UPKAR PRAKASHAN, AGRA–2

ByDr. N. K. Singh

(M.Sc. (Maths), B.Ed., Ph.D and U.T.I. (U.S.A.)

(According to the Syllabus)

© Publishers

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Code No. 1576

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Preface

This is an era of fierce competition, where only the best survive. Many students sitfor the AIEEE for admission. This examination enables those who qualify to takeadmission in prestigious engineering institutes of the country for the award of degreesin engineering.

An organised effort is required to succeed in the entrance examination of theseinstitutes, as is evident from the fact that, out of the many who appear for the examsonly a few manage to get through. This book is primarily meant for students appearingfor these examinations.

We have been guided by the following goals in writing this book.

* To stress problem-solving techniques rather than abstract formula andtheorems.

* To supplement the study material by use of examples.

* To solve objective type questions by appropriate reasoning.

* To make it possible for students with a high school mathematics education toextend their knowledge of mathematics and to be successful in highlychallenging and competitive entrance examinations.

The book thoroughly covers all the topics prescribed in the syllabi of the AIEEE.Each chapter begins with a review of important definitions, formulae, results andgeneral techniques of solving the problems. I am sure that the candidates appearing forthis examination will find this book extremely useful for achieving their goal.

Suggestions from teachers and students for the improvement of this book arewelcome.

—Dr. N. K. Singh

Contents

01. Elements of Set Theory 3–10

02. Relation and Function 11–18

03. Complex Numbers 19–41

04. Matrices and Determinants 42–53

05. Quadratic Equation 54–84

06. Permutation and Combination 85–98

07. Mathematical Induction 99–106

08. Binomial Theorem 107–118

09. Sequence and Series 119–136

10. Trigonometry 137–190

11. Two Dimensional Geometry 191–217

12. Three Dimensional Geometry 218–239

Calculus

13. Function 240–251

14. Limit, Continuity and Differentiability 252–277

15. Tangents and Normals 278–291

16. Maxima and Minima 292–301

17. Rolle’s Theorem, Mean Value Theorem, Taylor’s Theorem 302–315

18. Partial Differentiation 316–327

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19. Singular Points 328–339

20. Curvature 340–350

21. Asymptotes 351–361

22. Curve Tracing 362–372

23. Integration of Rational, Irrational and Trigonometric Functions 373–412

24. Differential Equations 413–429

25. Vector Algebra 430–449

26. Frequency Distribution, Mean, Median, Mode and Standard Deviation 450–470

27. Probability 471–487

Elementary Statics

28. Statics : Basic Concepts 488–540

29. Velocity, Acceleration and Rectilinear Motion 541–592

Mental Aptitude

● Part-I : Aptitude Test 594–639

● Part-II : Drawing Aptitude 640–652

SYLLABUS

Unit 1 : Sets, Relations and Functions

Sets and their representations, Union, inter-section and complements of sets and their algeb-raic properties, Relations, equivalence relations,mappings, one-one, into and onto mappings,composition of mappings.

Unit 2 : Complex NumbersComplex number in the form a + ib and their

representation in a plane. Argand diagram.Algebra of complex numbers, Modulus and Argu-ments (or amplitude) of a complex number,square root of a complex number. Cube roots ofunity, triangle inequality.

Unit 3 : Matrices and DeterminantsDeterminants and matrices of order two and

three, properties of determinants, Evaluation ofdeterminants. Area of triangles using determi-nants, Addition and multiplication of matrices,adjoint and inverse of matrix. Test of consistencyand solution of simultaneous linear equationsusing determinants and matrices.

Unit 4 : Quadratic Equations

Quadratic equation in real and complexnumber system and their solutions. Relation bet-ween roots and coefficients, nature of roots,formation of quadratic equations with given roots;Symmetric functions of roots, equations reducibleto quadratic equations—application to practicalproblems.

Unit 5 : Permutation and Combination

Fundamental principle of counting; Permuta-tion as an arrangement and combination asselection, Meaning of P(n, r) and C(n, r). Simpleapplications.

Unit 6 : Mathematical Induction and ItsApplications

Unit 7 : Binomial Theorem and Its Appli-cations

Binomial theorem for a positive integralindex; general term and middle term; Binomialtheorem for any index. Properties of Binomial co-efficients. Simple applications for approxima-tions.

Unit 8 : Sequences and Series

Arithmetic, Geometric and Harmonic pro-gressions. Insertion of Arithmetic, Geometric andHarmonic means between two given numbers.Relation between A.M., G.M. and H.M. specialseries : Σ n , Σ n2, Σ n3. Arithmetic-GeometricSeries, Exponential and Logarithmic series.

Unit 9 : Differential CalculusPolynomials, rational, trigonometric, logari-

thmic and exponential functions, Inverse func-tions. Graphs of simple functions. Limits, Conti-nuity; differentiation of the sum, difference, pro-duct and quotient of two functions. Differentiationof trigonometric, inverse trigonometric, logarith-mic, exponential, composite and implicit func-tions; derivatives of order upto two. Applicationsof derivatives : Rate of change of quantities,monotonic—increasing and decreasing functions,Maxima and minima of functions of one variable,tangents and normals, Rolle’s and Lagrange’smean value theorems.

