Relations and Functions (Class 12)

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

Let A = \2, 3, 6, 8, 9, 11\ and B = \1, 4, 5, 10, 15\. Let R be a relation on A B defined by (a, b)R(c, d) if and only if 3ad - 7bc is an even integer. Then the relation R is

1. reflexive but not symmetric.
2. an equivalence relation.
3. reflexive and symmetric but not transitive.
4. transitive but not symmetric.

Let the relations R1 and R2 on the set X = \1, 2, 3, , 20\ be given by R1 = \(x, y) : 2x - 3y = 2\ and R2 = \(x, y) : -5x + 4y = 0\. If M and N be the minimum number of elements required to be added in R1 and R2, respectively, in order to make the relations symmetric, then M + N equals

1. 16
2. 12
3. 8
4. 10

The function f(x) = x2 + 2x - 15x2 - 4x + 9, x R is

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

Let A = \1, 2, 3, 4, 5\. Let R be a relation on A defined by xRy if and only if 4x 5y. Let m be the number of elements in R and n be the minimum number of elements from A A that are required to be added to R to make it a symmetric relation. Then m + n is equal to :

1. 23
2. 26
3. 25
4. 24

Let A = \1, 3, 7, 9, 11\ and B = \2, 4, 5, 7, 8, 10, 12\. Then the total number of one-one maps f : A B, such that f(1) + f(3) = 14, is :

1. 120
2. 180
3. 240
4. 480

Let f, g : R R be defined as : f(x) = |x - 1| and g(x) = cases ex, & x 0 \ x + 1, & x 0. cases Then the function f(g(x)) is

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