If y=(x+y) and dy/dx=0, then prove that y=±1.

MathsCBSE Class 12 · Sample paper, session 2026-27 · 2 marks · Very short answer

CBSE Class 12 Mathematics sample paper 2026-27: question 22

From the official CBSE sample question paper for 2026-27, section B, question 22, for 2 marks. Try it first; the hints and our step-by-step solution are below.

Q22 · 2 marks · Very short answerContinuity and Differentiability

If $y = sin (x + y)$ and $\frac{d y}{d x} = 0$ , then prove that $y = \pm 1$ .

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

Differentiate both sides implicitly with respect to $x$ .

Hint 2

Substitute $\frac{d y}{d x} = 0$ and see what that forces on $cos (x + y)$ , then use $sin^{2} + cos^{2} = 1$ .

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.

Differentiate $y = sin (x + y)$ with respect to $x$ : $\frac{d y}{d x} = cos (x + y) (1 + \frac{d y}{d x})$ , so $\frac{d y}{d x} (1 - cos (x + y)) = cos (x + y)$ .
1 mark
Since $\frac{d y}{d x} = 0$ , the left side is $0$ , so $cos (x + y) = 0$ . Then $sin^{2} (x + y) = 1$ , so $sin (x + y) = \pm 1$ . Because $y = sin (x + y)$ , we get $y = \pm 1$ . Hence proved.
1 mark

Answer: $y = \pm 1$ (proved)

Where the marks are

Differentiates implicitly and collects the $\frac{d y}{d x}$ terms (1 mark)
Uses $\frac{d y}{d x} = 0$ to get $cos (x + y) = 0$ , hence $y = sin (x + y) = \pm 1$ (1 mark)

Revise the chapter

Chapter: Continuity and Differentiability (notes, formulas, where marks are lost and every past question)
Topic: Implicit Differentiation

Practise this chapterBoard-style questions from Continuity and Differentiability, one at a time, marked against the answer key and marking points.Back to the full paperSample paper 2026-27: every question, by section.

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CBSE Class 12 Mathematics sample paper 2026-27: question 22

Hints

Step-by-step solution

Revise the chapter

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