Let f(x) = x | x |. The set of points where f(x) is twice differentiable is ..................... [1992 - 2 Marks]

39.

Let f(x) =

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be a real-valued function. Then the set of points where f(x) is not differentiable is ....................... [1981 - 2 Marks] 40.

Let

the

function. Let

function

be and let

defined

by

be an arbitrary

be the product function defined by Then which of the following statements is/are

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TRUE?

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(b) (c) (d) 41.

[Adv. 2020] (a) If g is continuous at x = 1, then fg is differentiable at x = 1 If fg is differentiable at x = 1, then g is continuous at x = 1 If g is differentiable at x = 1, then fg is differentiable at x = 1 If fg is differentiable at x = 1, then g is differentiable at x = 1 given by Let

www.jeebooks.in

eb oo ks .in

[Adv. 2019]

Then which of the following options is/are correct ? (a) f’ has a local maximum at x = 1 (b) f is increasing on (–∞, 0) (c) f ′ is NOT differentiable at x = 1 (d) f is onto and be two non-constant differentiable 42. Let functions. If for all , , then which of the following statement (s) is (are)

and TRUE ?

[Adv. 2018]

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(a) (b) (c) (d)

43.

Let f :

and g :

be functions defined by f

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(x) = [x2–3] and g(x) = |x| f (x) + |4x–7 | f (x), where [y] denotes the greatest integer less than or equal to y for y R. Then [Adv. 2016]

w

(a) f is discontinuous exactly at three points in (b) f is discontinuous exactly at four points in

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(c) g is NOT differentiable exactly at four points in

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(d) g is NOT differentiable exactly at five points in 44. Let a, b and f : be defined by 3 f (x) = a cos (|x –x|) + b |x| sin (|x3 +x|). Then f is

[Adv. 2016]

(a) (b) (c) (d) 45.

differentiable at x=0 if a=0 and b=1 differentiable at x=1 if a=1 and b=0 NOT differentiable at x=0 if a=1 b=0 NOT differentiable at x=1 if a=1 and b=1 Let g : R → R be a differentiable function with g(0) = 0,

g’(0) = 0 and g’(1) ≠ 0. Let

w

w

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and h(x) = e|x| for all . Let (foh)(x) denote f(h(x)) and (hof)(x) denote h(f(x)). Then which of the following is (are) true?

www.jeebooks.in

[Adv. 2015] f is differentiable at x = 0 h is differentiable at x = 0 foh is differentiable at x = 0 hof is differentiable at x = 0 be a continuous function and letg : R → R be defined as Let f : [a, b] →

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(a) (b) (c) (d) 46.

[Adv. 2014]

(a) (b) (c) (d)

g(x) is continuous but not differentiable at a g(x) is differentiable on R g(x) is continuous but not differentiable at b g(x) is continuous and differentiable at either (a) or (b) but not both

If f (x) =

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47.

[2011]

(a) f (x) is continuous at x =

(b) f (x) is not differentiable at x = 0 (c) f (x) is differentiable at x = 1 (d) f (x) is differentiable at x = 48.

x, y

R. If f (x) is

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(a) (b) (c) (d) 49.

Let f : R →R be a function such that f (x + y) = f (x) + f (y), differentiable at x = 0, then [2011] f (x) is differentiable only in a finite interval containing zero x R f (x) is continuous f ′(x) is constant x R f (x) is differentiable except at finitely many points. If f(x) = min {1, x2, x3}, then

(a) f(x) is continuous ∀ x ∈ R (b) f(x) is continuous and differentiable everywhere. (c) f(x) is not differentiable at two points

[2006 - 5M, –1]

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(d) f(x) is not differentiable at one point 50. Let h(x) = min {x, x2}, for every real number of x, Then [1998 - 2 Marks] h is continuous for all x h is differentiable for all x h’(x) =1, for all x > 1 h is not differentiable at two values of x. The function f(x) = max {(1 – x), (1 + x), 2},

(a) (b) (c) (d)

continuous at all points differentiable at all points differentiable at all points except at x = 1 and x = – 1 continuous at all points except at x = 1 and x = –1, where it is discontinuous

eb oo ks .in

(a) (b) (c) (d) 51.

is

[1995]

52.

. At x = 0

Let g(x) = x f(x), where

[1994]

(a) (b) (c) (d)

g is differentiable but g’ is not continuous g is differentiable while f is not both f and g are differentiable g is differentiable and g’ is continuous

53.

Let

then for all x

[1994]

f ‘ is differentiable f is differentiable f ‘ is continuous f is continuous

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(a) (b) (c) (d)

54.

is [1988 - 2 Marks]

continous at x = 1 differentiable at x = 1 continous at x = 3 differentiable at x = 3.

w

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(a) (b) (c) (d)

The function f(x) =

55. The set of all points where the function f(x) =

is differentiable, is [1987 - 2 Marks]

(a)

(b)

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(c) (d) (e) None 56. Let [x] denote the greatest integer less than or equal to x. If f(x) = [x sin continuous at x = 0 continuous in (– 1, 0) differentiable at x = 1 differentiable in (– 1, 1) none of these The function f (x) = 1 + | sin x | is

(a) (b) (c) (d) (e) 58.

continuous nowhere continuous everywhere differentiable nowhere not differentiable at x = 0 not differentiable at infinite number of points. , then– If

eb oo ks .in

(a) (b) (c) (d) (e) 57.

x], then f(x) is [1986 - 2 Marks]

[1986 - 2 Marks]

[1985 - 2 Marks]

(a) (b) (c) (d) 59.

f (x) is continuous but not differentiable at x = 0 f (x) is differentiable at x = 0 f (x) is not differentiable at x = 0 none of these , then y as a function of x is If

[1984 - 3 Marks]

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(a) defined for all real x (b) continuous at x = 0 (c) differentiable for all x (d) such that

60.

Let f1 :

=

→

for x < 0

, f2 :

→

, f3 :

→

and f4 :

→

be functions

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defined by f1 (x) = sin

,

w

(i)

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(ii)

f2 (x) =

, where the inverse trigonometric function tan–1 x assumes values

,

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in

(iii) f3 (x) = [sin (loge (x + 2))], where, for t ∈ equal to t,

, [t] denotes the greatest integer less than or

.

(iv) f4 (x) =

LIST - I P. The function f1 is Q. The function f2 is differentiable at x = 0 R. The function f3 is is NOT continuous at x = 0 S. The function f4 is is continuous at x = 0

1.

LIST - II NOT continuous at x = 0 2. continuous at x = 0 and NOT

3.

differentiable at x = 0 and its derivative

4.

differentiable at x = 0 and its derivative

The correct option is: [Adv. 2018]

P → 2; Q → 3; R → 1; S → 4 P → 4; Q → 1; R → 2; S → 3 P → 4; Q → 2; R → 1; S → 3 P → 2; Q → 1; R → 4; S → 3 , Let

,

and

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(a) (b) (c) (d) 61.

f2(x)

=

x2 ;

be

defined by

and

P. f4 is Q. f3 is

1. 2.

[Adv. 2014] List-II Onto but not one-one Neither continuous nor one-one

R.

3.

Differentiable but not one-one

4.

Continuous and one-one

w

w

List-I

f2of1 is

S. f2 is (a) P → 3; Q → 1; R → 4; S → 2 (b) P → 1; Q → 3; R → 4; S → 2 (c) P → 3; Q → 1; R → 2; S → 4 (d) P → 1; Q → 3; R → 2; S → 4

www.jeebooks.in

62.

In the following [x] denotes the greatest integer less than or equal to x.

Match the functions in Column I with the properties in Column II and indicate your answer by darkening the appropriate bubbles in the 4 × 4 matrix given in the ORS. [2007 - 6 marks] (A) x | x | (B) (C) x + [x] (D) | x – 1 | + | x + 1 | in (–1, 1) 63.

(r) strictly increasing in (–1, 1) (s) not differentiable at least at one point

In this questions there are entries in columns I and II. Each entry in column I is related to exactly one entry in column II. Write the correct letter from column II against the entry number in column I in your answer book. [1992 - 2 Marks]

Column I (A) sin [x]) (B) sin (x–[x])

64.

Column II (p) continuous in (–1, 1) (q) differentiable in (–1, 1)

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Column I

If

Column II (p) differentiable everywhere (q) nowhere differentiable (r) not differentiable at 1 and – 1

and for all

.

If right hand derivative at x = 0 exists for f(x). Find derivative of g(x) at x = 0

[2005 - 4 Marks]

and f(x) is a differentiable function at x = 0 given

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65. If |c| ≤

.

by

Find the value of ‘a’ and prove that 64 b2 = 4 – c2

w

[2004 - 4 Marks] is an odd function such that f(x) = f(2a – x) for If a function and the left hand derivative at x = a is 0 then find the left hand derivative at x = – a\[2003 2 Marks]

w

66.

67.

Let

and [2002 - 5 Marks]

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where a and b arenon-negative real numbers. Determine

69.

eb oo ks .in

the composite function g o f. If (g o f) (x) is continuous for all real x, determine the values of a and b. Further, for these values of a and b, is g o f differentiable at x = 0? Justify your answer. 68. Let α ∈ R. Prove that a function f : R → R is differentiable at α if and only if there is a function g : R → R which is continuous at α and satisfies f(x) − f(α) = g(x) (x − α) for all x ∈ R. [2001 - 5 Marks] Let f (x), x ≥ 0, be a non−negative continuous function, and let F(x) =

, x ≥ 0. If for

some c > 0, f (x) ≤ cF(x) for all x ≥ 0, then show that f(x) = 0 for all x ≥ 0. 70.

[2001 - 5 Marks] Determine the values of x for which the following function fails to be continuous or differentiable: [1997 - 5 Marks] f (x)

71.

Let find f (2).

Justify your answer.

for all real x and y. If

[1995 - 5 Marks] A function f : R R satisfies the equation f (x + y) = f (x) f (y) for all x, y in R and f (x) 0 for any x in R. Let the function be differentiable at x = 0 and f′ (0) =2. Show that f′ (x) = 2 f (x) for all x in R. Hence, determine f (x).

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exists and equals – 1 and f (0) =1,

73. Draw a graph of the function y = [x] + | 1 – x |,

[1990 - 4 Marks] .

Determine the points, if any, where this function is not differentiable.

w

[1989 - 4 Marks] 74. Let f (x) be a function satisfying the condition f (– x)= f (x) for all real x. If (0) exists,

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find its value.

75.

[1987 - 2 Marks]

Let f (x) be defined in the interval [–2, 2] such that

and g(x) = f ( | x | ) + | f (x) |

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Test the differentiability of g(x) in (– 2, 2). [1986 - 5 Marks] 76.

and

Let

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[1985 - 5 Marks]

Discuss the continuity and differentiability of the function g(x) in the interval (0, 2).

