CBSE Class 12 · Chapter 1 · Relations and Functions · 2026-27

Relations and Functions Class 12: MCQ and case study questions

15 multiple-choice questions and 2 case-based questions on the current (rationalised) syllabus, in the style of the board paper's Sections A and E. Try each one first; the answer, a one-line reason and a worked solution open on a tap.

15 MCQs (1 mark each)
2 case studies (4 marks each)
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Multiple-choice questions

Tap an option to check it; the answer and a worked solution open. Section A of the board paper has 18 MCQs of 1 mark each, spread over all chapters.

Q1

·1 mark·Multiple choiceTypes of relations

On $A = {1, 2, 3}$ , consider $R = {(1, 1), (2, 2), (3, 3), (1, 2)}$ . Which property does $R$ fail to have?

Show answer

Answer: (b) Symmetry

Why: (1, 2) is in R but (2, 1) is not.

Reflexive: $(1, 1), (2, 2), (3, 3) \in R$ . Transitive: the only chains $(1, 1), (1, 2)$ and $(1, 2), (2, 2)$ give $(1, 2) \in R$ . Not symmetric: $(1, 2) \in R$ but $(2, 1) \notin R$ .

Q2

·1 mark·Multiple choiceTypes of relations

The number of symmetric relations on a set ${a, b}$ with two elements is

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Answer: (a)

8

Why: (a,a) and (b,b) are free; (a,b) and (b,a) must come together: 2 × 2 × 2.

Choices: include $(a, a)$ or not, include $(b, b)$ or not, include the pair ${(a, b), (b, a)}$ or not: $2^{3} = 8$ .

Q3

·1 mark·Multiple choiceOne-one and onto functions

Which of the following functions from $R$ to $R$ is a bijection?

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Answer: (c)

f (x) = 3 x - 5

Why: 3x − 5 is one-one, and every y has the pre-image (y + 5)/3.

$3 x_{1} - 5 = 3 x_{2} - 5 \Rightarrow x_{1} = x_{2}$ , and $x = \frac{y + 5}{3}$ maps to any $y$ . The others fail: $x^{2}$ and $| x |$ give $f (- 1) = f (1)$ , and $\sin x$ is never $2$ .

Q4

·1 mark·Multiple choiceOne-one and onto functions

Which of the following functions from $R$ to $R$ is neither one-one nor onto?

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Answer: (d)

f (x) = x^{2} + 1

Why: x² + 1 takes the same value at ±1 and never takes values below 1.

$x^{2} + 1$ : $f (- 1) = f (1) = 2$ (not one-one) and $f (x) \geq 1$ , so $0$ has no pre-image (not onto). $2 x + 1$ and $x^{3}$ are bijections; $e^{x}$ is one-one but not onto.

Q5

·1 mark·Multiple choiceEquivalence relations and classes

The number of ordered pairs in the smallest equivalence relation on ${a, b, c}$ that contains $(a, b)$ is

Show answer

Answer: (c)

5

Why: It needs (a,a), (b,b), (c,c), (a,b) and, for symmetry, (b,a).

Reflexivity needs $(a, a), (b, b), (c, c)$ ; $(a, b)$ is given; symmetry adds $(b, a)$ . This set of $5$ pairs is already transitive.

Q6

·1 mark·Multiple choiceEquivalence relations and classes

On $Z$ , $a R b$ if and only if $a - b$ is divisible by $5$ . The equivalence class $[2]$ is

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Answer: (b)

{5 k + 2 : k \in Z}

Why: b ∈ [2] exactly when b − 2 is a multiple of 5.

$b \in [2] \Leftrightarrow 5 ∣ (b - 2) \Leftrightarrow b = 5 k + 2$ : $[2] = {\dots, - 8, - 3, 2, 7, 12, \dots}$ .

Q7

·1 mark·Multiple choiceOne-one and onto functions

Which of the following functions $f : N \to N$ is one-one but not onto?

