For all Engineering Entrance Examinations held across India. Mathematics

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2 For all Engineering Entrance Eaminations held across India. JEE Main Mathematics Salient Features Ehaustive coverage of MCQs subtopic we. 946 MCQs including questions from various competitive eams. Prece theory for every topic. Neat, Labelled and authentic diagrams. Hints provided wherever relevant. Additional information relevant to the concepts. Simple and lucid language. Self evaluative in nature. Printed at: Repro India Ltd., Mumbai No part of th book may be reproduced or transmitted in any form or by any means, C.D. ROM/Audio Video Cassettes or electronic, mechanical including photocopying; recording or by any information storage and retrieval system without permsion in writing from the Publher. TEID : 750

3 PREFACE Mathematics the study of quantity, structure, space and change. It one of the oldest academic dcipline that has led towards human progress. Its root lies in man s fascination with numbers. Maths not only adds great value towards a progressive society but also contributes immensely towards other sciences like Physics and Chemtry. Interdciplinary research in the above mentioned fields has led to monumental contributions towards progress in technology. Target s Maths Vol. I has been compiled according to the notified syllabus for JEE (Main), which in turn has been framed after reviewing various national syllabus. Target s Maths Vol. I compres of a comprehensive coverage of theoretical concepts and multiple choice questions. In the development of each chapter we have ensured the inclusion of shortcuts and unique points represented as an Important Note for the benefit of students. The flow of content and MCQ s has been planned keeping in mind the weightage given to a topic as per the JEE (Main). MCQ s in each chapter are a mi of questions based on theory and numerical and their level of difficulty at par with that of various engineering competitive eaminations. Th edition of Maths Vol. I has been conceptualized with a complete focus on the kind of asstance students would require to answer tricky questions, which would give them an edge over the competition. Lastly, I am grateful to the publhers of th book for their perstent efforts, commitment to quality and their unending support to bring out th book, without which it would have been difficult for me to partner with students on th journey towards their success. All the best to all Aspirants! Yours faithfully, Author No. Topic Name Page No. Comple Numbers and Quadratic Equations 64 Permutations and Combinations 58 4 Mathematical Induction 97 5 Binomial Theorem and Its Simple Applications 08 6 Sequences and Series 6 7 Trigonometry 8 Co ordinate Geometry 50

4 TARGET Publications 0 Sets, Relations and Functions Syllabus For JEE (Main) Physics (Vol. II). Sets.. Sets and their representation, Power set.. Union, Intersection and Complement of sets and their algebraic properties. Relations.. Relation.. Types of relations. Functions.. Real valued functions, Algebra of functions and Kinds of functions.. One-one, Into and Onto functions, Composition of functions

5 Maths (Vol. I). Sets. Definition: Any collection of well defined and dtinct objects called a set. By Well-defined collection we mean that given a set and an object, it must be possible to decide whether or not the object belongs to the set. The objects in a set are called its members or elements. Sets are usually denoted by capital letters A, B, C,, Y, Z etc. Elements of the sets: The elements of the set are denoted by small letters i.e., a, b, c,, y, z etc. If an element of a set A, we write A and if not an element of A, we write A. If A = {,,, 4, 5}, then A but 6 A. Important Note Every set a collection of objects but every collection of objects not a set. Eamples of well defined collections: i. The collection of vowels in Englh alphabet a set containing five elements a, e, i, o, u. ii. The collection of first five prime nos. a set containing the elements,, 5, 7,. iii. The collection of rivers of India. iv. The collection of all states of India. v. The collection of the solutions of the equation = 0. vi. The set of all lines in a particular plane. Eamples of not well defined collections, hence not sets: i. The collection of good cricket players of India. ii. The collection of bright students in class I of a school. iii. The collection of beautiful girls of the world. iv. The collection of rich persons in India. TARGET Publications v. The collection of most talented writers of India. vi. The collection of most dangerous animals of the world.. Symbols: Symbol A B s.t (: or ) iff & a/b N I or Z R C Q. Representation of a set: Meaning Implies Belongs to A a subset of B Implies and implied by Does not belong to Such that For all or for every There ets if and only if And a a divor of b Set of natural nos. Set of integers Set of real nos. Set of comple nos. Set of rational nos. There are two methods for representing a set: i. Tabulation or Roster or Enumeration or Lting method: ii. In th method, we lt all the members of the set, separating them by commas and enclosing them in curly brackets {}. Egs. a. If A the set of all prime nos. less than 0, then A = {,, 5, 7}. b. If A the set of all even nos. lying between and 0, then A = {4, 6, 8, 0,, 4, 6, 8}. Set builder or Rule or Property method: In th method, we write the set by some special property and write it as A = { : P()} = {/ has the property P()} and read it as A the set of all elements such that has the property P.

6 TARGET Publications Maths (Vol. I) Egs. a. If A = {,,, 4}, then we can write A = { N : < 5}. b. If A the set of all odd integers lying between and 5, then A = { : < < 5, odd}. Important Notes The order of writing the elements of a set immaterial. {,, }, {,, }, {,, }, {,, } all denote the same set. An element of a set not written more than once. Thus the set {,,, 4,,,,,,, 4} can be written as {,,, 4}. 4. Null or Empty or Void set: A set having no element called a null set. It denoted by φ or { }. i. φ unique. ii. φ a subset of every set. iii. φ never written within brackets i.e., {φ} not a null set Egs. a. { : N, 4 < < 5 } = φ b. { : R, + = 0} = φ c. { : = 5, an even no.} = φ 5. Singleton set or Unit set: A set having one and only one element called singleton or unit set. Egs. i. { : = 4} = {7} a singleton set. ii. { : + 4 = 0, Z} = { 4} iii. { : = 7, N} = {7} 6. Finite and Infinite sets: A set called a finite set if it either void set or its elements can be lted (counted, labelled) by natural numbers,,,. and the process of lting or counting of elements surely comes to an end. And a set which not finite called an infinite set. Egs. i. A = {a, e, i, o, u} a finite set. ii. B = {,,, 4,..} an infinite set. 7. Cardinal number of a finite set: Number of elements in a finite set A called cardinal number of a finite set and denoted by n or o. It also called order of a finite set. If A = {,,, 4, 5, 6}, then o = 6 8. Equal sets: Two sets A and B are said to be equal if every element of A an element of B and every element of B an element of A. Symbolically: A = B if A B If A = {4, 8, 0} and B = {8, 4, 0}, then A = B. 9. Equivalent sets: Two finite sets A and B are equivalent if o = o(b). Sets A = {,, 5, 7}, B = {0,, 4, 6} are equivalent [ o = 4 = o(b)] Equal sets are always equivalent but equivalent sets may not be equal. In above e.g. A B although they are equivalent. 0. Subsets: If every element of A also an element of a set B, then A called a subset of B. We write A B, which read as A a subset of B or A contained in B. Thus, A B { A B} i. Every set a subset of itself i.e., A A. ii. φ a subset of every set. iii. iv. Important Note If A B and B C A C A = B iff A B and B A a. Proper subsets: If A a subset of B and A B, then A a proper subset of B. If a set A non-empty, then the null set a proper subset of A. We write th as A B. Important Note If A B, we may have B A but if A B, we cannot have B A.

