What is d² y/dx² equal to?

Correct Answer is : sec⁡3θaθ³θ/aθaθsec3θ

Correct Answer is : sec⁡3θaθ³θ/aθaθsec3θ

Test Index

UPSC NDA Math Model Paper 5

Comprehension:(Que No. 25 - 26)

Directions: For the next two (02) items that follow:

Consider the curve

x = a(\cos \theta + \theta \sin \theta)

and

y = a(\sin \theta - \theta \cos \theta)

Question : 26 of 120

Marks: +1, -0

What is

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

equal to?

Solution:

We have,

\frac{dy}{d\theta}=a(\theta\sin\theta)

and

\frac{dx}{d\theta}=a\theta\cos\theta

\frac{dy}{dx}=\tan\theta

\frac{d^2 y}{dx^2}=\frac{d}{dx}\left(\frac{dy}{dx}\right)

=\frac{d}{d\theta}\left(\frac{dy}{dx}\right)\times\frac{d\theta}{dx}

=\frac{d}{d\theta}(\tan\theta)\times\frac{1}{a\theta\cos\theta}

=\sec^2\theta\times\frac{\sec\theta}{a\theta}

=\frac{\sec^3\theta}{a\theta}

Hence, option (3) is correct.

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