If y√(x² + 1) = (√(x² + 1) - x ), show that (x² + 1)dy/dx + xy + 1 = 0.

MathsCBSE Class 12 · Board exam 2026, set 65/5/1 · 5 marks · Long answer

CBSE Class 12 Mathematics 2026: question 33(a)

Set in the CBSE Class 12 board exam of 2026, paper 65/5/1, section D, question 33(a), for 5 marks. Try it first; the hints and our step-by-step solution are below.

Q33(a) · 5 marks · Long answerContinuity and Differentiability

If $y x^{2} + 1 = lo g (x^{2} + 1 - x)$ , show that

$(x^{2} + 1) \frac{d y}{d x} + x y + 1 = 0$ .

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

Use the product rule on $y x^{2} + 1$ and the chain rule on $lo g (x^{2} + 1 - x)$ .

Hint 2

Write $\frac{x}{x ^{2} + 1} - 1$ as a single fraction; part of it cancels with $x^{2} + 1 - x$ .

Hint 3

Finally multiply the whole equation by $x^{2} + 1$ .

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 both sides with respect to $x$ (product rule on the left, chain rule on the right):
$x^{2} + 1 \frac{d y}{d x} + y \cdot \frac{x}{x ^{2} + 1} = \frac{1}{x ^{2} + 1 - x} (\frac{x}{x ^{2} + 1} - 1)$
2 marks
Simplify the right side: $\frac{1}{x ^{2} + 1 - x} \cdot \frac{x - x ^{2} + 1}{x ^{2} + 1}$
1 mark
$= - \frac{1}{x ^{2} + 1}$ , so $x^{2} + 1 \frac{d y}{d x} + \frac{x y}{x ^{2} + 1} = - \frac{1}{x ^{2} + 1}$
1 mark
Multiply by $x^{2} + 1$ : $(x^{2} + 1) \frac{d y}{d x} + x y = - 1$ , i.e. $(x^{2} + 1) \frac{d y}{d x} + x y + 1 = 0$ . Hence proved.
1 mark

Answer: $(x^{2} + 1) \frac{d y}{d x} + x y + 1 = 0$ (proved)

Where the marks are

Correct differentiation of both sides (2 marks)
Combines the bracket as $\frac{x - x ^{2} + 1}{x ^{2} + 1}$ (1 mark)
Simplifies the RHS to $- \frac{1}{x ^{2} + 1}$ (1 mark)
Multiplies through by $x^{2} + 1$ to get the result (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 paper2026 board paper, set 65/5/1: every question, by section.

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CBSE Class 12 Mathematics 2026: question 33(a)

Hints

Step-by-step solution

Revise the chapter

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