Relations and Functions (Class 12) · Question 6

For the function f : [1, ) [1, ) defined by f(x)=(x-1)4+1, among the two statements : (I) The set S=\x [1, ) : f(x)=f-1(x)\ contains exactly two elements, and (II) The set S=\x [1, ) : f(x)=f-1(x+1)\ is an empty set,

1. only (I) is TRUE
2. only (II) is TRUE
3. both (I) and (II) are TRUE
4. neither (I) nor (II) is TRUE

Let for some R, f : R R be a function satisfying f(x+y)=f(x)+2y2+y+ xy for all x, y R. If f(0)=-1 and f(1)=2, then the value of n=15 ( +f(n) ) is :

1. 110
2. 140
3. 150
4. 170

If g(x) = 3x2 + 2x - 3, f(0) = -3 and 4g(f(x)) = 3x2 - 32x + 72, then f(g(2)) is equal to:

1. 72
2. - 256
3. 256
4. - 72

Given below are two statements : In the light of the above statements, choose the correct answer from the options given below :

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

Let R be a relation defined on the set \1, 2, 3, 4\ \1, 2, 3, 4\ by R = \((a, b), (c, d)) : 2a + 3b = 3c + 4d\. Then the number of elements in R is

1. 6
2. 15
3. 12
4. 18

Let f be a function such that 3f(x) + 2f ( m19x ) = 5x, x 0, where m = i=19 (i)2. Then f(5) - f(2) is equal to

1. 36
2. 9
3. -9
4. 18

Let A = \-2, -1, 0, 1, 2, 3, 4\. Let R be a relation on A defined by xRy if and only if 2x + y 2. Let l be the number of elements in R. Let m and n be the minimum number of elements required to be added in R to make it reflexive and symmetric relations respectively. Then l + m + n is equal to :

1. 34
2. 32
3. 33
4. 35

Consider two sets A = \x Z : |(|x - 3| - 3)| 1\ and B = \x R - \1, 2\ : (x - 2)(x - 4)x - 1 e(|x - 2|) = 0 \. Then the number of onto functions f : A B is equal to :

1. 32
2. 81
3. 79
4. 62

Let A = \0, 1, 2, , 9\. Let R be a relation on A defined by (x, y) R if and only if |x - y| is a multiple of 3. Given below are two statements : In the light of the above statements, choose the correct answer from the options given below :

1. Statement I is correct but Statement II is incorrect
2. Both Statement I and Statement II are correct
3. Both Statement I and Statement II are incorrect
4. Statement I is incorrect but Statement II is correct

Let the relation R on the set M = \1, 2, 3, , 16\ be given by R = \(x, y) : 4y = 5x - 3, x, y M\. Then the minimum number of elements required to be added in R, in order to make the relation symmetric, is equal to

1. 4
2. 3
3. 1
4. 2