7 Maths (Vol. I) b. Improper subsets: The null set φ subset of every set and every set subset of itself, i.e., φ A and A A for every set A. They are called improper subsets of A. Thus, every non-empty set has two improper subsets. It should be noted that φ has only one subset φ, which improper. Let A = {, }. Then A has φ, {}, {}, {, } as its subsets out of which φ and {, } are improper and {} and {} are proper subsets.. Universal set: Superset of all the sets, i.e., all sets are contained in th set. Th usually denoted by Ω or S or U or.. Power set: The set of all the subsets of a given set A said to be the power set A and denoted by P. Important Note If A has n elements i.e., o = n, then o(p) = n Let A = {a, b, c}, then P = {φ, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}} Here, o = o(p) = = 8. Operations on sets: i. Union of sets: The union of two sets A and B the set of all those elements which are either in A or in B or in both. Th set denoted by A B or A + B [read as A union B or A join B ] Symbolically, A B = { : A or B} If A = {,, } and B = {,, 5, 7}, then A B = {,,, 5, 7} A A.. A n = ii. iii. iv. Important Note TARGET Publications n i = Intersection of sets: The intersection of two sets A and B the set of all elements which are common in A and B. Th set denoted by A B or AB [read as A intersection B or A meet B ] Symbolically, A B = { : A and B} If A = {,,, 4} and B = {, 4, 6}, then A B = {, 4}. Djoint sets: If two sets A and B have no common element i.e., A B = φ, then the two sets A and B are called djoint or mutually eclusive events. If A = {,, } and B = {a, b, c}, then A B = φ. A A A B B Difference of sets: Let A and B be two sets. The difference of A and B written as A B, the set of all those elements of A which do not belong to B. Thus, A B = { : A and B} Similarly, the difference B A the set of all those elements of B that do not belong to A. i.e., B A = { B and A} If A ={,,5,7,9} and B ={,,5,7,}, then A B = {, 9} and B A = {, }. B A U U i U U U A B A B B A A B A B A B 4

8 TARGET Publications Maths (Vol. I) A B = φ if A B A B B A The sets A B, B A and A B are djoint sets A B A and B A B A φ = A and A A = φ v. Symmetric difference of two sets: Let A and B be two sets. Then symmetric difference of two sets A and B the set (A B) (B A) or (A B) (A B) and denoted by A B or A B. i.e., A B or A B = (A B) (B A) = (A B) (A B) If A = {,, 5, 7, 9} and B = {,, 5, 7, }, then A B = (A B) (B A) = {, 9} {, } = {,, 9, } vi. Complement of a set: Let U be the universal set and A be a set such that A U, then the complement of A, denoted by A or A c or U A defined as A or A c = { : U and A} Let U = { : a letter in Englh alphabet} and A = { : a vowel}, then A = { : a consonant} U = φ φ = U A A = U A A = φ (A ) = A Important Notes A B A A B A B A U U Important Notes 4. Laws or properties of algebra of sets: i. Idempotent laws: For any set A, we have a. A A = A b. A A = A ii. iii. iv. Identity laws: For any set A, we have a. A φ = A b. A φ = φ c. A U = U d. A U = A Commutative laws: For any two sets A and B, we have a. A B = B A b. A B = B A c. A B = B A i.e., union, intersection and symmetric difference of two sets are commutative. But difference and cartesian product of two sets are not commutative. Associative laws: If A, B and C are any three sets, then a. (A B) C = A (B C) b. A (B C) = (A B) C c. (A B) C = A (B C) i.e. union, intersection and symmetric difference of three sets are associative. But difference and cartesian product of three sets are not associative. v. Dtributive laws: If A, B and C are any three sets, then a. A (B C)=(A B) (A C) b. A (B C)=(A B) (A C) vi. De-Morgan s law: If A, B and C are any three sets then a. (A B) = A B b. (A B) = A B c. A (B C) = (A B) (A C) d. A (B C) = (A B) (A C) vii. For any two sets A and B: a. P P(B) = P(A B) b. P P(B) P(A B) c. if P = P(B) A = B where P the power set of A 5

9 6 Maths (Vol. I) 5. More results on operations on sets: For any sets A and B, we have i. A A B, B A B, A B A, A B B ii. A B = A B, B A = B A iii. (A B) B = φ iv. (A B) B = A B v. A B B A vi. A B = B A vii. (A B) (A B ) = A viii. A B = (A B) (B A) (A B) i. A (A B) = A B. A B = B A A = B and A B = A B A = B 6. Results on cardinal number of some sets: If A, B and C are finite sets and U be the universal set, then i. n(a B) = n + n(b) if A and B are djoint sets. ii. n(a B) = n + n(b) n(a B) iii. n(a B) = n(a B) + n(b A) + n(a B) iv. n = n(a B) + n(a B) n(b) = n(b A) + n(a B) Here, n(a B) = n n(a B) and n(a B) = n(a B) n(b) v. n(a ) = n(u) n vi. vii. n(a B ) = n(a B) = n(u) n(a B) n(a B ) = n(a B) = n(u) n(a B) viii. n(a B ) = n n(a B) i. n(a B) = n(a B) n(a B ) n (A B). n(a B C) = n + n(b) + n n(a B) n(b C) n(c A) + n(a B C) i. If A, A, A,.., A n are djoint sets, then n(a A A.. A n ) = n(a ) + n(a ) + n(a ) + + n(a n ) ii. n(a B) = n + n(b) n (A B) TARGET Publications iii. n(a B C ) = n n(a B) n(a C) + n(a B C) n(b A C ) = n(b) n(b C) n(b A) + n(a B C) n(c A B ) = n n(c A) n(c B) + n (A B C) iv. n(a B C ) = n[(a B C) ] = n(u) n(a B C) 7. Ordered pair: If A be a set and a, b A, then the ordered pair of elements a and b in A are denoted by (a, b), where a called the first co-ordinate and b called the second co-ordinate. Ordered pairs (a, b) and (b, a) are different, i.e., (a, b) (b, a) Ordered pairs (a, b) and (c, d) are equal iff a = c and b = d i.e., (a, b) = (c, d) iff a = c, b = d. 8. Cartesian product of two sets: i. Let A and B be two non-empty sets. The cartesian product of A and B denoted by A B defined as the set of all ordered pairs (a, b), where a A and b B Symbolically, A B = {(a, b) : a A and b B} Similarly, B A = {(b, a) : b B and a A} If A = {,, } and B = {, y}, then A B = {(, ), (, y), (, ), (, y), (, ), (, y)} and B A = {(, ), (y, ), (, ), (y, ), (, ), (y, )} Important Notes Important Note If A B, then A B B A

