Class 11 Mathematics Relations and Functions MCQs Set D

Multiple Choice Questions

Question. The relation \( R \) in the set \( A = \{1, 2, 3, 4\} \) given by \( R = \{(1, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3, 3), (3, 2)\} \) 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
Answer: (b)

Question. If \( A = \{a, b, c, d\} \), then a relation \( R = \{(a, b), (b, a), (a, a)\} \) on \( A \) is
(a) symmetric only
(b) transitive only
(c) reflexive and transitive
(d) symmetric and transitive only
Answer: (d)

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 these
Answer: (a)

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)

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)

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 these
Answer: (b)

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)

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

Question. Let \( A \) and \( B \) be finite sets 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)

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)

Question. The function \( f : \mathbb{R} \rightarrow \mathbb{R} \) defined by \( f(x) = 2^x + 2^{|x|} \) is
(a) One-one and onto
(b) Many-one and onto
(c) One-one and into
(d) Many-one and into
Answer: (c)

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 these
Answer: (c)

Question. Which of the following functions from \( \mathbb{Z} \) into \( \mathbb{Z} \) 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)

Question. Let \( f : [2, \infty) \rightarrow \mathbb{R} \) 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)

Question. Let \( f : \mathbb{R} \rightarrow \mathbb{R} \) 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)

Question. Let \( f : \mathbb{R} \rightarrow \mathbb{R} \) be defined by \( f(x) = 3x^2 - 5 \) and \( g : \mathbb{R} \rightarrow \mathbb{R} \) by \( g(x) = \frac{x}{x^2 + 1} \). Then \( gof \) is
(a) \( \frac{3x^2 - 5}{9x^4 - 30x^2 + 26} \)
(b) \( \frac{3x^2 - 5}{9x^4 - 6x^2 + 26} \)
(c) \( \frac{3x^2}{x^4 + 2x^2 - 4} \)
(d) \( \frac{3x^2}{9x^4 + 30x^2 - 2} \)
Answer: (a)

Question. If \( f(x) = \sin^2 x \) and the composite function \( g(f(x)) = |\sin x| \), then \( g(x) \) is equal to
(a) \( \sqrt{x + 1} \)
(b) \( \sqrt{x - 1} \)
(c) \( \sqrt{x} \)
(d) \( -\sqrt{x} \)
Answer: (c)

Question. Let \( f : \mathbb{R} \rightarrow \mathbb{R} \) be the functions defined by \( f(x) = x^3 + 5 \). Then \( f^{-1}(x) \) is
(a) \( (x + 5)^{1/3} \)
(b) \( (x - 5)^{1/3} \)
(c) \( (5 - x)^{1/3} \)
(d) \( 5 - x \)
Answer: (b)

Question. Let \( f : \mathbb{R} - \{-\frac{4}{3}\} \rightarrow \mathbb{R} - \{\frac{4}{3}\} \) be a function defined as \( f(x) = \frac{4x}{3x+4} \). The inverse of \( f \) is the map \( g : \text{Range } f \rightarrow \mathbb{R} - \{-\frac{4}{3}\} \)
(a) \( g(y) = \frac{3y}{3 - 4y} \)
(b) \( g(y) = \frac{4y}{4 - 3y} \)
(c) \( g(y) = \frac{4y}{3 - 4y} \)
(d) \( g(y) = \frac{3y}{4 - 3y} \)
Answer: (b)

Question. Let \( f : \mathbb{R} - \{\frac{3}{5}\} \rightarrow \mathbb{R} - \{\frac{3}{5}\} \) be defined by \( f(x) = \frac{3x + 2}{5x - 3} \). Then
(a) \( f^{-1}(x) = f(x) \)
(b) \( f^{-1}(x) = -f(x) \)
(c) \( fof(x) = -x \)
(d) \( f^{-1}(x) = \frac{1}{19} f(x) \)
Answer: (a)

Solutions of Selected Multiple Choice Questions

Question. Since every element of \( A \) is related to itself in the given relation \( R \), therefore \( R \) is reflexive and as \( (1, 2) \in R \) and \( (2, 2) \in R \Rightarrow (1, 2) \in R \) also \( (1, 3) \in R \) and \( (3, 2) \in R \Rightarrow (1, 2) \in R \). Again \( (1, 3) \in R \) and \( (3, 3) \in R \Rightarrow (1, 3) \in R \). Thus \( R \) is also transitive. Hence relation \( R \) is reflexive and transitive but not symmetric because, \( (1, 2) \in R \) but \( (2, 1) \notin R \), also \( (1, 3) \in R \) but \( (3, 1) \notin R \) and \( (3, 2) \in R \) but \( (2, 3) \notin R \).
Answer: Since every element of \( A \) is related to itself in the given relation \( R \), therefore \( R \) is reflexive and as \( (1, 2) \in R \) and \( (2, 2) \in R \Rightarrow (1, 2) \in R \) also \( (1, 3) \in R \) and \( (3, 2) \in R \Rightarrow (1, 2) \in R \). Again \( (1, 3) \in R \) and \( (3, 3) \in R \Rightarrow (1, 3) \in R \). Thus \( R \) is also transitive. Hence relation \( R \) is reflexive and transitive but not symmetric because, \( (1, 2) \in R \) but \( (2, 1) \notin R \), also \( (1, 3) \in R \) but \( (3, 1) \notin R \) and \( (3, 2) \in R \) but \( (2, 3) \notin R \).

Question. On the set \( A = \{a, b, c, d\} \) given relation \( R = \{(a, b), (b, a), (a, a)\} \) is symmetric and transitive only. Since, \( (a, b) \in R \Rightarrow (b, a) \in R \), therefore it is symmetric. Also, \( (a, b) \in R \) and \( (b, a) \in R \Rightarrow (a, a) \in R \), so it is also transitive. As \( (b, b) \), \( (c, c) \) and \( (d, d) \) does not belong to \( R \) hence \( R \) is not reflexive. Hence relation \( R \) is symmetric and transitive only.
Answer: On the set \( A = \{a, b, c, d\} \) given relation \( R = \{(a, b), (b, a), (a, a)\} \) is symmetric and transitive only. Since, \( (a, b) \in R \Rightarrow (b, a) \in R \), therefore it is symmetric. Also, \( (a, b) \in R \) and \( (b, a) \in R \Rightarrow (a, a) \in R \), so it is also transitive. As \( (b, b) \), \( (c, c) \) and \( (d, d) \) does not belong to \( R \) hence \( R \) is not reflexive. Hence relation \( R \) is symmetric and transitive only.