CBSE Class 12 Mathematics 2024: question 27

Hints

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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. $\frac{dx}{d\theta }=3a{\sin }^2\theta \cos \theta$, $\frac{dy}{d\theta }=-3b{\cos }^2\theta \sin \theta$.

   1 mark

2. $\frac{dy}{dx}=\frac{-3b{\cos }^2\theta \sin \theta }{3a{\sin }^2\theta \cos \theta }=-\frac{b}{a}\cot \theta$.

   ½ mark

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

   1 mark

4. At $\theta =\frac{\pi }{4}$: $\sec \frac{\pi }{4}=\sqrt{2}$, ${\csc }^4\frac{\pi }{4}=4$, so $\frac{d^{2}y}{d{x}^2}=\frac{4\sqrt{2}b}{3a^2}$.

   ½ mark

Answer: $\frac{4\sqrt{2}b}{3a^2}$

Revise the chapter

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