Relations and Functions (Class 12)

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