77.

Let

[1983 - 2 Marks]

Discuss the continuity of f , f ′ and f ′′ on [ 0, 2]. 78.

Find the derivative of f (x) =

at x = 1

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[1979]

1.

The derivative of

(b)

(a)

w

(a) 3.

(c)

(b)

at

is :

[Main Sep. 05, 2020 (II)]

(d)

If

w

2.

with respect to

where a > b > 0, then

at

is :

[Main Sep. 04, 2020 (I)] (c)

(d)

If x = 2sinθ – sin2θ and y = 2cosθ – cos2θ, θ ∈ [0, 2π], then

at θ = π is : [Main Jan. 9, 2020 (II)]

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(a) 4.

(b)

(c)

(d)

If

then

is:

(a) 4 5.

eb oo ks .in

[Main Jan. 7, 2020 (I)]

(c) –4

(b)

(d)

Let y = y(x) be a function of x satisfying =k–

where k is a constant and

Then

at x = , is equal to:

[Main Jan. 7, 2020 (II)]

(b)

(a) 6.

(c)

(d)

If ey + xy = e, the ordered pair

equal to :

at x = 0 is

[Main April 12, 2019 (I)]

(b)

(a) (c)

The derivative of

, with respect t

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7.

(d)

(a) 1 8.

(b)

(c)

, x∈

[Main April 12, 2019 (II)]

is equal to : [Main April 08, 2019 (I)]

9.

Let S be the set of all points in (– , ) at which the function f(x) = min {sinx, cosx} is not differentiable. Then S is a subset of which of the following? [Main Jan. 12, 2019 (I)]

w

–x

(a)

(b)

(c)

(d)

–x

then

(a)

w

(c)

is :

(d) 2

If 2y =

(b) x –

where

(d) 2x –

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10.

For x > 1, if

then

is equal to : [Main Jan. 12, 2019 (I)]

eb oo ks .in

(b) loge 2x

(a)

(d) x loge 2x

(c)

11. Let f : R → R be a function such that f (x) = x3 + x2f(1) + xf ″(2) + f ″′(3), x∈R. Then f (2) equals:

[Main Jan 10, 2019 (I)]

(a) – 4 (b) 30 (c) – 2 (d) 8 12.

If x = 3 tan t and y = 3 sec t, then the value of

at

is:

[Main Jan. 09, 2019 (II)]

(b)

(a) 13.

If x =

(c)

and y =

(d)

(| t | ≥ 1), then

is equal to.

[Main Online April 16, 2018]

(b)

(c)

(d)

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(a)

14. Let f : [0, 2] → R be a function which is continuous on[0, 2] and is differentiable on (0, 2) with f (0) = 1. LetF(x) =

for

. If F′(x) = f ′(x) for all

, then F(2)

equals

[Adv. 2014]

(a) e – 1 (b) e4 – 1 (c) e – 1 (d) e4 15. If y is a function of x and log (x + y) – 2xy = 0, then the value of y’ (0) is equal to

w

w

2

(a) (b) (c) (d)

[2004S]

1 –1 2 0

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16. Let f : (0, ∞) → R and

. If F(x2) = x2(1 + x), then f(4) equals

(a) (b) (c) (d)

5/4 7 4 2

eb oo ks .in

[2001S]

17. If y = (sin x)tan x, then

is equal to[1994]

(a) (sin x)tan x (1 + sec2x log sin x) (b) tan x (sin x)tan x – 1.cos x (c) (sin x)tan x sec2x log sin x (d) tan x (sin x)tan x – 1 18. There exist a function f (x), satisfying f (0) = 1, f ′(0) = –1,f (x) > 0 for all x, and [1982 - 2 Marks] (a) f ‘’(x) > 0 for all x (b) –1 20 24. The derivative of

[1990 - 2 Marks] with respect to

at x =

is .................... [1986 - 2 Marks]

25. If f(x) = logx (ln x), then f ‘(x) at x = e is ....................

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[1985 - 2 Marks] , r = 1, 2, 3 are polynomials in x such that fr(a) = gr (a) = hr(a), r = 1,

26. If 2, 3

then F′’(x) at x = a is ....................

eb oo ks .in

and

[1985 - 2 Marks]

and

27. If

, then

= ....................

[1982 - 2 Marks]

28. The derivative of an even function is always an odd function.

[1983 - 1 Mark]

29. For any positive integer n, define fn : (0, ∞) →

for all x ∈ (0, ∞).

tan–1

fn (x) =

as

Then, which of the following statement(s) is (are) TRUE?

[Adv. 2018]

(a)

tan ( fj (0)) = 55

(b)

(1 + fj′ (0)) sec2 ( fj (0)) = 10

w .je

2

(c)

For any fixed positive integer n,

tan (fn (x)) =

(d)

For any fixed positive integer n,

sec2 (fn (x)) = 1

30. For every twice differentiable function

with

, which of

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the following statement(s) is (are) TRUE? There exist There exists

, where r < s, such that f is one-one on the open interval (r, s) such that

w

(a) (b)

[Adv. 2018]

(c)

(d) There exists such that and 31. Let f : → , g : → and h : → be differentiable functions such that f(x) = x3 + 3x + 2, g(f(x)) = x and h (g(g(x))) = x for all x ∈ . Then [Adv. 2016]

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g′(2) =

(b) (c) (d)

h′(1) = 666 h(0) = 16 h(g(3)) = 36

eb oo ks .in

(a)

32. Let f (x) = 2 + cos x for all real x.

(a) (b) (c) (d)

STATEMENT - 1 : For each real t, there exists a point c in[t, t + π] such that f ‘(c) = 0 because STATEMENT - 2 : f(t) = f(t + 2π) for each real t. [2007 -3 marks] Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1 Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1 Statement-1 is True, Statement-2 is False Statement-1 is False, Statement-2 is True.

33. If

y

=

,

prove

that

.

[1998 - 8 Marks]

at x = – 1, when

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34. Find

+

35. If x = se

– cos

+

and y =

tan (ln(x + 2)) = 0 [1991 - 4 Marks] , then show that

w

[1989 - 2 Marks] 36. If α be a repeated root of a quadratic equation f(x) = 0 and A(x), B(x) and C(x) be

w

polynomials of degree 3, 4 and 5 respectively, then show that

37.

is

divisible by f(x), where prime denotes the derivatives. [1984 - 4 Marks] . Find

[1981 - 2 Marks]

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+ cos2 (2x + 1) ; Find

38. Given y =

.

1. (a)

eb oo ks .in

[1980]

For all twice differentiable functios f : R→R, with f(0) = f(1) = f ’(0) = 0 Sep. 06, 2020 (II)] at every point

(b) f “(x) = 0, for some (c)

=0

[Main

(d) f “(x) = 0, at every point 2.

If

then :

[Main Sep. 03, 2020 (I)]

(b)

(a) (c) 3.

(d)

If c is a point at which Rolle’s theorem holds for the function, R, then f

(c) is equal to:

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interval [3, 4], where (a) 4.

5.

(b)

(c)

[Main Jan. 8, 2020 (I)]

(d)

= 0, then k is: [Main Jan. 7, 2020 (I)]

(d)

The value of c in the Lagrange’s mean value theorem for the function f(x) = x3 – 4x2 + 8x + 11, when x

w

(c)

Let xk + yk = ak, (a, k > 0) and

w

(a)

(b)

in the

[0,1] is: [Main Jan. 7, 2020 (II)]

(a)

(b)

(c)

(d)

www.jeebooks.in

6.

(a) (b) (c) (d) 8.

, then

is equal to [Main Online April 8,

2017] 12 y 224 y2 225 y2 225 y If f and g are differentiable functions in [0, 1] satisfying f (0) = 2 = g(1), g(0) = 0 and f (1) = 6, then for some [Main 2014] f ′(c) = g ′(c) f ′(c) = 2g ′(c) 2f ′(c) = g ′(c) 2f ′(c) = 3g ′(c) Let g (x) = log f (x) where f (x) is twice differentible positive function on (0, ) such that f (x + 1) = x f (x). Then, forN = 1, 2, 3, ........... [2008]

eb oo ks .in

(a) (b) (c) (d) 7.

If y =

(a) (b)

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(c)

(d)

equals

w

9.

(b)

(c)

(d)

w

(a)

10.

[2007 -3 marks]

If f(x) is a twice differentiable function and given that f(1) = 1; f(2) = 4, f(3) = 9, then [2005S]

www.jeebooks.in

(a) (b) (c) (d) 11.

f ‘’(x) = 2 for x ∈ (1, 3) f ‘’(x) = f ‘(x) = 5 for some x ∈ (2, 3) f ‘’(x) = 3 for x ∈ (2, 3) f ‘’(x) = 2 for some x ∈ (1, 3) If x2 + y2 = 1 then

(a) (b) (c) (d) 12.

(a) (b) (c) (d)

yy’’ – 2(y’) + 1 = 0 yy’’ + (y’)2 + 1 = 0 yy’’ + (y’)2 – 1 = 0 yy’’ + 2(y’)2 + 1 = 0 Let f (x) be a quadratic expression which is positive for all the real values of x. If g(x) = f (x) + f ‘(x) + f ‘’(x), then for any real x, [1990 - 2 Marks] g (x) < 0 g (x) > 0 g (x) = 0 g (x) 0

13.

If

eb oo ks .in

[2000]

2

, a polynomial of degree 3, then

equals

[1988 - 2 Marks]

P’ ‘ ‘ (x) + P’ (x) P’ ‘ (x) P’ ‘ ‘ (x) P(x) P’ ‘ ‘ (x) a constant

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(a) (b) (c) (d) 14.

For a polynomial g(x) with real coefficients, let mg denote the number of distinct real roots of g(x). Suppose S is the set of polynomials with real coefficients defined by

For a polynomial f, let f ‘ and f ‘’ denote its first and second order derivatives, respectively. Then where is ____ the minimum possible value of

_

Let f : (0, π) →

w

15.

w

If

be a twice differentiable function such that = sin2 x for all x ∈ (0, π).

, then which of the following statement(s) is (are) TRUE? [Adv. 2018]

(a)

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for all x ∈ (0, π)

(b)

(c) There exists α ∈ (0, π) such that f ′ (α) = 0

16.

eb oo ks .in

(d)

Let f and g be real valued functions defined on interval(–1, 1) such that g” (x) is continuous, g’(0) = 0, , and f (x) = g (x) sin x STATEMENT - 1 :

[g(x) cot x – g(0) cosec x] = f “(0) and

STATEMENT - 2 : f ‘(0) = g(0)

(c) (d)

[2008] Statement - 1 is True, Statement - 2 is True; Statement - 2 is a correct explanation for Statement - 1 Statement - 1 is True, Statement - 2 is True; Statement - 2 is NOT a correct explaination for Statement - 1 Statement - 1 is True, Statement - 2 is False Statement - 1 is False, Statement - 2 is True

17.