Unit 10 : Integral Calculus

Integral as an anti-derivative. Fundamentalintegrals involving algebraic, trigonometric,exponential and logarithmic functions. Integrationby substitution, by parts and by partial fractions.

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Integration using trigonometric identities. Integralas limit of a sum. Properties of definite integrals.Evaluation of definite integrals; Determiningareas of the regions bounded by simple curves.

Unit 11 : Differential Equations

Ordinary differential equations, their orderand degree. Formation of differential equations.Solution of differential equations by the methodof separation of variables. Solution ofhomogeneous and linear differential equations

and those of the type d2ydx2 = f(x).

Unit 12 : Two Dimensional Geometry

Recall of Cartesian system of rectangular co-ordinates in a plane, distance formula, area of atriangle, condition for the collinearity of threepoints and section formula, centroid and in-centreof a triangle, locus and its equation, translation ofaxes, slope of a line, parallel and perpendicularlines, intercepts of a line on the coordinate axes.

The Straight Line and Pair of StraightLines

Various forms of equations of a line, inter-section of lines, angles between two lines, condi-tions for concurrence of three lines, distance of apoint from a line. Equations of internal andexternal bisectors of angles between two lines,coordinates of centroid, orthocentre andcircumcentre of a triangle, equation of family oflines passing through the point of intersection oftwo lines, homogeneous equation of seconddegree in x and y, angle between pair of linesthrough the origin, combined equation of thebisectors of the angles between a pair of lines,condition for the general second degree equationto represent a pair of lines, point of intersectionand angle between two lines.

Circles and Family of Circles

Standard form of equation of a circle, generalform of the equation of a circle, its radius andcentre, equation of a circle in the parametric form,equation of a circle when the end points of adiameter are given points of intersection of a line

and a circle with the centre at the origin andcondition for a line to be tangent to the circle,length of the tangent, equation of the tangent,equation of a family of circles through the inter-section of two circles, condition for two intersect-ing circles to be orthogonal.

Conic Sections

Sections of cones, equations of conic sections(parabola, ellipse and hyperbola) in standardforms, condition for y = mx + c to be a tangentand point(s) of tangency.

Unit 13 : Three Dimensional Geometry

Coordinates of a point in space, distance bet-ween two points; Section formula, direction ratiosand direction cosines, angle between two inter-secting lines, Skew lines, the shortest distancebeween them and its equation. Equations of a lineand a plane in different forms; intersection of aline and a plane, coplanar lines, equation of asphere, its centre and radius. Diameter form of theequation of a sphere.

Unit 14 : Vector Algebra

Vectors and Scalars, addition of vectors,components of a vector in two dimensions andthree dimensional space, scalar and vector pro-ducts, scalar and vector triple product. Applicationof vectors to plane geometry.

Units 15 : Measures of Central Tendencyand Dispersion

Calculation of Mean, median and mode ofgrouped and ungrouped data. Calculation ofstandard deviation, variance and mean deviationfor grouped and ungrouped data.

Unit 16 : Probability

Probability of an event, addition and multi-plication theorems of probability and their appli-cations; Conditional probability; Bayes’ theorem,Probability distribution of a random variate;Binomial and Poisson distribution and theirproperties.

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Unit 17 : Trigonometry

Trigonometrical identities and equations.Inverse trigonometric functions and theirproperties. Properties of triangles, includingcentroid, incentre, circumcentre and orthocentre,solution of triangles. Heights and distances.

Unit 18 : Statics

Introduction, basic concepts and basic lawsof mechanics, force, resultant of forces acting at apoint, parallelogram law of forces, resolved parts

of a force. Equilibrium of a particle under threeconcurrent forces, triangle law of forces and itsconverse. Lami’s theorem and its converse. Twoparallel forces, like and unlike parallel forces,couple and its moment.

Unit 19 : Dynamics

Speed and velocity, average speed, instanta-neous speed, acceleration and retardation, resul-tant of two velocities. Motion of a particle alonga line moving with constant acceleration. Motionunder gravity. Laws of motion, Projectile motion.

MATHEMATICS

1Elements of Set Theory

1. Set—A well-defined collection of distinctobjects is called a set. When we say ‘well-defined’, we mean that we must be given a rule orrules with the help of which we should readily beable to say whether a particular object is amember of the set or is not a member of the set.

2. The collection of all honest persons inIndia is not a set, because the term ‘honesty’ is notwell-defined.

The members of a set are called its elements.The elements of a set are generally denoted bysmall letters a, b, c, …, x, y, z. The sets aregenerally denoted by capital letters A, B, C, …, X,Y, Z.

If an element x is in set A, then we say xbelong to A and we write x ∈ A. If an element x isnot in A, then we say x does not belong to A andwe write x ∉ A.

3. Finite and Infinite sets—Sets which havea finite number of elements are called finite setsand those having infinite number of elements arecalled infinite sets. If S is a set, then n (S) denotesthe number of elements in S. n (S) is also called ascardinal number of S.

