Relations and Functions (Class 12)

Consider the relations R1 and R2 defined as a R1 b a2+b2=1 for all a, b R and (a, b) R2(c, d) a+d=b+c for all (a, b),(c, d) N N. Then :

1. R1 and R2 both are equivalence relations
2. Only R1 is an equivalence relation
3. Only R2 is an equivalence relation
4. Neither R1 nor R2 is an equivalence relation

If f(x)= 4x+36x-4, x 23 and (f f)(x)=g(x), where g: R- \ 23 \ R- \ 23 \, then (g g g)(4) is equal to

1. -4
2. 1920
3. - 1920
4. 4

Let A=\1, 2, 3, 4\ and R=\(1, 2), (2, 3), (1, 4)\ be a relation on A. Let S be the equivalence relation on A such that R S and the number of elements in S is n. Then, the minimum value of n is .

Let A=\1, 2, 3, , 100\. Let R be a relation on A defined by (x, y) R if and only if 2x=3y. Let R1 be a symmetric relation on A such that R R1 and the number of elements in R1 is n. Then, the minimum value of n is .

Let A=\1,2,3, , 7\ and let P(A) denote the power set of A. If the number of functions f: A P(A) such that a f(a), a A is mn, m and n N and m is least, then m+n is equal to .

Let f: R-\0\ R be a function satisfying f ( xy )= f(x)f(y) for all x,y,f(y) 0. If f'(1)=2024, then

1. xf'(x)+2024f(x)=0
2. xf'(x)-2023f(x)=0
3. xf'(x)-2024f(x)=0
4. xf'(x)+f(x)=2024

The number of symmetric relations defined on the set \1,2,3,4\ which are not reflexive is .

Let R be a relation on Z Z defined by (a, b)R(c, d) if and only if ad-bc is divisible by 5. Then R is

1. Reflexive and transitive but not symmetric
2. Reflexive and symmetric but not transitive
3. Reflexive but neither symmetric nor transitive
4. Reflexive, symmetric and transitive

If f(x)= cases2+2x, & -1 x<0\ 1- x3, & 0 x 3 cases; g(x)= cases-x, & -3 x 0\ x, & 0<x 1 cases, then range of (f g)(x) is

1. [0,1)
2. [0,3)
3. (0,1]
4. [0,1]

If R is the smallest equivalence relation on the set \1,2,3,4\ such that \(1,2),(1,3)\ , then the number of elements in R is .

1. 15
2. 10
3. 12
4. 8