Relations and Functions (Class 12)

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