Examples :Finite set—

(i) Set of days in a week(ii) Set of dates in a month

(iii) Set of chairs in a classroom.

Infinite set—(i) Set of natural numbers

(ii) Set of points on a plane(iii) Set of lines passing through one point

4. Equivalent sets—Two sets A and B aresaid to be equivalent sets if the elements of A canbe paired with the elements of B, so that to eachelement of A these correspond exactly one

element of B, and to each element of B, thesecorresponds exactly one element of A.

Example :Let A = {a, b, c}and B = {4, 7, 10}then, A and B are equivalent sets.

5. Equal sets—Two sets are equal if theyhave exactly same elements.

Example :If A = {1, 4, 5}and B = {4, 1, 5}then, A = B.

6. Null or Empty set—A set is said to be anull set if it does not contain any element. A nullset is denoted by φ,

Example :Let A = {x : x ∈ N, 2 < x < 3}.A does not contain any element, because there

is no natural number between 2 and 3.

7. Subsets—A set A is said to be a subset ofset B, if every element of A is an element of B. IfA is a subset of B, then we write A ⊆ B.

A ⊆ B if x ∈ A ⇒ x ∈ B and for propersubset use ⊂.

Example :If A = {1, 2}then, subset of A are φ, {1}, {2}, {1, 2}.

8. Power set—The set of all subsets of a setA is called the power set of A and is denoted byP(A), i.e.

P(A) = {B : B ⊆ A}

Example :If A = {5, 7}then, P(A) = {φ, {5}, {7}, {5, 7}}

9. Universal set—Universal set U is supersetof all the set of all the sets under consideration.

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10. Complement of a set—Let U be anuniversal set and A be any element of it, then Ac

or A' is complement of A given by—Ac = {x : x ∉ A, x ∈ U}.

Clearly (Ac)c = Aor (A')' = ALet U = {1, 2, 3, 4, 5}and A = {1, 4}then, A' = {2, 3, 5}

11. Union of set—Let A and B be two sets. Aset consists of the elements of both A and B iscalled the union of A and B and is denoted byA ∪ B

∴ A ∪ B = {x : x ∈ A or x ∈ B}

Example :Let A = {a, b, c, d}

B = {a, e, f}then, A ∪ B = {a, b, c, d, e, f}

12. Intersection of sets—Let A and B be twosets then the set which consists the commonelements of A and B is called the intersection of Aand B and it is denoted by A ∩ B.

∴ A ∩ B]={x : x ∈ A and x ∈ B}

Example :If A = {a, b, c, d}

B = {a, e, f}then, A ∩ B = {a}

13. Disjoint sets—Two sets A and B are saidto be disjoint sets. If there is no common elementin A and B.

If A and B are disjoint sets,then, A ∩ B = φ.

Example :Let A = {a, b, c}

B = {x, y, z}then, A and B are disjoint sets because

A ∩ B = φ.

14. Difference of sets—The difference of twosets A and B in this order is the set of all thoseelements of A which are not in B. The differenceof A and B in this order is denoted by A – B.

∴ A – B = {x : x ∈ A and x ∉ B}and B – A = {x : x ∈ B and x ∉ A}.Example :Let A = {1, 2, 3, 4}

and B = {2, 3, 4, 5, 6}then A – B = {1}and B – A = {5, 6}.

15. Venn diagram—Most of the relationshipbetween the sets can be represented by diagramsknown as venn diagram. A universal set U isrepresented by points in interior of a rectangle andany of its non empty subsets by points in theinterior of closed curves.

16. Cartesian product of sets—Let A and Bbe two sets then,

A × B = {(a, b) : a ∈ A and b ∈ B}

A × B is called Cartesian product of sets. If Ahas m elements and B has n elements thenA × B has mn elements.

Example :Let A = {a, b}

B = {c, d, e}then, A × B = {(a, c), (a, d), (a, e), (b, c),

(b, d), (b, e)}∴ A × B has 2 × 3 i.e., 6 elements.

17. Some Results—(i) A ∪ φ = A(ii) A ∪ A = A(iii) A ∪ B = B ∪ A

(i.e. ∪ is commutative)

(iv) (A ∪ B) ∪ C = A ∪ (B ∪ C)(i.e. ∪ is associative)

(v) A ∪ U = UA ∪ A' = U

(vi) A ∩ A = AA ∩ A' = φ

(vii) A ∩ B = B ∩ Ai.e. ∩ is commutative

(viii) (A ∩ B) ∩ C = A ∩ (B ∩ C)i.e. is associative

(ix) A ∩ U = A(x) A ∩ φ = φ(xi) A ∩ (B ∪ C) = (A ∩ B) ∩ (A ∩ C)

i.e. is distributive over intersection

(xii) A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

(xiii) A – B = A ∩ B'= B' – A'

(xiv) A ⊆ B ⇔ B' ⊆ A'

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(xv) A Δ B = (A – B) ∪ (B – A)= (A ∪ B) – (A ∩ B)

(xvi) (A ∪ B)' = A' ∩ B'

