Relations and Functions (Class 12)

Let f: R - \0\ (- , 1) be a polynomial of degree 2, satisfying f(x)f ( 1x ) = f(x) + f ( 1x ). If f(K) = -2K, then the sum of squares of all possible values of K is:

1. 9
2. 1
3. 6
4. 7

Let f : [0, 3] A be defined by f(x) = 2x3 - 15x2 + 36x + 7 and g : [0, ) B be defined by g(x) = x2025x2025 + 1. If both the functions are onto and S = \x Z; x A or x B\, then n(S) is equal to :

1. 29
2. 31
3. 30
4. 36

Let f:R-\0\ be a function such that f(x)-6f ( 1x )= 353x- 52. If x 0 ( 1 x+f(x) )= ; , , then +2 is equal to

1. 6
2. 5
3. 3
4. 4

Let f(x)= 2x+2+1622x+1+2x+4+32. Then the value of 8 (f ( 115 )+f ( 215 )+ +f ( 5915 ) ) is equal to

1. 108
2. 92
3. 118
4. 102

The function f:(- , ) (- ,1), defined by f(x)= 2x-2-x2x+2-x is :

1. One-one but not onto
2. Onto but not one-one
3. Both one-one and onto
4. Neither one-one nor onto

Let R=\(1,2),(2,3),(3,3)\ be a relation defined on the set \1,2,3,4\. Then the minimum number of elements, needed to be added in R so that R becomes an equivalence relation, is:

1. 9
2. 8
3. 7
4. 10

Let X=R . Define a relation R on X as : (a1,b1)R(a2,b2) b1=b2 In the light of the above statements, choose the correct answer from the options given below :

1. Both Statement I and Statement II are true
2. Statement I is true but Statement II is false
3. Both Statement I and Statement II are false
4. Statement I is false but Statement II is true

Let f(x) be a real differentiable function such that f(0)=1 and f(x+y)=f(x)f'(y)+f'(x)f(y) for all x,y . Then n=1100 ef(n) is equal to :

1. 2406
2. 5220
3. 2525
4. 2384

The number of non-empty equivalence relations on the set \1,2,3\ is :

1. 7
2. 4
3. 5
4. 6

Let A=\1,2,3,4\ and B=\1,4,9,16\. Then the number of many-one functions f:A B such that 1 f(A) is equal to :

1. 151
2. 139
3. 163
4. 127

Let [t] be the greatest integer less than or equal to t. Let A be the set of all prime factors of 2310 and f : A Z be the function f(x) = [ 2 (x2 + [ x35 ] ) ]. The number of one-to-one functions from A to the range of f is

1. 20
2. 120
3. 25
4. 24

Let f(x) = cases -a & if -a x 0 \ x + a & if 0 < x a cases where a > 0 and g(x) = (f(|x|) - |f(x)|)/2. Then the function g : [-a, a] [-a, a] is

1. neither one-one nor onto.
2. both one-one and onto.
3. one-one.
4. onto

Let A = \2, 3, 6, 8, 9, 11\ and B = \1, 4, 5, 10, 15\. Let R be a relation on A B defined by (a, b)R(c, d) if and only if 3ad - 7bc is an even integer. Then the relation R is

1. reflexive but not symmetric.
2. an equivalence relation.
3. reflexive and symmetric but not transitive.
4. transitive but not symmetric.

Let the relations R1 and R2 on the set X = \1, 2, 3, , 20\ be given by R1 = \(x, y) : 2x - 3y = 2\ and R2 = \(x, y) : -5x + 4y = 0\. If M and N be the minimum number of elements required to be added in R1 and R2, respectively, in order to make the relations symmetric, then M + N equals

1. 16
2. 12
3. 8
4. 10

The function f(x) = x2 + 2x - 15x2 - 4x + 9, x R is

1. both one-one and onto.
2. onto but not one-one.
3. neither one-one nor onto.
4. one-one but not onto.