10 TARGET Publications Maths (Vol. I) ii. If there are three sets A, B, C and a A, b B, c C, then we form an ordered triplet (a, b, c). The set of all ordered triplets (a, b, c) called the cartesian product of these sets A, B and C. i.e., A B C ={(a, b, c): a A, b B, c C} 9. Order of A B: i. If o= m and o(b)= n, then o(a B)= mn ii. iii. If A = φ, B = φ, then A B = φ If A = φ, B = {a, b, c}, then A B = φ Similarly, If A = {a, b, c}, B = φ, then A B = φ 0. Some results on cartesian products of sets: i. A (B C) = (A B) (A C) ii. A (B C) = (A B) (A C) iii. A (B C) = (A B) (A C) iv. (A B) (C D) = (A C) (B D) v. If A B and C D, then (A C) (B D) vi. If A B, then vii. A A (A B) (B A) If A and B are non-empty subsets, then A B = B A A = B. viii. If A B, then (A C) (B C). Relations. Relations from a set A to a set B: A relation (or binary relation) R, from a nonempty set A to another non-empty set B, a subset of A B. i.e., R A B or R { (a, b): a A, b B} Now, if (a, b) be an element of the relation R, then we write a R b (read as a related to b ) i.e., (a, b) R a R b In particular, if B = A, then the subsets of A A are called relations from the set A to A. i.e., any subset of A A said to be a relation on A. Egs. i. Let A={,, 5, 7} and B={6, 8}, then R be the relation less than from A to B R6, R8, R6, R8, 5R6, 5R8, 7R8 R = {(, 6), (, 8), (, 6), (, 8), (5, 6), (5, 8), (7, 8)} ii. Let A = {,,,.., 4}, then R be the relation one fourth of on A R4, R8, R, 4R6, 5R0, 6R4, 7R8, 8R R = {(, 4), (, 8), (, ), (4, 6), (5, 0), (6, 4), (7, 8), (8, )}. Number of possible relations from A to B: If A has m elements and B has n elements, then A B has m n elements and total number of possible relations from A to B mn.. Domain and Range of a relation: i. Domain of R = {a : (a, b) R} i.e., if R a relation from A to B, then the set of first elements of ordered pairs in R called the domain of R. ii. Range of R = {b : (a, b) R} i.e., if R a relation from A to B, then the set of second elements of ordered pairs in R called the range of R. If R = {(4, 7), (5, 8), (6, 0)} a relation from the set A = {,,, 4, 5, 6} to the set B = {6, 7, 8, 9, 0}, then domain of R = {4, 5, 6} and range of R = {7, 8, 0}. Important Notes If R = A B, then domain of R A and range of R B. The domain as well as range of the empty set φ φ. If R a relation from the set A to the set B, then the set B called the co-domain of the relation R. i.e., Range Co-domain. 4. Inverse relation: If R a relation from a set A to a set B, then the inverse relation of R, to be denoted by R, a relation from B to A. Symbolically, R = {(b, a) : (a, b) R} Thus,(a, b) R (b, a) R a A, b B i. Domain (R ) = Range (R) and Range (R ) = Domain (R) ii. (R ) = R If R = {(, ), (, 4), (5, 6)}, then R = {(, ), (4, ), (6, 5)} (R ) = {(, ), (, 4), (5, 6)} = R Here, domain (R) = {,, 5}, range (R) = {, 4, 6} and domain (R ) = {, 4, 6}, range (R ) = {,, 5} Clearly, dom (R ) = range (R) and range (R ) = dom (R) 7

11 8 Maths (Vol. I) 5. Universal relation: A relation R in a set A called the universal relation in A if R = A A. If A = {a, b, c}, then A A = {(a, a), (a, b), (a, c), (b, a), (b, c), (b, b) (c, a), (c, b), (c, c)} the universal relation in A. 6. Identity relation: A relation R in a set A called identity relation in A, if R = {(a, a) : a A} = I A If A = {a, b, c}, then I A = {(a, a), (b, b), (c, c)} 7. Void relation: A relation R in a set A called void relation if R = φ. 8. Various types of relation: Let A be a non-empty set, then a relation R on A said to be i. Refleive: If ara a A i.e., (a, a) R a A If A = {, 4, 7}, then relation R = {(, ), (4, 4), (7, 7)} refleive. ii. Symmetric: If arb bra a, b A i.e., if (a, b) R (b, a) R a, b A If A = {, 4, 7}, then R = {(, 4), (4, ), (7, 7)} symmetric. iii. Transitive: If arb and brc arc a, b, c A i.e., if (a, b) R and (b, c) R (a, c) R a, b, c A. If A = {, 4, 7}, then relation R = {(, 4), (4, 7), (, 7), (4, 4)} transitive. iv. Anti-symmetric: If arb and bra a = b a, b A v. Equivalence relation: A relation R on a set A said to be an equivalence relation on A iff R i. Refleive ii. Symmetric and iii. Transitive i.e., for equivalence relation R in A i. ara a A ii. arb bra a, b A iii. arb and brc arc a, b, c A TARGET Publications 9. Composition of two relations: If A, B and C are three sets such that R A B and S B C, then (SoR) = R os. It clear that arb, bsc asorc. R S A B C a Th relation called the composition of R and S. If A = {,, }, B = {a, b, c, d}, C = {p, q, r, s} be three sets such that R = {(, a), (, b), (, c), (, d)} a relation from A to B and S = {(a, s), (b, r), (c, r)} a relation from B to C, then SoR a relation from A to C given by SoR = {(, s), (, r), (, r)} In th case, RoS does not et. In general, RoS SoR. 0. If R a relation on a set A, then i. R refleive R refleive ii. R symmetric R symmetric iii. R transitive R transitive. Functions or Mappings. Definition: Let A and B be any two non-empty sets. If to each element A a unique element y B under a rule f, then th relation called function from A into B and written as f f : A B or A B. The other terms used for functions are operators or transformations. A B A B b C SoR Important Notes y = f() If A, y = f() B, then (, y) f If (, y ) f and (, y ) f, then y = y c

12 TARGET Publications Maths (Vol. I) Real valued function: Let f : A B and A R & B R be defined by y = f(), where A, y B, then f called a real valued function of a real variable.. Domain, Co-domain and Range: i. Domain: The set of A called the domain of f i.e., all possible values of for which f() ets (denoted by D f ). ii. Co-domain: The set of B called the co-domain of f (denoted by C f ). iii. Range: The set of all f - images of the elements of A called the range of function f. i.e., all possible values of f(), for all values of (denoted by R f ) Range of f = {f() : A} The range of f always a subset of co-domain B. i.e., R f C f A From figure: Domain = {a, b, c, d} = A Co-domain = {,,, 4} = B Range = {,, } So, R f C f. Algebra of functions: Let f and g be two real valued functions with domains D f and D g, then i. Sum function defined by (f + g) () = f() + g () and domain of f() + g() D f D g. ii. Difference function defined by (f g)() = f() g() and domain of f() g() D f D g. iii. Multiplication by scalar defined by (α f)() = α f() iv. Product function defined by (fg) () = f(). g() and domain of f() g() D f D g. a Important Note f B b c d 4 v. Quotient function defined by f () = f( ), g() 0 and g g( ) domain of f( ) D f D g {g() = 0} g( ) vi. Domain of f( ) D f { : f() 0} 4. One-one function: A function f : A B said to be one-one if different elements of A have different images in B i.e., no two different elements of A have the same image in B. Such a mapping also known as injective mapping or an injection or monomorphm. Method to test one-one: If, A, then f( ) = f( ) = and f( ) f( ) A function one-one, if it increasing or decreasing. Let f : A B and g : Y be two functions represented by the following diagrams. Clearly, f : A B a one-one function. But g : Y not one-one function because two dtinct elements & have the same image under function g. 5. Onto function: Let f : A B, if every element in B has at least one pre-image in A, then f said to be onto function or surjective mapping or surjection. A a a a a 4 f Important Note B b b b b 4 b 5 If f (y) A, y B, then function onto. In other words, Range of f = Co-domain of f 4 Important Note g Y y y y y 4 y 5 9

