If x=a (θ+(θ)/(2) ) and y=θ, then find (d²y)/(dx²) at θ=(π)/(4).

.(d²y)/(dx²) θ=π/4 = (22)/(a²)

MathsCBSE Class 12 · Board exam 2025, set 65/1/1 · 5 marks · Long answer

CBSE Class 12 Mathematics 2025: question 32(b)

Set in the CBSE Class 12 board exam of 2025, paper 65/1/1, section D, question 32(b), for 5 marks. Try it first; the hints and our step-by-step solution are below.

Q32(b) · 5 marks · Long answerContinuity and Differentiability

If $x = a (cos θ + lo g tan \frac{θ}{2})$ and $y = sin θ$ , then find $\frac{d ^{2} y}{d x ^{2}}$ at $θ = \frac{π}{4}$ .

Source: official question paper from cbse.gov.in (PDF). See it in the full 2025 board paper, set 65/1/1.

Hints

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

Hint 1

Find $\frac{d x}{d θ}$ and $\frac{d y}{d θ}$ separately. The log tan term simplifies to $\frac{1}{s i n θ}$ .

Hint 2

Once you have $\frac{d y}{d x}$ in terms of $θ$ , differentiate it with respect to $θ$ and multiply by $\frac{d θ}{d x}$ .

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 $x$ : $\frac{d x}{d θ} = a (- sin θ + \frac{1}{tan \frac{θ}{2}} \cdot sec^{2} \frac{θ}{2} \cdot \frac{1}{2})$ .
½ mark
Since $\frac{sec ^{2} \frac{θ}{2}}{2 tan \frac{θ}{2}} = \frac{1}{2 sin \frac{θ}{2} cos \frac{θ}{2}} = \frac{1}{sin θ}$ , we get $\frac{d x}{d θ} = a (- sin θ + \frac{1}{sin θ}) = a (\frac{1 - sin ^{2} θ}{sin θ})$ .
½ mark
So $\frac{d x}{d θ} = a \frac{cos ^{2} θ}{sin θ} = a cot θ cos θ$ .
½ mark
Also, $y = sin θ \Rightarrow \frac{d y}{d θ} = cos θ$ .
½ mark
$\frac{d y}{d x} = \frac{d y / d θ}{d x / d θ} = \frac{cos θ sin θ}{a cos ^{2} θ} = \frac{tan θ}{a}$ .
1 mark
Differentiate again with respect to $x$ : $\frac{d ^{2} y}{d x ^{2}} = \frac{sec ^{2} θ}{a} \cdot \frac{d θ}{d x} = \frac{sec ^{2} θ}{a} \cdot \frac{sin θ}{a cos ^{2} θ} = \frac{sec ^{3} θ tan θ}{a ^{2}}$ .
1 mark
At $θ = \frac{π}{4}$ : $sec \frac{π}{4} = 2$ and $tan \frac{π}{4} = 1$ . So $\frac{d ^{2} y}{d x ^{2}} = \frac{( 2 ) ^{3} \cdot 1}{a ^{2}} = \frac{2 2}{a ^{2}}$ .
1 mark

Answer: $\frac{d ^{2} y}{d x ^{2}}_{θ = π /4} = \frac{2 2}{a ^{2}}$

Where the marks are

Differentiates $x$ with respect to $θ$ using the chain rule (½ mark)
Simplifies using $2 sin \frac{θ}{2} cos \frac{θ}{2} = sin θ$ (½ mark)
Gets $\frac{d x}{d θ} = a cot θ cos θ$ (½ mark)
Finds $\frac{d y}{d θ} = cos θ$ (½ mark)
Gets $\frac{d y}{d x} = \frac{t a n θ}{a}$ (1 mark)
Gets $\frac{d ^{2} y}{d x ^{2}} = \frac{s e c ^{3} θ t a n θ}{a ^{2}}$ (1 mark)
Evaluates $\frac{2 2}{a ^{2}}$ at $θ = \frac{π}{4}$ (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 paper2025 board paper, set 65/1/1: every question, by section.

The question is transcribed from the official paper; figures are described in words. The solution, hints and chapter tag are ours. Spotted an error? Write tosupport@questionbuddy.in.

CBSE Class 12 Mathematics 2025: question 32(b)

Hints

Step-by-step solution

Revise the chapter

More questions from Continuity and Differentiability