Let A = \1, 2, 3, 4, 5\. Let R be a relation on A defined by xRy if and only if 4x 5y. Let m be the number of elements in R and n be the minimum number of elements from A A that are required to be added to R to make it a symmetric relation. Then m + n is equal to :

1. 23
2. 26
3. 25
4. 24

Let A = \1, 3, 7, 9, 11\ and B = \2, 4, 5, 7, 8, 10, 12\. Then the total number of one-one maps f : A B, such that f(1) + f(3) = 14, is :

1. 120
2. 180
3. 240
4. 480

Let f, g : R R be defined as : f(x) = |x - 1| and g(x) = cases ex, & x 0 \ x + 1, & x 0. cases Then the function f(g(x)) is

1. neither one-one nor onto.
2. one-one but not onto.
3. both one-one and onto.
4. onto but not one-one.

Let a relation R on N N be defined as: (x1, y1) R (x2, y2) if and only if x1 x2 or y1 y2. Consider the two statements: Then which one of the following is true?

1. Only (II) is correct.
2. Both (I) and (II) are correct.
3. Neither (I) nor (II) is correct.
4. Only (I) is correct.

Let f:R and g:R be defined as f(x)= cases e x, & x>0\ e-x, & x 0 cases and g(x)= casesx, & x 0\ ex, & x<0 cases. Then, gof :R is :

1. one-one but not onto
2. neither one-one nor onto
3. onto but not one-one
4. both one-one and onto

Consider the relations R1 and R2 defined as a R1 b a2+b2=1 for all a, b R and (a, b) R2(c, d) a+d=b+c for all (a, b),(c, d) N N. Then :

1. R1 and R2 both are equivalence relations
2. Only R1 is an equivalence relation
3. Only R2 is an equivalence relation
4. Neither R1 nor R2 is an equivalence relation

If f(x)= 4x+36x-4, x 23 and (f f)(x)=g(x), where g: R- \ 23 \ R- \ 23 \, then (g g g)(4) is equal to

1. -4
2. 1920
3. - 1920
4. 4

Let f: R-\0\ R be a function satisfying f ( xy )= f(x)f(y) for all x,y,f(y) 0. If f'(1)=2024, then

1. xf'(x)+2024f(x)=0
2. xf'(x)-2023f(x)=0
3. xf'(x)-2024f(x)=0
4. xf'(x)+f(x)=2024

Let R be a relation on Z Z defined by (a, b)R(c, d) if and only if ad-bc is divisible by 5. Then R is

1. Reflexive and transitive but not symmetric
2. Reflexive and symmetric but not transitive
3. Reflexive but neither symmetric nor transitive
4. Reflexive, symmetric and transitive

If f(x)= cases2+2x, & -1 x<0\ 1- x3, & 0 x 3 cases; g(x)= cases-x, & -3 x 0\ x, & 0<x 1 cases, then range of (f g)(x) is

1. [0,1)
2. [0,3)
3. (0,1]
4. [0,1]

If R is the smallest equivalence relation on the set \1,2,3,4\ such that \(1,2),(1,3)\ , then the number of elements in R is .

1. 15
2. 10
3. 12
4. 8

Let S=\1,2,3, ,10\. Suppose M is the set of all the subsets of S, then the relation R=\(A, B):A B ; A, B M\ is :

1. symmetric only
2. reflexive only
3. symmetric and reflexive only
4. symmetric and transitive only

The function f:N-\1\ ; defined by f(n)= the highest prime factor of n, is :

1. one-one only
2. neither one-one nor onto
3. onto only
4. both one-one and onto

Let f:R- \ -12 \ and g:R- \ -52 \ be defined as f(x)= 2x+32x+1 and g(x)= |x|+12x+5. Then, the domain of the function fog is :

1. R- \- 74 \
2. R
3. R- \- 52,- 74 \
4. R- \- 52 \