Let f be a twice differentiable function such that

(a) (b)

Find h(10) if h(5) = 11

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w

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[1982 - 3 Marks]

www.jeebooks.in

eb oo ks .in w .je w

w www.jeebooks.in

eb oo ks .in

1.

The position of a moving car at time t is given by f (t) = at2 + bt + c, t > 0, where a, b and c are real numbers greater than 1. Then the average speed of the car over the time interval [t1, t2] is attained at the point : [Main Sep. 06, 2020 (I)] (a) (t2 – t1)/2

w .je

(b) a(t2 – t1) + b

(c) (t1 + t2)/2 (d) 2a(t1 + t2) + b

If the surface area of a cube is increasing at a rate of 3.6 cm2/sec, retaining its shape; then the rate of change of its volume (in cm3/sec.), when the length of a side of the cube is 10 cm, is : [Main Sep. 03, 2020 (II)] 18 10 20 9 If a function f (x) defined by [Main Sep. 02, 2020 (I)]

w

2.

w

(a) (b) (c) (d) 3.

www.jeebooks.in

and (a) (b) (c) (d) 4.

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be continuous for some then the value of a is :

A spherical iron ball of 10 cm radius is coated with a layer of ice of uniform thickness that melts at a rate of 50 cm3/min. When the thickness of ice is 5 cm, then the rate (in cm/min.) at which of the thickness of ice decreases, is: [Main Jan. 9, 2020 (I)]

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(a)

(b)

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(c)

(d)

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5.

A 2 m ladder leans against a vertical wall. If the top of the ladder begins to slide down the wall at the rate 25 cm/sec., then the rate (in cm/sec.) at which the bottom of the ladder slides away from the wall on the horizontal ground when the top of the ladder is 1 m above the ground is:

www.jeebooks.in

[Main April 12, 2019 (I)]

(c) (d) 25 6.

eb oo ks .in

(b)

(a)

A spherical iron ball of radius 10 cm is coated with a layer of ice of uniform thickness that melts at a rate of 50 cm3/min. When the thickness of the ice is 5 cm, then the rate at which the thickness (in cm/min) of the ice decreases, is : [Main April 10, 2019 (II)]

(a) (b) (c)

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(d)

Two ships A and B are sailing straight away from a fixed point O along routes such that ∠AOB is always 120°. At a certain instance, OA = 8 km, OB = 6 km and the ship A is sailing at the rate of 20 km/hr while the ship B sailing at the rate of 30 km/hr. Then the distance between A and B is changing at the rate (in km/hr): [Main Online April 11, 2014]

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7.

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(a)

(b)

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(c)

8.

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(d) A spherical balloon is being inflated at the rate of 35cc/min. The rate of increase in the surface area (in cm2/min.) of the balloon when its diameter is 14 cm, is :

[Main Online April 25, 2013]

(a) 10 (b) (c) 100 (d)

1.

The function,

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in :

is increasing for all x lying [Main Sep. 03, 2020 (I)]

(a)

(b)

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Let f be any function continuous on [a, b] and twice differentiable on (a, b). If for all x ∈ (a, b), f 2 (x) > 0 and f ″(x) < 0, then for any c ∈

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(a, b),

is greater than:

(a) (b) 1 (c) (d) 3.

eb oo ks .in

[Main Jan. 9, 2020 (I)]

Let f(x) = x cos–1 (–sin |x|), x is true?



then which of the following

[Main Jan. 8, 2020 (I)]

(a) f ’ is increasing in (b) f ’ (0) = –

and decreasing in

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(c) f ’ is not differentiable at x = 0 (d) f ’ is decreasing in 4.

and increasing in

Let f(x) = ex – x and g(x) = x2 – x, x ∈ R. Then the set of all x ∈ R, where the function h(x) = (fog) (x) is increasing, is : [Main April 10, 2019 (I)]

w

(a)

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(b) (c)

(d)

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5.

If the function f : R – {1, –1} → A defined by f(x) =

is

surjective, then A is equal to: (a) R – {–1}

eb oo ks .in

[Main April 09, 2019 (I)] (b)

[0, “)

(c) R – [–1, 0) (d) R – (–1, 0) 6.

where a, b and d are

Let

non-zero real constants. Then :

[Main Jan. 11, 2019 (II)]

f is an increasing function of x f is a decreasing function of x f ′ is not a continuous function of x f is neither increasing nor decreasing function of x The function f defined by f(x) = x3 – 3x2 + 5x + 7, is : [Main Online April 9, 2017] (a) increasing in R. (b) decreasing in R. (c) decreasing in (0, ∞) and increasing in (– ∞ , 0). (d) increasing in (0, ∞) and decreasing in (– ∞, 0). 8. The real number k for which the equation, 2x3 + 3x + k = 0 has two distinct real roots in [0, 1] [Main 2013] (a) lies between 1 and 2 (b) lies between 2 and 3 (c) lies between .1 and 0 (d) does not exist.

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(a) (b) (c) (d) 7.

9.

Let

the

be

function

given

by

. Then, g is

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[2008] even and is strictly increasing in (0, ∞) odd and is strictly decreasing in (–∞, ∞) odd and is strictly increasing in (–∞, ∞) neither even nor odd, but is strictly increasing in (–∞, ∞) , then f (x) is If f (x) =

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(a) (b) (c) (d) 10.

[2001S]

(a) (b) (c) (d) 11.

increasing on [−1/2, 1] decreasing on R increasing on R decreasing on [−1/2, 1] For all

[2000S]

(a) (b) (c) (d) 12.

ex < 1 + x loge(1 + x) < x sin x > x loge x > x If the normal to the curve y = f(x) at the point (3,4) makes an angle

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with the positive x-axis, then

[2000S]

(a) –1 (b)

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(c)

(d) 1

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13.

Let

(a) (–∞, –2) (b) (–2, –1)

. Then f decreases in the interval [2000S]

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(c) (1, 2) (d) (2, +∞) 14. Consider the following statments in S and R

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[2000S] S : Both sin x and cos x are decreasing functions in the interval

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R : If a differentiable function decreases in an interval(a, b), then its derivative also decreases in (a, b). Which of the following is true ? (a) Both S and R are wrong (b) Both S and R are correct, but R is not the correct explanation of S (c) S is correct and R is the correct explanation for S (d) S is correct and R is wrong 15. The function f(x) = sin4 x + cos4 x increases if [1999 - 2 Marks] (a) 0 < x < π/8 (b) π/4 < x < 3π/8 (c) 3π/8 < x < 5π/8 (d) 5π/8 < x < 3π /4 16. The slope of the tangent to a curve y = f(x) at [x, f(x)] is2x + 1. If the curve passes through the point (1, 2), then the area bounded by the curve, the x-axis and the line x = 1 is [1995S] (a)

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(b) (c)

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(d) 6 17.

The function f (x) =

is [1995S]

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increasing on (0, ∞) decreasing on (0, ∞) increasing on (0, π/e), decreasing on (π/e, ∞) decreasing on 0, π/e), increasing on (π/e, ∞) The function defined by f(x) = (x + 2) e–x is

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(a) (b) (c) (d) 18.

[1994]

(a) (b) (c) (d) 19. (a) (b) (c) (d)

decreasing for all x decreasing in (–∞, –1) and increasing in (–1, ∞) increasing for all x decreasing in (–1, ∞) and increasing in (–∞, –1) If a + b + c = 0, then the quadratic equation 3ax2 + 2bx + c = 0 has [1983 - 1 Mark] at least one root in [0, 1] one root in [2, 3] and the other in [ –2, –1] imaginary roots none of these

20.

A vertical line passing through the point (h, 0) intersects the ellipse

at the points P and Q. Let the tangents to the ellipse at P

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and Q meet at the point R. If ∆(h) = area of the triangle PQR, ∆1 and ∆2

then [Adv. 2013]

The set of all x for which ln(1 + x)

22.

The function

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21.

x is equal to .............. [1987 - 2 Marks] is monotonically increasing for values

of x(≠ 0) satisfying the inequalities ....... and monotonically decreasing for values of x satisfying the inequalities .................. [1983 - 2 Marks]

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23.

If x – r is a factor of the polynomial

24.

(a) (b) (c) (d) 25.

then r is a root of f′(x) = 0 repeated m times. [1983 - 1 Mark]

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m times

, repeated

If f : R → R is a differentiable function such that f ′(x) > 2f(x) for all x ∈ R, and f(0) = 1, then [Adv. 2017] f(x) is increasing in (0,∞) f(x) is decreasing in (0, ∞) f(x) > e2x in (0, ∞) f ′(x) < e2x in (0, ∞) → R be given by

Let f :

. Then

[Adv. 2014]

(a) f (x) is monotonically increasing on

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(b) f (x) is monotonically decreasing on (0, 1) , for all

(c)

(d) f (2x) is an odd function of x on R For the function, f (x) = x cos

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26.

(a)

for at least one x in the interval [1,

[2009] ), f (x + 2) – f (x) < 2

(b)

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(c) for all x in the interval [1, ), f (x + 2) – f (x) > 2 (d) f ‘ (x) is strictly decreasing in the interval [1, ) 27. Let h(x) = f(x) – (f(x))2 + (f(x))3 for every real number x. Then [1998 - 2 Marks] (a) h is increasing whenever f is increasing

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(b) h is increasing whenever f is decreasing (c) h is decreasing whenever f is decreasing (d) nothing can be said in general. then:

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28. If

[1993 - 2 Marks]

(a) (b) (c) (d) 29. (a) (b) (c) (d) (e)

f (x) is increasing on [ –1, 2] f(x) is continues on [ –1, 3] f ‘(2) does not exist f (x) has the maximum value at x = 2 Let f and g be increasing and decreasing functions, respectively from [0, ) to [0, ). Let h(x) = f (g(x)). Ifh(0) = 0, then h(x) – h (1) is [1987 - 2 Marks] always zero always negative always positive strictly increasing None of these.

