CBSE Class 12 Mathematics sample paper 2025-26: question 26(B)

Hints

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

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.

1. $\frac{dx}{d\theta }=a(1-\cos \theta )$, $\frac{dy}{d\theta }=a\sin \theta$.

   1 mark

2. $\frac{dy}{dx}=\frac{a\sin \theta }{a(1-\cos \theta )}=\frac{2\sin \frac{\theta }{2}\cos \frac{\theta }{2}}{2{\sin }^2\frac{\theta }{2}}=\cot \frac{\theta }{2}$.

   1 mark

3. $\frac{d^{2}y}{d{x}^2}=-\frac{1}{2}{\csc }^2\frac{\theta }{2}\cdot \frac{d\theta }{dx}=-\frac{1}{2}{\csc }^2\frac{\theta }{2}\cdot \frac{1}{2a{\sin }^2\frac{\theta }{2}}=-\frac{1}{4a}{\csc }^4\frac{\theta }{2}$.

   1 mark

Answer: $\frac{d^{2}y}{d{x}^2}=-\frac{1}{4a}{\csc }^4\frac{\theta }{2}$

Revise the chapter

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