(A ∩ B)' = A' ∪ B'

are known as De Morgan's Law

(xvii) (a) A Δ φ = A

(b) A Δ A = φ

(c) A Δ B = B Δ A

(d) A Δ B = φ ⇒ A = B

(e) A Δ B = A ∪ B ⇔ A ∩ B = φ

(xviii) (a) A ⊂ B and C ⊂ D ⇒ A × C ⊂ B × D

(b) A × (B ∩ C) = (A × B) ∩ (A × C)

(c) A × (B ∪ C) = (A × B) ∪ (A × C)

(d) A × (B – C) = A × B – A × C

(e) (A × B) ∩ (C × D) = (A ∩ C) × (B ∩ D)

(f) (A × B) ∪ (C × D) = (A ∪ C) × (B ∪ D)

(xix) (a) n (A ∪ B) = n (A) + n (B)– n (A ∩ B)

(b) n (A ∪ B ∪ C) = n (A) + n (B)

+ n (C) – n (A ∩ B) – n (B ∩ C)

– n (C ∩ A) + n (A ∩ B ∩ C)

(c) n (A') = n (U) – n (A)

(d) n (A ∩ B') = n (A) – n (A ∩ B)

ILLUSTRATIONSExample 1.

If A = {x : x2 + 6x – 7 = 0}

and B = {x : x2 + 9x + 14 = 0}

then, A – B is equal to—

(A) {1, – 7} (B) {1}

(C) {–7} (D) {1, 2, –7}

Sol.A = {x : x2 + 6x – 7 = 0}

= {x : x = 1, – 7}

and B = {x : x2 + 9x + 14 = 0}

= { x : (x + 2) ( x + 7)}

= {x : x = –2, –7}

∴ A – B = {1}

Example 2. If A and B are sets with 10 and 6elements respectively with 4 common elements,

then the number of elements in (A × B) ∩ (B × A)is—

(A) 60 (B) 36(C) 16 (D) 4

Sol. Let A and B are two sets have 10 and 6elements respectively with 4 common elements,then number of elements

(A × B) ∩ (B × A) = 24 = 16

Example 3. Which one of the followingstatement for set A, B, C correct ?

(A) A – (B ∪ C) = (A – B) ∪ (A – C)

(B) A ∪ (B – C) = (A ∪ B) – (A ∪ C)

(C) A – (B ∩ C) = (A – B) ∩ (A – C)

(D) A – (B ∪ C) = (A – B) ∩ (A – C)

Sol. Let x ∈ A – (B ∪ C)

⇒ x ∈ A and x ∉ (B ∪ C)

⇒ x ∈ A and (x ∉ B, x ∉ C)

⇒ (x ∈ A and x ∉ B)

and (x ∉ A and x ∉ C)

⇒ x ∈ (A – B) and x ∈ (A – C)

⇒ x ∈ (A – B) ∩ (A – C)

∴ A – (B ∪ C) ⊆ (A – B) ∩ (A – C)

Similarly, (A – B) ∩ (A – C) ⊆ A – (B ∪ C)

Hence, A – (B ∪ C) = (A – B) ∩ (A – C)

Example 4. If A = {1, 2, 5, 6} and B ={1, 2, 3} then

(A × B) ∩ (B × A) is—

(A) {(1, 1) (1, 2) (2, 1) (2, 2)}

(B) {(1, 1) (2, 2) (5, 1) (1, 6)}

(C) {(1, 1) (2, 1) (6, 1) (3, 2)}

(D) {(2, 3) (3, 1) (3, 2) (5, 3)}

Sol. A = {1, 2, 5, 6}

B = {1, 2, 3}

A × B = {(1,1) (1, 2) (1, 3) (2,1)

(2, 2) (2, 3) (5,1) (5, 2)

(5, 3) (6, 1) (6, 2) (6, 3)}

B × A = {(1,1) (1, 2) (1, 5) (1, 6)

(2,1) (2, 2) (2, 5) (2, 6)

(3, 1) (3, 2) (3, 5) (3, 6)}

(A × B) ∩ (B × A) = {(1, 1) (1, 2) (2, 1)

(2, 2)}

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Objective Questions6 Each of the following question has four alter-native answer one or more than one of them iscorrect. Tick mark the correct answers—

1. For any three sets A, B and C which one ofthe following statement is correct ?

(A) A ∩ B = φ ⇒ A = φ or B = φ(B) A – B = φ ⇒ A ⊆ B

(C) A ∪ B = φ ⇒ A ⊆ B

(D) A ∩ B ≠ φ, A ∩ C ≠ φ ⇒ A ∩ B ∩ C ≠ φ

2. If A = {a , b, c} then the number of propersubsets of A is—(A) 5 (B) 6(C) 7 (D) 8

3. Let U = {1, 2, 3, …, 8} be a universal set andA = {1, 2, 3, 4} and B = {2, 4, 5, 7} besubsets of U, then AC ∪ BC is equal to—

