Topic 1Types of Sets, OperationandCartesian Product20101. Let S { , , , } 1 2 34 . The total number of unordered pairsof disjoint subsets of S is equal to [IITJEE](a) 25 (b) 34 (c) 42 (d) 412. The shaded region in the figurerepresents [Kerala CEE](a) A B (b) A B (c) B A (d) A B (e) ( ) ( ) A B B A 20093. If AB , andC are three sets such that A B A C andA B A C , then [AIEEE](a) A C (b) B C (c) A B (d)A B 4. Two finite sets Aand Bhave mand nelementsrespectively. If the total number of subsets ofA is 112more than the total number of subsets ofB, then thevalue of mis [Kerala CEE](a) 7 (b) 9(c) 10 (d) 12(e) 135. For any two setsA andB, if A X B X andA X B X for some setX, then [AMU](a) A B A B (b) A B (c) B A A B (d) None of these6. x Rxx x xR + +';):2 14 33 2equals[EAMCET](a) R { } 0 (b) R { , , } 0 1 3(c) R { , , } 0 1 3 (d) R +';)0 1 312, , ,7. For any two setsA and B,A A B ( ) equals(a) B (b) A B [WB JEE](c) A B (d) A BC C8. If S is a set with 10 elements andA x y x y S x y {( , ) : , , }, then the number ofelements inA is [J&K CET](a) 100 (b) 90 (c) 50 (d) 4520089. Let A B and be two sets, then ( ) ( ) A B A B isequal to [DCE](a) A (b) A(c) B (d) None of theseSets, Relations andFunctions1ChapterUA B10. A survey shows that 63% of the Americans like cheesewhereas 76% like apples. Ifx% of the Americans likeboth cheese and apples, then [UP SEE](a) x 39 (b) x 63(c)39 63 x (d) None of these11. In a certain town 25% families own a cell phone, 15%families own a scooter and 65% families own neither acell phone nor a scooter. If 1500 families own both acell phoneandascooter, thenthetotal number offamilies in the town is [Kerala CEE](a) 10000 (b) 20000 (c) 30000 (d) 40000(e) 5000012. If A { , , }, 1 2 3 B { , } 34 , C { , , } 4 5 6 . Then,A B C ( ) is [OJEE](a) {1, 2} (b) {}(c) {4, 5} (d) {1, 2, 3, 4}13. Three sets ABC , , are such that A B C andB C A , then [WB JEE](a) A B (b) A B (c) A B (d) A B 14. If A a bc {, , }, B bcd { , , }and C a d c {, , }, then( ) ( ) A B B C is equal to [Jamia Millia Islamia](a){( , ), ( , )} a c a d (b) {( , ), ( , ) a b cd }(c) {( , ), ( , ) ca d a } (d) {( , ), ( , ), ( , ) a c a d bd }15. If nA ( ) denotes the number of elements in the setA andif nA ( ) , 4 nB ( ) 5 and nA B ( ) , 3 thenn A B B A [( ) ( )] is equal to [J&KCET](a) 8 (b) 9 (c) 10 (d) 11200716. The value of ( ) ( ) A B C A B C Cc c c c is(a) B Cc (b) B Cc c[DCE](c) B C (d) A B C 17. Let Z denote the set of all integers andA a b a b + {( , ) :2 23 28, a b Z , } andB a b a ba b Z > {( , ) : , , }. Then, the number ofelements in A B is [Kerala CEE](a) 2 (b) 3(c) 4 (d) 5(e) 618. Which of the following is true ? [OJEE](a) A A (b) A (c) A U (d) A A 19. If setsA B and are defined asA x y yxx R ';)( , ) : ,10 ,B x y y x x R {( , ) : , }, then [Guj. CET](a) A B A (b) A B B (c) A B (d) None of these200620. Universal set, U x x x x x + { : }5 4 3 26 11 6 0and A x x x + { : }25 6 0B x x x + { : }23 2 0Then, ( ) A B is equal to [BITSAT](a) {1, 3} (b) {1, 2, 3} (c) {0, 1, 3} (d) {0, 1, 2, 3}21. IfA x y { , }, then the power set ofA is[UP SEE, Guj. CET](a){ , } x yy x(b){ , , } x y(c) {, { }, { } x y 2 } (d) {, {x}, { y}, {x y , }}22. Let A { , , , }, 1 2 34 B { , , } 24 6 . Then, the number of setsC such thatA B C A B is [Kerala CEE](a) 6 (b) 9 (c) 8 (d) 10(e) 1223. Let X Y and be the sets of all positive divisors of 400and1000respectively(including1andthenumber).Then, nX Y ( ) is equal to [Kerala CEE](a) 4 (b) 6 (c) 8 (d) 10(e) 1224. Let A x x { : isamultipleof 3}and B x x { : isamultiple of 5}. Then, A B is given by [AMU](a) {3, 6, 9, } (b) {5, 10, 15, 20, }(c) {15, 30, 45, } (d) None of these25. IfA B and are two sets, thenA A B ( ) is equal to(a) B (b) A B [J&K CET](c) A B (d) B A 200526. If nA ( ) , 4 nB ( ) , 3 nA B C ( ) , 24 thennC ( ) isequal to [Kerala CEE](a) 288 (b) 1 (c) 12 (d) 17(e) 227. The number of elements in the set{( , ) : , a b a b 2 3 352 2+ a b Z , }, whereZ is the set ofall integers, is [Kerala CEE](a) 2 (b) 4 (c) 8 (d) 12(e) 1628. {nn n n Z ( )( ) : + + 1 2 1 } [EAMCET](a) {6k k Z : } (b) {12k k Z : }(c) {18k k Z : } (d) {24k k Z : }29. If A B , then B A is equal to [BCECE](a) B A (b) A(c) B (d) None of these30. In a class of 30 pupils12 take needlswork, 16 takephysics and 18 take history. If all the 30 students take atleast one subject and no one takes all three, then thenumber of pupils taking 2 subjects is [J&KCET](a) 16 (b) 6 (c) 8 (d) 202|Chapterwise & Topicwise Solved Papers Sets, Relations and Functions200431. IfA { , , }, 1 2 3 B ab {, }, then A B mapped on Ato B is(a) {( , ), ( , ), ( , ) 1 2 3 a b b } [DCE](b) {( , ), ( , ) 1 2 b a }(c) {( , ), ( , ), ( , ), ( , ), ( , ), ( , ) 1 1 2 2 3 3 a b a b a b }(d) {( , ), ( , ), ( , ), ( , ) 1 2 2 3 a a b b }32. If two setsA B and are having 99 elements in common,then the number of elements common to each of the setsA B B A and are [Kerala CEE](a)299(b)992(c) 100 (d) 18(e) 933. Given n U nA nB ( ) , ( ) , ( ) , 20 12 9 nA B ( ) , 4where Uis the universal set, A B and are subsets of U,then n A Bc[( ) ] equals to [Kerala CEE](a) 17 (b) 9 (c) 11 (d) 3(e) 1634. Two finite sets havem n and elements. The number ofelements in the power set of first set is 48 more than thetotal number of elements in the power set of the secondset. Then, the value of m n and are [Kerala CEE](a) 7, 6 (b) 6, 3 (c) 6, 4 (d) 7, 4(e) 3, 735. Suppose A A A1 2 30, , . . . , arethirtysets, eachhaving5elementsand B B Bn 1 2, , . . . , are nsets eachwith3 elements, let iijnjA B S1301and each element ofS belongs to exactly 10 of theAi s and exactly 9 of theBj s. Then, nis equal to [AMU](a) 115 (b) 83(c) 45 (d) None of these36. If A { , , , , }, 1 2 34 5 B { , , } 24 6 , C { , , }, 34 6 then( ) A B C is [OJEE](a) {3, 4, 6} (b) {1, 2, 3}(c) {1, 4, 3} (d) None of these200337. A class has 175 students. The following data shows thenumber of students opting one or more subjects.Mathematics 100; Physics 70; Chemistry 40;Mathematics and Physics 30; Mathematics andChemistry 28; Physics and Chemistry 23; Mathematics,Physics andChemistry 18. How many studentshaveoffered Mathematics alone ? [Kerala CEE](a) 35 (b) 48(c) 60 (d) 22(e) 3038. An investigator interviewed 100 students to determinethe performance of three drinks milk, coffee and tea.The investigator reported that 10 students take all threedrinks milk, coffee and tea; 20 students take milk andcoffee, 30 students take coffee and tea, 25 students takemilk and tea, 12 students take milk only, 5 students takecoffeeonlyand8studentstaketeaonly. Then, thenumber of students who did not take any of the threedrinks, is [OJEE](a) 10 (b) 20(c) 25 (d) 3039. Let Y { , , , , } 1 2 34 5 , A { , } 1 2 , B { , , } 34 5 and denotes null set. If ( ) A B denotes cartesian product ofthe sets A B and ; then ( ) ( ) Y A Y B is [OJEE](a) Y (b) A(c) B (d)Topic 2Relation, Equivalence Relation20101. Consider thefollowingrelations R x y x y {( , ) | , arereal numbers and x wy for some rational number w};Smnpqmn

