Relations and Functions (Class 12)

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.

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