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(Qs. 30-32) : By appropriately matching the information given in the three columns of the following table. Let f(x) = x + loge x – x loge x, x

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Column 1 contains information about zeros of f(x), f ′(x) and f ′′(x) . Column 2 contains information about the limiting behaviour of f (x), f ′(x) and f ′′(x) at infinity. Column 3 contains information about increasing/decreasing nature of f (x) and f ′(x) . Column 1 Column 2 (I) f (x) = 0 for some x(i) ∈ (1, e2) (II) f ′ (x) = 0 for some x(ii) ∈ (1, e)

Column 3 (P) f is increasing in (0, 1) (Q) f is increasing in (e, e2)

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(III) f ′(x) = 0 for some x(iii) (R) f ′ is increasing in ∈ (0, 1) (0, 1) (S) f ′ is decreasing (IV) f ′′(x) = 0 for some x(iv) in (e, e2) ∈ (1, e) 30. Which of the following options is the only correct combination? [Adv. 2017] (I) (i) (P) (b) (II) (ii) (Q) (III) (iii) (R) (d) (IV) (iv) (S) Which of the following options is the only correct combination? [Adv. 2017] (a) (I) (ii) (R) (b) (II) (iii) (S) (c) (III) (iv) (P) (d) (IV) (i) (S) 32. Which of the following options is the only incorrect combination? [Adv. 2017] (a) (I) (iii) (P) (b) (II) (iv) (Q) (c) (III) (i) (R) (d) (II) (iii) (P) 33. In this questions there are entries in columns I and II. Each entry in column I is related to exactly one entry in column II. Write the correct letter from column II against the entry number in column I in your answer book.

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(a) (c) 31.

Let the functions defined in column I have domain

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Column I (A) x + sin x (p) (B) sec x (q) (r)

[1992 - 2 Marks]

Column II

increasing decreasing neither increasing decreasing

nor

Passage

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Let f (x) = (1 – x)2 sin2 x+ x2 for all x ∈ IR and let for all x ∈ (1, ).

34.

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[2012]

Consider the statements: P : There exists some x ∈ R such that, f (x) + 2x = 2(1 + x2) Q : There exists some x ∈ R such that, 2f (x) + 1 = 2x(1 + x)

Then (a) both P and Q are true (b) P is true and Q is false (c) P is false and Q is true(d) both P and Q are false 35. Which of the following is true? (a) g is increasing on (1, ∞) (b) g is decreasing on (1, ∞) (c) g is increasing on (1, 2) and decreasing on (2, ∞) (d) g is decreasing on (1, 2) and increasing on (2, ∞)

36.

If P(1) = 0 and

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for all x > 1.

for all

37.

then prove that P(x) > 0

[2003 - 4 Marks] R is differentiable then show that (i) For If the function f : [0,4] 2 a, b (0,4), (f(4)) – (f(0))2 = 8f ′(a) f(b)

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(ii)

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38. Using the relation 2(1 – cos x) < x , x

39.

(tan x)

2

[2003 - 4 Marks] 0 or otherwise, prove that sin

[2003 - 4 Marks]

Let −1 ≤ p ≤ 1. Show that the equation 4x3 − 3x − p =0 has a unique root in the interval[1/2, 1] and identify it. [2001 - 5 Marks]

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40.

Suppose p(x) = a0 + a1x+ a2x2 +....... + anxn. If

for all

, prove that .

If

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41.

[2000 - 5 Marks] Let a + b = 4, where a < 2, and let g(x) be a differentiable function. > 0 for all x, prove that increases.

dx increases as (b – a)

[1997 - 5 Marks]

42.

Let f(x) =

[1996 - 3 Marks] (x) is Where a is a positive constant. Find the interval in which increasing. 43. Let

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[1993 - 5 Marks] Find all possible real values of b such that f(x) has thesmallest value at x = 1.

44.

Show that

3x where 0

x
0) at a point (c, f(c)) is parallel to the line segement joining the points (1, 0) and (e, e), then c is equal to: [Main Sep. 06, 2020 (II)]

eb oo ks .in

1.

(a) (b) (c) (d) 2.

Which of the following points lies on the tangent to the curve at the point (1, 0)?

[Main Sep. 05, 2020 (II)]

(2, 2) (2, 6) (– 2, 6) (– 2, 4) The length of the perpendicular from the origin, on the normal to the curve, x2 + 2xy – 3y2 = 0 at the point (2, 2) is: [Main Jan. 8, 2020 (II)]

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(a) (b) (c) (d) 3.

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(a) (b) (c) 2 (d) 4.

If the tangent to the curve

,

, at a point (a,

b) (0, 0) on it is parallel to the line 2x + 6y – 11 = 0, then :

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[Main April 10, 2019 (II)]

w .je

(a)

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(a) |6α + 2β| = 19 (b) |6α + 2β| = 9 (c) |2α + 6β| = 19 (d) |2α + 6β| = 11 5. If the tangent to the curve, y = x3 + ax – b at the point (1, –5) is perpendicular to the line, – x + y + 4 = 0, then which one of the following points lies on the curve? [Main April 09, 2019 (I)] (a) (–2, 1) (b) (–2, 2) (c) (2, –1) (d) (2, –2) , 1) to the circle x2 6. The tangent and the normal lines at the point ( + y2 = 4 and the x-axis form a triangle. The area of this triangle (in square units) is : [Main April 08, 2019 (II)]

(b) (c)

(d)

The maximum area (in sq. units) of a rectangle having its base on the x-axis and its other two vertices on the parabola, y = 12 – x2 such that the rectangle lies inside the parabola, is: [Main Jan. 12, 2019 (I)] (a) 36 (b) (c) 32

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7.

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(d) 8.

The tangent to the curve,

passing through the point (1, e)

also passes through the point: (a) (2, 3e) (b) (c) (d) (3, 6e) 9.

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[Main Jan. 10, 2019 (II)]

A helicopter is flying along the curve given byy – x3/2 = 7, (x ≥ 0). A soldier positioned at the point

wants to shoot down the

helicopter when it is nearest to him. Then this nearest distance is: [Main Jan. 10, 2019 (II)]

w .je

(a) (b) (c)

(d)

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10. If q denotes the acute angle between the curves, y = 10 – x2 and y = 2 + x2 at a point of their intersection, then |tan q| is equal to: [Main Jan. 09, 2019 (I)] (a)

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(b)

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(c) (d) 11.

(a) (b) (c) (d) 12.

Let P be a point on the parabola, x2 = 4y. If the distance of P from the centre of the circle, x2 + y2 + 6x + 8 = 0 is minimum, then the equation of the tangent to the parabola at P, is [Main Online April 16, 2018] x + 4y – 2 = 0 x + 2y = 0 x+y+1=0 x–y+3=0 The normal to the curve y(x – 2)(x – 3) = x + 6 at the point where the curve intersects the y-axis passes through the point: [Main 2017]

(a)

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(b) (c)

(d)

The eccentricity of an ellipse whose centre is at the origin is

w

13.

. If

w

one of its directices is x = – 4, then the equation of the normal to it at is : [Main 2017]

(a) x + 2y = 4

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(b) 2y – x = 2 (c) 4x – 2y = 1 (d) 4x + 2y = 7 14. Consider

[Main 2016]

A normal to y = f(x) at (a) (b) (c) (0, 0) (d)

If the tangent at a point P, with parameter t, on the curvex = 4t2 + 3, y = 8t3 – 1, t ∈ R, meets the curve again at a point Q, then the coordinates of Q are : [Main Online April 9, 2016] (16t2 + 3, –64t3 – 1) (4t2 + 3, –8t3 – 2) (t2 + 3, t3 – 1) (t2 + 3, – t3 – 1) The normal to the curve, x2 + 2xy – 3y2 = 0, at (1, 1) [Main 2015] meets the curve again in the third quadrant. meets the curve again in the fourth quadrant. does not meet the curve again. meets the curve again in the second quadrant. The equation of a normal to the curve,

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15.

also passes through the point:

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(a) (b) (c) (d) 16.

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(a) (b) (c) (d) 17.

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sin y = x sin

at x = 0, is :

(a) 2x –

=0

(b) 2x +

=0

(c) 2y – (d) 2y +

=0 =0

18.

For the curve y = 3 sinθ cosθ, x = eθ sin θ, 0 ≤ θ ≤ π, the tangent is parallel to x-axis when θ is: [Main Online April 11, 2014]

(a) (b) (c) (d)

If an equation of a tangent to the curve, y – cos(x + y),– 1 is x + 2y = k then k is equal to : [Main Online April 25, 2013]

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19.

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[Main Online April 11, 2015]

(a) l

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(b) 2 (c)

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(d)

20.

Consider the two curves C1 : y2 = 4x, C2 : x2 + y2 – 6x + 1 = 0. Then, [2008]

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(a) (b) (c) (d) 22.

w .je

(a) (b) (c) (d) 23.

C1 and C2 touch each other only at one point. C1 and C2 touch each other exactly at two points C1 and C2 intersect (but do not touch) at exactly two points C1 and C2 neither intersect nor touch each other The tangent to the curve y = ex drawn at the point (c, ec) intersects the line joining the points (c – 1, ec–1) and (c + 1, ec + 1) [2007 -3 marks] on the left of x = c on the right of x = c at no point at all points If P(x) is a polynomial of degree less than or equal to 2 and S is the set of all such polynomials so that P(0) = 0, P(1) = 1 and P’(x) > 0 x ∈ [0, 1], then [2005S] S=φ S = ax + (1 – a) x2 a ∈ (0, 2) a ∈ (0, ∞) S = ax + (1 – a) x2 a ∈ (0, 1) S = ax + (1 – a) x2 α If f (x) = x log x and f (0) = 0, then the value of α for which Rolle’s theorem can be applied in [0, 1] is [2004S] –2 –1 0 1/2 In [0,1] Lagranges Mean Value theorem is NOT applicable to [2003S]

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(a) (b) (c) (d) 21.

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(a) (b) (c) (d) 24.

(a)

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(c) (d) 25.

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(b)

The point(s) on the curve y3 + 3x2 = 12y where the tangent is vertical, is (are) [2002S]

(a) (b) (c) (0, 0) (d)

The triangle formed by the tangent to the curve f(x) = x2 + bx − b at the point (1, 1) and the coordinate axes, lies in the first quadrant. If its area is 2, then the value of b is [2001S] (a) −1 (b) 3 (c) −3

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26.

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The slope of the tangent to the curve (y – x5)2 = x(1 + x2)2 at the point (1, 3) is [Adv. 2014]

w .je

29.

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(d) 1 27. Which one of the following curves cut the parabola y2 = 4ax at right angles? [1994] 2 2 2 (a) x + y = a (b) y = e–x/2a (c) y = ax (d) x2 = 4ay 28. The normal to the curve x = a (cos θ + θ sin θ),y = a (sin θ – θ cos θ) at any point ‘θ’ is such that [1983 - 1 Mark] (a) it makes a constant angle with the x– axis (b) it passes through the origin (c) it is at a constant distance from the origin (d) none of these

30.

If the lines x + y = a and x – y = b touch the curve y = x2 – 3x + 2 at the points where the curve intersects the x-axis, then

is equal to

_____.

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31.

