NCERT Solutions · Class 12 · Chapter 1: Relations and Functions

NCERT Solutions for Class 12 Maths Chapter 1 Miscellaneous Exercise

The Miscellaneous Exercise on Relations and Functions. Mixed practice: prove one-one by showing \(f(x_1) = f(x_2) \Rightarrow x_1 = x_2\) (split into cases where the rule changes), onto by solving \(f(x) = y\) inside the domain; count relations or functions on a small set by listing them.

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Miscellaneous Exercise questions and solutions

Miscellaneous Exercise, Question 1

Show \(f : \mathbb R \to \{x \in \mathbb R : -1 < x < 1\}\), \(f(x) = \dfrac{x}{1 + |x|}\), is one-one and onto.

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For \(x \ge 0\), \(f(x) = \dfrac{x}{1 + x} \in [0, 1)\); for \(x < 0\), \(f(x) = \dfrac{x}{1 - x} \in (-1, 0)\). So \(f(x)\) has the sign of x, and equal outputs come from the same branch.
One-one: \[\begin{aligned}&\dfrac{x_1}{1 + x_1} = \dfrac{x_2}{1 + x_2} \\ \Rightarrow\ &x_1 = x_2\end{aligned}\] (similarly for the negative branch).
Onto: for \(0 \le y < 1\) take \(x = \dfrac{y}{1 - y} \ge 0\); for \(-1 < y < 0\) take \(x = \dfrac{y}{1 + y} < 0\).

Answer: Shown: f is a bijection onto \((-1, 1)\)

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Miscellaneous Exercise, Question 2

Show \(f : \mathbb R \to \mathbb R\), \(f(x) = x^3\), is injective.

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\[\begin{aligned}&x_1^3 = x_2^3 \\ \Rightarrow\ &(x_1 - x_2)(x_1^2 + x_1x_2 + x_2^2) = 0\end{aligned}\]
The second factor is \[\left(x_1 + \tfrac{x_2}{2}\right)^2 + \tfrac34x_2^2 > 0\] unless \(x_1 = x_2 = 0\); so \(x_1 = x_2\).

Answer: Shown

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Miscellaneous Exercise, Question 3

For a non-empty set X and subsets A, B of X, define A R B iff \(A \subset B\). Is R an equivalence relation on P(X)?

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Reflexive: \(A \subset A\). Transitive: \(A \subset B\), \(B \subset C \Rightarrow A \subset C\).
Not symmetric: \(\emptyset \subset X\) but \(X \not\subset \emptyset\) (X is non-empty).

Answer: No: R is reflexive and transitive but not symmetric

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Miscellaneous Exercise, Question 4

Find the number of onto functions from \(\{1, 2, \ldots, n\}\) to itself.

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A function from a finite set to itself is onto exactly when it is one-one, i.e. a permutation of the n elements.

Answer: \(n!\)

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Miscellaneous Exercise, Question 5

\(A = \{-1, 0, 1, 2\}\), \(B = \{-4, -2, 0, 2\}\), \(f(x) = x^2 - x\), \(g(x) = 2\left|x - \tfrac12\right| - 1\). Are f and g equal?

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\(f(-1) = 2, f(0) = 0, f(1) = 0, f(2) = 2\).
\(g(-1) = 2 \cdot \tfrac32 - 1 = 2\), \(g(0) = 0\), \(g(1) = 0\), \(g(2) = 2\). Same values at every point of A (and the same domain and codomain).

Answer: Yes, f and g are equal

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Miscellaneous Exercise, Question 6

\(A = \{1, 2, 3\}\). The number of relations containing \((1, 2)\) and \((1, 3)\) that are reflexive and symmetric but not transitive is: (A) 1 (B) 2 (C) 3 (D) 4

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Any such R contains \[(1,1), (2,2), (3,3), (1,2), (2,1), (1,3), (3,1)\]; the only other possible pairs are \((2,3), (3,2)\), which symmetry forces together.
Without them: \((2,1), (1,3) \in R\) but \((2,3) \notin R\), not transitive. With them: R is the universal relation, transitive. So exactly one.

Answer: (A) 1

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Miscellaneous Exercise, Question 7

\(A = \{1, 2, 3\}\). The number of equivalence relations containing \((1, 2)\) is: (A) 1 (B) 2 (C) 3 (D) 4

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An equivalence relation is a partition into classes; 1 and 2 must share a class.
Either \(\{1, 2\}, \{3\}\) or \(\{1, 2, 3\}\).

Answer: (B) 2

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Done the NCERT exercises? The board paper asks more

Relations and Functions has 39 original board-style questions (MCQ, assertion–reason, short and long answers, case studies) with step mark schemes, revision notes and a four-step Route to 95. Three sample questions are open to everyone; a free account opens the rest.

Relations and Functions: notes and questions Create a free account Teachers: set it as homework

Relations and Functions in our sample papers: Sample paper 1 (questions 1, 38) · Sample paper 2 (questions 1, 21, 36) · Sample paper 3 (questions 1, 21, 38) · Sample paper 4 (questions 1, 21, 36) · Sample paper 5 (questions 1, 21, 38).

Also useful: free MCQs and case studies for Relations and Functions · formulas for this chapter (Class 12 formula sheet, free PDF) · official CBSE board and sample papers · our sample papers with marking scheme · the Route to 95 plan

Textbook: NCERT Mathematics Class 12, Parts I and II (rationalised edition, 2023-24 reprint onward), free from ncert.nic.in. Question statements are shortened to the minimum needed; the solutions and tips are our own. CBSE Math Revision is independent and not affiliated with NCERT or CBSE. Spotted a slip? Tell us and it goes in the corrections log.