01 - SETS, RELATIONS AND FUNCTIONS Page 1 ( Answers at the end of all questions )

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1 0 SETS, RELATIONS AND FUNCTIONS Page ( ) Let R = { ( 3, 3 ) ( 6, 6 ) ( ( 9, 9 ) (, ), ( 6, ) ( 3, 9 ) ( 3, ), ( 3, 6 ) } be a relation on the set A = { 3, 6, 9, }. The relation ( a ) refleive and transitive ( b ) refleive only ( c ) an equivalence relation ( d ) refleive and symmetric only [ AIEEE 005 ] ( ) Let f : (, ) B be a function defined by f ( ) = oneone and onto when B the interval tan, then f both π π π π π π ( a ) 0, ( b ) 0, ( c ), ( d ), [ AIEEE 005 ] ( 3 ) If a real valued function f ( ) satfies the fun tional equation f ( y ) = f ( ) f ( y ) f ( a ) f ( a y ), whe e a a given constant and f ( 0 ) =, then f ( a ) equal to ( a ) f ( ) ( b ) f ( ) ( c ) f ( ) f ( a ) ( d ) f ( ) [ AIEEE 005 ] ( 4 ) Let R = { (, 3 ), ( 4, ) (, 4 ), (, 3 ), ( 3, ) } be a relation on the set A = {,, 3, 4 }. The relation R ( a ) a function ( b ) transitive ( c ) not symmetric ( d ) refleive [ AIEEE 004 ] 7 ( 5 ) The ran e o the function f ( ) = P 3 ( a ) {, 3 } ( b ) {,, 3, 4, 5, 6 } c ) {,, 3, 4 } ( d ) {,, 3, 4, 5 } [ AIEEE 004 ] ( 6 ) If f : R S, defined by f( ) = sin 3 cos onto, then the interval of S ( a ) [ 0, 3 ] ( b ) [, ] ( c ) [ 0, ] ( d ) [, 3 ] [ AIEEE 004 ] ( 7 ) The graph of the function f ( ) symmetrical about the line =, then ( a ) f ( ) = f ( ) ( b ) f ( ) = f ( ) ( c ) f ( ) = f ( ) ( d ) f ( ) = f ( ) [ AIEEE 004 ]

2 0 SETS, RELATIONS AND FUNCTIONS Page ( 8 ) The domain of the function f ( ) = sin ( 3 ) 9 ( a ) [, 3 ] ( b ) [, 3 ) ( c ) [, ] ( d ) [, ) [ AIEEE 004 ] ( 9 ) If f : {,, 3,. } { 0, ±, ±,.. } defined by f ( ) =, if even ( ), if odd then value of f ( 00 ) ( a ) 00 ( b ) 99 ( c ) 00 ( d ) 0 [ AIEEE 003 ] ( 0 ) Domain of definition of the function f ( ) = 3 4 log 0 ( 3 ) ( a ) (, ) ( b ) (, 0 ) (, ) ( c ) (, ) (, ) ( d ) ( 0 ) (, ) (, ) [ AIEEE 003 ] ( ) The function f ( ) = log ( ) a / an ( a ) even function b ) odd function ( c ) periodic function ( d ) none of these [ AIEEE 003 ] ( ) The functi n f : R R defined by f ( ) = sin ( a ) into ( b ) onto ( c ) oneone ( d ) manyone [ AIEEE 00 ] 3 The range of the function f ( ) =, ( a ) R ( b ) R { } ( c ) R { } ( d ) R { } [ AIEEE 00 ] ( 4 ) If f ( ) =, 0, Q Q and g ( ) = 0,, Q, then ( f g ) Q ( a ) oneone, onto ( b ) neither oneone nor onto ( c ) oneone but not onto ( d ) onto but not oneone [ IIT 005 ]

3 0 SETS, RELATIONS AND FUNCTIONS Page 3 ( 5 ) If f ( ) = sin cos and g ( ) =, then g [ f ( ) ] will be invertible for the domain π π π π ( a ) [ 0, π ] ( b ), ( c ) 4 4 0, ( d ), 0 [ I T 004 ] ( 6 ) The range of the function f ( ) = ( a ) [, ) ( b ) (, ( 7 ) f : [ 0, ) [ 0, ), f ( ) =, (, ) 7 7 ) ( c ) (, ] ( d ) [, ] [ IIT 003 ] ( a ) oneone and onto ( b ) oneone bu no onto ( c ) onto but not oneone ( d ) neither one one nor onto [ IIT 003 ] ( 8 ) If f ( ) = ( ) for and g ( ) the function whose graph reflection of the graph of f ( ) with respect o the line y =, then g ( ) equals ( ) ( a ), 0 ( b, > ( c ), ( d ), 0 [ IIT 00 ] ( 9 ) If function f : R R defined as f ( ) = sin for R, then f ( a ) one on and onto ( b ) oneone but not onto ( c ) ont bu not oneone ( d ) neither oneone nor onto [ IIT 00 ], < 0 ( 0 ) If g ( ) = [ ] and f ( ) = 0, = 0,, > 0 then for all, f [ g ( ) ] = ( a ) ( b ) ( c ) f ( ) ( d ) g ( ) [ IIT 00 ] ( ) If f : [, ) [, ) given by f ( ) =, then f ( ) equals ( a ) 4 ( b ) ( c ) 4 ( d ) 4 [ IIT 00 ]