(A) {1, 2, 3, 5, 6, 7, 8}

(B) {1, 3, 4, 5, 6, 7, 8}

(C) {1, 3, 5, 6, 8}

(D) {1, 3, 5, 6, 7, 8}

4. Let A and B be two sets having 5 commonelements. The numbers of elements commonto A × B and B × A is—(A) 0 (B) 52

(C) 25 (D) None of these

5. The number of non-empty subset of a setconsisting of 8 elements is—

(A) 256 (B) 255(C) 128 (D) None of these

6. Let A and B have 4 and 8 elements respec-tively. What can be the maximum andminimum number of element in A × B ?

(A) 16 and 64 (B) 32 and 32

(C) 32 and 64 (D) 64 and 64

7. If A = {a, c , d, g} and B = {b, d, j, k} thenwhich one of the following is true ?

(A) A ∩ B is a null set(B) A and B are disjoint set(C) A ∩ B is a singleton set(D) All of the above are true

8. If A and B be two subsets of a set U, thenwhich of the following is false ?

(A) A ∩ φ = φ (B) A ∩ U = A

(C) A ∩ B ⊂ A (D) B ⊃ A ∩ B

9. It is given that n {P(S)} = 64 where P(S) ispower set of S, then n (S) is—

(A) 2 (B) 4(C) 8 (D) 6

10. If A and B are two subset of universal set Uthen A–B is equal to—(A) A ∩ B (B) A ∩ B'(C) A' ∩ B (D) A' ∩ B'

11. Let A = {(x, y) : y = 1x, 0 ≠ x ∈[R}

B = {(x, y) : y –x, x ∈[R}[R

being the set of reals, then which one of thefollowing is true ?(A) A ∩ B = A (B) A ∩ B = B(C) A ∩ B = A ∪ B (D) A ∩ B = φ

12. If P, Q ⊂ U, then P ∩ (P ∪ Q)' is equal to—(A) P (B) Q(C) φ (D) None of these

13. If A = {1, 2, 3, 4, 5} and B = {2, 3, 6, 7} thenthe number of elements in (A × B) ∩ (B × A)—(A) 0 (B) 4(C) 6 (D) 18

14. If A = {4, 5, 8, 12}, B = {1, 4, 6, 9} then A –(B – A) is—

(A) {4, 5, 8, 12} (B) {4, 5, 6, 8}

(C) {4, 5, 8, 9} (D) {4, 5, 1, 9}

15. If A = {x : x = 4n + 1, n ≤ 5, n ∈ N} and B ={3n : n ≤ 8 . n ∈ N} then A – (A ∩ B) isequal to—

(A) {5, 13, 17} (B) {13, 17, 16}

(C) {5, 13, 15} (D) {17, 16, 13}

16. Let A and B be two finite sets such thatn(A – B) = 15, n(A ∪ B) = 90, n(A ∩ B) = 30then n(B) is equal to—(A) 45 (B) 75(C) 60 (D) 30

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17. Which of the following sets are null sets ?(A) The set A of all prime numbers lying

between 15 and 19(B) A = {x : x < 5, x > 6}(C) A = {x : x2 = 16 x ∈ N}(D) A = {x : | x| > – 4, x ∉ N}

18. Let A = {x : x ∈ N, 0 < x < 5} and B = {x : x ∈N, 4 < x < 6} then A ∪ B is equal to—

(A) {1, 2, 3, 4, 5} (B) {2, 3, 4, 5}

(C) {5, 3, 2, 6} (D) {1, 2, 4, 5}

Directions—(Q. 19–21) are based on thefollowing—

Out of a total of 130 students appearing in aexam. at a centre, 60 had Computer Science asone of the subjects at degree level, 51 hadMathematics and 30 had both Computer Scienceand Mathematics. Out of the 50 students who hadPhysics as one of the subjects, 26 had ComputerScience, 21 had Maths and 12 had both ComputerScience and Mathematics. Every one who hadneither Computer Science. Nor Mathematics hadChemistry as one of the subject.

19. How many students had Chemistry as one ofthe subjects ?(A) 16 (B) 84(C) 49 (D) 56

20. How many had Computer Science but neitherMathematics nor Physics as one of thesubjects ?(A) 44 (B) 42(C) 16 (D) 56

21. How many students did not have ComputerScience, Mathematics or Physics as one of thesubjects ?(A) 30 (B) 16(C) 4 (D) 12

22. In a town of 840 persons, 450 persons readHindi, 300 read English and 200 read both.The number of persons who read neither is—(A) 210 (B) 260(C) 290 (D) 180

23. In a group of 1000 people, there are 750 whocan speak Hindi and 450 can speak English.How many can speak Hindi only ?(A) 750 (B) 150(C) 350 (D) 600

24. In a survey 60% of those surveyed owned acar and 80% of those surveyed who owned aT.V. If 55% owned both a car and a T.V.,then percentage of those surveyed who owneda car or a T.V. but not both will be—

(A) 30% (B) 5%(C) 25% (D) 15%

25. In a city, three daily newspapers A, B, C arepublished, 42% of the people in that city readA, 51% read B and 68% read C. 30% read Aand B; 28% read B and C; 36% read A and C;8% do not read any of the three newspapers.The percentage of persons who read all thethree paper is—

(A) 20% (B) 25%(C) 18% (D) 30%

26. Consider the following statements—

Assertion (A) : If A = [1, 2} then P(A) = [φ,{1}, {2}, {1, 2}] whereP(A) is the power set of A.

