Preview A Complete Resource Book in Mathematics for JEE Main 2019 by Dinesh Khattar (2018)

Preview A Complete Resource Book in Mathematics for JEE Main 2019 by Dinesh Khattar (2018) Preview A Complete Resource Book in Mathematics for JEE Main 2019 by Dinesh Khattar (2018) Preview A Complete Resource Book in Mathematics for JEE Main 2019 by Dinesh Khattar (2018) Preview A Complete Resource Book in Mathematics for JEE Main 2019 by Dinesh Khattar (2018) Preview A Complete Resource Book in Mathematics for JEE Main 2019 by Dinesh Khattar (2018)

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About Pearson

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MatheMaticsA Complete Resource Book in

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Dr Dinesh Khattar

Kirori Mal College, University of Delhi

A Complete Resource Book in

The aim of this publication is to supply information taken from sources believed to be valid and reliable This is not an

attempt to render any type of professional advice or analysis, nor is it to be treated as such While much care has been

taken to ensure the veracity and currency of the information presented within, neither the publisher nor its authors bear

any responsibility for any damage arising from inadvertent omissions, negligence or inaccuracies (typographical or

factual) that may have found their way into this book

Copyright © 2018 Pearson India Education Services Pvt Ltd

This book is sold subject to the condition that it shall not, by way of trade or otherwise, be lent, resold, hired out,

or otherwise circulated without the publisher’s prior written consent in any form of binding or cover other than that

in which it is published and without a similar condition including this condition being imposed on the subsequent

purchaser and without limiting the rights under copyright reserved above, no part of this publication may be reproduced,

stored in or introduced into a retrieval system, or transmitted in any form or by any means (electronic, mechanical,

photocopying, recording or otherwise), without the prior written permission of both the copyright owner and the

publisher of this book

ISBN 978-93-530-6217-0

eISBN 9789353063436

First Impression

Published by Pearson India Education Services Pvt Ltd, CIN: U72200TN2005PTC057128

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Preface ix

JEE Mains 2018 Paper                                                   xiii

JEE Mains 2017 Paper                                                   xxv

Chapter 1 Set Theory                                              11–118

Chapter 2 Functions                                               21–260

Chapter 3 Complex Numbers                                          31–368

Chapter 4 Quadratic Equations and Expressions                                41–452

Chapter 5 Matrices                                               51–530

Chapter 6 Determinants                                             61–656

Chapter 7 Permutations and Combinations                                   71–746

Chapter 8 Mathematical Induction                                        81–84

Chapter 9 Binomial Theorem                                          91–946

Chapter 10 Sequence and Series                                       101–1062

Chapter 11 Limits                                               111–1148

Chapter 12 Continuity and Differentiability                                  121–1250

Chapter 13 Differentiation                                           131–1344

Chapter 14 Applications of Derivatives                                    141–1462

Chapter 15 Indefi nite Integration                                       151–1552

Chapter 16 Defi nite Integral and Area                                     161–1686

Chapter 17 Differential Equations                                       171–1748

Chapter 18 Coordinates and Straight Lines                                 181–1854

Chapter 19 Circles                                              191–1956

Chapter 20 Conic Sections (Parabola, Ellipse and Hyperbola)                        201–2062

Chapter 21 Vector Algebra                                          211–2146

Chapter 22 Three Dimensional Geometry                                  221–2236

Chapter 23 Measures of Central Tendency and Dispersion                         231–2324

Contents

viii Contents

Chapter 24 Probability                                            241–2460

Chapter 25 Trigonometric Ratios and Identities                               251–2530

Chapter 26 Trigonometric Equations                                     261–2622

Chapter 27 Inverse Trigonometric Functions                                 271–2736

Chapter 28 Heights and Distances                                      281–2836

Chapter 29 Mathematical Reasoning                                      291–296

About the Series

A Complete Resource Book for JEE Main series is a must-have resource for students preparing for JEE Main examination