_,

', , , p and q are integers such thatn,q 0 and qm pn }. Then [AIEEE](a)R is an equivalence relation but S is not anequivalence relation(b) Neither R nor S is an equivalence relation(c)S is an equivalence relation but R is not anequivalence relation(d)R and S both are equivalence relations2. Let A x y z { , , } andB a bcd {, , , }. Which one of thefollowing is not a relation from Ato B ? [Kerala CEE](a){( , ), ( , )} x a x c (b){( , ), ( , )} yc yd(c) {( , ), ( , )} z a z d (d) {( , ), ( , ), ( , )} z b yb a d(e){( , )} x c20093. Let R and S be two non-void relations on a set A. Whichof the following statements is false ? [DCE](a) R and S are transitive implies R S is transitive.(b)R and S are transitive implies R S is transitive.(c) R and S are symmetric implies R S is symmetric.(d)R and S are reflexive implies R S is reflexive.Chapterwise& Topicwise Solved Papers Sets, Relations and Functions|34. Let r be a relation fromR (set of real numbers) toRdefinedby r a b a b R {( , ) | , and a b + 3is anirrational number}. The relationr is [AMU](a) an equivalence relation(b) reflexive only(c) symmetric only(d) transitive only5. Let a relationRon the set Nof naturalnumbers bedefined as ( , ) x y x xy y x + 2 24 3 0 , y N . Therelation R is [AMU](a)reflexive (b) symmetric(c) transitive (d) an equivalence relation20086. Let R be the real line. Consider the following subsets ofthe plane R R S x y y x + {( , ) : 1and 0 2 < < x }T x y x y {( , ) : is an integer}Which one of the following is true ? [AIEEE](a) T is an equivalence relation onR but S is not(b) Neither S nor T is an equivalence relation onR(c) Both S T and are equivalence relations onR(d) S is an equivalence relation onR but T is not7. R is a relation onN given by R x y x y + {( , ) : } 4 3 20 .Which of the following belongs to R ? [KCET](a) ( 4, 12) (b) (5, 0)(c) (3, 4) (d) (2, 4)8. Let A { , , } 1 2 3 and B { , , } 2 34 , thenwhichof thefollowing relations is a function fromA to B ?(a) {(1, 2), (2, 3), (3, 4), (2, 2)} [WB JEE](b) {(1, 2), (2, 3), (1, 3)}(c) {(1, 3), (2, 3), (3, 3)}(d) {(1, 1), (2, 3), (3, 4)}9. If R be a relation defined as aRb iff | | , a b > 0 then therelation is [VITEEE](a) reflexive (b) symmetric(c) transitive (d) symmetric and transitive10. R is a relation from {11, 12, 13} to {8, 10, 12} definedbyy x 3.Then, R1is [Jamia Millia Islamia](a) {(8, 11), (10, 13)} (b) {(11, 18), (13, 10)}(c) {(10, 13), (8, 11)} (d) None of these200711. On the set N of all natural numbers define the relation Rby aRbif and only if the GCDof a b and is 2, then R is(a) reflexive, but not symmetric [Kerala CEE](b) symmetric only(c) reflexive, and transitive(d) reflexive, symmetric and transitive(e) not reflexive, not symmetric and not transitive12. If R is an equivalence relation on a set A, then R1is(a) reflexive only [AMU](b) symmetric but not transitive(c) equivalence(d) None of the above13. The relation R defined on the set of natural numbers as{( , ) : a b a differs frombby 3} is given by [AMU](a) {(1, 4), (2, 5), (3, 6), }(b) {(4, 1), (5, 2), (6, 3), }(c) {(1, 3), (2, 6), (3, 9), }(d) None of the above14. Let R {( , ), ( , ), ( , ), ( , ), ( , ), ( , ) 3 3 6 6 9 9 12 12 6 12 3 9 ,(3, 12), (3, 6)} be a relation on the set A { , , , } 3 6 9 12 .The relation is [OJEE](a) reflexive and symmetric only(b) an equivalence relation(c) reflexive only(d) reflexive and transitive only15. Let R {( , ), ( , ), ( , ), ( , ), ( , )} 1 3 4 2 24 2 3 3 1 be a relation onthe setA { , , , } 1 2 34 . The relation R is[Jamia Millia Islamia](a) a function (b) transitive(c) not symmetric (d) reflexive200616. Let Wdenotes