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: 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

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

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

Let A be the set of all functions f : Z Z and R be a relation on A such that R = \(f, g) : f(0) = g(1) and f(1) = g(0)\. Then R is :

1. Symmetric and transitive but not reflexive
2. Symmetric but neither reflexive nor transitive
3. Transitive but neither reflexive nor symmetric
4. Reflexive but neither symmetric nor transitive

Define a relation R on the interval [0, 2 ) by x R y if and only if 2 x - 2 y = 1. Then R is :

1. both reflexive and symmetric but not transitive
2. both reflexive and transitive but not symmetric
3. reflexive but neither symmetric nor transitive
4. an equivalence relation

If f(x) = 2x2x + 2, x R, then k=181 f ( k82 ) is equal to

1. 82
2. 81 2
3. 41
4. 812

The relation R = \(x, y) : x, y Z and x + y is even\ is:

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