13 0 Maths (Vol. I) In the following diagrams: A B f. f : A B onto function. But g : Y not onto funtion because Range Co-domain. 6. Into function: A funtion f : A B an into function, if there ets an element in B having no pre-image in A. If f B i.e., Range Co-domain, then the function into or f : A B an into function, if it not an onto function. The following diagrams show into functions: Because in both the diagrams R f C f. 7. Bijection (one-one onto function): A function f : A B a bijection or bijective, if it one-one as well as onto. In other words, a function f : A B a bijection if i. it one-one i.e., f() = f(y) = y, y A ii. it onto i.e., y B, there ets A such that f() = y A B f \ a a a A a a a f a a a a 4 b b b Clearly, f a bijection, since it both injective as well as surjective. 4 Important Note B b b b b b b b 4 g g Y y y y y 4 Y y y y y 4 TARGET Publications 8. Many-one function: A function f : A B said to be a many-one function, if two or more elements of set A have the same image in B. In other words, f : A B a many-one function, if it not a one-one function. f : A B a many-one function, if there ets, A such that but f( ) = f( ) It can also be defined as a function many-one, if it has local maimum or local minimum. The following diagrams show many-one functions: A a a a a 4 9. Inverse of a function: If f : A B be one-one and onto function, then the mapping f (B) A such that f (b) = a (where a A & b B) called inverse function of the function f : A B. or Let f : A B be a one-one and onto function, then there ets a unique function, g : B A such that f() = y g (y) =, A and y B. Then g said to be inverse of f. Thus, g = f : B A={(f (), ) (, f ()) f} f B b b b b 4 b 5 a 5 b 6 Important Notes Let us consider one-one function with domain A and range B, where A = {,,, 4} and B = {, 4, 6, 8} and f : A B given by f() =, then write f and f as a set of ordered pairs. So, f = {(, ), (, 4), (, 6), (4, 8)} and f = {(, ), (4, ), (6, ), (8, 4)} 4 5 g Y y y y y 4 y 5

14 TARGET Publications Maths (Vol. I) A 4 In above function, Domain of f = {,,, 4} = range of f Range of f = {, 4, 6, 8} = domain of f Which represents for a function to have its inverse, it must be one-one onto or bijective. 0. Graph of a function: If f : A B be a function defined by y = f(), then graph of f defined as a subset of A B given by G(f) = {(, f ()) : A}. Some particular functions with their graphs: i. Constant function: A function f : Y said to be constant function, if its range a singleton set i.e., f() = c, where c some constant. f : R R defined by y = f() = 7 a constant function [ f() = 7, f() = 7, f() = 7,..] Here, D f = R and R f = 7 = c ii. Identity function: The function f defined by f() = R called the identity function. Y Here, D f = R and R f = R f B Y O 7 B Important Notes y = 7 O f A 4 iii. iv. Polynomial function: A function f defined by f() = a 0 + a + a +. + a n n ; where a 0, a, a,., a n are real constants and n non-negative integer, called a polynomial function. Rational function: A function f() which can be epressed as g( ) h( ), where g() and h() are polynomials and h() 0 called a rational function. v. Modulus function or Absolute value or Numerical function: A function f : R R defined by, 0 f() = or f() =, < 0 called the absolute value or modulus function. Here, D f = R and R f = R + = [0, ) vi. Properties of Modulus of a real number:, y R, we have a. = ma (, ) b. = = c. y = y d. =, [y 0] y y e. + y + y f. y + y g. y y h. + y y i. k k k, (k > 0) j. k k or k, (k > 0) Signum function: The function f defined by, 0 f() = 0, = 0 Y O or f() =, if > 0 0, if = 0, if < 0

15 Maths (Vol. I) TARGET Publications vii. called the signum function. Here, D f = R and R f = {, 0, } Y y = O y = Greatest Integer function or Step function or Floor function: The function f defined by f() = [], R called greatest integer function. [] indicates the integral part of which nearest and smaller integer. Thus, [] = (if an integer) = an integer immediately on the left of (if not an integer) Here, D f = R and R f = I For graph of f, we construct the table of values y = [] < < 0 0 < 0 < < Y viii. Fractional part function: The function defined by the rule f() = [], where [] indicates the integral part of called the fractional part function. Here, D f = R R and R f = [0, ) [ ] < [ ] + 0 [ ] < 0 f( ) < Some facts about the function f() = []: a. f() = 0 iff an integer. b. f() = iff 0 <. c. 0 < f () < iff not an integer. d. f( + ) = f() R i.e., f a periodic function with period. For graph of f, construct the table of values: [] y = [] < + < < 0 < < Y i. Y Reciprocal function: The function f defined by f() = Y Some facts about the function f() = []: a. [] = iff I b. [] < iff I c. [] = k, (k I) iff k < k + d. [ + I] = [] + I, if I an integer and R e. [ ] = [], if I f. [ ] = [], if I called reciprocal function. Here, D f = R {0} and R f = R {0} Y y =

16 TARGET Publications Maths (Vol. I). Eponential function: If a > 0, then the function defined by f() = a R, called the general eponential function with base a. Here, {} if a = D f = R and R f = (0, ) if a > 0, a In particular, f() = e, R called the natural eponential function. Here, D f = R and R f = (0, ) a increases if a > 0 and a decreases if 0 < a <. If 0 < a < If a > Y Y O Important Note O ii. e. log a y = log a log a y f. log a ( n ) = n log a g. log a n = n log a h. log a = log log a i. For 0, log a not defined. j. log a decreases if 0 < a < and increases if a >. Power function: A function f : R R defined by f() = α, α R called a power function. iii. Trigonometric functions: a. Sine function: f() = sin, Y O, i. Y Logarithmic function: The function defined by f() = y = log a iff = a y (a > 0, a ), > 0 called logarithmic function. Here, D f = (0, ) and R f = R. In particular, the function f() = log e called natural logarithmic function and f() = log 0 called common logarithmic function. If 0 < a < If a > Y, Y, The domain of sine function R and the range [, ]. b. Cosine function: f() = cos Y (0, ) O Y Y O Y Y O (, ) (, ) Y The domain of cosine function R and the range [, ]. c. Tangent function: f() = tan Y Some properties of logarithmic function: a. y = log a iff = a y, > 0, y R b. log a = 0 and log a a = c. a log a =, for > 0 d. log a (y) = log a + log a y, > 0, y > 0. O Y

