Relations and Functions (Class 12)

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