4 0 SETS, RELATIONS AND FUNCTIONS Page 4 ( ) The domain of definition of f ( ) = log ( 3 ) 3 ( a ) R {, } ( b ) (, ) ( c ) R {,, 3 } ( d ) ( 3, ) {, } [ IIT 00 ] ( 3 ) If E = {,, 3, 4 } and F = {, }, then the number of onto func ions om E to F ( a ) 4 ( b ) 6 ( c ) ( d ) 8 [ IIT 00 ] ( 4 ) If f ( ) = α,, then for which value of α f [ f ( ) ] =? ( a ) ( b ) ( c ) ( d ) [ IIT 00 ] ( 5 ) Let f : R R be any function. Define g : R R by g ( ) = l f ( ) l for all. Then g ( a ) onto if f onto ( b oneone if f oneone ( c ) continuous if f continuous ( d ) differentiable if f differentiable [ IIT 000 ] ( 6 ) The domain of definition of the function y ( ) as given by the equation y = ( a ) 0 < ( b ) 0 ( c ) < 0 ( d ) < < [ IIT 000 ] ( 7 ) If the function f : [, ) [, ) defined by f ( ) = ( ), then f ( ) ( a ) ( c ) ( ) ( b ) ( 4 log ) ( 4 log ) ( d ) not defined [ IIT 999 ] ( 8 ) In a college of 300 students, every student reads 5 newspapers and every newspaper read by 60 students. The number of newspapers ( a ) at least 30 ( b ) at most 0 ( c ) eactly 5 ( d ) none of these [ IIT 998 ] ( 9 ) If f ( ) =, for every real number, then the minimum value of f ( a ) does not et as f unbounded ( b ) equal to ( c ) not attained even though f bounded ( d ) equal to [ IIT 998 ]

5 0 SETS, RELATIONS AND FUNCTIONS Page 5 ( 30 ) If f ( ) = 3 5, then f ( ) 5 ( a ) given by ( b ) given by ( c ) does not et because f not oneone ( d ) does not et because f not onto [ IIT 998 ] ( 3 ) If g [ f ( ) ] = l sin l and f [ g ( ) ] = ( sin ), then ( a ) f ( ) = sin, g ( ) = ( b ) f ( ) = sin g ( ) = l l ( c ) f ( ) =, g ( ) = sin ( d ) f and g cann t be determined [ IIT 998 ] ( 3 ) If f ( ) = ( ), ( ), then the se S = { : f ( ) = f ( ) } 3 i 3 3 i 3 ( a ) 0,,, ( b ) { 0,, } ( c ) { 0, } ( d ) empty [ IIT 995 ] ( 33 ) The number log 7 ( a ) an integer ( b ) a rational number ( c ) an irrationa number ( d ) a prime number [ IIT 990 ] ( 34 ) If S the set of all real such that ( a ), ( b ), ( c ), 4 4 ( d ), 3 positive, then S contains ( e ) none of these [ IIT 986 ] ( 35 ) If y = f ( ) =, then ( a ) = f ( y ) ( b ) f ( ) = 3 ( c ) y increases with for < ( d ) f a rational function of [ IIT 984 ]

6 0 SETS, RELATIONS AND FUNCTIONS Page 6 ( 36 ) Let f ( ) = l l. Then ( a ) f ( ) = [ f ( ) ] ( b ) f ( y ) = f ( ) f ( y ) ( c ) f ( l l ) = l f ( ) l ( d ) None of these [ IT 983 ] ( 37 ) The domain of definition of the function y = log0 ( ) ( a ) ( 3, ) ecluding.5 ( b ) [ 0, ] ecluding 0.5 ( c ) [, ] ecluding 0 ( d ) None of these [ IIT 983 ] ( 38 ) Which of the following functions periodic? ( a ) f ( ) = [ ] where [ ] denotes the largest integer less than or equal to the real number ( b ) f ( ) = sin for 0, f ( 0 ) = 0 ( c ) f ( ) = cos ( d ) None of these [ IIT 983 ] ( 39 ) If X and Y are two se s, then X ( X Y ) c equals ( a ) X ( b ) Y c ) φ ( d ) none of these [ IIT 979 ] ( 40 ) Let R be the set of real numbers. If f : R R a function defined by f ( ) =, then f i ( a ) inject ve but not subjective ( b ) subjective but not injective ( ) bijective ( d ) none of these [ IIT 979 ] Answers a d a c a d b b d b a d b a b c b d a b s a d a d c d b c d b a c c a,d a,d d c a c d

BRAIN TEASURES FUNCTION BY ABHIJIT KUMAR JHA EXERCISE I. log 5. (ii) f (x) = log 7. (iv) f (x) = 2 x. (x) f (x) = (xii) f (x) =

BRAIN TEASURES FUNCTION BY ABHIJIT KUMAR JHA EXERCISE I. log 5. (ii) f (x) = log 7. (iv) f (x) = 2 x. (x) f (x) = (xii) f (x) = EXERCISE I Q. Find the domains of definitions of the following functions : (Read the symbols [*] and {*} as greatest integers and fractional part functions respectively.) (i) f () = cos 6 (ii) f () = log