17 4 Maths (Vol. I) Here, domain of tangent function ( n + ) R,n I and range R. d. Cosecant function: f() = cosec e. Secant function: f() = sec Y (, ) = (, ) = (0, ) Here, domain R ( n + ) range R (, ). f. Cotangent function: f() = cot Y,0,0 Y y =,, O,, y = = = Y = = Here, domain R (n+ ),n I and range R (, ). O = O,0 (, ) (, ) y = y = = and,0 = = = = TARGET Publications Here, domain R {n / n I} and range R.. Domain and range of some standard functions: Function Domain Range Polynomial function R R Identity function R R Constant function K R {K} Reciprocal function R {0} R {0}, R [0, ), R R Signum function R {, 0, } + R [0, ) R R {0} [] R I [] R [0, ) [0, ) [0, ) a R R + log R + R sin R [, ] cos R [, ] tan R ±, ±,... R cot R {0, ±, ±,...} R sec R cosec ±, ±,... R {0, ±, ±,..} sin [, ] (, ] [, ) (, ] [, ), cos [, ] [0, ] tan R, cot R (0, ) sec R (, ) [0, ] cosec R (, ), {0}

18 TARGET Publications Maths (Vol. I). Even and odd functions: A function y = f() said to be i. Even if f( ) = f() ii. Odd if f( ) = f() iii. Neither even nor odd if f( ) ± f() Egs. i. f() = e + e, f() =, f() = sin, f() = cos, f() = cos all are even functions. ii. f() = e e, f() = sin, f() =, f() = cos, f() = sin all are odd functions. Properties of even and odd functions: i. The product of two even or two odd functions an even function. ii. The product of an even function by an odd function or vice versa an odd iii. function. The sum of even and odd function neither even nor odd function. iv. Zero function f() = 0 the only function which even and odd both. v. Every function f() can be epressed as the sum of even and odd function. i.e., f() = [f() + f( )] + [f() f( )] = F() + G() Here, F() even and G() odd. [ F( ) = F() and G( ) = G()] 4. Periodic function: A function said to be periodic function, if there ets a constant T > 0 such that f( + T) = f( T) = f() domain. Here, the least positive value of T called the period of the function. i. Periodic functions Functions Period sin n, cos n ; (if n = even) sec n, cosec n ; (if n odd and fraction) sin, cos, tan, cot, cosec, sec [], sin( []), sin( [ ]), [ ] sin (sin), cos (cos) sin, cos, cos, + cos sin + cos ( sin + cos ), sin 4 + cos 4 cos + cos + cos + cos +. + cos n + cos n cos(cos) + cos(sin) sin(sin) + sin(cos) n sin sin + cos + sin + cos = 4 sin + cos n sin + cos ii. Some non-periodic functions: sin, cos, cos, sin, ± cos, ± sin, sin, cos, (cos + cos ), (sin + {}), (sin + []), [where {} fractional part function & [] a greatest integer function] Properties of periodic function: i. If f() periodic with period T, then a. a.f() periodic with period T. b. f ( + a) periodic with period T. c. f () ± a periodic with period T. where a any constant. We know sin has period. Then f() = 5(sin ) + 7 also periodic with period. i.e., If constant added, subtracted, multiplied or divided in periodic function, period remains same. ii. If f() periodic with period T, then kf (a + b) has period T a. i.e., period only affected by coefficient of, where k, a, b constant. We know f() = 5sin + 7 has the period =, as sin periodic with period. 5

19 6 Maths (Vol. I) iii. If f (), f () are periodic functions with periods T, T respectively, then we have, h() = f () + f () has period L.C.M. of {T,T }; If f ( ) and f ( ) are complementary = pair we comparable evenfunctions L.C.M. of {T,T }; otherwe While taking L.C.M. we should always remember. a c e L.C.M.of (a,c,e) a. L.C.M. of,, = b d f H.C.F. of (b,d, f ) L.C.M. of,, 6 L.C.M. of (,, ) = = H.C.F.of (, 6, ) L.C.M. of,, 6 = b. L.C.M. of rational with rational possible. L.C.M. of irrational with irrational possible. But L.C.M. of rational and irrational not possible. L.C.M. of (,, 6) not possible as, 6 irrational and rational. 5. Some special functions: i. If f( + y) = f() + f(y), then f() = k. ii. If f(y) = f() + f(y), then f() = log. iii. If f( + y) = f(). f(y), then f() = e. iv. If f().f = f() + f, then f() = n ±. 6. Composite function: Let f : A B be defined by b = f(a) and g : B C be defined by c = f(b), then h : A C be defined by h(a) = g[f(a)] called composite function. We write h = gof Thus, gof : A C will be defined as gof() = g[f()], A. f g A B f() C g(f()) A B C gof TARGET Publications i. gof defined, if R f D g ii. gof one-one f one-one. iii. gof onto f onto. iv. if f, g are one-one onto, then gof also one-one onto. v. f even, g even fog even function. vi. f odd, g odd fog odd function. vii. f even, g odd fog even function. viii. f odd, g even fog even function. i. fog gof i.e., composite of functions not commutative.. (fog)oh = fo(goh) i.e., composite of functions associative. i. (gof) = (f og ) Formulae. Sets All functions are relation but all relations may not be a function. If A, B and C are finite sets and U be the universal set, then. A B iff { A B}. A = B iff A B and B A. i. P = {B : B a subset of A} ii. Important Note If A has n elements i.e., o = n, then o(p) = n. 4. A B = { : A or B} 5. A B = { : A and B} 6. A and B are djoint iff A B = φ. 7. A B = { : A and B} 8. A B = B A iff A = B 9. A B = A iff A B = φ 0. A B or A B = (A B) (B A) or (A B) (A B). A (or A c ) = { : U and A}. A A = A and A A = A These are called Idempotent laws.. A B = A iff B A and A B = A iff A B 4. A φ = A, A φ = φ, A U = U and A U = A These are called Identity laws. 5. A (B C) = (A B) (A C) and A (B C) = (A B) (A C) These are called Dtributive laws.

