Relations and Functions (Class 12)

Consider the relation R on the set \-2, -1, 0, 1, 2\ defined by (a, b) R if and only if 1+ab>0. Then, among the statements :

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 f be a polynomial function such that 2(f(x))= ( 2 (2+ 23+ 29+ ) ) 3 (1+ f(x)f ( 1x ) ), x>0 and f(6)=37. Then n=110f(n) is equal to .

Let f: R R be defined as f(x)= 2x2-3x+23x2+x+3. Then f is :

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

Let R= \(x, y) N N : e(x+y) 2 \. Then the minimum number of elements, required to be added in R to make it a transitive relation, is .

Let A=\1, 2, 3, 4, 5, 6\. The number of one-one functions f : A A such that f(1) 3, f(3) 4 and f(2)+f(3)=5, is .

The number of functions f : \1, 2, 3, 4\ \a, b, c\, which are not onto, is :

1. 48
2. 45
3. 51
4. 35

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

Let f and g be functions satisfying f(x + y) = f(x)f(y), f(1) = 7 and g(x + y) = g(xy), g(1) = 1, for all x, y N. If x = 1n ( f(x)g(x) ) = 19607, then n is equal to :

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

The number of strictly increasing functions f from the set \1, 2, 3, 4, 5, 6\ to the set \1, 2, 3, , 9\ such that f(i) i for 1 i 6, is equal to :

1. 21
2. 28
3. 27
4. 22

The number of relations, defined on the set \a, b, c, d\, which are both reflexive and symmetric, is equal to:

1. 16
2. 64
3. 256
4. 1024

Let A = \2, 3, 5, 7, 9\. Let R be the relation on A defined by xRy if and only if 2x 3y. Let l be the number of elements in R, and m be the minimum number of elements required to be added in R to make it a symmetric relation. Then l + m is equal to:

1. 21
2. 25
3. 23
4. 27

Let A = \0, 1, 2, 3, 4, 5\. Let R be a relation on A defined by (x, y) R if and only if \x, y\ \3, 4\. Then among the statements (S1): The number of elements in R is 18, and (S2): The relation R is symmetric but neither reflexive nor transitive

1. both are false
2. only (S1) is true
3. only (S2) is true
4. both are true

The number of relations on the set A = \1, 2, 3\, containing at most 6 elements including (1, 2), which are reflexive and transitive but not symmetric, is .

Let f : R R be a continuous function satisfying f(0) = 1 and f(2x) - f(x) = x for all x R. If n \f(x) - f ( x2n ) \ = G(x), then r=110 G (r2 ) is equal to

1. 215
2. 420
3. 385
4. 540

Consider the sets A = \(x, y) R R : x2 + y2 = 25 \, B = \(x, y) R R : x2 + 9y2 = 144 \, C = \(x, y) Z Z : x2 + y2 4 \ and D = A B. The total number of one-one functions from the set D to the set C is:

1. 15120
2. 18290
3. 17160
4. 19320

Let f, g : (1, ) R be defined as f(x) = 2x + 35x + 2 and g(x) = 2 - 3x1 - x. If the range of the function f g : [2, 4] R is [ , ], then 1 - is equal to

1. 56
2. 2
3. 29
4. 68

Let A = \-3, -2, -1, 0, 1, 2, 3\ and R be a relation on A defined by xRy if and only if 2x - y \0, 1\. 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. 17
2. 18
3. 15
4. 16

Let A = \-3, -2, -1, 0, 1, 2, 3\. Let R be a relation on A defined by x R y if and only if 0 x2 + 2y 4. Let l be the number of elements in R and m be the minimum number of elements required to be added in R to make it a reflexive relation. Then l + m is equal to

1. 18
2. 20
3. 17
4. 19

Let f be a function such that f(x) + 3f ( 24x ) = 4x, x 0. Then f(3) + f(8) is equal to

1. 13
2. 11
3. 10
4. 12

Let A = \-2, -1, 0, 1, 2, 3\. Let R be a relation on A defined by xRy if and only if y = \x, 1\. 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. 11
2. 12
3. 14
4. 13

Let f : R R be a twice differentiable function such that ( x y)(f(2x + 2y) - f(2x - 2y)) = ( x y)(f(2x + 2y) + f(2x - 2y)), for all x, y R. If f'(0) = 12, then the value of 24f'' ( 5 3 ) is :

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