# Relations and Functions VBQs Class 12 Mathematics

VBQs Relations and Functions Class 12 Mathematics with solutions has been provided below for standard students. We have provided chapter wise VBQ for Class 12 Mathematics with solutions. The following Relations and Functions Class 12 Mathematics value based questions with answers will come in your exams. Students should understand the concepts and learn the solved cased based VBQs provided below. This will help you to get better marks in class 12 examinations.

## Relations and Functions VBQs Class 12 Mathematics

Question. For x ∈ R – {0, 1}, let f1(x) = 1/x , f2(x) = 1 – x and f3(x) = 1/(1 – x) be three given functions. If a function, J(x) satisfies (f2oJof1) (x) = f3(x) then J(x) is equal to:
(a) f3(x)
(b) 1/x (f3(x))
(c) f2(x)
(d) f1(x)

A

Question. Let N denote the set of all natural numbers. Define two binary relations on N as R1 = {(x, y) ∈ N × N : 2x + y = 10} and R2 = {(x, y) ∈ N × N : x + 2y = 10}. Then
(a) Both R1 and R2 are transitive relations
(b) Both R1 and R2 are symmetric relations
(c) Range of R2 is {1, 2, 3, 4}
(d) Range of R1 is {2, 4, 8}

C

Question. Let A = {x∈R : x is not a positive integer}. Define a function ƒ: A → R as ƒ(x) = 2x/(x – 1), then f is:
(a) not injective
(b) neither injective nor surjective
(c) surjective but not injective
(d) injective but not surjective

D

Question. Let A = {x1, x2, ………, x7} and B = {y1, y2, y3} be two sets containing seven and three distinct elements respectively. Then the total number of functions f : A → B that are onto, if there exist exactly three elements x in A such that f(x) = y2, is equal to :
(a) 14.7C3
(b) 16.7C3
(c) 14.7C2
(d) 12.7C2

A

Question. Let R = {(x, y) : x, y ∈ N and x2 – 4xy + 3y2 = 0}, where N is the set of all natural numbers. Then the relation R is :
(a) reflexive but neither symmetric nor transitive.
(b) symmetric and transitive.
(c) reflexive and symmetric,
(d) reflexive and transitive.

D

Question. Let A = {a, b, c} and B = {1, 2, 3, 4}. Then the number of elements in the set C = { ƒ : A → B | 2 ∈ ƒ (A) and ƒ is not one-one} is ___________.

19

Question. Let a funct ion ƒ : (0,∞) → (0,∞) be defined by ƒ(x) = | 1 – 1/x |. Then ƒ is:
(a) not injective but it is surjective
(b) injective only
(c) neither injective nor surjective
(d) both injective as well as surjective

None

Question. If P(S) denotes the set of all subsets of a given set S, then the number of one-to-one functions from the set S = {1, 2, 3} to the set P(S) is
(a) 24
(b) 8
(c) 336
(d) 320

C

Question. Let f : (– 1, 1) → B, be a function defined by f(x) = tan-1(2x/1 – x2), then f is both one – one and onto when B is the interval
(a) (0, π/2)
(b) (0, – π/2)
(c) ( π/2, π/2)
(d) (- π/2, π/2)

D

Question. Let R = {(1,3), (4, 2), (2, 4), (2,3), (3,1)} be a relation on the set A = {1, 2,3, 4}. . The relation R is|
(a) reflexive
(b) transitive
(c) not symmetric
(d) a function

C

Question. If A = {x ∈ z+ : x < 10 and x is a multiple of 3 or 4}, where z+ is the set of positive integers, then the total number of symmetric relations on A is
(a) 25
(b) 215
(c) 210
(d) 220

B

Question. Let A and B be non empty sets in R and f : A → B is a bi ective function.
Statement 1: f is an onto function.
Statement 2: There exists a function g : B → A such that fog = IB.
(a) Statement 1 is true, Statement 2 is false.
(b) Statement 1 is true, Statement 2 is true; Statement 2 is a correct explanation for Statement 1.
(c) Statement 1 is false, Statement 2 is true.
(d) Statement 1 is true, Statement 2 is true, Statement 2 is not the correct explanation for Statement 1.