20 TARGET Publications Maths (Vol. I) 6. (A B) = A B, (A B) = A B, A (B C) = (A B) (A C) and A (B C) = (A B) (A C) These are called De-Morgan s law. 7. i. P P(B) = P (A B) ii. P P(B) P (A B) iii. if P = P(B) A = B where P the power set of A 8. A B = A B = A (A B) 9. n(a B) = n + n(b), if A and B are djoint sets. 0. n(a B) = n + n(b) n(a B). n(a B) = n(a B) + n(b A) + n(a B). n(a B) = n n(a B). n (A B C) = n + n(b) + n n(a B) n (B C) n(c A) + n (A B C) 4. n (A B C ) = n n(a B) n(a C) + n(a B C) 5. A B = {(, y) : A and y B} 6. A B = B A iff A = B 7. n(a B) = n.n(b) 8. If A B, then A A (A B) (B A) 9. (A B) (C D) = (A C ) (B D). Relations If A and B are finite sets and R be the relation, then. R A B i.e., R { (a, b) : a A, b B}. i. If n = m and n(b) = n, then total number of possible relations from A to B = mn. ii. The number of relations on finite set A n having n elements.. Domain of R = { a : (a, b) R} Range of R = {b : (a, b) R} 4. R = {(b, a) : (a, b) R} called Inverse relation. 5. R = A A called Universal relation. 6. R = {(a, a) : a A} = I A called Identity relation. 7. R = φ called Void relation. 8. If A be a non-empty set, then a relation R on A said to be i. Refleive : If (a, a) R a A ii. Symmetric : If (a, b) R (b, a) R a, b A iii. Transitive : If (a, b) R and (b, c) R (a, c) R a, b, c A iv. Anti-symmetric : If (a, b) R and (b, a) R a = b a, b A v. Equivalence : iff it refleive, symmetric and transitive.. Functions. If f : A B a function, then = y f () = f(y), y A. A function f : A B a one-one function or an injection, if f( ) = f( ) =, A or f( ) f( ), A. A function f : A B an onto function or a surjection if range (f) = co-domain (f). 4. A function f : A B an into function, if range (f) co-domain (f). 5. A function f : A B a bijection or bijective, if it one-one as well as onto. 6. A function f : A B many-one function, if f( ) = f( ), A 7. For domain and range, if function in the form i. f( ), take f() 0 ii., take f() > 0 f( ) iii. Shortcuts. Sets, take f() 0 f( ). The total number of subsets of a finite set containing n elements n.. Number of proper subsets of A containing n elements n.. Number of non-empty subsets of A containing n elements n. 4. Let A, B, C be any three sets, then i. n (A only) = n n(a B) n (A C) + n(a B C) ii. n (B only) = n(b) n (B C) n(a B) + n (A B C) iii. n(c only) = n n(c A) n(b C) + n (A B C) 7

21 8 Maths (Vol. I) 5. Number of elements in eactly two of the sets A, B and C = n(a B) + n (B C) + n(c A) n (A B C) 6. Number of elements in eactly one of the sets A, B and C = n + n(b) + n n(a B) n(b C) n (A C) + n (A B C) 7. Number of elements which belong to eactly one of A or B. i.e., n(a B) = n + n(b) n (A B) 8. If o(a B) = n, then o[(a B) (B A)] = n 9. If N a = {an : n N}, then N b N c = N (L.C.M. of b and c) where a, b, c N. Relations. The identity relation on a set A an anti-symmetric relation.. The relation congruent to on the set T of all triangles in a plane a transitive relation.. If R and S are two equivalence relations on a set A, then R S also an equivalence relation on A. 4. The union of two equivalence relations on a set not necessarily an equivalence relation on the set. 5. The inverse of an equivalence relation an equivalence relation. 6. If a set A has n elements, then the number of n binary relations on A = n. 7. Empty relation always symmetric and transitive. 8. A relation R on a non-empty set A symmetric iff R = R. 9. Total number of refleive relations in a set with n elements = n.. Functions. The number of functions from a finite set A into a finite set B = [n(b)] n. i. The domain of ii. The domain of a [ a, a]. ( a, a). a iii. iv. The domain of a (, a] [a, ). The domain of a (, a) (a, ). TARGET Publications. i. The domain of ( a) ( b ) when a < b [a, b]. ii. The domain of ( a)(b ) when a < b (a, b). iii. The domain of ( a)( b) when a < b (, a] [b, ). iv. The domain of ( a)( b) when a < b (, a) (b, ). a 4. i. The domain of b when a < b (, a] (b, ). a ii. The domain of b when a > b (, b) [a, ). a iii. The domain of b when a < b [a, b). a iv. The domain of b when a > b (b, a]. 5. i. The domain of log(a ) ( a, a). ii. The domain of log ( a ) (, a) (a, ). iii. The domain of log[( a) (b )] when a < b (a, b). 6. i. Range of f() = a [0, a]. ii. Range of f() = acos + bsin + c [c a + b, c + a + b ]. 7. The domain of the function f() = + c R { c} and range = {, }. + c

22 TARGET Publications Maths (Vol. I) 8. If y = f() = ( a ) a + b, then fof() =. 9. Any polynomial function f : R R onto if degree of f odd and into if degree of f even. 0. If f() periodic with period a, then also periodic with same period a. f ( ). If f() periodic with period a, f( ) also periodic with same period a.. Period of algebraic functions,, + 5 etc. doesn t et.. i. If A and B have n and m dtinct elements respectively, then the number of mappings from A to B = m n. ii. If A = B, then the number of mapping = n n. 4. The number of one-one functions that can be defined from a set A into a finite set B n(b) P n ; if n(b) n 0 ; otherwe 5. The number of onto functions that can be defined from a finite set A containing n elements onto a finite set B containing elements = n. 6. The number of onto functions from A to B, where o = m, o(b) = n and m n n n r ( ) n C r (r) m. r= 7. The number of bijections from a finite set A onto a finite set B n! ; if n = n (B) 0 ; otherwe 8. If any line parallel to -a, cuts the graph of the function atmost one point, then the function one-one. 9. If any vertical line does not meet the graph of the function f(), then the function onto. Multiple Choice Questions. Sets.. Sets and their representation, Power set. The set of intelligent students in a class [AMU 998] a null set (B) a singleton set a finite set (D) not a well defined collection. The set B = { : a positive prime < 0} in the tabular form {,, 5, 7} (B) {, 5, 7, 9} {,, 5, 6} (D) {,, 7, 8}. If A the set of numbers obtained by adding to each of the even numbers, then its set builder notation [DCE 00] A = { : odd and > } (B) A = { : odd and I} A = { : even} (D) A = { : an integer} 4. In rule method the null set represented by [Karnataka CET 998] {} (B) φ { : = } (D) { : } 5. The set of all prime numbers a finite set (B) a singleton set an infinite set (D) a null set 6. Which set the subset of all given sets? {,,, 4,.} (B) {} {0} (D) { } 7. Which of the following a true statement? [UPSEAT 005] a {a, b, c} (B) a {a, b, c} φ {a, b, c} (D) none of these 8. Which of the following a singleton set? { : = 8, Z} (B) { : = 4, N} { : = 7, N} (D) { : + + = 0, N} 9. If a set A has n elements, then the total number of subsets of A [Roorkee 99; Karnataka CET 99,000] n (B) n n (D) n 9

