CBSE Class 12 Mathematics Relations and Functions MCQs Set 08

Multiple Choice Questions [1 mark]

Choose and write the correct option in the following questions.

Question. Let \( R \) be a relation on the set \( N \) of natural numbers defined by \( nRm \) if \( n \) divides \( m \). Then \( R \) is
(a) reflexive and symmetric
(b) transitive and symmetric
(c) equivalence
(d) reflexive, transitive but not symmetric
Answer: (d) reflexive, transitive but not symmetric
Solution: Since \( n \) divides \( n, \forall n \in N, R \) is reflexive. \( R \) is not symmetric as for \( 2, 4 \in N, 2 \) divides 4 but 4 does not divide 2 i.e., \( 2R4 \) but 4 is not related to 2. \( R \) is transitive since for \( n, m, r \) whenever \( n/m \) and \( m/r \)
\( \implies \) \( n/r \), i.e., \( n \) divides \( m \) and \( m \) divides \( r \), then \( n \) will divide \( r \).

Question. Set \( A \) has 3 elements and the set \( B \) has 4 elements. Then the number of injective mapping that can be defined from \( A \) to \( B \) is
(a) 144
(b) 12
(c) 24
(d) 64
Answer: (c) 24
Solution: The total number of injective mappings from the set containing \( n \) elements into the set containing \( m \) elements is \( ^mP_n \). So here it is \( ^4P_3 = 4! = 24 \).

Question. Let \( f : \mathbb{R} \to \mathbb{R} \) be defined by \( f(x) = \sin x \) and \( g : \mathbb{R} \to \mathbb{R} \) be defined by \( g(x) = x^2 \), then \( f \circ g \) is
(a) \( x^2 \sin x \)
(b) \( (\sin x)^2 \)
(c) \( \sin x^2 \)
(d) \( \frac{\sin x}{x^2} \)
Answer: (c) \( \sin x^2 \)

Question. Let \( f : \mathbb{R} \to \mathbb{R} \) be defined by \( f(x) = 3x - 4 \). Then \( f^{-1}(x) \) is given by
(a) \( \frac{x + 4}{3} \)
(b) \( \frac{x}{3} - 4 \)
(c) \( 3x + 4 \)
(d) none of the options
Answer: (a) \( \frac{x + 4}{3} \)

Question. Consider the non-empty set consisting of children in a family and a relation \( R \) defined as \( aRb \) if \( a \) is brother of \( b \). Then \( R \) is
(a) symmetric but not transitive
(b) transitive but not symmetric
(c) neither symmetric nor transitive
(d) both symmetric and transitive
Answer: (b) transitive but not symmetric
Solution: Given, \( aRb \)
\( \implies \) \( a \) is brother of \( b \)
This does not mean that \( b \) is also a brother of \( a \) because \( b \) can be a sister of \( a \).
Hence, \( R \) is not symmetric.
Again, \( aRb \)
\( \implies \) \( a \) is brother of \( b \) and \( bRc \)
\( \implies \) \( b \) is brother of \( c \).
So, \( a \) is brother of \( c \).
Hence, \( R \) is transitive.

Question. The maximum number of equivalence relation on the set \( A = \{1, 2, 3\} \) are
(a) 1
(b) 2
(c) 3
(d) 5
Answer: (d) 5

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 the options
Answer: (b) transitive

Question. Let \( A = \{1, 2, 3\} \) and consider the relation \( R = \{(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1, 3)\} \). Then \( R \) is
(a) reflexive but not symmetric
(b) reflexive but not transitive
(c) symmetric and transitive
(d) neither symmetric nor transitive
Answer: (a) reflexive but not symmetric

Question. Let \( A \) and \( B \) be finite set containing \( m \) and \( n \) elements respectively. The number of relations that can be defined from \( A \) to \( B \) is
(a) \( 2^{mn} \)
(b) \( 2^{m+n} \)
(c) \( mn \)
(d) 0
Answer: (a) \( 2^{mn} \)

Question. If the set \( A \) contains 5 elements and the set \( B \) contains 6 elements, then the number of one-one and onto mapping from \( A \) to \( B \) is
(a) 720
(b) 120
(c) 0
(d) none of the options
Answer: (c) 0

Question. Let \( f : \mathbb{R} \to \mathbb{R} \) be defined by \( f(x) = \frac{1}{x} \forall x \in \mathbb{R} \). Then \( f \) is
(a) one-one
(b) onto
(c) bijective
(d) \( f \) is not defined
Answer: (d) \( f \) is not defined