thewordsintheEnglishdictionary.Define the relation R byR x y W W {( , ) :thewordsx y and haveatleastone letter in common}. Then, R is [AIEEE](a) reflexive, symmetric and not transitive(b) reflexive, symmetric and transitive(c) reflexive, not symmetric and transitive(d) not reflexive, symmetric and transitive17. Which of the following statements is not correct for therelation R defined by aRb, if and only, if b lives withinon kilometre froma ? [BITSAT](a) R is reflexive (b) R is symmetric(c) R is anti-symmetric (d) None of these18. Let Rbearelationontheset of integersgivenbyaRb a bk 2 for some integer k. Then, R is(a) an equivalence relation [Kerala CEE](b) reflexive but not symmetric(c) reflexive and transitive but not symmetric(d) reflexive and symmetric but not transitive(e) symmetric and transitive but not reflexive200519. x xy2 is a relation which is [BITSAT](a) symmetric (b) reflexive and transitive(c) transitive (d) None of these4|Chapterwise & Topicwise Solved Papers Sets, Relations and Functions20. The relation R {( , ), ( , ), ( , )} 1 1 2 2 3 3 on the set {1, 2, 3}is [Guj. CET](a) symmetric only (b) reflexive only(c) an equivalence relation (d) transitive only200421. Thenumberofreflexiverelationsofasetwithfourelements is equal to [UPSEE](a)216(b)212(c)28(d)2422. LetR be the relation on the setR of all real numbersdefined by aRbif | | , a b 1 then R is [J&KCET](a) reflexive and symmetric (b) symmetric only(c) transitive only (d) anti-symmetric only200323. LetS be the set of all real numbers. Then, the relationR a b ab + > {( , ) : } 1 0on S is [Kerala CEE](a) reflexive and symmetric but not transitive(b) reflexive and transitive but not symmetric(c) symmetric and transitive but not reflexive(d) reflexive, transitive and symmetric(e) None of the above24. LetA { , , , , . . . , , , } 2 34 5 16 17 18 . Let be the equivalencerelation on A A , cartesian product of A A and , definedby ( , ) ( , ) a b cd if ad bc , then the number of orderedpairs of the equivalence class of (3, 2) is [OJEE](a) 4 (b) 5(c) 6 (d) 7Topic 3Types of Mapping20101. A B { , , , }, { , , , , , } 1 2 34 1 2 34 5 6 are two sets, andfunction f A B : is defined by f x x x A ( ) + 2 ,then the functionf is [WB JEE](a) bijective (b) onto (c) one-one(d) many-one20092. For real x, let f x x x ( ) + +35 1, then[AIEEE](a) f is one-one but not ontoR(b) f is onto R but not one-one(c) f is one-one and ontoR(d) f is neither one-one nor ontoR3. Let f x x x ( ) ( ) , + 1 1 12[AIEEE]Statement I The set { : ( ) ( )} { , } x f x f x 10 1Statement IIf is a bijection.(a) Statement I is true, Statement II is true; Statement IIis a correct explanation for Statement I(b) Statement I is true, Statement II is true; Statement IIis not a correct explanation for Statement I(c) Statement I is true, Statement II is false(d) Statement I is false, Statement II is true4. On the set of integers Z, define f Z Z : asf nnnn( ),,, '20is evenis oddthen f is [KCET](a) injective but not surjective(b) neither injective nor surjective(c) surjective but not injective(d) bijective5. Let nA ( ) 4 andnB ( ) 6. The number of one to onefunctions fromA to B is [AMU](a) 24 (b) 60(c) 120 (d) 3606. Let R andC denote the set of real numbers and complexnumbers respectively. The functionf C R : definedbyf z z ( ) | | is [AMU](a) one to one(b) onto(c) bijective(d) neither one to one nor onto20087. Let f N N : defined by f x x x ( ) , + +21 x N ,thenf is [DCE](a) One-one onto(b) Many-one onto(c) One-one but not onto(d) None of these8. Which one of the following functions is one-to-one ?(a) f x x x ( ) sin , [ , ) [Kerala CEE](b) f x x x ( ) sin , ,