D

Question. Let f be a function defined by f(x) = (x – 1)2 +1, (x ≥ 1).
Statement – 1 : The set {x : f(x) = f-1(x)} = {1,/2).
Statement – 2 : f is a bijection and f-1(x)} = 1 + √(x – 1), x ≥ 1.
(a) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
(b) Statement-1 is true, Statement-2 is true; Statement-2 is NOT a correct explanation for Statement-1.
(c) Statement-1 is true, Statement-2 is false.
(d) Statement-1 is false, Statement-2 is true.

A

Question. Let R be the set of real numbers.
Statement-1: A = {(x, y) ∈ R × R : y – x is an integer} is an equivalence relation on R.
Statement-2: B = {(x, y) ÎR × R : x = ay for some rational number a} is an equivalence relation on R.
(a) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
(b) Statement-1 is true, Statement-2 is false.
(c) Statement-1 is false, Statement-2 is true.
(d) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.

A

Question. Consider the following relations:
R = {(x, y) | x, y are real numbers and x = wy for some rational number w}; S = {(m/n, p/q) | m,n, p and q are integers such that n, q ≠ 0 and qm = pn}. Then
(a) Neither R nor S is an equivalence relation
(b) S is an equivalence relation but R is not an equivalence relation
(c) R and S both are equivalence relations
(d) R is an equivalence relation but S is not an equivalence relation

B

Question. Let R = {(3, 3) (5, 5), (9, 9), (12, 12), (5, 12), (3, 9), (3, 12), (3, 5)} be a relation on the set A = {3, 5, 9, 12}. Then, R is :
(a) reflexive, symmetric but not transitive.
(b) symmetric, transitive but not reflexive.
(c) an equivalence relation.
(d) reflexive, transitive but not symmetric.

D

Question. Let A = {1, 2, 3, 4} and R : A→ A be the relation defined by R = {(l, 1), (2, 3), (3, 4), (4, 2)}. The correct statement is :
(a) R does not have an inverse.
(b) R is not a one to one function.
(c) R is an onto function.
(d) R is not a function.

C

Question. Let R = {(3, 3), (6, 6), (9, 9), (12, 12), (6, 12), (3, 9), (3, 12), (3, 6)} be a relation on the set A = {3, 6, 9, 12}. The relation is
(a) reflexive and transitive only
(b) reflexive only
(c) an equivalence relation
(d) reflexive and symmetric only

A

Question. Let f (x) = 210·x + 1 and g(x) = 310·x – 1. If (fog)(x)=x, then x is equal to :

D

Question. For x ∈ R, x ≠ 0 , let f0(x) = 1/(1- x) and fn+1(x) = f0(fn(x)), n = 0, 1, 2, …. Then the value of f100(3) + f1(2/3) + f2(3/2) is equal to :
(a) 8/3
(b) 4/3
(c) 5/3
(d) 1/3

C

Question. If f : R → S, defined by f (x) = sin x – √3 cos x +1, is onto, then the interval of S is
(a) [ 1, 3]
(b) [–1, 1]
(c) [ 0, 1]
(d) [0, 3]

A

Question. Let f(x) = (x +1)2 –1, x ≥ –1
Statement -1 : The set {x : f(x) = f–1(x) = {0, –1}
Statement-2 : f is a bijection.
(a) Statement-1 is true, Statement-2 is true. Statement-2 is not a correct explanation for Statement-1.
(b) Statement-1 is true, Statement-2 is false.
(c) Statement-1 is false, Statement-2 is true.
(d) Statement-1 is true, Statement-2 is true. Statement-2 is a correct explanation for Statement-1.

D

Question. Let f: N → Y be a function defined as f(x) = 4x + 3 where Y = {y ∈ N : y = 4x + 3 for some x ∈ N}. Show that f is invertible and its inverse is
(a) g(y) = (3y + 4)/3
(b) g(y) = 4 + (y + 3)/4
(c) g(y) = (y + 3)/4
(d) g(y) = (y – 3)/4

D

Question. A function f from the set of natural numbers to integers defined by

(a) neither one -one nor onto
(b) one-one but not onto
(c) onto but not one-one
(d) one-one and onto both.

D

Question. Let f : R → R be defined by f(x) = (|x|-1)/(|x|+1) then f is:
(a) both one-one and onto
(b) one-one but not onto
(c) onto but not one-one
(d) neither one-one nor onto.