There are three separate books on Physics, Chemistry, and Mathematics; the main objective of this series is to strengthen the fundamental concepts and prepare students for various engineering entrance examinations It provides class-tested course material and numerical applications that will supplement any ready material available as student resource

To ensure high level of accuracy and practicality, this series has been authored by highly qualifi ed and experienced faculties for all three titles

About the Book

This book, A Complete Resource Book in Mathematics for JEE Main 2019, covers both the text and various types of

prob-lems required as per the syllabus of JEE Main examination It also explains various short-cut methods and techniques to solve objective questions in lesser time

me to treat my work as worship

Anuj Agarwal from IIT-Delhi, Ankit Katial from National Institute of Technology (Kurukshetra) and Raudrashish Chakraborty from Kirori Mal College, University of Delhi, with whom I have had fruitful discussions, deserve special mention

I earnestly hope that the book will help the students grasp the subject well and respond with a commendable score in the JEE Main examination There are a plethora of options available to students for Mathematics, however, ever grateful to them and to the readers for their candid feedback

Despite of our best eff orts, some errors may have crept into the book Constructive comments and suggestions to further improve the book are welcome and shall be acknowledged gratefully

Best of luck!

Dinesh KhattarPreface

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Mathematics Trend Analysis

(2007 to 2018)

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1 Two sets A and B are as under:

(B) contains exactly one element

(C) contains exactly two elements

5 If the system of linear equations, x + ky + 3z = 0,

3x + ky – 2z = 0, 2x + 4y – 3z = 0 has a non-zero

solu-tion (x, y, z), then xz

y2 is equal to

(A) –10 (B) 10 (C) –30 (D) 30

6 From 6 different novels and 3 different

dictionar-ies, 4 novels and 1 dictionary are to be selected and

arranged in a row on a shelf so that the dictionary is

always in the middle The number of such

9 Let A be the sum of the first 20 terms and B be the sum

of the first 40 terms of the series 12 + 2 · 22 + 32 + 2 ·

42 + 52 + 2 · 62 + … If B – 2A = 100λ, then λ is equal to

12 If the curves y2 = 6x, 9x2 + by2 = 16 intersect each

other at right angles, then the value of b is

(A) 6 (B) 7

2 (C) 4 (D)

92

xiv JEE Mains 2018 Paper

15 The value of sin

16 Let g(x) = cos x2, f(x) = x , and α, β(α < β) be the

roots of the quadratic equation 18x2 – 9πx + π2 = 0

Then the area (in sq units) bounded by the curve y =

(gof) (x) and the lines x = α, x = β and y = 0, is

89

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

(A) 3x + 2y = 6 (B) 2x + 3y = xy

(C) 3x + 2y = xy (D) 3x + 2y = 6xy

19 Let the orthocentre and centroid of a triangle be

A(–3,  5) and B(3, 3), respectively If C is the

cir-cumcentre of this triangle, then the radius of the circle

having line segment AC as diameter, is

21 Tangent and normal are drawn at P(16, 16) on the

parabola y2= 16x, which intersect the axis of the ola at A and B, respectively If C is the centre of the circle through the points P, A and B and ∠CPB = θ,

parab-then a value of tan θ is (A) 1

2 (B) 2 (C) 3 (D)

43

22 Tangents are drawn to the hyperbola 4x2 – y2= 36 at

the points P and Q If these tangents intersect at the point T(0, 3) then the area (in sq units) of ΔPTQ is