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Answer: (b)

f (n) = 2 n

Why: 2m = 2n forces m = n, but odd numbers are never reached.

$2 n$ : one-one, and $3$ has no pre-image. $f (n) = n$ is a bijection; $| n - 3 | + 1$ is onto but $f (2) = f (4)$ ; the constant $5$ is neither.

Q8

·1 mark·Multiple choiceOne-one and onto functions

The number of one-one functions from ${1, 2}$ to ${a, b, c, d, e}$ is

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Answer: (d)

20

Why: 1 has 5 choices, then 2 has 4 remaining.

$5 \times 4 = 20$ . (All functions: $5^{2} = 25$ ; one-one ones need different images.)

Q9

·1 mark·Multiple choiceOne-one and onto functions

How many functions from $A = {1, 2, 3, 4}$ onto $B = {a, b}$ (every element of $B$ is an image) are there?

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Answer: (a)

14

Why: All 2⁴ = 16 functions except the 2 constant ones.

There are $2^{4} = 16$ functions; only the two constant functions miss a value. Onto functions: $16 - 2 = 14$ .

Q10

·1 mark·Multiple choiceTypes of relations

$R = {(x, y) : x + 2 y = 8}$ is a relation on $N$ . The domain of $R$ is

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Answer: (b)

{2, 4, 6}

Why: y = 1, 2, 3 give x = 6, 4, 2 (y = 4 gives x = 0 ∉ ℕ).

$x = 8 - 2 y$ with $x, y \in N$ : $y = 1, 2, 3$ give $x = 6, 4, 2$ . Domain $= {2, 4, 6}$ .

Q11

·1 mark·Multiple choiceTypes of relations

On the set of all lines in a plane, the relation $L_{1} R L_{2}$ if $L_{1}$ is perpendicular to $L_{2}$ is

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Answer: (c) symmetric but neither reflexive nor transitive

Why: L₁ ⟂ L₂ gives L₂ ⟂ L₁, but no line is ⟂ itself, and L₁ ⟂ L₂ ⟂ L₃ makes L₁ ∥ L₃.

Symmetric: yes. Not reflexive: a line is not perpendicular to itself. Not transitive: if $L_{1} ⊥ L_{2}$ and $L_{2} ⊥ L_{3}$ then $L_{1} ∥ L_{3}$ , not perpendicular.

Q12

·1 mark·Multiple choiceTypes of relations

The number of relations on $A = {1, 2, 3}$ that contain both $(1, 2)$ and $(2, 1)$ is

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Answer: (a)

128

Why: Two of the 9 pairs are fixed; the other 7 are free.

$A \times A$ has $9$ pairs; $(1, 2)$ and $(2, 1)$ must be in, and each of the other $7$ may be in or out: $2^{7} = 128$ .

Q13

·1 mark·Multiple choiceOne-one and onto functions

For $f : R - {- 1} \to R$ , $f (x) = \frac{x}{x + 1}$ , which real number is not in the range of $f$ ?

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Answer: (b)

1

Why: x/(x + 1) = 1 would need x = x + 1; so f is not onto.

$\frac{x}{x + 1} = y \Rightarrow x = \frac{y}{1 - y}$ , defined for every $y \neq 1$ . For $y = 1$ : $x = x + 1$ , impossible. So $1$ is not in the range (and $f$ is not onto).

Q14

·1 mark·Multiple choiceOne-one and onto functions

For $f : R \to R$ , $f (x) = x^{2} - 2$ , the set of pre-images of $7$ is

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Answer: (b)

{- 3, 3}

Why: x² − 2 = 7 gives x² = 9.

$x^{2} - 2 = 7 \Rightarrow x^{2} = 9 \Rightarrow x = \pm 3$ . ( $47 = f (7)$ is the image of $7$ , not a pre-image.)

Q15

·1 mark·Multiple choiceEquivalence relations and classes

On ${1, 2, 3, 4}$ , $R = {(1, 1), (2, 2), (3, 3), (4, 4), (1, 2), (2, 1)}$ . The number of distinct equivalence classes of $R$ is

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Answer: (c)

3

Why: The classes are {1, 2}, {3} and {4}.

