Question

If \( x = \sin \theta, y = \sin^3 \theta \), then \( \frac{d^2y}{dx^2} \) at \( \theta = \frac{\pi}{6} \) is

• A. \( \frac{1}{2} \)
• B. \( \frac{\sqrt{3}}{2} \)
• C. 3
• D. 6

Hint:

If $y$ can be expressed directly in terms of $x$ (here $y=x^3$), use that instead of parametric differentiation to save time.

Answer 1

The Correct Option is C

Step 1: Simplify Relation
Since $x = \sin \theta$, then $y = x^3$.
Step 2: Differentiate
$dy/dx = 3x^2$.
$d^2y/dx^2 = 6x$.
Step 3: Substitute value
At $\theta = \pi/6$, $x = \sin(\pi/6) = 1/2$.
$d^2y/dx^2 = 6(1/2) = 3$.
Step 4: Conclusion
The value is 3.
Final Answer:(C)