(A) 45 5 (B) 54 3 (C) 60 3 (D) 36 5

23 If L1 is the line of intersection of the planes 2x – 2y + 3z – 2 = 0, x – y + z + 1 = 0 and L2 is the line of inter-

section of the planes x + 2y – z – 3 = 0, 3x – y + 2z –

1 = 0, then the distance of the origin from the plane,

containing the lines L1 and L2, is (A) 1

24 The length of the projection of the line segment

join-ing the points (5, –1, 4) and (4, –1, 3) on the plane, x +

y + z = 7 is (A) 2

25 Let u be a vector co-planar with the vectors

a=2iˆ+3j kˆ ˆ and − b= +ˆj k ˆ If u is perpendicular

to a and  

u b ⋅ = 24, then u2 is equal to (A) 336 (B) 315 (C) 256 (D) 84

26 A bag contains 4 red and 6 black balls A ball is drawn

at random from the bag, its color is observed and this ball along with two additional balls of the same color are returned to the bag If now a ball is drawn at ran-dom from the bag, then the probability that this drawn ball is red, is

then the

stan-dard deviation of the 9 items x1, x2, …, x9 is (A) 9 (B) 4 (C) 2 (D) 3

HINts ANd solutIoNs

JEE Mains 2018 Paper xv

28 If sum of all the solutions of the equation

8

1

2 1cosx⋅⎛cos⎛⎝⎜ +x⎞⎠⎟ ⋅cos⎛⎝⎜ −x⎞⎠⎟−

29 PQR is a triangular park with PQ = PR = 200 m A T.V

tower stands at the midpoint of QR If the angles of

elevation of the top of the tower at P, Q and R are

respectively 45°, 30° and 30°, then the height of the tower (in m) is

(A) 100 (B) 50 (C) 100 3 (D) 50 2

30 The Boolean expression: ∼ (p ∨ q) ∨ (∼ p ∧ q) is

equiv-alent to

(A) p (B) p (C) q (D) ∼p

11 (A) 12 (D) 13 (D) 14 (A) 15 (D) 16 (A) 17 (C) 18 (C) 19 (C) 20 (D)

21 (B) 22 (A) 23 (B) 24 (D) 25 (A) 26 (B) 27 (C) 28 (B) 29 (A) 30 (D)

Even through x and y ≠ 4, 6 We will check whether the

boundary lies in the ellipse B

xvi JEE Mains 2018 Paper

Then the equation has infinitely many solution.

45 18

5 2

y= y= k k

Δ Δ

k x k

2 4 2

5 2 4

⇒ 6

4 3 1

JEE Mains 2018 Paper xvii

∴ Total number of arrangement are

2 1 17

1 17

1 17

9 We are required to find the sum of 20 terms and 40 terms,

therefore the number of terms are even

15 16

xviii JEE Mains 2018 Paper

0 π

Hence, the function is differentiable at x = π

Therefore, set S is an empty set of ϕ

1 1

1 2

and

JEE Mains 2018 Paper xix

t t

sinn x dx n

n

n n

∫ = ∫cossin = ln sin =sin

⇒ yxsinx=∫4 cosecx xsinx dx C+

xx JEE Mains 2018 Paper

Radius is 100− c and centre (–8, –6)

We can calculate the radius of tangent 2x – y + 5 = 0 (Normal

JEE Mains 2018 Paper xxi

4 3

2 2 2

2 2 2

1 2 1

(x1, y1) are point of contact

∴ First point of contact is x1 = − 3 5 and y1= –12

2 2 2

xxii JEE Mains 2018 Paper

24 We have to find the projection of AB on the plane, i.e., length

‘PQ’ equation of the plane x + y + z = 7

Q P

R n = i + j + k B

2 2

JEE Mains 2018 Paper xxiii

Case II: Black Ball is drawn in 1st draw

Let R1 be Red ball drawn in 1st draw

Let B1 be Black ball drawn in 1st draw

P R

B

2 1

4 12

6 12

6 10

4 12

48 120

n

2 1

n = 9

i i

1 9

⇒ x i x i

i i

i

2

1 9

1 9

1 9

1 9

1 9

4 cos [ cos cosx2 A B− = 1 ] 1

2cos cosA B= cos(A B+ ) cos( + A B− )