Question. Let \( f : \mathbb{R} \to \mathbb{R} \) be given by \( f(x) = \tan x \). Then \( f^{-1}(1) \) is
(a) \( \frac{\pi}{4} \)
(b) \( \left\{ n\pi + \frac{\pi}{4} : n \in Z \right\} \)
(c) does not exist
(d) none of the options
Answer: (a) \( \frac{\pi}{4} \)
Solution: Given that, \( f(x) = \tan x \)
Let \( y = \tan x \)
\( \implies \) \( x = \tan^{-1} y \)
\( f^{-1}(x) = \tan^{-1} x \)
\( \implies \) \( f^{-1}(1) = \tan^{-1} 1 = \frac{\pi}{4} \)

Question. Let \( A = \{1, 2, 3\} \). Then number of relations containing (1, 2) and (1, 3) which are reflexive and symmetric but not transitive is
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (a) 1

Question. Let \( A = \{1, 2, 3\} \). Then number of equivalence relations containing (1, 2) is
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (b) 2

Question. Which of the following function from \( Z \) into \( Z \) is bijection?
(a) \( f(x) = x^3 \)
(b) \( f(x) = x + 2 \)
(c) \( f(x) = 2x + 1 \)
(d) \( f(x) = x^2 + 1 \)
Answer: (b) \( f(x) = x + 2 \)

Question. Let \( L \) denotes the set of all straight lines in a plane. Let a relation \( R \) be defined by \( lRm \) if and only if \( l \) is perpendicular to \( m \forall l, m \in L \). Then \( R \) is
(a) reflexive
(b) symmetric
(c) transitive
(d) none of the options
Answer: (b) symmetric

Question. Let \( N \) be the set of natural numbers and the function \( f : N \to N \) be defined by \( f(n) = 2n + 3 \forall n \in N \). Then \( f \) is
(a) surjective
(b) injective
(c) bijective
(d) none of the options
Answer: (b) injective

Question. Let \( f : \mathbb{R} \to \mathbb{R} \) be defined by \( f(x) = x^2 + 1 \). Then, pre-images of 17 and –3, respectively, are
(a) \( \phi, \{4, -4\} \)
(b) \( \{3, -3\}, \phi \)
(c) \( \{4, -4\}, \phi \)
(d) \( \{4, -4\}, \{2, -2\} \)
Answer: (c) \( \{4, -4\}, \phi \)
Solution: Since \( f^{-1}(17) = x \)
\( \implies \) \( f(x) = 17 \) or \( x^2 + 1 = 17 \)
\( \implies \) \( x = \pm 4 \) or \( f^{-1}(17) = \{4, -4\} \)
and \( f^{-1}(-3) = x \)
\( \implies \) \( f(x) = -3 \)
\( \implies \) \( x^2 + 1 = -3 \)
\( \implies \) \( x^2 = -4 \) and hence \( f^{-1}(-3) = \phi \).

Question. For real numbers \( x \) and \( y \), define \( xRy \) if and only if \( x - y + \sqrt{2} \) is an irrational number. Then the relation \( R \) is
(a) reflexive
(b) symmetric
(c) transitive
(d) none of the options
Answer: (a) reflexive

Question. Let \( f : [2, \infty) \to \mathbb{R} \) be the function defined by \( f(x) = x^2 - 4x + 5 \), then the range of \( f \) is
(a) \( \mathbb{R} \)
(b) \( [1, \infty) \)
(c) \( [4, \infty) \)
(d) \( [5, \infty) \)
Answer: (b) \( [1, \infty) \)
Solution: Given that, \( f(x) = x^2 - 4x + 5 \)
Let \( y = x^2 - 4x + 5 \)
\( y = x^2 - 4x + 4 + 1 = (x - 2)^2 + 1 \)
\( (x - 2)^2 = y - 1 \)
\( \implies \) \( x - 2 = \sqrt{y - 1} \)
\( x = 2 + \sqrt{y - 1} \)
\( \therefore y - 1 \ge 0, y \ge 1 \)
Range = \( [1, \infty) \)

Assertion-Reason Questions [1 mark]

The following questions consist of two statements—Assertion(A) and Reason(R). Answer these questions selecting the appropriate option given below:
(a) Both A and R are true and R is the correct explanation for A.
(b) Both A and R are true and R is not the correct explanation for A.
(c) A is true but R is false.
(d) A is false but R is true.

Question. Assertion (A) : Let \( L \) be the collection of all lines in a plane and \( R_1 \) be the relation on \( L \) as \( R_1 = \{(L_1, L_2) : L_1 \perp L_2\} \) is a symmetric relation.
Reason (R) : A relation \( R \) is said to be symmetric if \( (a, b) \in R \)
\( \implies \) \( (b, a) \in R \).
(a) Both A and R are true and R is the correct explanation for A.
(b) Both A and R are true and R is not the correct explanation for A.
(c) A is true but R is false.
(d) A is false but R is true.
Answer: (a) Both A and R are true and R is the correct explanation for A.
Solution: If a line \( L_1 \perp L_2 \) then it is also true that \( L_2 \perp L_1 \).
Hence, \( A \) is true.
Clearly \( R \) is the definition of a symmetric relation, so it is also true, and gives the correct explanation of \( A \).
Hence, (a) is the correct option.