23 Maths (Vol. I) 0. If A = {, y}, then the power set of A [Pb.CET 004, UPSEAT 000] {, y y } (B) {φ,, y} {φ, {}, {y}} (D) {φ, {}, {y}, {, y}}. The number of proper subsets of the set {,, } [JMIEE 000] 5 (B) 6 7 (D) 8. The number of non-empty subsets of the set {,,, 4} [Karnataka CET 997; AMU 998] 4 (B) 6 5 (D) 7. Which of the following the empty set? [Karnataka CET 990] { : a real number and = 0} (B) { : a real number and + = 0} { : a real number and 9 = 0} (D) { : a real number and = + } 4. If A = { : a multiple of 4} and B = { : a multiple of 6}, then A B consts of all multiples of [UPSEAT 000] 6 (B) 8 (D) 4 5. If = {64n : n N} and Y = { n+ 8n 9 : n N}, then Y (B) Y = Y (D) none of these 6. Which of the following not true? 0 {0, {0}} (B) {0} {0, {0}} {0} {0, {0}} (D) 0 {0, {0}} 7. Power set of the set A = {φ, {φ}} {φ, {φ}, {{φ}}} (B) {φ, {φ}, {{φ}}, A} {φ, {φ}, A} (D) none of these 8. Two finite sets have m and n elements. The total number of subsets of the first set 48 more than the total number of subsets of the second set. The values of m and n are [M.N.R.E.C. Allahabad 988,9; Kerala P.E.T. 00] 7, 6 (B) 6, 6, 4 (D) 7, 4 0 TARGET Publications 9. The set A = { : R, = 6 and = 6} equals [Karnataka CET 995] φ (B) {4,, 4} {4, 4} (D) {4} 0. If a set contains (n + ) elements, then the number of subsets of th set containing more than n elements equal to [UPSEAT 00,04] n (B) n n+ (D) n.. Union, Intersection and Complement of sets and their algebraic properties. If A B = B, then A B (B) B A A = B (D) A B = φ. (A B) c = A c B c (B) A c B c A c B c (D) None of these. If A and B are djoint, then n(a B) equal to n (B) n(b) n + n(b) (D) n.n(b) 4. If A, B and C are any three sets, then A (B C) equal to (A B) (A C) (B) (A B) (A C) (A B) C (D) (A B) C 5. If A, B and C are any three sets, then A (B C) equal to (A B) (A C) (B) (A B) (A C) (A B) (A C) (D) none of these 6. If A any set and U be the universal set, then A A = φ (B) A A = U A A = U (D) none of these 7. If A = {,, 4, 5, 6} and B = {,, 4, 5, 6}, then A B equal to {,, 4} (B) {,, } {, 4, 5, 6} (D) {,, 5, 6} 8. If A = {,, 5, 8, 0}, B = {, 4, 5, 0, } and C = {4, 5, 6,, 4}, then (A B) (A C) equal to {, 5, 0} (B) {, 7, 0} {4, 5, 6} (D) {, 5, }

24 TARGET Publications Maths (Vol. I) 9. If A = { a, b, c}, B = {b, c, d}, C = {a, b, d, e}, then A (B C) [Kurukshetra CEE 997] {a, b, c} (B) {b, c, d} {a, b, d, e} (D) {e} 0. If A = {,, 4}, B = {, 4, 5}, C = {, 5}, then (A B) (B C) {,, } (B) {,, 5} {(, 5)} (D) {(, 4)}. If n = 0, n(b) = 7 and n = 6 for three djoint sets A, B and C, then n(a B C) = 7 (B) 9 (D). A B = φ if A B (B) B A A = B (D) A B = φ. If Q a set of rational numbers and P a set of irrational numbers, then P Q = φ (B) P Q Q P (D) P Q = φ 4. If the sets A and B are defined as A = {(, y) : y = e, R}; B = {(, y) : y =, R}, then [UPSEAT 994,99,00] B A (B) A B A B = A (D) A B = φ 5. In a city 0 percent of the population travels by car, 50 percent travels by bus and 0 percent travels by both car and bus. Then persons travelling by car or bus [Kerala (Engg.) 00] 40 percent (B) 60 percent 80 percent (D) 70 percent 6. If A = {,, } and B = {, 8}, then (A B) (A B) {(, ), (, ), (, ), (, 8)} (B) {(, ), (, ), (, ), (8, )} {(, ), (, ), (, ), (8, 8)} (D) {(8, ), (8, ), (8, ), (8, 8)} 7. In a class of 00 students, 55 students have passed in Mathematics and 67 students have passed in Physics, then the number of students who have passed in Physics only [DCE 99; ISM Dhanbad 994] (B) 45 (D) Of the members of three athletic teams in a school are in the cricket team, 6 are in hockey team and 9 are in the football team. Among them, 4 play hockey and cricket, 5 play hockey and football and play football and cricket. Eight play all the three games. The total number of members in the three athletic teams 4 (B) (D) If A and B are any two sets, then A B equal to (A B) (A B) (B) A B A B (D) B A 40. If A and B are any two sets, then A (A B) equal to [Karnataka CET 996] A (B) A c B (D) B c 4. If A and B are any two sets, then (A B) (A B) equal to A B (B) B A (A B) (B A) (D) none of these 4. If the set A has p elements, B has q elements, then the number of elements in A B [Karnataka CET 999] p (B) p + q pq (D) p + q + 4. If A and B are two sets, then A (A B) equal to φ (B) A B (D) none of these 44. If A and B are two sets, then A B A B (B) A B A B A B = A B (D) none of these 45. If N a = {an : n N}, then N 5 N 7 = [Kerala (Engg.) 005] N 5 (B) N 7 N (D) N If A, B, C are three sets such that A B = A C and A B = A C, then [Roorkee 99] A = B (B) B = C A = C (D) A = B = C

25 Maths (Vol. I) 47. If U = {,,, 4, 5, 6, 7, 8, 9, 0}, A = {,, 5} and B = {6, 7}, then A B B (B) B A (D) A 48. If n = and n(b) = 6, then the minimum number of elements in A B [MNR 987; Karnataka CET 996] (B) 9 6 (D) 49. If n(u) = 700, n = 00, n(b) = 00 and n(a B) = 00, then n(a c B c ) = [Kurukshetra CEE 999] 00 (B) (D) If A and B are two sets, then A (A B) c equal to [AMU 998, K.U.K.C.E.E.T. 999] A (B) B φ (D) A B c 5. If A = {a, b}, B = {c, d}, C ={d, e}, then {(a, c), (a, d), (a, e), (b, c), (b, d), (b, e)} equal to [AMU 999, Him. CET 00] A (B C) (B) A (B C) A (B C) (D) A (B C) 5. If n = 4, n(b) =, n(a B C) = 4, then n = [Kerala (Engg.) 005] (B) (D) 7 5. If A = { : = 0}, B = {, 4}, C = {4, 5}, then A (B C) [Kerala PET 00] {(, 4), (, 4)} (B) {(4, ), (4, )} {(, 4), (, 4), (4, 4)} (D) {(, ), (, ), (4, 4)} 54. A B = B A if A B (B) B A A = B (D) A B = φ 55. If A B = A and A B = A, then A B (B) B A A = B (D) none of these 56. If A and B are non-empty sets and A B = B A, then A a proper subset of B (B) B a proper subset of A A = B (D) none of these TARGET Publications 57. If A = { C : = } and B = { C : 4 = }, then A B {, } (B) {,, i, i} { i, i} (D) none of these 58. If P, Q and R are subsets of a set A, then R (P c Q c ) c = [Karnataka CET 99] (R P) (R Q) (B) (R Q) c (R P) c (R P) (R Q) (D) none of these 59. The shaded region in the given figure [NDA 000] A (B C) (B) A (B C) A A (B C) (D) A (B C) C B 60. If n(u) = 0, n =, n(b) = 9, n(a B) = 4, where U the universal set, A and B are subsets of U, then n((a B) c ) = [Kerala CET 004, Him. CET 007] (B) 6 9 (D) 6. If U = {,,, 4, 5, 6, 7, 8, 9}, A = { N : 0 < < 70}, B = { : a prime number less than 0}, then which of the following false? A B = {,, 5, 6, 7, 8} (B) A B = {7, 8} A B = {6, 8} (D) A B = {,, 5, 6, 8} 6. If A = {(, y) : + y = 5} and B = {(, y) : + 9y =44}, then A B contains [AMU 996; Pb. CET 00] one point (B) two points three points (D) four points 6. If two sets A and B are having 99 elements in common, then the number of elements common to each of the sets A B and B A are [Kerala (Engg.) 004] 99 (B) (D) If U the universal set and A B C = U, then [(A B) (B C) (C A)] equal to A B C (B) A (B C) A B C (D) A (B C)