Reason (R) : P(A) is the super set of itssubsets of these statements.

Code :(A) Both A and R are true and R is the

correct explanation of A

(B) Both A and R are true but R is not thecorrect explanation of A

(C) A is true but R is false(D) A is false but R is true

27. Which one of the following is false ?

(A) If n(A) = 20, n(B) = 25 and n(A ∩ B) =10 then n(A ∪ B) = 35

(B) n(A ∩ B') = n(A) – n(A ∩ B)

(C) If n(A) = 15, n(B) = 20, n(A ∪ B) = 30then n(A ∩ B') = 10

(D) At least one of the above is false

28. If A = {1, 2, 3, 4, 5} then the number ofsubsets of A which contain 2 but not 4 is—(A) 2 (B) 4(C) 6 (D) 8

29. If z denotes the set of integers and A, B, C areits subsets given by—A = {x ∈ z : x is divisible by 2}B = {x ∈ z : x is divisible by 4}C = {x ∈ z : x is divisible by 6}

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Then A ∩ B ∩ C is—(A) {x ∈ z : x is divisible by 4}(B) {x ∈ z : x is divisible by 12}(C) {x ∈ z : x is divisible by 24}(D) The empty set

Answers with Hints1. (D) Let A = (1, 2, 3)

B = (2, 3, 4)C = (1, 2, 5)

A ∩ B = {2, 3} ⇒ A ∩ B ≠ φA ∩ C = {1, 2) ⇒ A ∩ C ≠ φ

A ∩ B ∩ C = {2} ⇒ A ∩ B ∩ C ≠ φ∴ A ∩ B ≠ φ, A ∩ C ≠ φ ⇒ A ∩ B ∩ C ≠ φ

2. (C) Given that A = {a, b, c}

Since A consists 3 elements

Therefore number of subsets of A

= 23 = 8

∴ Total number of proper subsets

= 8 – 1 = 7

3. (D) Given that

U = {1, 2, 3, 4, 5, 6, 7, 8}

A = {1, 2, 3, 4}

B = {2, 4, 5, 7}

Ac (complement of A) = {5, 6, 7, 8}

and Bc = {1, 3, 6, 8}

∴ A c∪ Bc = {1, 3, 5, 6, 7, 8}

4. (B) If two sets X and Y have n elements incommon, then the total no. of commonelements in X × Y and Y × X are n2. Giventhat sets A and B have 5 elements incommon, therefore, the total no. of commonelements in A × B and B × A are 52

5. (B) Total no. of subset = 28 = 256 but in thesesubsets also include an empty set φ.

Hence, the total no. of non empty subset= 256 – 1= 255

6. (B) If sets A and B have m and n elementsrespectively, then A × B has always mnelements.

Since given that sets A and B have 4 and 8elements respectively. Therefore, A × B have4 × 8 i.e., 32 elements.

7. (C) Given that A = {a, c, d, g}and B = {b, d, j, k}∴ A ∩ B = {d} which is a singleton set.

8. (A) ˙.˙ φ ⊂ A ⇒ A ∩ φ = φ

9. (D) Let number of elements in S be ‘n’ then

n [P (S)] = 64

= 2n = 64 = 2n

= 26 = 2n

⇒ n = 6

10. (B) A – B = A ∩ B'

11. (D) A = {(x, y) : y = 1x , 0 ≠ x ∈ R}

and B = {(x, y) : y = – x, x ∈ R}

Solving y =1x and y = – x

We get1x

= – x ⇒ x2 = – 1

∴ A ∩ B = φ

12. (C) (P ∪ Q)' = P' ∩ Q', hence (P ∪ Q)' has noelements of P.

Hence P ∩ (P ∪ Q)' = φ13. (B) A = {1, 2, 3, 4, 5}

B = {2, 3, 6, 7}

(A × B) ∩ (B × A) = {(2, 2), (2, 3),

(3, 2), (3, 3)}

14. (A) Given A = {4, 5, 8, 12}

B = {1, 4, 6, 9}

B – A = {1, 4, 6, 9} – {4, 5, 8, 12}

= {1, 6, 9}

∴ A – (B – A) = {4, 5, 8, 12} – {1, 6, 9}

= {4, 5, 8, 12}

15. (A) (A ∩ B) = {5, 9, 13, 17, 21}

∩ {3, 6, 9, 12, 15, 18, 21, 24}

= {9, 21}

∴ A – (A ∩ B) = {5, 9, 13, 17, 21}

– {9, 21}= {5, 13, 17}

16. (B) We have n (A ∪ B) = n (A – B) + n (A ∩B) + n (B – A)

⇒ 90 = 15 + 30 + n (B – A)

∴ n (B – A) = 90 – 45 = 45

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Now n (B) = n (A ∩ B) + n (B – A)

∴ n (B) = 30 + 45= 75

17. (B) If a number is less than 5 then it cannot begreater than 6.

∴ There is no ‘x’ for which x < 5 and x > 6.

∴ Given set is null set.

