CBSE Class 12 Mathematics Relations and Functions MCQs Set 03

Question. The function f : R → R defined by f (x) = (x – 1) (x – 2) (x – 3) is
(a) one-one but not onto
(b) onto but not one-one
(c) both one-one and onto
(d) neither one-one nor onto

Answer : B

Question. Let f : (2, 3) → (0, 1) be defined by f(x) = x – [x]. Then, f–1(x) equals to
(a) x – 2
(b) x + 1
(c) x – 1
(d) x + 2

Answer : D

Question. Let f : N → R be the function defined by f(x) = 2x-1/2 and g : Q → R be another function defined by g(x) = x + 2. Then (gof) 3/2 is
(a) 1
(b) 0
(c) 7/2
(d) 3

Answer : D

Question. Consider the following statements on a set A = {1, 2, 3} I. R = {(1, 1), (2, 2)} is reflexive relation on A
II. R = {(3, 3)} is symmetric and transitive but not a reflexive relation on A
Which of the statements given above is/are correct ?
(a) Only I
(b) Only II
(c) Both I and II
(d) Neither I nor II

Answer : A

Question. Consider the following statements
Statement – I : An onto function f : {1, 2, 3} → {1, 2, 3} is always one-one.
Statement – II : A one-one function f :{1, 2, 3} → {1, 2, 3} must be onto.
(a) Only I is true
(b) Only II is true
(c) Both I and II are true
(d) Neither I nor II is true

Answer : C

Question. Let f : N → R be the function defined by f (x) = 2x – 1/2 and g : Q → R be another function defined by g (x) = x + 2. Then (gof) 3/2 is
(a) 1
(b) – 1
(c) 7/2
(d) 3

Answer : D

Question. The function f : R → R defined by f(x) = x² + x is.
(a) one-one
(b) onto
(c) many-one
(d) None of the above

Answer : C

Question. Range of the function f(x) = x²+x+2/x²+x+1
(a) (1, ∞ )
(b) (1,11/7]
(c) (1, 7/3]
(d) (1, 7/5]

Answer : C

Question. The binary operation * defined on N by a * b = a + b + ab for all a, b ∈ N is
(a) commutative only
(b) associative only
(c) both commutative and associative
(d) None of these

Answer : C

Question. In the set N of natural numbers, define the binary operation * by m * n = GCD (m, n), m, n ∈ N. Then, which of the following is true?
I. * is not a binary operation
II. * is a binary operation
III. Inverse of each element of N exist
IV. Inverse of each element of N does not exist
(a) I and IV are true
(b) II and III are true
(c) Only I is true
(d) II and IV are true

Answer : D

Question. The maximum number of equivalence relations on the set A = {2, 3, 4} are
(a) 1
(b) 27
(c) 3
(d) 5

Answer : D

Question. If a relation R on the set {1, 2, 3} be defined by R = {(1, 2)}, then R is
(a) reflexive
(b) transitive
(c) symmetric
(d) none of these

Answer : B

Question. If f : A → B and g : B → C be the bijective functions, then (gof)^–1 is
(a) f^–1og^–1
(b) fog
(c) g^–1of^–1
(d) gof

Answer : A

Question. If f : R – {3/5} → R be defined by f (x) = 3x + 2 / 3x – 3 then
(a) f^–1(x) = f (x)
(b) f^–1(x) = – f (x)
(c) fof (x) = – x
(d) f^–1(x) = 1/19 f (x)

Answer : A

Question. Consider the following statements I. For an arbitrary binary operation x on a set N, a x a = a ∀ a ∈ N.
II. If * is a commutative binary operation on N, then a x (b x c) = (c x b) x a.
(a) Only I is true
(b) Only II is true
(c) Both I and II are true
(d) Neither I nor II is true

Answer : B

Question. Let S be a finite set containing n elements. Then the total number of binary operations on S is:
(a) n²ⁿ
(b) nⁿ
(c) 2n²
(d) n²

Answer : A

Question. The function f : R → R defined by f (x) = sin x is :
(a) into
(b) onto
(c) one-one
(d) many one

Answer : D
 

Assertion Reason Type Questions :
(a) Assertion is correct, reason is correct; reason is a correct explanation for assertion.
(b) Assertion is correct, reason is correct; reason is not a correct explanation for assertion
(c) Assertion is correct, reason is incorrect
(d) Assertion is incorrect, reason is correct.

Question. Assertion : If f is even function, g is odd function, then f/g , (g ≠ 0) is an odd function .
Reason : If f(–x) = –f(x) for every x of its domain, then f(x) is called an odd function and if f(–x) = f(x) for every x of its domain, then f(x) is called an even function.

Answer : A

Question. Assertion : The binary operation * : R × R → R given by a * b → a + 2b is associative.
Reason : A binary operation*: A × A → A is said to be associative, if (a * b) * c = a * (b * c) for all a, b, c ∈ A.

Answer : D

Question. Assertion : If the relation R defined in A = {1, 2, 3} by aRb, if |a² – b²| ≤ 5, then R– 1 = R
Reason : For above relation, domain of R–1 = Range of R.