[Main Sep. 05, 2020 (II)] If the tangent to the curve, y = e at a point (c, ec) and the normal to the parabola, y2 = 4x at the point (1, 2) intersect at the same point on the x-axis, then the value of c is ____________. [Main Sep. 03, 2020 (II)] x

32.

If

then

at x = 0 is

___________. [Main Sep. 02, 2020 (II)]

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33.

Let the normal at a point P on the curve y2 – 3x2 + y + 10 = 0 intersect the y-axis at

. If m is the slope of the tangent at P to the curve,

then |m| is equal to ______.

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[Main Jan. 8, 2020 (I)]

34.

Let C be the curve y3 – 3xy + 2 = 0. If H is the set of points on the curve C where the tangent is horizontal and V is the set of the point on the curve C where the tangent is vertical then H =............ and V = ............. [1994 - 2 Marks]

35.

Let f, g: [–1, 2] → be continuous functions which are twice differentiable on the interval (–1, 2). Let the values of f and g at the points –1, 0 and 2 be as given in the following table:

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In each of the intervals (–1, 0) and (0, 2) the function (f – 3g)” never vanishes. Then the correct statement(s) is(are) [Adv. 2015] (a) f ′(x) – 3g′(x) = 0 has exactly three solutions in (–1, 0) ∪ (0, 2) (b) f ′(x) – 3g′(x) = 0 has exactly one solution in (–1, 0) (c) f ′(x) – 3g′(x) = 0 has exactly one solution in (0, 2) (d) f ′(x) – 3g′(x) = 0 has exactly two solutions in (–1, 0) and exactly two solutions in (0, 2) 36. If the line ax + by + c = 0 is a normal to the curve xy =1, then [1986 - 2 Marks] (a) a > 0, b > 0 (b) a > 0, b < 0 (c) a < 0, b > 0 (d) a < 0, b < 0 (e) none of these.

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If a continuous function f defined on the real line R, assumes positive and negative values in R then the equation f(x) = 0 has a root in R. For example, if it is known that a continuous function f on R is positive at some point and its minimum value is negative then the equation f(x) = 0 has a root in R. Consider f(x) = kex – x for all real x where k is a real constant. 37. The line y = x meets y = kex for k 0 at [2007 - 4 marks] (a) no point (b) one point (c) two points (d) more than two points 38. The positive value of k for which kex – x = 0 has only one root is [2007 - 4 marks] (a)

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(b) 1 (c) e (d) loge2 39. For k > 0, the set of all values of k for which kex – x = 0 has two distinct roots is [2007 - 4 marks] (a)

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(b) (c)

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(d) (0, 1) 40.

If |f (x1) – f (x2)| < (x1 – x2)2, for all x1, x2 ∈ R. Find the equation of tangent to the cuve y = f (x) at the point (1, 2).

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42.

43.

44.

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45.

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41.

[2005 - 2 Marks] Using Rolle’s theorem, prove that there is at least one root in (451/100, 46) of the polynomial P(x) = 51x101 – 2323(x)100 – 45x + 1035. [2004 - 2 Marks] A curve C has the property that if the tangent drawn at any point P on C meets the co-ordinate axes at A and B, then P is the mid-point of AB. The curve passes through the point (1, 1). Determine the equation of the curve. [1998 - 8 Marks] A curve y = f(x) passes through the point P(1,1). The normal to the curve at P is a(y – 1) + (x – 1) = 0. If the slope of the tangent at any point on the curve is proportional to the ordinate of the point, determine the equation of the curve. Also obtain the area bounded by the y-axis, the curve and the normal to the curve at P. [1996 - 5 Marks] 3 2 The curve y = ax + bx + cx + 5, touches the x-axis atP(–2, 0) and cuts the y axis at a point Q, where its gradientis 3. Find a, b, c. [1994 - 5 Marks] Find the equation of the normal to the curve at x = 0

46.

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47.

[1993 - 3 Marks] What normal to the curve y = x forms the shortest chord? [1992 - 6 Marks] , that Find all the tangents to the curve are parallel to the line x + 2y = 0. [1985 - 5 Marks]

48.

2

Find the coordinates of the point on the curve

where the

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tangent to the curve has the greatest slope.

49.

[1984 - 4 Marks] Find the shortest distance of the point (0, c) from the parabola y = x2 where 0≤ c≤ 5 . [1982 - 2 Marks]

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If f(x) and g(x) are differentiable function for such that f(0) = 2, g(0) = 0, f(1) = 6; g(1) = 2, then show that there exist c satisfying 0< c < 1 and f ′ (c) = 2g′ (c). [1982 - 2 Marks]

1.

Let m and M be respectively the minimum and maximum values of [Main Sep. 06, 2020 (I)]

eb oo ks .in

50.

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Then the ordered pair (m, M) is equal to : (a) (– 3, 3) (b) (– 3, – 1) (c) (– 4, – 1) (d) (1, 3) 2. The set of all real values of λ for which the function ,

, has exactly one maxima

and exactly minima, is:

[Main Sep. 06, 2020 (II)]

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(a)

(b)

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(c)

(d)

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3.

If x = 1 is a critical point of the function f (x) = (3x2 + ax – 2 – a)ex, then : [Main Sep. 05, 2020 (II)]

(b) x = 1 and

are local minima of f.

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(a) x = 1 and

are local maxima of f.

(c) x = 1 is a local maxima and

is a local minima of f.

(d) x = 1 is a local minima and

is a local maxima of f.

4.

The area (in sq. units) of the largest rectangle ABCD whose vertices A and B lie on the x-axis and vertices C and D lie on the parabola, y = x2 – 1 below the x-axis, is : [Main Sep. 04, 2020 (II)]

(a)

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(b) (c)

(d)

Suppose f(x) is a polynomial of degree four, having critical points at – then the sum of squares of all the 1, 0, 1. If elements of T is : [Main Sep. 03, 2020 (II)]

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5.

(a) (b) (c) (d)

4 6 2 8

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(a) (b) (c) (d) 7.

Let f(x) be a polynomial of degree 5 such that x = ±1 are its critical points. If = 4, then which one of the following is not true ? [Main Jan. 7, 2020 (II)] f is an odd function. f(l) – 4f(–l) = 4. x = 1 is a point of maxima and x = –1 is a point of minima of f. x = 1 is a point of minima and x = –1 is a point of maxima of f. Consider all rectangles lying in the region [Adv. 2020]

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6.

and having one side on the x-axis. The area of the rectangle which has the maximum perimeter among all such rectangles, is (a) (b) π (c)

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(d) 8.

If m is the minimum value of k for which the function is increasing in the interval [0,3] and M is the

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maximum value of f in [0,3] when k = m, then the ordered pair (m, M) is equal to : [Main April 12, 2019 (I)]

(a)

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(b) (c)

(d)

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9.

Let a1, a2, a3, …. be an A. P. with a6 = 2. Then the common difference of this A.P., which maximises the product a1 a 4 a 5, is : [Main April 10, 2019 (II)]

eb oo ks .in

(a) (b) (c) (d) 10.

(a) (b) (c) (d)

The maximum value of

w .je

11.

If S1 and S2 are respectively the sets of local minimum and local maximum points of the function, f(x) = 9x4 + 12x3 – 36x2 + 25, x ∈ R, then : [Main April 08, 2019 (I)] S1 = {–2}; S2 = {0, 1} S1 = {–2, 0}; S2 = {1} S1 = {–2, 1}; S2 = {0} S1 = {–1}; S2 = {0, 2} for any real value of

is:

[Main Jan. 12, 2019 (I)]

(a)

w

(b)

w

(c)

(d)

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12.

The

Let x, y be positive real numbers and m, n positive integers. maximum value of the expression

is :

eb oo ks .in

[Main Jan. 11, 2019 (II)]

(a) 1 (b) (c) (d) 13.

The maximum volume (in cu.m) of the right circular cone having slant height 3 m is: [Main Jan. 09, 2019 (I)] (a) 6 p (b)

w .je

(c) (d)

14.

Let

and

,

.

If

w

, then the local minimum value of h(x) is : [Main 2018]

w

(a) – 3 (b) (c) (d) 3 15. Twenty metres of wire is available for fencing off a flower-bed in the form of a circular sector. Then the maximum area (in sq. m) of the

www.jeebooks.in

flower-bed, is : [Main 2017] 30 12.5 10 25 If f : R → R is a twice differentiable function such thatf ′′ (x) > 0 for

eb oo ks .in

(a) (b) (c) (d) 16.

all x ∈ R, and

, f(1) = 1, then

[Adv. 2017]

(a) f ′ (1) ≤ 0 (b) (c) (d) f ′ (1) > 1

A wire of length 2 units is cut into two parts which are bent respectively to form a square of side = x units and a circle of radius = r units. If the sum of the areas of the square and the circle so formed is minimum, then: [Main 2016] x = 2r 2x = r 2x = (π + 4)r (4 – π) x = πr The minimum distance of a point on the curve y = x2 – 4 from the origin is [Main Online April 9, 2016]

w .je

17.

w

(a) (b) (c) (d) 18.

w

(a)

(b)

www.jeebooks.in

(c)

19.

eb oo ks .in

(d) The least value of

for which 4αx2 +

, for all x > 0, is

[Adv. 2016]

(a) (b) (c) (d) 20.

Let k and K be the minimum and the maximum values of the function f(x) =

w .je

equal to :

in [0, 1] respectively, then the ordered pair (k, K) is

(2 , 1) (2–0.4, 20.6) (2–0.6, 1) (1, 20.6) If x = –1 and x = 2 are extreme points of then

w

(a) (b) (c) (d) 21.

[Main Online April 11, 2015]

–0.4

[Main 2014]

w

(a)

(b)

www.jeebooks.in

(c)

22.

(a) (b) (c) (d) 23.

eb oo ks .in

(d) The cost of running a bus from A to B, is `

where v km/h is

the average speed of the bus. When the bus travels at 30 km/h, the cost comes out to be ` 75 while at 40 km/h, it is ` 65. Then the most economical speed (in km/ h) of the bus is : [Main Online April 23, 2013] 45 50 60 40 The total number of local maxima and local minima of the function is

[2008]

0 1 2 3

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(a) (b) (c) (d) 24. (a)

[2004S]

f (x) has a local maxima f (x) is a strictly decreasing function f (x) is bounded Tangent is drawn to ellipse

w

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(b) (c) (d) 25.

If f (x) = x3 + bx2 + cx + d and 0 < b2 < c, then in (–∞, ∞) f (x) is a strictly increasing function

www.jeebooks.in

eb oo ks .in

Then the value of such that sum of intercepts on axes made by this tangent is minimum, is [2003S] (a) π/3 (b) π/6 (c) π/8 (d) π/4 26. The length of a longest interval in which the function3 sin x – 4 sin3x is increasing, is [2002S] (a) (b) (c) (d)

Let f(x) = (1 + b2)x2 + 2bx + 1 and let m(b) be the minimum value of f(x). As b varies, the range of m(b) is [2001S] (a) [0, 1] (b) (0, 1/2] (c) [1/2, 1] (d) (0, 1]

w .je

27.