18. (A) If x ∉ A ∪ B then it means that x isneither in A nor in B; otherwise it would havebeen in A ∪ B.

∴ A ∪ B = {1, 2, 3, 4} ∪ (5)}

= {1, 2, 3, 4, 5}

19. (C) Let C, M, P and CH denote the set ofthose has Computer Science, Mathematics,Physics and Chemistry respectively.

∴ n (C) = 60

n (M) = 51

n (C ∩ M) = 30

n (P) = 54

n (P ∩ C) = 26

n (P ∩ M) = 21

and n (P ∩ C ∩ M) = 12

U

Comp.Sc.16

18

129

19Physics

30 Chemistry

Maths12

14

From Venn diagram the student who hadChemistry as one of the subject

= 19 + 30= 49

20. (C) From the above diagram student who hadComputer Science, but neither Mathematicsnor Physics as one of the subject = 16

21. (A) From the above diagram students who didnot have Computer Science, Mathematics orPhysics as one of the subject = 30.

22. (C) n (U) = 840n (H) = 450n (E) = 300

n (H ∩ E) = 200

∴ The required number is= 840 – n (H ∪ E)= 840 – {n (H) + n (E) – n (H ∩ E)}= 840 – {450 + 300 – 200}= 840 – 550= 290

23. (D) Let H and E denote the set of those speakHindi and English respectively. We have

n (H ∪ E) = 1000n (H) = 750n (E) = 400

∴ n (H ∪ E) = n (H) + n (E) – n (H ∩ E)⇒ 100 = 750 + 400 – n (H ∩ E)∴ n (H ∩ E) = 150The people who can speak Hindi only i.e.n (H ∩ E)∴ n (H ∩ E) = n (H) – n (H ∩ E)

= 750 – 150= 600

24. (A) Let n (U) = 100, n = (Car) = 60,n (T. V.) = 80and n (Car ∩ T. V.) = 55

∴ From Venn diagram, the percentage ofthose, who owned either Car or T. V. but notboth

= 5 + 25 = 30%

25. (B) n (A ∪ B ∪ C) = n (A) + n (B)– n (A ∩ B)

– n (B ∩ C) – n (C ∩ A) + n (A ∩ B ∩ C)

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92 = 42 + 51 + 68 – 30 – 28 – 36 + x92 = 161 – 94 + x

= 67 + xx = 92 – 67x = 25%

26. (B) ˙.˙ A = {1, 2}Subset of A areφ, {1}, {2}, {1, 2}Since the power set is the set of subset of A.i.e., P (A) = {φ, {1}, {2}, {1, 2}}∴ A is true.Since every set is the super set of subsets ofitself.∴ R is also true.But R is not a correct explanation of A.∴ The correct answer is (B).

27. (B) Statement (A) is true

Again n (A ∩ B) = n (A) + n (B)– n (A ∪ B)

n (A ∩ B) = 15 + 20 – 30= 5

∴ n (A ∩ B') = n (A) – n (A ∩ B)

= 15 – 5= 10

Hence, the statements (B) and (C) both aretrue.

28. (D) The number of subsets of the set {1, 3, 5}is 23 i.e. 8.

29. (B) L.C.M of 2, 4 and 6.

= 2 × 2 × 3 = 12

∴ A ∩ B ∩ C = {x ∈ Z : x is divisible by 12}

●●●

2Relation and Function

1. Relation—Let A and B be two non emptysets. A relation from set A to set B is a subset of A× B. Thus if R is a relation from A to B, then R ⊆A × B.

Also if (a , b) ∈ R then we say that a is Rrelated to b and denote this by aRb.

In particular if B = A then subsets of A × Aare called relations from the set A to the set A, orsimply as relation in the set A.

2. Identity relation—Let A be a non emptyset then the relation {(x, y) : x, y ∈ A and x = y} iscalled an identity relation.

Example—Let A = {a , b , c , d} then theidentity relation

IA = {(a, a), (b, b), (c, c), (d, d)}

3. Void relation—Relation φ is called thevoid relation.

Void relation φ is the smallest relation.

4. Universal relation—The relation {(x, y) :x , y ∈ A} is called the universal relation on A.Then, R is universal iff R = A × A.

Example—Let A = {1, 2, 3} then

R = {(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3),(3, 1) (3, 2), (3, 3)} is universal.

5. Reflexive relation—A relation R on A iscalled reflexive relation.

If (x, x) ∈ R ∀ x ∈ A

i.e., x R x ∀ x ∈ A

Example—Let X be a set of all straight linesin a plane. The relation R in X defined by ‘x’ isparallel to ‘y’ is reflexive because every straightline is parallel to itself.

6. Symmetric relation—A relation R in A iscalled symmetric relation.

iff (x, y) ∈ R ⇒ (x, y) ∈ R

i.e., xRy ⇒ yRx ∀ xy, ∈ A

Example—Let P be a set of all straight linesin a plane. The relation R defined by ‘x isperpendicular to y’ is symmetric as x ⊥ y ⇒ y ⊥ x.

Note—A necessary and sufficient conditionfor a relation to be symmetric is R = R–1.

