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MHTCET 23 Apr 2025 Shift 2 Paper

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Question : 144 of 150

Marks: +1, -0

If

x=\sin \theta, y=\sin ^{3} \theta

\frac{d^{2} y}{d x^{2}}

\theta=\frac{\pi}{6}

is

$\frac{1}{2}$
$\frac{\sqrt{3}}{2}$
3
6

Solution:

Concept:

Since

y

is expressed directly in terms of

x

, simplify the relation before differentiating.

Explanation:

Given

x = \sin \theta

and

y = \sin^3 \theta

.

Therefore,

y = x^3

.

Differentiate with respect to

x

:

\frac{dy}{dx} = 3x^2

.

Differentiate again:

\frac{d^2y}{dx^2} = 6x

.

At

\theta = \frac{\pi}{6}

, we have

x = \sin \frac{\pi}{6} = \frac{1}{2}

.

So,

\left(\frac{d^2y}{dx^2}\right)_{\theta = \frac{\pi}{6}} = 6\left(\frac{1}{2}\right) = 3

.

Answer:

3

— Option C.

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