Relations and Functions (Class 12)

Let X=R . Define a relation R on X as : (a1,b1)R(a2,b2) b1=b2 In the light of the above statements, choose the correct answer from the options given below :

1. Both Statement I and Statement II are true
2. Statement I is true but Statement II is false
3. Both Statement I and Statement II are false
4. Statement I is false but Statement II is true

Let f(x) be a real differentiable function such that f(0)=1 and f(x+y)=f(x)f'(y)+f'(x)f(y) for all x,y . Then n=1100 ef(n) is equal to :

1. 2406
2. 5220
3. 2525
4. 2384

The number of non-empty equivalence relations on the set \1,2,3\ is :

1. 7
2. 4
3. 5
4. 6

Let A=\1,2,3,4\ and B=\1,4,9,16\. Then the number of many-one functions f:A B such that 1 f(A) is equal to :

1. 151
2. 139
3. 163
4. 127

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 .

Let [t] be the greatest integer less than or equal to t. Let A be the set of all prime factors of 2310 and f : A Z be the function f(x) = [ 2 (x2 + [ x35 ] ) ]. The number of one-to-one functions from A to the range of f is

1. 20
2. 120
3. 25
4. 24

Let f(x) = cases -a & if -a x 0 \ x + a & if 0 < x a cases where a > 0 and g(x) = (f(|x|) - |f(x)|)/2. Then the function g : [-a, a] [-a, a] is

1. neither one-one nor onto.
2. both one-one and onto.
3. one-one.
4. onto