Relations and Functions (Class 12)

Let A be the set of all functions f : Z Z and R be a relation on A such that R = \(f, g) : f(0) = g(1) and f(1) = g(0)\. Then R is :

1. Symmetric and transitive but not reflexive
2. Symmetric but neither reflexive nor transitive
3. Transitive but neither reflexive nor symmetric
4. Reflexive but neither symmetric nor transitive

Define a relation R on the interval [0, 2 ) by x R y if and only if 2 x - 2 y = 1. Then R is :

1. both reflexive and symmetric but not transitive
2. both reflexive and transitive but not symmetric
3. reflexive but neither symmetric nor transitive
4. an equivalence relation

If f(x) = 2x2x + 2, x R, then k=181 f ( k82 ) is equal to

1. 82
2. 81 2
3. 41
4. 812

The relation R = \(x, y) : x, y Z and x + y is even\ is:

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

Let f: R - \0\ (- , 1) be a polynomial of degree 2, satisfying f(x)f ( 1x ) = f(x) + f ( 1x ). If f(K) = -2K, then the sum of squares of all possible values of K is:

1. 9
2. 1
3. 6
4. 7

Let f : [0, 3] A be defined by f(x) = 2x3 - 15x2 + 36x + 7 and g : [0, ) B be defined by g(x) = x2025x2025 + 1. If both the functions are onto and S = \x Z; x A or x B\, then n(S) is equal to :

1. 29
2. 31
3. 30
4. 36

Let f:R-\0\ be a function such that f(x)-6f ( 1x )= 353x- 52. If x 0 ( 1 x+f(x) )= ; , , then +2 is equal to

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

Let f(x)= 2x+2+1622x+1+2x+4+32. Then the value of 8 (f ( 115 )+f ( 215 )+ +f ( 5915 ) ) is equal to

1. 108
2. 92
3. 118
4. 102

The function f:(- , ) (- ,1), defined by f(x)= 2x-2-x2x+2-x is :

1. One-one but not onto
2. Onto but not one-one
3. Both one-one and onto
4. Neither one-one nor onto

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 R=\(1,2),(2,3),(3,3)\ be a relation defined on the set \1,2,3,4\. Then the minimum number of elements, needed to be added in R so that R becomes an equivalence relation, is:

1. 9
2. 8
3. 7
4. 10

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

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