Relations and Functions (Class 12)

Let f be a polynomial function such that 2(f(x))= ( 2 (2+ 23+ 29+ ) ) 3 (1+ f(x)f ( 1x ) ), x>0 and f(6)=37. Then n=110f(n) is equal to .

Let R= \(x, y) N N : e(x+y) 2 \. Then the minimum number of elements, required to be added in R to make it a transitive relation, is .

Let A=\1, 2, 3, 4, 5, 6\. The number of one-one functions f : A A such that f(1) 3, f(3) 4 and f(2)+f(3)=5, is .

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 .

Number of functions f:\1,2, ,100\ \0,1\, that assign 1 to exactly one of the positive integers less than or equal to 98, is equal to .

Let A=\1,2,3\. The number of relations on A, containing (1,2) and (2,3), which are reflexive and transitive but not symmetric, is .

Let A = \2, 3, 6, 7\ and B = \4, 5, 6, 8\. Let R be a relation defined on A B by (a1, b1)R(a2, b2) if and only if a1 + a2 = b1 + b2. Then the number of elements in R is .

If a function f satisfies f(m + n) = f(m) + f(n) for all m, n N and f(1) = 1, then the largest natural number such that k = 12022 f( + k) (2022)2 is equal to .

Let A = \(x, y) : 2x + 3y = 23, x, y N\ and B = \x : (x, y) A\. Then the number of one-one functions from A to B is equal to .

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 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 .

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 .

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