1]132 4 (c) f x x x ( ) cos , ,

1]1 2 2(d) f x x x ( ) cos , ,

_,

32(e) f x x x ( ) cos , ,

1]1 4 4Chapterwise& Topicwise Solved Papers Sets, Relations and Functions|59. If f R C : is defined by f x eix( ) 2forx R , thenf is (where C denotes the set of all complex numbers)(a) one-one [AMU, EAMCET](b) onto(c) one-one and onto(d) neither one-one nor onto10. A mapping f N N : , whereNis the set of naturalnumbers is defined asf nn nn n( ),,+'22 1for oddfor evenfor n N . Then, f is [WB JEE](a) surjective but not injective(b) injective but not surjective(c) bijective(d) neither injective nor surjective11. The mappingf N N : given byf n n ( ) , + 12n N whereN is the set of natural number, is [WB JEE](a) one-to-one and onto(b) onto but not one-to-one(c) one-to-one but not onto(d) neither one-to-one nor onto12. Afunction f A B : , where A x x { : } 1 1 andB y y { : } 1 2 is defined by the ruley f x x + ( ) 12. Which of the following statement istrue ? [WB JEE](a) f is injective but not surjective(b) f is surjective but not injective(c) f is both injective and surjective(d) f is neither injective nor surjective13. The functionf R R : given byf x x ( ) 31is(a) a one-one function [J&K CET](b) an onto function(c) a bijection(d) neither one-one nor onto200714. LetA [ , ] 1 1 andf A A : be defined asf x x x ( ) | | for all x A , thenf x ( ) is [BITSAT](a) many-one into function(b) one-one into function(c) many-one onto function(d) one-one onto function15. The functionf R R : defined byf x x x x ( ) ( )( )( ) 1 2 3is [AMU](a) one-one but not onto(b) onto but not one-one(c) both one-one and onto(d) neither one-one nor onto200616. The function f X Y : defined by f x x ( ) sin isone-one but not onto, ifX Y and are respectively equalto [KCET](a) R R and(b)[ , ] and [ , ] 0 0 1 (c) 02,

1]1 and [1, 1](d)

1]1 2 2, and [ , ] 1 117. If R denotes the set of all real numbers, then the functionf R R : defined byf x x ( ) | | is [AMU](a) one-one only(b) onto only(c) both one-one and onto(d) neither one-one nor onto200518. f xx xx( ),, 'if is rationalif is irrational 0andgxxx x( ),, '0 if is rationalif is irrational. Then, f g is[IIT JEE](a) one-one and into(b) neither one-one nor onto(c) many one and onto(d) one-one and onto19. LetA be a set containing 10 distinct elements, then thetotal number of distinct function fromA toA is [DCE](a) 1010(b) 101(c)210(d)2 110200420. The number of onto mappings from the setA { , , . . . , } 1 2 100to set B { , } 1 2is [AMU](a)2 2100 (b)2100(c)2 299 (d)29921. Let A R { } 3 , B R { } 1 . Let f A B : be definedbyf xxx( ) . 23Then, [AMU](a) f is bijective(b) f is one-one but not onto(c) f is onto but not one-one(d) None of the above22. f x x x ( ) +2is a function from R to R, thenf x ( ) is(a) injective (b) surjective [OJEE](c) bijective (d) None of these6|Chapterwise & Topicwise Solved Papers Sets, Relations and Functions200323. If f :[ , ) [ , ) 0 0 andf xxx( ) , + 1thenf is(a) one-one and onto [IIT JEE](b) one-one but not onto(c) onto but not one-one(d) neither one-one nor onto24. A functionf fromthe set of natural numbers to integersdefined byf nnnnn( ),,'122when is oddwhen is evenis [AIEEE](a) one-one but not onto (b) onto but not one-one(c) one-one and onto both (d) neither one-one nor ontoTopic 4Domain-Range, Odd-Even andPeriodic Function20101. The domain of the functionf xx( ) cos| |