28.

Let

then at x = 0, f has

a local maximum no local maximum a local minimum no extremum

29.

On the interval [0, 1] the function

w

w

(a) (b) (c) (d)

[2000S]

takes its maximum

value at the point [1995S]

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(a) 0

(c) (d) 30.

If y = a ln x + bx2 + x has its extremum values at x = –1 and x = 2, then [1983 - 1 Mark]

(a) a = 2, b = – 1 (b) a = 2, b = (c) a = – 2, b = (d) none of these

AB is a diameter of a circle and C is any point on the circumference of the circle. Then [1983 - 1 Mark] the area of ∆ ABC is maximum when it is isosceles the area of ∆ ABC is minimum when it is isosceles the perimeter of ∆ ABC is minimum when it is isosceles none of these

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31.

eb oo ks .in

(b)

w

(a) (b) (c) (d)

w

32.

A cylindrical container is to be made from certain solid material with the following constraints: It has a fixed inner volume of V mm3, has a 2 mm thick solid wall and is open at the top. The bottom of the container is a solid circular disc of thickness 2 mm and is of radius equal to the outer radius of the container.

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If the volume of the material used to make the container is minimum when the inner radius of the container is 10 mm, then the value of

is

eb oo ks .in

[Adv. 2015] 33. Let p(x) be a real polynomial of least degree which has a local maximum at x = 1 and a local minimum at x = 3. Ifp(1) = 6 and p(3) = 2, then p’(0) is [2012] The total number 34. let f : IR → IR be defined as f (x) =

w .je

of points at which f attains either a local maximum or a local minimum is [2012] 35. Let f be a function defined on R (the set of all real numbers) such that f ′(x) = 2010 (x–2009) (x–2010)2 (x–2011)3(x–2012)4 .for all x ∈R. If g is a function defined on R with values in the interval (0, ∞) such that f (x) = ln ( g (x)), for all x ∈ R then the number of points in R at which g has a local maximum is [2010] 36. Let f be a real-valued differentiable function on R (the set of all real numbers) such that f(1) = 1. If the y-intercept of the tangent at any point P(x, y) on the curve y = f(x) is equal to the cube of the abscissa of P, then find the value of f(–3) [2010] 37. Let p(x) be a polynomial of degree 4 having extremum at x = 1, 2 and

w

Then the value of p (2) is

[2009]

w

38. The maximum value of the function f (x) = 2x3 – 15x2 + 36x – 48 on the set A= {x | x2 + 20 9x}is [2009]

www.jeebooks.in

Let AD and BC be two vertical poles at A and B respectively on a horizontal ground. If AD = 8 m, BC = 11 m and AB = 10 m; then the distance (in meters) of a point M on AB from the point A such that MD2 + MC2 is minimum is ______. [Main Sep. 06, 2020 (I)] 40. Let f(x) be a polynomial of degree 3 such that f(–1) = 10, f(1)= –6, f(x) has a critical point at x = –1 and f (x) has a critical point at x = 1. Then f(x) has a local minima at x = ________. [Main Jan. 8, 2020 (II)] be defined by 41. Let the function

eb oo ks .in

39.

Suppose the function f has a local minimum at θ precisely when where Then the value of is _____ [Adv. 2020]

Let P be a variable point on the ellipse

w .je

42.

with foci F1 and

F2 . If A is the area of the triangle PF1F2 then the maximum value of A is .............. [1994 - 2 Marks]

The larger of cos ( ln θ ) and ln (cos θ) if

w

43.

w

44.

For 0 < a < x, the minimum value of the function 2.

is ................. [1983 - 1 Mark] is [1984 - 1 Mark]

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45.

Let

for every n

(b)

for every n

(c) (d) x1 < y1 46.

eb oo ks .in

Let x1 < x2 < x3 < ….. < xn < …. be all the points of local maximum of f and y1 < y2 < y3< …. < yn < ….. be all the points of local minimum of f. Then which of the following options is/are correct? [Adv. 2019] (a) xn+1 – xn > 2

Let

be given by f (x) = (x – 1) (x – 2) (x – 5). Define .

Then which of the following options is/are correct?

[Adv. 2019]

F has a local maximum at x = 2 F has a local minimum at x = 1 F has two local maxima and one local minimum in (0, ∞) F (x) < 0 for all x ∈ (0,5) Define the collections {E1, E2, E3, ……} of ellipses and {R1, R2, R3, …..} of rectangles as follows :

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(a) (b) (c) (d) 47.

;

w

w

R1 : rectangle of largest area, with sides parallel to the axes, inscribed in E1; En : ellipse

of largest area inscribed in Rn – 1, n > 1;

Rn : rectangle of largest area, with sides parallel to the axes, inscribed in En, n > 1. Then which of the following options is/are correct?

www.jeebooks.in

[Adv. 2019] (a) The eccentricities of E18 and E19 are NOT equal

(c)

eb oo ks .in

(b) The length of latus rectum of E9 is (area of Rn) < 24, for each positive integer N

(d) The distance of a focus from the centre in E9 is

48.

If f(x) =

, then

[Adv. 2017]

(a) f ′(x) = 0 at exactly three points in (–π, π) (b) f ′(x) = 0 at more than three points in (–π, π) (c) f(x) attains its maximum at x = 0 (d) f(x) attains its minimum at x = 0 (0, ∞) and g : be twice differentiable functions 49. Let f : . Suppose f ‘(2) = such that f” and g” are continuous functions on 0 and g’(2)

w .je

g(2)= 0, f”(2)

0. If

, then

[Adv. 2016]

f has a local minimum at x=2 f has a local maximum at x=2 f “(2) > f (2) f (x) – f “(x) = 0 for at least one x A rectangular sheet of fixed perimeter with sides having their lengths in the ratio 8 : 15 is converted into an open rectangular box by folding after removing squares of equal area from all four corners. If the total area of removed squares is 100, the resulting box has maximum volume. Then the lengths of the sides of the rectangular sheet are [Adv. 2013] (a) 24 (b) 32

w

w

(a) (b) (c) (d) 50.

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(c) 45 (d) 60 51.

for all x ∈ (0,∞), then

If

(a) (b) (c) (d) 52.

eb oo ks .in

[2012]

f has a local maximum at x = 2 f is decreasing on (2, 3) there exists some c ∈ (0, ∞), such that f ′′(c) = 0 f has a local minimum at x = 3 Let

and

then g(x) has

[2006 - 5M, –1]

local maxima at x = 1 + ln 2 and local minima at x = e local maxima at x = 1 and local minima at x = 2 no local maxima no local minima f(x) is cubic polynomial with f(2) = 18 and f(1) = –1. Alsof(x) has local maxima at x = –1 and f ‘(x) has local minima atx = 0, then [2006 - 5M, –1] (a) the distance between (–1, 2) and (a f(a)), where x = a is the point of local minima is

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(a) (b) (c) (d) 53.

w

(b) f(x) is increasing for x ∈ [1, 2

]

w

(c) f(x) has local minima at x = 1 (d) the value of f(0) = 15 54. The function f (x) =

dt has a local

minimum at x = [1999 - 3 Marks]

www.jeebooks.in

0 1 2 3 The number of values of x where the function ) attains its maximum is f(x) = cos x + cos (

eb oo ks .in

(a) (b) (c) (d) 55.

[1998 - 2 Marks]

(a) (b) (c) (d) 56.

0 1 2 infinite If f(x) =

, for every real number x, then the minimum value of f

[1998 - 2 Marks]

does not exist because f is unbounded is not attained even though f is bounded is equal to 1 is equal to –1 The smallest positive root of the equation, tan x – x = 0 lies in [1987 - 2 Marks]

w .je

(a) (b) (c) (d) 57. (a)

(b)

w

(c)

w

(d)

(e) None of these 58. Let P(x) = a0 + a1x2 + a2x4 + ...... + anx2n be a polynomial in a real variable x with < < < ..... < . The function P(x) has 0
0, where b

0 is a constant.

w

[1996 - 5 Marks] 67. Let (h, k) be a fixed point, where h > 0, k > 0. A straight line passing through this point cuts the positive direction of the coordinate axes at the points P and Q. Find the minimum area of the triangle OPQ, O being the origin. [1995 - 5 Marks]

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The circle x2 + y2 = 1 cuts the x-axis at P and Q. Another circle with centre at Q and variable radius intersects the first circle at R above the x-axis and the line segment PQ at S. Find the maximum area of the triangle QSR. [1994 - 5 Marks] 69. A cubic f (x) vanishes at x = 2 and has relative minimum /maximum

eb oo ks .in

68.

at x = –1 and x =

if

, find thecubic f(x).

w .je

[1992 - 4 Marks] 70. A window of perimeter P (including the base of the arch) is in the form of a rectangle surmounded by a semi circle. The semi-circular portion is fitted with coloured glass while the rectangular part is fitted with clear glass transmits three times as much light per square meter as the coloured glass does. What is the ratio for the sides of the rectangle so that the window transmits the maximum light ? [1991 - 4 Marks] 71. A point P is given on the circumference of a circle of radius r. Chord QR is parallel to the tangent at P. Determine the maximum possible area of the triangle PQR. [1990 - 4 Marks] 72. Find all maxima and minima of the function

Also determine the area bounded by the curve y = x

[1989 - 5 Marks] , the y-axis and

w

w

the line y = 2. 73. Investigate for maxima and minima the function

74.

[1988 - 5 Marks]

f(x) =

Find the point on the curve

that is

farthest from the point (0, – 2).

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[1987 - 4 Marks] . Find the intervals in which

75. Let

76.

If

eb oo ks .in

λ should lie in order that f(x) has exactly one minimum and exactly one maximum. [1985 - 5 Marks] for all positive x where a > 0 and b > 0 show that

.

77.

Use the function numbers

[1982 - 2 Marks] , x > 0. to determine the bigger of the two

and

[1981 - 4 Marks] 78. Let x and y be two real variables such that x > 0 and xy = 1. Find the minimum value of x+y. [1981 - 2 Marks] 79.

,

Prove that the minimum value of

.

[1979]

w

w

w .je

a, b > c, x > – c is

www.jeebooks.in

eb oo ks .in w .je w

w www.jeebooks.in

eb oo ks .in

1.

[Main Sep. 06, 2020 (I)]

w .je

(a) is equal to

(b) is equal to 1 (c) is equal to –

(d) does not exist If

=

, where c is a

w

2.

constant of integeration, then g (0) is equal to: [Main Sep. 05, 2020 (I)]

w

(a) (b) (c) (d)

e e2 1 2

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3.

