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COMEDK 2025 Afternoon Solved Paper

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Question : 134 of 180

Marks: +1, -0

If

y=\left(\sin ^{-1} x\right)^2+\left(\cos ^{-1} x\right)^2

\left(1-x^2\right) \frac{d^2 y}{d x^2}-x \frac{d y}{d x}=

2
3
0
4

Solution:

Concept:

Use the identity

\sin^{-1}x + \cos^{-1}x = \frac{\pi}{2}

to combine the two inverse trigonometric functions and express

y

as a quadratic in

\sin^{-1}x

.

Explanation:

Let

A = \sin^{-1}x

. Then

\cos^{-1}x = \frac{\pi}{2} - A

.

Thus

y = A^2 + \left(\frac{\pi}{2} - A\right)^2 = 2A^2 - \pi A + \frac{\pi^2}{4}

.

Differentiate with respect to

x

:

\frac{dy}{dx} = (4A - \pi)\frac{dA}{dx} = \frac{4\sin^{-1}x - \pi}{\sqrt{1-x^2}}

.

Differentiate again:

\frac{d^2y}{dx^2} = 4\frac{1}{1-x^2} + (4\sin^{-1}x - \pi)\frac{x}{(1-x^2)^{3/2}}

.

Now compute

(1-x^2)\frac{d^2y}{dx^2} = 4 + \frac{x(4\sin^{-1}x - \pi)}{\sqrt{1-x^2}}

.

Subtract

x\frac{dy}{dx} = \frac{x(4\sin^{-1}x - \pi)}{\sqrt{1-x^2}}

.

The two terms with

x(4\sin^{-1}x - \pi)

cancel, leaving

4

.

Answer:

4

(Option D)

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