_,

112is(a) ( , ) 3 3 (b) [ , ] 3 3 [WB JEE](c) ( , ) ( , ) 3 3 (d) ( , ] [ , ) 3 32. The functionf x x x ( ) sec [log ( )] + + 12is(a) odd (b) even [BITSAT](c) neither odd nor even (d) constant3. The domain of sin log

_,

1]11212xis[Kerala CEE](a) [ , ] 2 12 (b) [ , ] 1 1(c)1324 ,

1]1(d)2324 ,

1]1(e) [6, 24]4. The period of the functionf ( ) sin sin + 4 4 33is[BITSAT](a) 23(b) 3(c) 2(d) (e) 220095. Range of the functionf xxx( ) + 12 is [DCE](a) ( , ) (b)[ , ] 1 1(c)

1]11212, (d)[ , ] 2 26. Let f be a function with domain[ , 3 5] and letgx x ( ) | | + 3 4 . Then the domain of ( ) ( ) fog x is(a)

_,

313, (b)

_,

313, [BITSAT](c)

1]1313, (d)

1]1313,7. The domain of the functionf x x ( ) log (log (log ) 2 3 4)is [Kerala CEE](a) ( , ) 4 (b) ( , ) 4(c) (0, 4) (d) ( , ) 1 (e) ( , ) 18. If f R :[ , ] 2 3 is defined by f x x x ( ) + 33 2, thenthe rangef x ( ) is contained in the interval [EAMCET](a) [1, 12] (b) [12, 34](c) [35, 50] (d)[ , ] 12 129. Iff xx x( )| |1, then domain off x ( ) is [OJEE](a) ( , ) 0(b) ( , ) 2(c) ( , ) (d) None of the above10. The domain of definition of the functionf x xe( ) log ( ) + 1 1 is [WB JEE](a) < x 0(b) < xee1(c) < x 1(d) x e 111. The period of the functionf x x x ( ) cot + cosec23 4 is(a)3(b)4[BCECE](c)6(d) 12. The domain of the real functionf xx( ) 142is(a) the set of all real numbers [J&K CET]Chapterwise& Topicwise Solved Papers Sets, Relations and Functions|7(b) the set of all positive real numbers(c) ( , ) 2 2(d)[ , ] 2 2200813. The domain of sin log

_,

1]1133xis[DCE, Jamia Millia Islamia](a) [1, 9] (b) [1, 9] (c) [ 9, 1] (d) [ 9, 1]14. If f R R : andg R R : are defined by f x x ( ) | | and gx x ( ) [ ] 3 for x R , then g f x x ( ( )) : < and , then domain of the functionf x ax bx c x ( ) log {( )( )} + + +21 is [RPET](a) Rba

_,

2(b) R ( , ) 1(c) ( , ) ';)12ba(d) Rba ';)

