Let A=N× N, where N is the set of natural numbers and a relation R is defined on A such that (a,b)R(c,d) if and only if a²+d²=b²+c². Check whether R is an equivalence relation or not?

MathsCBSE Class 12 · Sample paper, session 2026-27 · 5 marks · Long answer

CBSE Class 12 Mathematics sample paper 2026-27: question 32(A)

From the official CBSE sample question paper for 2026-27, section D, question 32(A), for 5 marks. Try it first; the hints and our step-by-step solution are below.

Q32(A) · 5 marks · Long answerRelations and Functions

Let $A = N \times N$ , where $N$ is the set of natural numbers and a relation $R$ is defined on $A$ such that $(a, b) R (c, d)$ if and only if $a^{2} + d^{2} = b^{2} + c^{2}$ . Check whether $R$ is an equivalence relation or not?

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2026-27.

Hints

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

Hint 1

Check reflexive, symmetric and transitive one at a time, using the definition $a^{2} + d^{2} = b^{2} + c^{2}$ .

Hint 2

For transitivity, add the two given equations and cancel the common terms.

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.

Reflexive: For any $(p, q) \in N \times N$ , $p^{2} + q^{2} = q^{2} + p^{2}$ , so $(p, q) R (p, q)$ . Hence $R$ is reflexive.
1½ marks
Symmetric: Let $(p, q) R (r, s)$ . Then $p^{2} + s^{2} = q^{2} + r^{2}$ , which can be rewritten as $r^{2} + q^{2} = s^{2} + p^{2}$ . So $(r, s) R (p, q)$ , and $R$ is symmetric.
1½ marks
Transitive: Let $(p, q) R (r, s)$ and $(r, s) R (u, v)$ . Then $p^{2} + s^{2} = q^{2} + r^{2}$ …(i) and $r^{2} + v^{2} = s^{2} + u^{2}$ …(ii). Adding (i) and (ii) and cancelling $r^{2} + s^{2}$ gives $p^{2} + v^{2} = q^{2} + u^{2}$ . So $(p, q) R (u, v)$ , and $R$ is transitive.
1½ marks
Since $R$ is reflexive, symmetric and transitive, $R$ is an equivalence relation.
½ mark

Answer: Yes, $R$ is an equivalence relation.

Where the marks are

Proves reflexivity (1½ marks)
Proves symmetry (1½ marks)
Proves transitivity by adding the two conditions (1½ marks)
Concludes that $R$ is an equivalence relation (½ mark)

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

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CBSE Class 12 Mathematics sample paper 2026-27: question 32(A)

Hints

Step-by-step solution

Revise the chapter

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