xxiv JEE Mains 2018 Paper

Summing the three roots

2 3

13 9

200 m

200 m 45°

30°

30°

m h

(B) Surjective but not injective

(C) Neither injective nor surjective

5 If S is the set of distinct values of b for which the

fol-lowing system of linear equations

6 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

com-mon 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

(A) 468 (B) 469 (C) 484 (D) 485

7 The value of (21C1 – 10C1) + (21C2 – 10C2) + (21C3 –

10C3) + (21C4 – 10C4) + … + (21C10 – 10C10) is (A) 221 - 210 (B) 220 - 29

(C) 220 - 210 (D) 221 - 211

8 For any three positive real numbers a, b and c, 9(25a2 +

b2) + 25(c2 – 3ac) = 15b(3a + c) Then (A) b, c and a are in A.P.

(B) a, b and c are in A.P.

(C) a, b and c are in G.P.

(D) b, c and a are in G.P.

9 Let a, b, c ∈ R If f(x) = ax2 + bx + c is such that a +

b + c = 3 and f(x + y) = f(x) + f(y) + xy, ∀ x, y ∈ R, then

10 limcot cos

11 If for x ∈ 0 1

4,

1 9 3

x x x

3

1 9 3

x x

− (C) 3

⎛

12

13,−

⎛

⎝⎜ ⎞⎠⎟ (D) ⎛⎝⎜− − ⎞⎠⎟

12

12,

13 Twenty meters 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 flower-bed, is

(A) 10 (B) 25 (C) 30 (D) 12.5

JEE MAINs 2017 pApEr

xxvi JEE Mains 2017 Paper

14 Let I n = tan∫ n xdx n,( >1 If I) 4 + I6 = a tan5 x +

bx5 + C, where C is a constant of integration, then the

ordered pair (a, b) is equal to

π

is equal to (A) 2 (B) 4 (C) –1 (D) –2

16 The area (in sq units) of the region {(x, y) : x ≥ 0,

(C)

4

3 (D)

13

18 Let k be an integer such that the triangle with vertices

(k, – 3k), (5, k) and (– k, 2) has area 28 sq units Then

the orthocentere of this triangle is at the point

(A) 1 3

4,

⎛

⎝⎜ ⎞⎠⎟ (B) 1

34,−

⎛

⎝⎜ ⎞⎠⎟

(C) 2 1

2,

⎛

⎝⎜ ⎞⎠⎟ (D) 2

12,−

⎛

⎝⎜ ⎞⎠⎟

19 The radius of a circle, having minimum area, which

touches the curve y = 4 – x2 and the lines, y = | x | is

has foci at (± 2, 0) Then the tangent to this hyperbola

at P also passes through the point

(A) (2 2 3 3 (B) , ) ( 3, 2 )

(C) (− 2,− 3 ) (D) (3 2 2 3, )

22 The distance of the point (1, 3, –7) from the plane

passing through the point (1, –1, –1), having normal perpendicular to both the lines x− = y+ z

− =

−1

1

22

43and x− = y+ z

− =

+

−

22

11

7

1 is (A) 19

23 If the image of the point P(1, –2, 3) in the plane,

2x + 3y – 4z + 22 = 0 measured parallel to the line,

x y z

1= = is Q, then PQ is equal to4 5 (A) 2 42 (B) 42 (C) 6 5 (D) 3 5

a=2iˆ+ −ˆj 2k and bˆ = +iˆ j ˆ Let c be a vector

such that |c a − =| 3, | (a b × × =) c| 3 and the angle between c and a b × be 30º Then  a c is equal to (A) 2 (B) 5 (C) 1

8 (D)

258

25 A box contains 15 green and 10 yellow balls If 10 balls

are randomly drawn, one-by-one, with replacement, then the variance of the number of green balls drawn is (A) 6 (B) 4 (C) 6/25 (D) 12/5

26 For three events A, B and C, P(Exactly one of A or B

occurs) = P(Exactly one of B or C occurs)

