If x=³t, y=³t, Evaluate (d²y)/(dx²) at ((1)/(2√(2)),(1)/(2√(2)) ).

MathsCBSE Class 12 · Sample paper, session 2026-27 · 3 marks · Short answer

CBSE Class 12 Mathematics sample paper 2026-27: question 28(A)

From the official CBSE sample question paper for 2026-27, section C, question 28(A), for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q28(A) · 3 marks · Short answerContinuity and Differentiability

If $x = sin^{3} t$ , $y = cos^{3} t$ , Evaluate $\frac{d ^{2} y}{d x ^{2}}$ at $(\frac{1}{2 2}, \frac{1}{2 2})$ .

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

Find $\frac{d y}{d x}$ as $\frac{d y / d t}{d x / d t}$ .

Hint 2

For the second derivative, differentiate $\frac{d y}{d x}$ with respect to $t$ and multiply by $\frac{d t}{d x}$ ; at the given point, $sin t = cos t$ .

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.

$\frac{d x}{d t} = 3 sin^{2} t cos t$ and $\frac{d y}{d t} = - 3 cos^{2} t sin t$ , so $\frac{d y}{d x} = \frac{- 3 c o s ^{2} t s i n t}{3 s i n ^{2} t c o s t} = - cot t$
1 mark
$\frac{d ^{2} y}{d x ^{2}} = \frac{d}{d t} (- cot t) \cdot \frac{d t}{d x} = csc^{2} t \cdot \frac{1}{3 s i n ^{2} t c o s t} = \frac{1}{3} sec t csc^{4} t$
1 mark
At $(\frac{1}{2 2}, \frac{1}{2 2})$ : $sin^{3} t = cos^{3} t = \frac{1}{2 2} \Rightarrow t = \frac{π}{4}$ . Then $\frac{d ^{2} y}{d x ^{2}} = \frac{1}{3} \cdot 2 \cdot (2)^{4} = \frac{4 2}{3}$
1 mark

Answer: $\frac{d ^{2} y}{d x ^{2}} = \frac{4 2}{3}$

Where the marks are

Finds $\frac{d y}{d x} = - cot t$ (1 mark)
Finds $\frac{d ^{2} y}{d x ^{2}} = \frac{1}{3} sec t csc^{4} t$ (1 mark)
Identifies $t = \frac{π}{4}$ and evaluates $\frac{4 2}{3}$ (1 mark)

Revise the chapter

Chapter: Continuity and Differentiability (notes, formulas, where marks are lost and every past question)
Topic: Parametric and Second Order Derivatives

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 28(A)

Hints

Step-by-step solution

Revise the chapter

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