Tiêu đề 15.RELATIONS AND FUNCTIONS -II.pdf ✅

15. RELATIONS AND FUNCTIONS II-MCQS TYPE (1.) Let A 1,3,4,6,9 =   and B 2,4,5,8,10 =   . Let R be a relation defined on A B such that R a , b , a , b = (( 1 1 2 2 ) ( )): 1 2 a b  and b a 1 2   . Then the number of elements in the set R is [11 APRIL 2023 (AFTERNOON)] (a.) 52 (b.) 160 (c.) 26 (d.) 180 (2.) Let A 1, 2,3, 4,5,6,7 =   . Then the relation R x, y A A : x y 7 =   + = ( )  is [08 APRIL 2023 (EVENING)] (a.) Symmetric but neither reflexive nor transitive (b.) Transitive but neither symmetric nor reflexive (c.) An equivalence relation (d.) Reflexive but neither symmetric nor transitive (3.) Let the number of elements in sets A and B be five and two respectively. Then the number of subsets of A B each having at least 3 and at most 6 elements is: [08 APRIL 2023 (MORNING)] (a.) 752 (b.) 772 (c.) 782 (d.) 792 (4.) Let f : 2,6 R R − →   be real valued function defined as ( ) 2 2 2 1 8 12 x x f x x x + + = − + . Then range of f is [31 Jan 2023(Evening)] (a.)  ) 21 , 0, 4     − −    (b.) ( ) 21 , 0, 4       − −    (c.) 21 21 , , 4 4         − −       (d.)  ) 21 , 1, 4     − −    (5.) If the domain of the function ( )   2 x f x 1 x = + , where  x is greatest integer  x , is 2,6) , then its range is [31 Jan 2023(Morning)] (a.) 5 2 9 27 18 9 , , , , 26 5 29 109 89 53       −       (b.) 5 2 , 26 5       (c.) 5 2 9 27 18 9 , , , , 37 5 29 109 89 53       −       (d.) 5 2 , 37 5       (6.) The domain of ( ) ( ) ( ) ( ) 1 2log log 2 , 2 3 e x x x f x x R e x + − =  − + is [29 Jan 2023(Morning)] (a.) R − − 1 3 (b.) (2, 3  ) −  (c.) (− − 1, 3  )   (d.) R −3
(7.) Let f R R : → be a function such that ( ) 2 2 2 1 1 x x f x x + + = + . Then [29 Jan 2023(Morning)] (a.) f x( ) is many-one in (− − , 1) (b.) f x( ) is many-one in (1, ) (c.) f x( ) is one-one in 1, ) but not in (− , ) (d.) f x( ) is one-one in (− , ) (8.) Let ( ) n f x 2x , , n = +     R N , and f 4 133 ( ) = , f (5 255 ) = . Then the sum of all the positive integer divisors of (f 3 f 2 ( ) − ( )) is [25 Jan 2023(Evening)] (a.) 61 (b.) 60 (c.) 58 (d.) 59 (9.) The number of functions f : 1,2,3,4 a : a 8   →    Z  satisfying f (n)+ ( )   1 f n 1 1, n 1,2,3 n + =   is [25 Jan 2023(Evening)] (a.) 3 (b.) 4 (c.) 1 (d.) 2 (10.) Let f :R R → be a function defined by f x( ) = log 2 sin cos 2 m  ( x x m − + − )  , for some m , such that the range of f is 0, 2 . Then the value of m is [25 Jan 2023(Evening)] (a.) 5 (b.) 3 (c.) 2 (d.) 4 (11.) Let f x( ) be a function such that f x y f x f y ( + =  ) ( ) ( ) for all x y N ,  . If f (1 3 ) = and 1 ( ) 3279 n k f k  = = , then the value of n is [24 Jan 2023(Morning)] (a.) 6 (b.) 8 (c.) 7 (d.) 9 (12.) If ( ) 2 2 2 , 2 2 x x f x x R =  + , then 1 2 2022 f f . f 2023 2023 2023             + + +       is equal to [24 Jan 2023(Morning)] (a.) 2011 (b.) 1010 (c.) 2010 (d.) 1011 (13.) Let f : R 0,1 R − →   be a function such that ( ) 1 f x f 1 x 1 x   + = +     − . Then f 2( ) is equal to :[01 Feb 2023(Evening)] (a.) 9 2 (b.) 9 4 (c.) 7 4 (d.) 7 3
