Relations and Functions (Class 12)

Let a relation R on N N be defined as: (x1, y1) R (x2, y2) if and only if x1 x2 or y1 y2. Consider the two statements: Then which one of the following is true?

1. Only (II) is correct.
2. Both (I) and (II) are correct.
3. Neither (I) nor (II) is correct.
4. Only (I) is correct.

Consider the function f : R R defined by f(x) = 2x 1 + 9x2. If the composition of f, (f f f f)10 times(x) = 210x 1 + 9 x2, then the value of 3 + 1 is equal to .

Let f:R and g:R be defined as f(x)= cases e x, & x>0\ e-x, & x 0 cases and g(x)= casesx, & x 0\ ex, & x<0 cases. Then, gof :R is :

1. one-one but not onto
2. neither one-one nor onto
3. onto but not one-one
4. both one-one and onto

Let A=\1, 2, 3, , 20\. Let R1 and R2 be two relations on A such that R1=\(a, b): b is divisible by a\ R2=\(a, b): a is an integral multiple of b\. Then, number of elements in R1-R2 is equal to .

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