If

where C is a constant can be :

eb oo ks .in

of integration, then

[Main Sep. 05, 2020 (II)]

(a) (b) (c) (d) 4.

The integral

is equal to

(where C is a constant of integration) :

[Main Sep. 04, 2020 (I)]

w .je

(a) (b) (c)

w

(d)

w

5.

Let

Then f (3) – f (1) is equal to : [Main Sep. 04, 2020 (I)]

(a)

www.jeebooks.in

(b)

(d) 6.

If

eb oo ks .in

(c)

wher e

C

is

a

constant of integration, then the ordered pair (A(x), B(x)) can be : [Main Sep. 03, 2020 (II)] (a) (b) (c) (d) 7.

The integral

is equal to:

w .je

(where C is a constant of integration)

[Main Jan. 9, 2020 (I)]

(a)

(b)

w

(c)

w

(d)

www.jeebooks.in

(a) (b) (c) (d) 9.

If

= λtanθ + 2loge|f(θ)| + C where C is a

constant of integration, then the ordered pair (λ, f(θ)) is equal to: [Main Jan. 9, 2020 (II)] (1, 1 – tanθ) (–1, 1 – tanθ) (–1, 1 + tanθ) (1, 1 + tanθ) The integral

eb oo ks .in

8.

is equal to :

(Here C is a constant of integration)

[Main April 12, 2019 (I)]

(a) (b)

w .je

(c) (d)

10.

If

w

=

where C is a constant of integration,

w

then :

[Main April 10, 2019 (I)]

(a)

and f(x) = 3 (x – 1)

(b)

and f(x) = 3 (x – 1)

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and f(x) = 9 (x – 1)

(d)

and f(x) = 9 (x – 1)2

11.

The integral

eb oo ks .in

(c)

is equal to:

[Main April 09, 2019 (I)]

(a) –3 tan

–1/3

x+C

tan–4/3 x + C

(b) –

(c) –3 cot–1/3 x + C (d) 3 tan–1/3 x + C (Here C is a constant of integration) 12. The integral

is equal to :

(where C is a constant of integration)

[Main Jan. 12, 2019 (I)]

(a) (b)

w .je

(c)

(d)

13.

If

for a suitable chosen integer m

w

and a function A (x), where C is a constant of integration, then (A(x))m equals : [Main Jan. 11, 2019 (I)]

w

(a)

(b)

www.jeebooks.in

(c)

14.

The integral

eb oo ks .in

(d)

is equal to:

[Main 2018]

(a) (b) (c) (d)

(where C is a constant of integration) = x + 2, x ≠

If

w .je

15.

, and

f (x) dx = A log |1 – x | + Bx + C, then the ordered pair(A, B) is

equal to :

[Main Online April 9, 2017]

(where C is a constant of integration)

w

(a)

w

(b) (c)

www.jeebooks.in

(d) The integral K)

equals (for some arbitrary constant

eb oo ks .in

16.

[2012]

(a)

(b)

(c)

(d) Let

w .je

17.

.

Then,

for

an

arbitrary constant C, the value of J – I equals [2008]

(a)

w

(b)

w

(c)

www.jeebooks.in

(d) If

, then

is

eb oo ks .in

18.

[2005S]

(a) (b) (c) 3 (d) 19.

The value of the integral

dx is

[1995S]

sin x – 6 tan (sin x) + c sin x – 2(sinx)–1 + c sin x – 2(sinx)–1 – 6tan–1(sin x) + c sin x – 2(sinx)–1 + 5tan–1(sin x) + c

w .je

(a) (b) (c) (d)

–1

20.

If

dx = Ax + B log

+ C, thenA = ......., B =

w

....... and C = .......

w

21.

[1990 - 2 Marks]

Let b be a nonzero real number. Suppose is a differentiable function such that f(0) = 1. If the derivative f ' of f satisfies the equation

www.jeebooks.in

then which of the following statements is/are TRUE? [Adv. 2020] (a) If b > 0, then f is an increasing function (b) If b < 0, then f is a decreasing function for all (c) (d) 22.

Let

for all

eb oo ks .in

for all

for all and

If

be functions satisfying and

then which of the following statements

is/are TRUE?

[Adv. 2020]

f is differentiable at every If g(0) = 1, then g is differentiable at every The derivative f ' (1) is equal to 1 The derivative f ' (0) is equal to 1

23.

Let F(x) be an indefinite integral of sin2x.

w .je

(a) (b) (c) (d)

STATEMENT-1 : The function F(x) satisfies F(x + π) = F(x) for all real x. because

STATEMENT-2 : sin2(x + π) = sin2x for all real x.

w

[2007 -3 marks] (a) Statement-1 is True, statement-2 is True; Statement-2 is a correct explanation for Statement-1.

w

(b) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1 (c) Statement-1 is True, Statement-2 is False (d) Statement-1 is False, Statement-2 is True.

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24.

For any natural number m, evaluate , x > 0.

25.

Evaluate

eb oo ks .in

[2002 - 5 Marks]

.

[2001 - 5 Marks]

26. Evaluate

[1996 - 2 Marks]

27. Find the indefinite integral

[1994 - 5 Marks]

28.

Find the indefinite integral

[1992 - 4 Marks]

Evaluate

w .je

29.

[1989 - 3 Marks]

30. Evaluate :

[1987 - 6 Marks]

w

w

31. Evaluate the following

32.

[1985 - 2½ Marks]

Evaluate the following

33. Evaluate

[1984 - 2 Marks] .

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[1981 - 2 Marks] 34. Evaluate

35. Evaluate

eb oo ks .in

[1979]

[1978]

1.

The integral

equals:

[Main Sep. 06, 2020 (II)]

e (4e +1) 4e2 – 1 e (4e–1) e (2e – 1) A value of a such that

w .je

(a) (b) (c) (d) 2.

is : [Main April 12, 2019 (II)]

(a) –2

w

(b) (c)

w

(d) 2 3.

If

where C is a constant of

integration, then f (x) is equal to: [Main Jan. 10, 2019 (II)]

www.jeebooks.in

4.

– 2x3 – 1 – 4x3 – 1 – 2x3 + 1 4x3 + 1 The integral

eb oo ks .in

(a) (b) (c) (d)

is equal to :

(where C is a constant of integration)

[Main Online April 10, 2016]

(a) (b) (c) (d) The integral

equals :

w .je

5.

[Main 2015]

(a)

w

(b)

w

(c)

(d)

www.jeebooks.in

6.

If m is a non-zero number and

,

then f(x) is:

(a)

(b)

(c)

(d)

The integral

equals :

w .je

7.

eb oo ks .in

[Main Online April 19, 2014]

[Main Online April 23, 2013]

(a)

(b)

w

(c)

(d)

w

8.

[2006 - 3M, –1]

www.jeebooks.in

(a)

(c) (d)

9.

Integrate

eb oo ks .in

(b)

[1999 - 5 Marks]

10. Evaluate :

w .je

[1983 - 2 Marks]

1.

If I1 =

and I2 =

such that I2 = αI1

then α equals to :

[Main Sep. 06, 2020 (I)]

w

(a)

w

(b) (c)

www.jeebooks.in

(d) The value of

is:

eb oo ks .in

2.

[Main Sep. 05, 2020 (I)]

(a) (b) π (c) (d) 3.

Let Then

and

is equal to :

[Main Sep. 04, 2020 (I)]

w .je

(a) 1 (b) 0 (c)

(d)

The integral

w

4.

w

is equal to :

[Main Sep. 04, 2020 (II)]

(a)

(b)

www.jeebooks.in

(c)

5.

eb oo ks .in

(d) is equal to :

[Main Sep. 03, 2020 (I)]

(a) (b) (c) (d) 6.

If the value of the integral :

is

then k is equal to

[Main Sep. 03, 2020 (II)]

w .je

(a) (b) (c) (d) 7.

If for all real triplets (a, b, c), f(x) = a + bx + cx2; then

is

w

equal to:

[Main Jan. 9, 2020 (I)]

w

(a)

(b)

www.jeebooks.in

(c)

8.

The value of

eb oo ks .in

(d) is equal to:

[Main Jan. 9, 2020 (I)]

(a) (b) (c) (d) 9.

2π 2π2 π2 4π If I =

, then:

[Main Jan. 8, 2020 (II)]

(a)

w .je

(b) (c)

(d)

10.

If f (a + b + 1 – x) = f(x), for all x, where a and b are fixed positive

w

real numbers, then

x(f (x) + f (x + l))dx is equal to: [Main Jan. 7, 2020 (I)]

w

(a)

(b)

www.jeebooks.in

(c)

11.

The value of

eb oo ks .in

(d) for which 4

, is :

[Main Jan. 7, 2020 (II)]

(a) loge2 (b) loge (c) loge (d) loge 12.

If

1

and

2

be respectively the smallest and the largest values of

in (0,2 ) – { } which satisfy the equation, 3

d is equal to:

w .je

then

,

[Main Jan. 7, 2020 (II)]

(a)

(b)

w

(c)

(d)

w

13.

Let f : R → R be a continuously differentiable function such that f(2) = 6 and f ’(2) =

.

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If

4t3dt = (x – 2) g (x), then

g(x) is equal to :

(a) (b) (c) (d)

18 24 12 36 14.

eb oo ks .in

[Main April 12, 2019 (I)]

, then m.n is equal to :

If

[Main April 12, 2019 (I)]

(a) (b) 1 (c) (d) –1 15.

The value of

, where [t] denotes the greatest

w .je

integer function, is: (a) (b) (c) (d)

p –p –2p 2p

The value of

w

16.

[Main April 10, 2019 (I)]

is: [Main April 9, 2019 (I)]

w

(a)

(b)

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(c)

17.

Let f(x) = g(x), then

eb oo ks .in

(d) , where g is a non-zero even function. If f(x + 5) = equals :

[Main April 08, 2019 (II)]

(a) (b) (c) 2

w .je

(d) 5

18. Let f and g be continuous functions on [0, a] such that f(x) = f(a – x) is equal to: [Main Jan. 12, 2019 (I)]

w

and g(x) + g(a – x) = 4, then

w

(a)

(b)

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(d)

19.

The integral

eb oo ks .in

(c)

loge x dx is equal to :

[Main Jan. 12, 2019 (II)]

(a) (b) (c) (d)

The value of the integral

w .je

20.

w

(where [x] denotes the greatest integer less than or equal to x) is : [Main Jan. 11, 2019 (I)] (a) 0 (b) sin 4 (c) 4 (d) 4 –sin 4

w

21.

If

then

is: [Main Jan. 10, 2019 (II)]

(a)

www.jeebooks.in

(b)

(d) 22.