(14.) Suppose f : R 0, → (  ) be a differentiable function such that 5 , , f x y f x f y x y R ( + =    ) ( ) ( ) . If f 3 320 ( ) = , then ( ) 5 n 0f n  = is equal to :[30 Jan 2023(Morning)] (a.) 6875 (b.) 6575 (c.) 6825 (d.) 6528 (15.) Consider a function f :N R → , satisfying f f f xf x x x f x x (1 2 2 3 3 1 ; 2 ) + + ++ = +  ( ) ( ) ( ) ( ) ( ) with f 1 1 ( ) = . Then ( ) ( ) 1 1 f 2022 f 2028 + is equal to [29 Jan 2023(Evening)] (a.) 8200 (b.) 8000 (c.) 8400 (d.) 8100 (16.) Let ( ) 2 f x x x = + −  − ( 1) 1, 1. Statement-1: The set  ( ) ( )   1 x f x f x : 0, 1 − = = − . Statement-2: f is a bijection. [AIEEE-2009] (a.) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation forStatement-1 (b.) Statement-1 is true, Statement-2 is false (c.) Statement-1 is false, Statement-2 is true (d.) Statement-1 is true, Statement- 2 is true; Statement-2 is a correct explanation for Statement-1 (17.) For real x , let ( ) 3 f x x x = + + 5 1 , then [AIEEE-2009] (a.) f is onto R but not one-one (b.) f is one-one and onto R (c.) f is neither one-one nor onto R (d.) f is one-one but not onto R (18.) Let y be an implict function of x defined by 2 2 cot 1 0 x x x x y − − = . Then y (1) equals [AIEEE-2009] (a.) 1 (b.) log2 (c.) −log2 (d.) -1 (19.) Let f be a function defined by ( ) ( ) 2 f x x x = − +  ( 1) 1, 1 . Statement - 1: The set  ( ) ( )   1 x f x f x : 1,2 − = = . Statement - 2 : f is a bijection and ( ) 1 f x x x 1 1, 1 − = + −  . [AIEEE-2011] (a.) Statement-1 is true, Statement-2 is false (b.) Statement-1 is false, Statement-2 is true (c.) Statement-1 is true, Statement-2 is true; Statement-2 is the correct explanation for Statement-1 (d.) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1
(20.) The equation sin sin 4 0 x x e e − − − = has [AIEEE-2012] (a.) No real roots (b.) Exactly one real root (c.) Exactly four real roots (d.) Infinite number of realroots (21.) If a R  and the equation   (  ) 2 2 − − + − + = 3( ) 2 0 x x x x a (where  x denotes the greatest integer  x ) has no integral solution, then all possible values of a lie in the interval [JEE (Main)-2014] (a.) (− − 2, 1) (b.) (− −    , 2 2, ) ( ) (c.) (−  1,0 0,1 ) ( ) (d.) (1, 2) (22.) If g is the inverse of a function f and ( ) 5 1 1 f x x = +  , then g x ( ) is equal to [JEE (Main)-2014] (a.) ( ) 5 1 1 { } + g x (b.) ( ) 5 1 { } + g x (c.) 5 1+ x (d.) 4 5x (23.) If ( ) 1 f x f x x 2 3 , 0 x   + =      , and S x R f x f x =  = −  : ; ( ) ( ) then S [JEE (Main)-2016] (a.) Contains exactly one element (b.) Contains exactly two elements (c.) Contains more thantwo elements (d.) Is an empty set (24.) The function 1 1 : , 2 2 f R   → −    defined as ( ) 2 1 x f x x = + , is [JEE (Main)-2017] (a.) Injective but not surjective (b.) Surjective but not injective (c.) Neither injective nor surjective (d.) Invertible (25.) For x R  −0,1 , let 1 2 ( ) ( ) 1 f x f x x , 1 x = = − and 3 ( ) 1 1 f x x = − be three given functions. If a function, J x( ) satisfies ( f f x f x 2 1 3  )( ) = ( ) then J x( ) is equal to [JEE (Main)-2019] (a.) ( ) 1 f x (b.) 3 ( ) 1 f x x (c.) ( ) 2 f x (d.) ( ) 3 f x (26.) Let A x R x =  { : is not a positive integer } . Define a function f A R : → as ( ) 2 1 x f x x = − , then f is [JEE (Main)-2019] (a.) Injective but not surjective (b.) Neither injective nor surjective (c.) Surjective but not injective (d.) Not injective