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UPSC NDA I 2022 Mathematics Solved Paper

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Question : 37 of 120

Marks: +1, -0

If

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

x

equal to?

0
$\frac{1}{2}$
$\frac{1}{\sqrt{2}}$
$\frac{1}{\sqrt{3}}$

Solution:

Concept:

Use the standard identity

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

for

x \in [-1, 1]

.

Explanation:

Given:

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

.

Rewrite

3\sin^{-1} x

as

2\sin^{-1} x + \sin^{-1} x

.

Then the equation becomes

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

.

Substitute the identity:

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

.

Subtract

\frac{\pi}{2}

from both sides:

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

.

Divide by 2:

\sin^{-1} x = \frac{\pi}{4}

.

Therefore,

x = \sin \frac{\pi}{4} = \frac{1}{\sqrt{2}}

.

Answer:

\frac{1}{\sqrt{2}}

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