Question. Assertion (A) : Let \( R \) be the relation on the set of integers \( Z \) given by \( R = \{(a, b) : 2 \text{ divides } (a - b)\} \) is an equivalence relation.
Reason (R) : A relation \( R \) in a set \( A \) is said to be an equivalence relation if \( R \) is reflexive, symmetric and transitive.
(a) Both A and R are true and R is the correct explanation for A.
(b) Both A and R are true and R is not the correct explanation for A.
(c) A is true but R is false.
(d) A is false but R is true.
Answer: (a) Both A and R are true and R is the correct explanation for A.
Solution:
Reflexivity
Clearly \( (a, a) \in R \) as \( a - a = 0 \) which is an even integer and is divisible by 2. So, it is reflexive.
Symmetry
Let \( (a, b) \in R \)
\( \implies \) 2 divides \( (a - b) \)
\( \implies \) 2 divides \( -(a - b) \)
\( \implies \) 2 divides \( b - a \)
\( \implies \) \( (b, a) \in R \)
So, it is symmetric.
Transitivity
Let \( (a, b) \in R \) and \( (b, c) \in R \)
\( \implies \) 2 divides \( a - b \) and 2 divides \( b - c \)
\( \implies \) 2 divides \( a - b + b - c = a - c \)
\( \implies \) \( (a, c) \in R \)
So, it is transitive.
\( \implies \) Relation \( R \) is an equivalence relation. So \( A \) is true.
Clearly \( R \) is also true and gives the correct explanation of \( A \).
Hence, (a) is the correct option.

Question. Assertion (A) : Let \( f : \mathbb{R} \to \mathbb{R} \) given by \( f(x) = x \), then \( f \) is a one-one function.
Reason (R) : A function \( g : A \to B \) is said to be onto function if for each \( b \in B, \exists a \in A \) such that \( g(a) = b \).
(a) Both A and R are true and R is the correct explanation for A.
(b) Both A and R are true and R is not the correct explanation for A.
(c) A is true but R is false.
(d) A is false but R is true.
Answer: (b) Both A and R are true and R is not the correct explanation for A.
Solution: Let \( x_1, x_2 \in \mathbb{R} \) such that \( f(x_1) = f(x_2) \)
\( \implies \) \( x_1 = x_2 \)
\( \implies \) \( f \) is a one-one function.
Clearly \( A \) is true and \( R \) is also true. But \( R \) does not give correct explanation of \( A \).
Hence, (b) is the correct option.

Question. Assertion (A) : Let function \( f : \{1, 2, 3\} \to \{1, 2, 3\} \) be an onto function. Then it must be one-one function.
Reason (R) : A one-one function \( g : A \to B \), where \( A \) and \( B \) are finite set and having same number of elements, then it must be onto and vice-versa.
(a) Both A and R are true and R is the correct explanation for A.
(b) Both A and R are true and R is not the correct explanation for A.
(c) A is true but R is false.
(d) A is false but R is true.
Answer: (a) Both A and R are true and R is the correct explanation for A.
Solution: Suppose \( f \) is not one-one function.
i.e., \( \exists \) two elements say 1 and 2 in the domain mapped to a single element of the co-domain.
Then 3 can be mapped to any one of two remaining element.
So, range set has only two elements.
\( \implies \) \( R(f) \neq \{1, 2, 3\} \) which contradict the fact that \( f \) is an onto function.
Thus \( f \) must be a one-one function.
So, \( A \) and \( R \) gives the correct explanation of \( A \).
Hence, (a) is the correct option.

Question. Assertion (A) : Let \( f : \mathbb{R} \to \mathbb{R} \) such that \( f(x) = x^2 \). The function \( f \) is an onto function.
Reason (R) : A function \( g : A \to B \) is said to be onto function if \( g(A) = B \) i.e., range of \( g = B \).
(a) Both A and R are true and R is the correct explanation for A.
(b) Both A and R are true and R is not the correct explanation for A.
(c) A is true but R is false.
(d) A is false but R is true.
Answer: (d) A is false but R is true.
Solution: Clearly \( R(f) = [0, \infty) \)
Here \( R(f) \neq \mathbb{R} = (-\infty, \infty) \)
\( \implies \) \( f \) is not an onto function.
So, \( A \) is false but \( R \) is true.
Hence, (d) is the correct option.