27 If two different numbers are taken from the set {0, 1, 2,

3, …, 10}; then the probability that their sum as well

as absolute difference are both multiple of 4, is (A) 12

28 If 5(tan2x – cos2x) = 2 cos 2x + 9, then the value of cos 4x is

29 Let a vertical tower AB have its end A on the level

ground Let C be the mid-point of AB and P be a point

on the ground such that AP = 2AB If ∠BPC = β then

tan β is (A) 1

30 The following statement (p → q) → [( ~ p → q) → q] is (A) Equivalent to ~ p → q

(B) Equivalent to p → ~ q

(C) A fallacy (D) A tautology

HINts ANd solutIoNs

JEE Mains 2017 Paper xxvii

x

x x

f ′(x) changes sign in different intervals.

1 2 ,

∴ Surjective but not injective.

Hence, the correct option is (B)

0 0

2 2

2 2

2 2

11 (D) 12 (A) 13 (B) 14 (A) 15 (A) 16 (C) 17 (D) 18 (C) 19 (B) 20 (A)

21 (A) 22 (A) 23 (A) 24 (A) 25 (D) 26 (A) 27 (D) 28 (C) 29 (B) 30 (D)

Answer keys

HINts ANd solutIoNs

xxviii JEE Mains 2017 Paper

Eq (1) and (2) are identical i.e., x + y + z = 1

To have no solution with x + by + z = 0

Hence, the correct option is (C)

8. 9(25a2 + b2) + 25(c2 – 3ac) = 15b(3a + c)

⇒ (15a)2 + (3b)2 + (5c)2 – 45ab – 15bc – 75ac = 0

= 116

Hence, the correct option is (A)

11 f(x) = 2tan –1 (3x x)

For x ∈ 0 1

4 ,

JEE Mains 2017 Paper xxix

2 2

1

2

2 4

π

= 1

2

2 1

2 1

2 1

2 1 1

3 8

4

4 2

x x

m = –2 E

m =

8

xxx JEE Mains 2017 Paper

3 2 3

2 3 ⇒ 4x – 2y – 1 = 0

Hence, the correct option is (A)

JEE Mains 2017 Paper xxxi

21 x

a

y b

22 Let the plane be a(x – 1) + b(y + 1) + c(z + 1) = 0

It is perpendicular to the given lines

3 5

var(X) = n.p.q

= 10 6

25 = 125

16

Hence, the correct option is (A)

27 Total number of ways = 11C2 = 55

Favourable ways are

xxxii JEE Mains 2017 Paper

C

B x x

tan (θ + β ) = 1

2

∴

1 4

4

1 2

+

tan tan

β β

Set Theory 1.1

Set, Representation of a set, Types of sets, Operations on sets, Algebra of sets, Cartesian product of two sets,

Relations, Types of relations on a set, Equivalence relation, Congruence modulo m.

A set is a well defi ned collection of objects such that given

an object, it is possible to determine whether that object

belongs to the given collection or not

For example, the collection of all students of Delhi

University is a set, whereas, collection of all good books

on mathematics is not a set, since a mathematics book

con-sidered good by one person might be concon-sidered bad or

average by another

notations

The sets are usually denoted by capital letters A, B, C, etc

and the members or elements of the set are denoted by

lower-case letters a, b, c etc If x is a member of the set A,

we write x ∈ A (read as ‘x belongs to A’) and if x is not a

member of the set A, we write x ∉ A (read as ‘x does not

belong to A’) If x and y both belong to A, we write x, y ∈ A.

rePresenTaTion of a seT

Usually, sets are represented in the following two ways:

