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MHTCET 15 May 2024 Shift 2 Paper

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Question : 114 of 150

Marks: +1, -0

If

\sin ^{-1}\left(\frac{x}{13}\right)+\operatorname{cosec}^{-1}\left(\frac{13}{12}\right)=\frac{\pi}{2}

, then the value of

4
12
5
11

Solution:

Concept:

Use the identity

\sin^{-1}(a) + \operatorname{cosec}^{-1}(a) = \frac{\pi}{2}

and convert the inverse functions.

Explanation:

Given

\sin^{-1}\left(\frac{x}{13}\right) + \operatorname{cosec}^{-1}\left(\frac{13}{12}\right) = \frac{\pi}{2}

.

Rewrite

\operatorname{cosec}^{-1}\left(\frac{13}{12}\right)

as

\sin^{-1}\left(\frac{12}{13}\right)

.

So,

\sin^{-1}\left(\frac{x}{13}\right) + \sin^{-1}\left(\frac{12}{13}\right) = \frac{\pi}{2}

.

This implies

\sin^{-1}\left(\frac{x}{13}\right) = \frac{\pi}{2} - \sin^{-1}\left(\frac{12}{13}\right) = \cos^{-1}\left(\frac{12}{13}\right)

.

Since

\cos^{-1}\left(\frac{12}{13}\right) = \sin^{-1}\left(\frac{\sqrt{13^2 - 12^2}}{13}\right) = \sin^{-1}\left(\frac{5}{13}\right)

.

Therefore,

\frac{x}{13} = \frac{5}{13}

, giving

x = 5

.

Answer:

x = 5

, so the correct option is C.

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