C

Question. Let P be the relation defined on the set of all real numbers such that P = {(a, b) : sec2a – tan2b = 1}. Then P is:
(a) reflexive and symmetric but not transitive.
(b) reflexive and transitive but not symmetric.
(c) symmetric and transitive but not reflexive.
(d) an equivalence relation.

D

Question. The inverse function of

is _________________ .

A

Question. If g(x) = x2 + x – 1 and (goƒ) (x) = 4x2 – 10x + 5, then ƒ(5/4) is equal to:
(a) 3/2
(b) – 1/2
(c) 1/2
(d) – 3/2

B

Question. Let 2 f(x) = x2 , x ∈ R . For any A ⊆ R , define g(A) = {x ∈ R : f (x) ∈ A} . If S = [0, 4], then which one of the following statements is not true ?
(a) g(f (S)) ≠ S
(b) f(g (S)) = S
(c) g(f (S)) = g (S)
(d) f(g (S)) ≠ f (S)

C

Question. Consider the following two binary relations on the set A = {a, b, c} : R1 = {(c, a) (b, b) , (a, c), (c, c), (b, c), (a, a)} and R2 = {(a, b), (b, a), (c, c), (c, a), (a, a), (b, b), (a, c). Then
(a) R2 is symmetric but it is not transitive
(b) Both R1 and R2 are transitive
(c) Both R1 and R2 are not symmetric
(d) R1 is not symmetric but it is transitive

A

Question. The number of functions ƒ from {1, 2, 3, …, 20} onto {1, 2, 3, …., 20} such that ƒ(k) is a multiple of 3, whenever k is a multiple of 4 is :
(a) 65 × (15)!
(b) 5! × 6!
(c) (15)! × 6!
(d) 56 × 15

C

Question. Let N be the set of natural numbers and two functions ƒ and g be defined as ƒ, g : N → N such that

and g(n) = n – (– 1)n. Then fog is:
(a) onto but not one-one.
(b) one-one but not onto.
(c) both one-one and onto.
(d) neither one-one nor onto.

A

Question. Let f : A → B be a function defined as f(x) = (x – 1)/(x – 2), where A = R – {2} and B = R – {1}. Then f is
(a) invertible and f–1(y) = (2y+1)/(y – 1)
(b) invertible and f–1(y) = (3y – 1)/(y – 1)
(c) no invertible
(d) invertible and f–1(y) = (2y – 1)/(y – 1)

D

Question. If g is the inverse of a function f and f'(x) = 1/(1+x5), then g'(x) is equal to:
(a) 1/(1+{g(x)}5
(b) (1+{g(x)}5
(c) 1 + x5
(d) 5x4

B

Question. For a suitably chosen real constant a, let a function, f : R –{– a} → R be defined by f(x) = (a – x)/(a + x) Further suppose that for any real number x ≠ – a and f (x) ≠ – a, ( fof ) (x) = x. Then f(– 1/2) is equal to:
(a) 1/3
(b) – 1/3
(c) – 3
(d) 3

D

Question. For x ∈ (0, 3/2), let f(x) = √x , g(x) = tan x and h(x) = (1 -x2)/(1+ x2). If Φ (x) = ((hof)og) (x), then Φ(π/3) is equal to:
(a) tan π/12
(b) tan 11π/12
(c) tan 7π/12
(d) tan 5π/12

B

Question. The function f : R → [- 1/2, 1/2] defined as f(x) = x/(1+x2), is :
(a) neither injective nor surjective
(b) invertible
(c) injective but not surjective
(d) surjective but not injective

D

Question. The function f : N → N defined by f(x) = x – 5(x/58), where N is set of natural numbers and [x] denotes the greatest integer less than or equal to x, is :
(a) one-one and onto.
(b) one-one but not onto.
(c) onto but not one-one.
(d) neither one-one nor onto.

D

Question. Let R be the real line. Consider the following subsets of the plane R × R:
S ={(x, y): y = x + 1 and 0 < x < 2}
T ={(x, y): x – y is an integer},
Which one of the following is true?
(a) Neither S nor T is an equivalence relation on R
(b) Both S and T are equivalence relation on R
(c) S is an equivalence relation on R but T is not
(d) T is an equivalence relation on R but S is not