1 Roster form or tabular form

2 Set builder form or rule method

roster form

In this form, we list all the members of the set within braces

(curly brackets) and separate these by commas

For example, the set A of all odd natural numbers less

than 10 in the roster form is written as:

i M P o R t a n t P o i n t S

set-builder form

In this form, we write a variable (say x) representing any

member of the set followed by a property satisfi ed by each member of the set

For example, the set A of all prime numbers less than

10 in the set-builder form is written as

A = {x | x is a prime number less than 10}

The symbol ‘|’ stands for the words ‘such that’ Sometimes,

we use the symbol ‘:’ in place of the symbol ‘|’

TYPes of seTs empty set or null set

A set which has no element is called the null set or empty set It is denoted by the symbol Φ

For example, each of the following is a null set:

1 The set of all real numbers whose square is –1.

2 The set of all rational numbers whose square is 2.

3 The set of all those integers that are both even and odd.

A set consisting of atleast one element is called a non-empty set.

1.2 Chapter 1

singleton set

A set having only one element is called singleton set.

For example, {0} is a singleton set, whose only

member is 0

finite and infi nite set

A set which has fi nite number of elements is called a fi nite

set Otherwise, it is called an infi nite set.

For example, the set of all days in a week is a fi nite

set whereas, the set of all integers, denoted by {…, – 2, – 1,

0, 1, 2, …} or {x | x is an integer}, is an infi nite set.

An empty set Φ which has no element, is a fi nite set

The number of distinct elements in a fi nite set A is

called the cardinal number of the set A and it is denoted

by n (A).

equal sets

Two sets A and B are said to be equal, written as A = B, if

every element of A is in B and every element of B is in A.

For example, the sets A = {4, 5, 3, 2} and B = {1, 6,

8, 9} are equivalent but are not equal.

c a u t i o n

subset

Let A and B be two sets If every element of A is an element

of B, then A is called a subset of B and we write A ⊆ B or

B ⊇ A (read as ‘A is contained in B’ or B contains A’) B is

called superset of A.

 If A ⊆ B and A ≠ B, we write A ⊂ B or B ⊃ A (read as : A

is a proper subset of B or B is a proper superset of A).

 Every set is a subset and a superset of itself.

 If A is not a subset of B, we write A ⊄ B.

 The empty set is the subset of every set.

 If A is a set with n (A) = m, then the number of subsets

of A are 2 m and the number of proper subsets of A are

The set of all subsets of a given set A is called the power set

of A and is denoted by P(A).

For example, if A = {1, 2, 3}, then

In Venn diagrams, the universal set U is represented

by the rectangular region and its subsets are represented by closed bounded circles inside this rectangular region

oPeraTions on seTs Union of Two sets

The union of two sets A and B, written as A ∪ B (read as

‘A union B’), is the set consisting of all the elements which are either in A or in B or in both Thus,

intersection of Two sets

The intersection of two sets A and B, written as A ∩ B (read

as ‘A intersection B’) is the set consisting of all the

com-mon elements of A and B Thus,

Two sets A and B are said to be disjoint, if A ∩ B = f, i.e., A

and B have no element in common.

For example, if A = {1, 2, 5} and B = {2, 4, 6}, then

A ∩ B = f, so A and B are disjoint sets.

U

B A

Fig 1.3

difference of Two sets

If A and B are two sets, then their diff erence A – B is defi ned as

symmetric difference of Two sets

The symmetric diff erence of two sets A and B, denoted by

6 For any two sets A and B, we have

(a) P(A) ∩ P(B) = P(A ∩ B)

(b) P(A) ∪ P(B) ⊆ P(A ∪ B), where P(A) is the power

set of A.

7 If A is any set, we have (A ′)′ = A.

8 Demorgan’s Laws: For any three sets A, B and C, we

some results about cardinal number

If A, B and C are fi nite sets and U be the fi nite universal

10 If A1, A2, A3, … A n are disjoint sets, then

n (A1∪ A2∪ A3∪ … ∪ A n) = n (A1) + n (A2) + n (A3)

+ … + n (A n)

11 n (A D B) = number of elements which belong to exactly one of A or B

carTesian ProdUcT of Two seTs

If A and B are any two non-empty sets, then cartesian product of A and B is defi ned as

A × B = [(a, b) : a ∈ A and b ∈ B]

A × B ≠ B × A

c a u t i o n

Trick(s) for Problem solving

key results on cartesian Product

If A, B, C are three sets, then

9 If A and B are two non-empty sets having n elements

in common, then A × B and B × A have n2 elements in

2 Let A = {2, 3, 4} and X = {0, 1, 2, 3, 4}, then which of

the following statements is correct?