26 TARGET Publications Maths (Vol. I) 65. In a class of 0 pupils, take needle work, 6 take Physics and 8 take Htory. If all the 0 students take at least one subject and no one takes all three, then the number of pupils taking subjects [J AND K 005] 6 (B) 6 8 (D) If A and B are any two sets, then A B not equal to A B c (B) B A c (A c B) c (D) A (A B) 67. If N a = {an : n N} and N b N c = N d, where a, b, c, d N and b, c are relatively prime, then d = b + c (B) d = b c d = bc (D) d = b c 68. If a set A contains 4 elements and a set B contains 8 elements, then maimum number of elements in A B 4 (B) 8 (D) The set (A B C) (A B C ) C equal to B C (B) B C A C (D) A C 70. If A = {(, y) : y = e, R} and B = {(, y) : y = e, R}, then A B = φ (B) A B φ A B = R (D) none of these 7. If A and B are two sets, then (A B) (A B) equal to [DCE 008] A (B) A B (D) none of these 7. If = {4 n n : n N} and Y = {9 (n ) : n N}, then Y equal to [Karnataka CET 997] (B) Y N (D) none of these 7. In a town of 0,000 families, it was found that 40% family buy newspaper A, 0% buy newspaper B and 0% families buy newspaper C, 5% families buy A and B, % buy B and C and 4% buy A and C. If % families buy all the three newspapers, then number of families which buy A only [Roorkee 997] 00 (B) (D) In a class of 55 students, the number of students studying different subjects are in Mathematics, 4 in Physics, 9 in Chemtry, in Mathematics and Physics, 9 in Mathematics and Chemtry, 7 in Physics and Chemtry and 4 in all three subjects. The number of students who have taken eactly one subject [UPSEAT 990] 6 (B) 9 7 (D) 75. Out of 800 boys in a school, 4 played cricket, 40 played hockey and 6 played basketball. Of the total, 64 played both basketball and hockey; 80 played cricket and basketball and 40 played cricket and hockey; 4 played all the three games. The number of boys who did not play any game [DCE 995; MP PET 996] 8 (B) 6 40 (D) A survey shows that 6% of the Americans like cheese whereas 76% like apples. If % of the Americans like both cheese and apples, then = 9 (B) = (D) none of these 77. Which of the following an empty set? The set of prime numbers which are even (B) The set of reals which satfy + i + i = 0 (A B) (B A), where A and B are djoint (D) The solution set of the equation ( + ) + = 0, R If A = { : + > 0} and B = { : 4 + 0}, then A B (, ) (B) [, ] (, ) (D) (, ) (, ) 79. If A = :sin and B =,, then A B equal to 5 5, 6 6 (B), 6 6 0, 6 (D), 6 6

27 Maths (Vol. I) 80. If = ( y, ): y=, R 4 and Y = {(, y) : y =, R}, then = Y (B) Y = φ Y φ (D) none of these 8. Suppose A, A,., A 0 are thirty sets each with five elements and B, B,., B n are n sets each with three elements such that 0 A i = n B j = S. If each element of S belongs i= j= to eactly 0 of the A i s and eactly 9 of the B j s, then the value of n [DCE 009] 5 (B) 0 40 (D) If A = {θ : cos θ + sin θ } and B = θ: θ, then A B equal to θ: θ (B) 5 θ: θ 6 5 θ: θ θ: θ 6 (D) none of these. Relations.. Relation 8. If R a relation from a non-empty set A to a non-empty set B, then R = A B (B) R = A B R = A B (D) R A B 84. If R a relation from a finite set A having m elements to a finite set B having n elements, then the number of relations from A to B mn (B) mn mn (D) m n 85. If R a relation on a finite set A having n elements, then the number of relations on A n n (B) n (D) n n 86. The relation R defined on the set of natural numbers as {(a, b) : a = b}. Then, R given by {(, ), (4, ), (6, ),.} (B) {(, ), (, 4), (, 6),.} 4 (D) R not defined none of these TARGET Publications 87. If A = {,, }, then domain of the relation R = {(, ), (, ), (, )} defined on A {, } (B) {, } {, } (D) {,, } 88. If P = {, 4, 5}, then range of the relation R = {(, ), (, 4), (5, 4)} defined on P {, 4} (B) {, 5} {4, 5} (D) {, 4, 5} 89. Let A = {,, }, B = {,, 5}. If relation R from A to B given by R = {(, ), (, 5), (, )}, then R {(, ), (, ),(5, )} (B) {(, ), (, 5), (, )} {(, ), (5, )} (D) {(, ), (, 5)} 90. If A and B are two finite sets such that n =, n(b) =, then total number of relations from A to B 64 (B) 8 6 (D) 9. If A the set of even natural numbers less than 8 and B the set of prime numbers less than 7, then the number of relations from A to B [NDA 00] 9 (B) 9 (D) 9 9. If R a relation from {,, } to {8, 0, } defined by y =, then R {(8, ), (0, )} (B) {(, 8), (8, 0)} {(, 8), (, 0)} (D) {(, ), (8, 0)} 9. If R = {(, y) : N, y N and + y = 5}, then the range of R {,,, 5} (B) {,,, 4} {,, 4, 5} (D) {,, 4, 5} 94. Number of relations that can be defined on the set A = {,, } (B) 6 (D) If P = {a, b, c, d} and Q = {,, }, then which of the following a relation from A to B? R = {(, a), (, b), (, c)} (B) R = {(a, ), (, b), (c, )} R = {(a, ), (d, ), (b, ), (b, )} (D) R 4 = {(a, ), (b, ), (c, ), (, d)}