$R$ is an equivalence relation with classes $[1] = [2] = {1, 2}$ , $[3] = {3}$ , $[4] = {4}$ : three classes.

Case study questions

Section E of the board paper has three case-based questions of 4 marks: a real-life passage, then parts of 1, 1 and 2 marks, with a choice (OR) on the 2-mark part. All Class 12 case study questions by chapter.

Case study 1: School houses (4 marks)

A school allots its $12$ new students, with roll numbers $1$ to $12$ , to three houses by roll number: students whose roll numbers leave the same remainder on division by $3$ go to the same house. On $A = {1, 2, \dots, 12}$ define $a R b$ if and only if $a - b$ is divisible by $3$ .

(i) Write the equivalence class

[2]

. [1 mark]

Show answer

Answer:

{2, 5, 8, 11}

[2] = {b \in A : 3 ∣ (b - 2)} = {2, 5, 8, 11}

. A1

(ii) Is

(4, 10) \in R

? Give a reason. [1 mark]

Show answer

Answer: Yes

Yes:

4 - 10 = - 6

is divisible by

3

. A1

(iii) Show that

R

is an equivalence relation. [2 marks]

Show answer

Answer: Reflexive, symmetric and transitive

Reflexive:

a - a = 0

is divisible by

3

. Symmetric: if

3 ∣ (a - b)

then

3 ∣ (b - a)

. M1 Transitive: if

3 ∣ (a - b)

and

3 ∣ (b - c)

then

3 ∣ (a - b) + (b - c) = a - c

. A1

OR How many ordered pairs does

R

contain? [2 marks]

Show answer

Answer:

48

There are

3

classes of

4

students each M1; each class gives

4 \times 4 = 16

pairs, so

3 \times 16 = 48

. A1

Case study 2: Temperature converter (4 marks)

A weather app converts a temperature of $x$ degrees Celsius into degrees Fahrenheit using $f (x) = \frac{9 x}{5} + 32$ . Treat $f$ as a function from $R$ to $R$ .

(i) Find

f (25)

. [1 mark]

Show answer

Answer:

77

f (25) = \frac{9 \times 25}{5} + 32 = 45 + 32 = 77

. A1

(ii) Is

f

one-one? Justify. [1 mark]

Show answer

Answer: Yes

\frac{9 x_{1}}{5} + 32 = \frac{9 x_{2}}{5} + 32 \Rightarrow x_{1} = x_{2}

, so

f

is one-one. A1

(iii) Show that

f

is onto, and find the Celsius temperature whose image is

212

. [2 marks]

Show answer

Answer:

100

For any real

y

x = \frac{5 (y - 32)}{9}

gives

f (x) = y

, so

f

is onto. M1 For

y = 212

x = \frac{5 \times 180}{9} = 100

. A1

OR Find the temperature

x

for which

f (x) = x

. [2 marks]

Show answer

Answer:

- 40

\frac{9 x}{5} + 32 = x \Rightarrow \frac{4 x}{5} = - 32

\Rightarrow x = - 40

. A1

Next steps for Relations and Functions

This free set is separate from the chapter's question bank. On the Relations and Functions chapter page the revision notes and the first step of the Route to 95 are free for everyone. CBSE Pro adds all 42 questions in the chapter bank (short and long answers, assertion–reason and more case studies) with their full step-marking schemes, the Route to 95 checkpoints with your progress saved, Skill Builders, the full common-mistakes library and AI marking of your written working.

Relations and Functions: important questions What Pro includes

Original questions written by CBSE Math Revision for the CBSE 2026-27 syllabus; not taken from NCERT, NCERT Exemplar or CBSE papers. Every answer was checked twice, the second time by an independent reviewer. CBSE Math Revision is independent and not affiliated with CBSE or NCERT. Spotted a slip? Tell us and it goes in the corrections log.