7. Antisymmetric—A relation R on A iscalled antisymmetric iff (x, y) and (y, x) ∈ R ⇒ x= y.

i.e. xRy and yRx ⇒ x = y ∀ x, y ∈ A.Example— Let N be the set of natural

numbers and R be relation defined by ‘x divides y’for x, y ∈ N. Then R is antisymmetric relation as xdivides y and y divides x ⇒ x = y.

8. Equivalence relations—A relation R on anon-empty set A is called an equivalence relationiff.

(a) R is reflexive (xRx ∀ x ∈ A)(b) R is symmetric (xRy ⇒ yRx ∀ x, y ∈ A)(c) R is transitive (xRy and yRz ⇒ xRz ∀ x,

y, z ∈ A)9. Transitive—A relation R on A is called

transitive iff.(x, y), (y, z) ∈ R ⇒ (x, y) ∈ R

i.e., xRy and yRz ⇒ xRz ∀ x, y, z ∈ A.

10. Mapping—A mapping is a relationbetween the elements of A and those of B havingno ordered pairs with the same first component.Mapping are also called function. Function iswritten as f : A → B. It means f is a mapping thattakes all elements of A and sends to uniqueelements of B. A is domain of f and B is co-domain of ‘f ’. Range of ‘f ’ is given by{f (x) : x ∈ A}.

11. Mapping f : A →→→→ B is called welldefined if—

(a) All the elements of A are mapped onsome member of B.

(b) A member of A should be mapped onunique element of B. If any of above two

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conditions fail we say f : A → B is not welldefined.

12. One-one mapping—A function f : A →B is said to be one-one if different element of Ahave different f-images in B. i.e.

f (x1) = f (x2)

⇒ x1 = x2

or equivalentlyx1 ≠ x2

⇒ f (x1) ≠ f (x2)

One-one mapping are also called injection.

13. Many one mapping—A function f : A →B is said to be many one iff two or more differentelements in A have the same f-image in B.

14. Into mapping—The mapping f is said tobe into iff there is at least one element in B whichis not the f-image of any element in A.

In this case : f (A) ⊂ Bi.e., range of A is proper subset of B.

15. Onto mapping—The mapping f is said tobe onto iff every elements in B is the f image of atleast one element in A.

In this case f (A) = B

i.e. the range of f = co-domain

Onto mapping is also called surjection

When mapping f is one-one and onto, then itis called bijection.

16. Identity mapping—If A is a non-emptyset then f : A → B such that f (x) = x ∀ x ∈ A iscalled identity mapping. It is denoted by I.

17. If f : A → B is one-one and onto, then wesay that A and B are in one-one correspondence.

18. Inverse function—It f is one-one andonto from A → B, then f–1 exists and it carrieselements of B back to ‘A’.

19. Two functions f and g on same domain‘A’ are equal if f (x) = g (x) ∀ x ∈ A.

20. Symmetric function—If f and f–1 areequal then f is said to be symmetric function. Forexample :

Let f = {(2, 7), (3, 8), (7, 2), (8, 3)}

Then f–1 = {(7, 2), (8, 3), (2, 7), (3, 8)}

Here f = f–1

Hence f is symmetric function.

21. Binary operation—Consider a set ‘A’and an operation denoted by ‘o’ which whenplaced between two elements a and b produces. aresult denoted by aob which may or may notbelong to A, the binary operation just combinesany two elements of a set A at one time.

22. If ‘o’ is binary operation on A and aob εA ∀ a, b ε A then o is said to be closed and we say‘A’ is closed with respect to the binary operation‘o’.

Thus, if o is closed, then ‘o’ is a functionfrom A × A to A.

23. Types of Binary operation—

(a) The binary operation o is said to becommutative if

aob = boa, for a, b ∈ A

(b) The binary operation o is said to beassociative if

(aob) °C = ao (boc), for a, b, c ∈ A.

(c) An element e ∈ A is said to be an identityelement for the binary operation o if aoe = a = eoafor a ∈ A.

(d) For a ∈ A, an element b ∈ A is said to bean inverse of a if

aob = e = boa.

ILLUSTRATIONS

Example 1. Let A = {2, 3, 4, 5, 6}. Let R bethe relation in A defined by {(a, b) : a ∈ A, b ∈ A,a divides b}. Then its domain is—

(A) {2, 3, 4, 5, 6} (B) {4, 3, 2, 1}

(C) {5, 6, 7, 3} (D) {2, 3, 4, 6}

Sol. We haveA = {2, 3, 4, 5, 6}

Domain of R = {a ! (a, b) ∈ R}

∴ Domain of R = {2, 3, 4, 5, 6}

Example 2. The mapping between the set ofinteger and the set of natural numbers is/are—

(1) Identity (2) Injective

(3) Surjective (4) Bijective

Choose the correct answer using the codesgiven below :

(A) (1) and (2) (B) (1) and (3)

(C) (1) and (4) (D) (2), (3) and (4)

CBSE All India Engineering EntranceExam. Mathematics

Publisher : Upkar Prakashan ISBN : 9788174822017 Author : Dr. N. K. Singh

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