Relations and Functions (Class 12)

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

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