MathsCBSE Class 12 · Board exam 2026, set 65/5/1 · 4 marks · Case-based

CBSE Class 12 Mathematics 2026: question 36

Set in the CBSE Class 12 board exam of 2026, paper 65/5/1, section E, question 36, for 4 marks. Try it first; the hints and our step-by-step solution are below.

Q36 · 4 marks · Case-basedCompetency-basedRelations and Functions

A school wants the students of class XII to do a project on 'Sustainability' keeping the world environment in mind. They select the student participants on the basis of an essay writing competition.

7 students out of 80 are selected for the project and are categorized into two sets such that :

Girl students belong to Set A = {G $_{1}$ , G $_{2}$ , G $_{3}$ , G $_{4}$ },

Boy students belong to Set B = {B $_{1}$ , B $_{2}$ , B $_{3}$ }.

Based on the above information, answer the following questions :

(i)
How many relations are possible from Set A $\to$ Set B ?
1 mark
(ii)
Let R be a relation from A $\to$ B such that R = {(G $_{1}$ , B $_{1}$ ), (G $_{2}$ , B $_{2}$ ), (G $_{3}$ , B $_{2}$ ), (G $_{4}$ , B $_{3}$ ), (G $_{1}$ , B $_{2}$ )}. Is R an injective function ? Justify your answer.
1 mark
(iii)(a)
Let the relation R from A $\to$ A be such that R = {(x, y), x, y $\in$ A, x and y are students from the same colony in the city}. Verify if R is an equivalence relation.
2 marks
OR
(iii)(b)
Verify if any function f : B $\to$ A is bijective. Give reason to support your answer.
2 marks

Source: official question paper from cbse.gov.in (PDF). See it in the full 2026 board paper, set 65/5/1.

Hints

Open one hint at a time and try again before the next.

Hint 1

A relation from A to B is any subset of $A \times B$ . Count the elements of $A \times B$ first.

Hint 2

For a function, each element of A must have exactly one image. Check $G_{1}$ in R.

Hint 3

For (iii), test reflexive, symmetric and transitive one by one using 'same colony'. For the bijection, compare the number of elements in B and A.

Step-by-step solution

Each step shows the marks it earns, as an examiner would award them. Checked by an independent re-solve from the question paper.

(i) $n (A \times B) = 4 \times 3 = 12$ . Every subset of $A \times B$ is a relation from A to B, so the number of relations is $2^{12} = 4096$ .
1 mark
(ii) No, R is not an injective function.
½ mark
(ii) R is not even a function, because $G_{1}$ is related to two images, $B_{1}$ and $B_{2}$ .
½ mark
(iii)(a) Reflexive: every student x lives in the same colony as himself or herself, so $(x, x) \in R$ for all $x \in A$ .
½ mark
(iii)(a) Symmetric: if $(x, y) \in R$ , then x and y live in the same colony. So y and x live in the same colony, and $(y, x) \in R$ .
½ mark
(iii)(a) Transitive: if $(x, y) \in R$ and $(y, z) \in R$ , then x, y live in the same colony and y, z live in the same colony. So x and z live in the same colony, and $(x, z) \in R$ .
½ mark
(iii)(a) R is reflexive, symmetric and transitive, hence R is an equivalence relation.
½ mark
(iii)(b) No function $f : B \to A$ is bijective.
1 mark
(iii)(b) Reason: $n (B) = 3 < n (A) = 4$ . The range of f has at most 3 elements, so at least one girl in A has no pre-image. Thus f cannot be onto, and so it cannot be bijective.
1 mark

Answer: (i) $2^{12} = 4096$ ; (ii) No, R is not a function ( $G_{1}$ has two images), so not injective; (iii)(a) R is an equivalence relation; (iii)(b) No $f : B \to A$ is bijective, since $n (B) < n (A)$ so f cannot be onto

Where the marks are

(i) Number of relations $2^{12} = 4096$ (1 mark)
(ii) States R is not injective (½ mark)
(ii) Justifies that R is not a function since $G_{1}$ has two images (½ mark)
(iii)(a) Proves R is reflexive (½ mark)
(iii)(a) Proves R is symmetric (½ mark)
(iii)(a) Proves R is transitive (½ mark)
(iii)(a) Concludes R is an equivalence relation (½ mark)
(iii)(b) States no function from B to A is bijective (1 mark)
(iii)(b) Reason: $n (B) < n (A)$ , so f cannot be onto (1 mark)

Revise the chapter

Chapter: Relations and Functions (notes, formulas, where marks are lost and every past question)

Practise this chapterBoard-style questions from Relations and Functions, one at a time, marked against the answer key and marking points.Back to the full paper2026 board paper, set 65/5/1: every question, by section.

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CBSE Class 12 Mathematics 2026: question 36

Hints

Step-by-step solution

Revise the chapter

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