Answer : B

Question. Assertion : Let A = {–1, 1, 2, 3} and B = {1, 4, 9}, where f : A → B given by f(x) = x², then f is a many-one function.
Reason : If x₁ ≠ x₂ ⇒ f(x₁) ≠ f(x₂), for every x₁, x₂ ∈ domain, then f is one-one or else many-one.

Answer : A

Question. Assertion : f : R → R is a function defined by f(x) = 2x+1/3 . Then f–1(x) = 3x-1/2 .
Reason : f(x) is not a bijection.

Answer : C
 

Case Based Questions

A relation R on a set A is said to be an equivalence relation on A if it is
• Reflexive i.e., (a, a) ∈ R V a ∈ A.
• Symmetric i.e., (a, b) ∈ R ⇒ (b, a) ∈ R V a, b ∈ A.
• Transitive i.e., (a, b) ∈ R and (b, c) ∈ R
⇒ (a, c) ∈ R ∀ a, b, c ∈A.

Based on the above information, answer the following questions:

Question. If the relation R = {(1, 1), (1, 2), (1, 3), (2, 2),
(2, 3), (3, 1), (3, 2), (3, 3)} defined on the set A = {1, 2, 3}, then R is
(a) reflexive
(b) symmetric
(c) transitive
(d) equivalence

Answer : A

Question. If the relation R = {(1, 2), (2, 1), (1, 3), (3, 1)}
defined on the set A = {1, 2, 3}, then R is
(a) reflexive
(b) symmetric
(c) transitive
(d) equivalence

Answer : B

Question. If the relation R on the set N of all natural numbers defined as R = {(x, y) : y = x + 5 and (x < 4), then R is
(a) reflexive
(b) symmetric
(c) transitive
(d) equivalence

Answer : C

Question. If the relation R on the set A = {1, 2, 3, … 13, 14} defined as R = {(x, y) : 3x – y = 0}, then R is
(a) reflexive
(b) symmetric
(c) transitive
(d) equivalence

Answer : D

Question. If the relation R on the set A = {1, 2, 3} defined as R = {(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)}, then R is
(a) reflexive only
(b) symmetric only
(c) transitive only
(d) equivalence

Answer : D
 

Sherlin and Danju are playing Ludo at home during Covid-19. While rolling the dice, Sherlin’s sister Raji observed and noted the possible outcomes of the throw every time belongs to set {1, 2, 3, 4, 5, 6}. Let A be the set of players while B be set of all possible outcomes.
A = {S, D}, B = {1, 2, 3, 4, 5, 6

Based on the above information answer the following:

Question. Let R : B → B be defined by R = {(x, y) : y is divisible by x} is
(a) Reflexive and transitive but not symmetric
(b) Reflexive and symmetric but not transitive
(c) Not reflexive but symmetric and transitive
(d) Equivalence

Answer : A

Question. Raji wants to know the number of functions from A to B. How many number of functions are possible?
(a) 62
(b) 26
(c) 6!
(d) 212

Answer : A

Question. Let R be a relation on B defined by R = {(1, 2), (2, 2), (1, 3), (3, 4), (3, 1), (4, 3), (5, 5)}. Then R is
(a) Symmetric
(b) Reflexive
(c) Transitive
(d) None of these three

Answer : D

Question. Raji wants to know the number of relations possible from A to B. How many numbers of relations are possible?
(a) 62
(b) 26
(c) 6!
(d) 212

Answer : D

Question. Let R : B → B be defined by R = {(1, 1), (1, 2), (2, 2)(3, 3), (4, 4), (5, 5), (6, 6)}, then R is
(a) Symmetric
(b) Reflexive and Transitive
(c) Transitive and symmetric
(d) Equivalence

Answer : B
 

Students of Grade 9, planned to plant saplings along straight lines, parallel to each other to one side of the playground ensuring that they had enough play area. Let us assume that they planted one of the rows of the saplings along the line y = x – 4. Let L be the set of all lines which are parallel on the ground and R be a relation on L.

Answer the following using the above information:

Question. Let relation R be defined by R = {(L₁, L₂) : L₁ || L₂ where L₁, L₂ ∈ L}, then R is ____ relation.
(a) Equivalence
(b) Only reflexive
(c) Not reflexive
(d) Symmetric but not transitive

Answer : A

Question. Let R = {(L₁, L₂) : L₁ ⊥ L₂ where L₁, L₂ ∈ L} which of the following is true?
(a) R is Symmetric but neither reflexive nor transitive.
(b) R is Reflexive and transitive but not symmetric
(c) R is Reflexive but neither symmetric nor transitive.
(d) R is an Equivalence relation.

Answer : A

Question. The function f : R → R defined by f(x) = x – 4 is _______ .
(a) Bijective
(b) Surjective but not injective
(c) Injective but not Surjective
(d) Neither Surjective nor Injective

Answer : A

Question. Let f : R → R be defined by f(x) = x – 4. Then the range of f(x) is ______ .
(a) R
(b) Z
(c) W
(d) Q

Answer : A

Question. Let R = {(L₁, L₂) : L₁ is parallel to L₂ and L₁ : y = x – 4} then which of the following can be taken as L₂?
(a) 2x – 2y + 5 = 0
(b) 2x + y = 5
(c) 2x + 2y + 7 = 0
(d) x + y = 7

Answer : A