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UPSC NDA Math Model Paper 9

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Question : 50 of 120

Marks: +1, -0

If

x=\cos (2 t)

y=\sin^{2} t

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

equal to?

0
sin(2t)
-cos(2t)
-1/2

Solution:

Given, x = cos (2t)

\Rightarrow \frac{dx}{dt}=-2 \sin (2t)

and

y=\sin^{2} t

\Rightarrow \frac{dy}{dt}=2 \sin t \cos t

\Rightarrow \frac{dy}{dt}=\sin 2t

By Chain Rule, we have

\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}

\frac{dy}{dx}=\frac{\sin 2t}{-2 \sin 2t}

\frac{dy}{dx}=\frac{-1}{2}

Differentiating with respect to x, we get

\frac{d^{2} y}{d x^{2}}=0

Hence, if

x=\cos (2 t)

and

y=\sin^{2} t

, then

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

equal to 0

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