The value of

eb oo ks .in

(c)

is:

[Main Jan 9, 2019 (I)]

(a) 0 (b) (c) (d) 23.

If I1 =

cos2 x dx; I2 =

cos2 x dx and I3 =

dx; then

[Main Online April 15, 2018]

I2 > I3 > I1 I3 > I1 > I2 I2 > I1 > I3 I3 > I2 > I1

w .je

(a) (b) (c) (d)

The integral

w

24.

w

(a) (b) (c) (d)

is equal to : [Main 2017]

–1 –2 2 4

www.jeebooks.in

25.

Let In =

. I4 + I6 = a tan5x + bx5 + C, where C is

(b)

(a) (c) (d) 26.

For

eb oo ks .in

constant of integration, then the ordered pair (a, b) is equal to : [Main 2017]

, if y(x) is a differentiable function such that

=

, then y(x) equals :

(where C is a constant)

[Main Online April 10, 2016]

(a)

w .je

(b) (c)

w

(d)

w

27.

The value of

is equal to [Adv. 2016]

(a)

www.jeebooks.in

(b)

(d) 28.

The integral

eb oo ks .in

(c)

[2015]

is equal to :

(a) (b) (c) (d) 29.

1 6 2 4 The integral

equals:

[Main 2014]

(a)

w .je

(b) (c)

(d)

The following integral

w

w

30.

is equal to

[Adv. 2014]

(a)

www.jeebooks.in

(c)

(d) 31.

For

eb oo ks .in

(b)

the value of

equals :

[Main Online April 25, 2013]

(a)

w .je

(b) 0 (c) 1 (d)

32.

Let f :

(the set of all real number) be a positive, non-

constant and differentiable function such that

w

f ′(x) < 2f(x) and

Then the value of

lies in the interval [Adv. 2013]

w

(a) (2e – 1, 2e) (b) (e – 1, 2e – 1) (c)

www.jeebooks.in

(d) The value of the integral

is

eb oo ks .in

33.

[2012]

(a) 0 (b) (c) (d) 34.

The value of

is

[2011]

w .je

(a) (b) (c)

Let f be a real-valued function defined on the interval(–1, 1) such that

w

35.

(d)

w

e–x f (x)=

(a) 1

dt, for all x∈(–1,1) , and let f –1 be the inverse

function of f. Then (f –1)′ (2) is equal to [2010] (b)

(c)

(d)

www.jeebooks.in

36.

Let f be a non-negative function defined on the interval [0, 1]. If

eb oo ks .in

and f (0) = 0, then

[2009]

(a) (b) (c) (d) 37.

is equal to

[2005S]

–4 0 4 6

w .je

(a) (b) (c) (d)

38. The value of the integral

is [2004S]

w

(a)

w

(b)

(c) – 1 (d) 1

www.jeebooks.in

39.

If f (x) is differentiable and

equals

(a) (b) (c) (d)

2/5 –5/2 1 5/2

40. If

eb oo ks .in

[2004S]

then f (x) increases in

[2003S]

(a) (b) (c) (d)

(–2, 2) no value of x (0, ) (– , 0)

41.

The integral

equal to

[2002S]

w

w

w .je

(a)

www.jeebooks.in

(b) 0 (c) 1

Let T > 0 be a fixed real number. Suppose f is a continuous function such , f(x + T) = f(x). that for all

eb oo ks .in

42.

(d)

then the value of

If

is

[2002S]

(a) (b) (c) (d)

3/2I 2I 3I 6I

43. Let 0 are

. Then the real roots of the equation x2 –

=

[2002S]

(a) (b)

w .je

(c)

(d) 0 and 1 44.

dx, a> 0, is [2001S]

π aπ π/2 2π

w

w

(a) (b) (c) (d)

The value of

45. The value of the integral

is: [2000S]

www.jeebooks.in

3/2 5/2 3 5

46.

If

eb oo ks .in

(a) (b) (c) (d)

then

[2000S]

(a) (b) (c) (d)

0 1 2 3

47. Let

, where f is such that and

.

Then g(2) satisfies the inequality

[2000S]

(a)

w .je

(b) (c)

(d) 48.

If for a real number y, [y] is the greatest integer less than or equal to y,

w

then the value of the integral

w

(a) – π (b) 0 (c) – π / 2 49.

[1999 - 2 Marks] (d) π /2 is equal to

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(a) (b) (c) (d)

2 –2 1/2 –1/2

50.

If g(x) =

eb oo ks .in

[1999 - 2 Marks]

t dt, then g(x+π) equals

[1997 - 2 Marks]

(a) g(x) + g(π) (b) g(x) – g(π) (c) g(x) g(π) (d) 51.

The value of function is

where [.] represents the greatest integer

[1995S]

(a)

w .je

(b) – (c)

(d) – 2 52.

If f(x) = A

+ B,

and

then

w

constants A and B are

[1995S]

w

(a)

(b)

(c) 0 and

www.jeebooks.in

(d)

and 0 is

eb oo ks .in

53. The value of

[1993 - 1 Marks]

(a) (b) (c) (d) 54.

0 1 π/2 π/4 R and g : R Let f : R the integral

R be continous functions. Then the value of is

[1990 - 2 Marks]

w .je

(a) (b) 1 (c) – 1 (d) 0 55. For any integer n the integral ––

has the value

π 1 0 none of these

w

(a) (b) (c) (d)

w

56.

[1985 - 2 Marks]

The value of the integral

(a) π/4 (b) π/2

is [1983 - 1 Mark]

www.jeebooks.in

(c) π (d) none of these 57. Let a, b, c be non-zero real numbers such that

eb oo ks .in

.

Then the quadratic equation

has

[1981 - 2 Marks]

(a) (b) (c) (d)

no root in (0, 2) at least one root in (0, 2) a double root in (0, 2) two imaginary roots

58. The value of the definite integral

is

[1981 - 2 Marks]

(a) – 1 (b) 2 (c)

w .je

(d) none of these 59.

Let be a differentiable function such that its derivative f ' is continuous and is defined by

and if

w

If

w

then the value of f(0) is ______ 60.

If I =

[Adv. 2020] , then 27 I2 equals ____, [Adv. 2019]

www.jeebooks.in

61.

The value of the integral

[Adv. 2018]

eb oo ks .in

is _____ .

62. Let f : R → R be a differentiable function such that f(0) = 0,

=3

and f ′(0) = 1. If g(x) =

for

, then

=

[Adv. 2018]

63.

The total number of distinct x ∈ [0, 1] for which

= 2x – 1 is

[Adv. 2016]

If α =

w .je

64.

dx where tan–1x takes only principal

values, then the value of

[Adv. 2015]

Let f : R → R be a function defined by

w

65.

is

w

where [x] is the greatest integer less than or equal to x, if , then the value of (4I – 1) is [Adv. 2015]

www.jeebooks.in

66.

The value of

is [Adv. 2014]

Then the value of

eb oo ks .in

67. For any real number x, let [x] denote the largest integer less than or equal to x. Let f be a real valued function defined on the interval [–10, 10] by

is

[2010]

68. Let f: R → R be a continuous function which satisfies

Then the value of f (ln 5) is

w .je

[2009]

69.

Let {x} and [x] denote the fractional part of x and the greatest integer respectively of a real number x. If

and 10(n2 –

w

are three consecutive terms of a G.P., then n is equal to n), _______________. [Main Sep. 04, 2020 (II)]

w

70. The integral

71.

is equal to ________.

[Main Sep. 02, 2020 (I)] Let [t] denote the greatest integer less than or equal to t. Then the value of

is ___________.

www.jeebooks.in

[Main Sep. 02, 2020 (II)] 72.

The value of the integral

eb oo ks .in

equals _____ [Adv. 2019]

73.

Let

F(x) =

, x > 0. If

dx = F(k) –F(1) then one of

the possible values of k is .......

[1997 - 2 Marks]

74. The value of

dx is .......

[1997 - 2 Marks]

75.

For n > 0,

......

[1996 - 1 Mark]

If for nonzero x, af(x) + bf

w .je

76.

The value of

w

77.

w

78. The value of

=

– 5 where a

b, then

[1996 - 2 Marks] is ....... [1994 - 2 Marks] is ....... [1993 - 2 Marks]

79. The value of [1989 - 2 Marks]

www.jeebooks.in

80.

The integral [1988 - 2 Marks]

eb oo ks .in

Where [ ] denotes the greatest integer function, equals ....... .

81.

Then

[1987 - 2 Marks]

82.

The value of the integral

is equal to a.

[1988 - 1 Mark]

Which of the following inequalities is/are TRUE?

w .je

83.

[Adv. 2020]

(a)

(b)

w

(c)

w

(d)

84.

If I =

, then [Adv. 2017]

www.jeebooks.in

(a) l > loge 99 (b) l < loge 99

(d) l > 85.

If

eb oo ks .in

(c) 1
0 such that 91.

and f ′ is continuous on

, but not

for all for all

The value(s) of

is (are)

w .je

[2010]

(a)

(b)

(c) 0

w

(d)

w

92.

If

n = 0, 1, 2, ..., then [2009]

(a)

www.jeebooks.in

(c) (d) In = In + 1 93.

eb oo ks .in

(b)

Let f (x) be a non-constant twice differentiable function definied on such that f (x) = f (1 – x) and

. Then,

[2008]

(a) f "(x) vanishes at least twice on [0, 1] (b) (c)

w .je

(d) 94.

Let f(x) = x – [x], for every real number x, where [x] is the integral f (x) dx is

part of x. Then

1 2 0 1/2

w

(a) (b) (c) (d)

95.

dt, then the value of f(1) is

If

(a) 1/2

w

[1998 - 2 Marks]

[1998 - 2 Marks] (b)

0

(c)

1

(d)

–1/2

96.

www.jeebooks.in

List

-

I

eb oo ks .in

List - II P. The number of polynomials f (x) with non-negative integer coefficients 1. 8 , is

of degree ≤ 2, satisfying f (0) = 0 and Q. The

number

of

points

in

the

interval

2. 2 at which f (x) = sin(x2) + cos(x2) attains its maximum value, is R. equals 4

3.

w .je

S.

w

P

w

(a) (b) (c) (d)

3 2 3 2

Q R

2 3 2 3

4 4 1 1

4.

0 [Adv. 2014]

S

1 1 4 4

PASSAGE - 1

www.jeebooks.in

Let F :

be a thrice differentiable function. Suppose that F(1) = 0, F(3)

= –4 and F(x) < 0 for all x ∈

. Let f(x) = xF(x) for all x

. [Adv. 2015]

= 0 for some x ∈ (1, 3)

(d) 98.

eb oo ks .in

97. The correct statement(s) is(are)