_,

21 ( , )70. Iff xx( ) 1, then domain offof is [J&KCET](a) ( , ) 0 (b) ( , ) 0(c){ } 0 (d) {}Topic 5Inverse, Composition andDifferent Types of Functions20101. Let Rbetheset of real numbersandthemappingf R R : andg R R : be defined by f x x ( ) 52and gx x ( ) 3 4, then the value of ( )( ) fog 1is(a) 44 (b) 54 [WB JEE](c) 32 (d) 642. If f x x ( ) 21and gx x ( ) ( ) +12, then ( )( ) gof x is(a) ( ) x + 1 14(b) x41 [Kerala CEE](c) x4(d) ( ) x +14(e) ( ) x 1 143. Let f xxxx ( ) , + 211. The value of for whichf a a a ( ) , ( ) 0is [BITSAT](a) 11a(b) 1a(c) 11+a(d) 11a(e) 1a4. If f is a real valued function such thatf x y f x f y ( ) ( ) ( ) + + andf ( ) 1 5 , then the value ofChapterwise& Topicwise Solved Papers Sets, Relations and Functions|11f ( ) 100is [Kerala CEE](a) 200 (b) 300 (c) 350 (d) 400(e) 50020095. Letf :[ , [ [ , [ 4 4 be defined by f xx x( )( )54thenf x1( ) [DCE](a)2 45 + log x (b)2 45+ + log x(c)154

_,

x x ( )(d) not defined6. If f x x x ( ) sin sin + +

_,

2 23+ +

_,

cos cos x x3andg541

_,

, then gof x ( ) is equal to[UP SEE, Jamia Millia Islamia](a) 1 (b) 1 (c) 2 (d) 27. If f xxx( ) +44 2, thenf f f1972979697

_,

+

_,

+ +

_,

....is equal to [UP SEE](a) 1 (b) 48 (c) 48 (d) 18. Ifafunction f satisfies f f x x { ( )} +1 for all realvalues of x and if f ( ) 012 , thenf ( ) 1is equal to(a)12(b) 1 [Kerala CEE](c)32(d) 2(e) 09. If f R R : and g R R : are defined byf x x ( ) 3and gx x ( ) +21, then the values of x for whichg f x { ( )} 10 are [Kerala CEE](a)0 6 , (b)2 2 , (c) 1 1 , (d) 0, 6(e) 0, 210. If f x ( ) satisfies the relation 2 12f x f x x ( ) ( ` ) + forall real x, thenf x ( ) is [Kerala CEE](a)x x22 16+ (b)x x22 13+ (c)x x24 13+ (d)x x23 16 +(e)x x23 13+ 11. If 2 311222f x fxx ( ) +

_,

forall x R { } 0 , thenf x ( )4is [AMU](a)( ) ( ) 1 2 354 44 + x xx(b)( ) ( ) 1 2 354 44+ x xx(c)( ) ( ) 1 2 354 44 x xx(d) None of these12. Let a andb be two integers such that 10 5 a b + andPx x ax b ( ) + + . The integer n such thatP P Pn ( ) ( ) ( ) 10 11 is [AMU](a) 15 (b) 65 (c) 115 (d) 16513. Iff x x x ax b ( ) + + 2 134 2is divisible by x x23 2 + ,then ( , ) a bis equal to [EAMCET](a) ( , ) 9 2 (b) (6, 4)(c) (9, 2) (d) (2, 9)14. If f x yx x y xy ( , , ) + 2 2 , thenf x y ( , ) equals[Jamia Millia Islamia](a)x y2 28(b)x y2 24(c)x y2 24+(d)x y2 2215. If D30isthesetofthedivisorsof 3030, , x y D , wedefine x y x y + LCM ( , ), x y x y GCD ( , ), xx 30and f x y z x y y z ( , , ) ( ) ( ) + + , then f ( , , ) 2 5 15 isequal to [MHT CET](a) 2 (b) 5 (c) 10 (d) 1516. If the functionf N N : is defined byf x x ( ) , thenff f( )( ) ( )2516 1 +is equal to [MPPET](a)56(b)57(c)53(d) 117. Iff R R : is defined asf x x ( ) ( )/ 113, thenf x1( )is [RPET](a) ( )/113x (b) ( ) 13 x (c) 13 x (d) 113 x/200818. Let f N Y : be a function defined as f x x ( ) + 4 3where Y y N y x + { : 4 3 for somex N }. Showthatf is invertible and its inverse is [AIEEE](a) g yy( ) 34(b) g yy( ) + 3 43(c) g yy( ) ++434(d) g yy( ) + 3419. If f R R : isdefinedby f x x ( ) , 3then f18 ( ) isequal to [KCET](a) {2} (b) {2 2 22, , }(c) {2, 2} (d) {2, 2}20. If f xxxx ( ) log , , +

_,