{0} ⊂ A′ in X is correct, because the only element

of {0} namely 0 also belongs to A ′ in X.

0 ⊂ A′ in X is false, because 0 is not a set.

3 If X = {8n – 7 n – 1/n ∈ N} and Y = {49 (n – 1)/n ∈ N},

then

(A) X ⊂ Y (B) Y ⊂ X (C) X = Y (D) None of these

0 for n = 1 Hence, X consists of all positive integral multiples of 49 of the form 49 K n where K n= n C2 +

n C37 + … + n C n7n – 2 together with zero Also, Y

con-sists of all positive integral multiples of 49 including

6 Let A and B two non-empty subsets of a set X such that

A is not a subset of B then

(A) A is subset of the complement of B

(C) B is a subset of A

(C) A and B are disjoint

(D) A and the complement of B are non-disjoint

Solution: (D)

Since A ⊄ B, $ x ∈ A such that x ∉ B

Then x ∈ B ′

\ A ∩ B ′ ≠ f

7 Two finite sets have m and n elements, then total number

of subsets of the first set is 56 more that the total number

of subsets of the second The values of m and n are,

(A) 7, 6 (B) 6, 3 (D) 5, 1 (D) 8, 7

Solution: (B)

Since the two finite sets have m and n elements, so

number of subsets of these sets will be 2m and 2n

respectively According to the question

(A) The smallest set of Y is {3, 5, 9}

(B) The smallest set of Y is {2, 3, 5, 9}

(C) The largest set of Y is {1, 2, 3, 4, 9}

(D) The largest set of Y is {2, 3, 4, 9}

Solution: (A and C)

Since the set on the right hand side has 5 elements,

\ smallest set of Y has three elements and largest set of Y has five elements,

\ smallest set of Y is {3, 5, 9}

and largest set of Y is {1, 2, 3, 4, 9}.

10 If A has 3 elements and B has 6 elements, then the

minimum number of elements in the set A ∪ B is

mini-in B, that is, 6.

11 Suppose A1, A2, … A30 are thirty sets, each with five

elements and B1, B2, …, B n are n sets each with three

If there are m distinct elements in S and each element

of S belongs to exactly 10 of the A i′s, we have

n A i

i

( )

= 1 30

[(A ∪ B) ∩ B′] ∪ A′ = A ∪ A′ = N

13 If X and Y are two sets and X′ denotes the complement

of X, then X ∩ (X ∪ Y)′ equals

14 In a group of 65 people, 40 like cricket, 10 like both

cricket and tennis The number of persons liking tennis

only and not cricket is

(C) 15 (D) None of these

Solution: (B)

Let A be the set of people who like cricket and B the

set of people who like tennis

= n(B) – n (A ∩ B) = 35 – 10 = 25

\ Number of people who like tennis only and not cricket = 25

15 In a group of 1000 people, there are 750 people who

can speak Hindi and 400 who can speak English Then number of persons who can speak Hindi only is

16 If f : R → R, defined by f (x) = x2 + 1, then the values of

f –1(17) and f –1(–3) respectively are (A) f, {4, –4} (B) {3, –3}, f (C) f, {3, –3} (D) {4, –4}, f

Solution: (D)

Let y = x2 + 1 Then for y = 17,

we have x = ± 4 and for y = –3, x becomes imaginary that is, there is no value of x.

Hence, f (17) = {–4, 4}

and f–1(–3) = f