Question

Assertion :The equation ${sin}^{-1}x=3{sin}^{-1}\left(a\right)$has a solution for $-\frac{1}{2}\le a\le \frac{1}{2}$ Reason: $\forall x\in [-1,1],{sin}^{-1}x\in [0,2\pi ]$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

Given

${\sin }^{-1}x=3{\sin }^{-1}a$

The domain of ${\sin }^{-1}x$ is $[-1,1]$

The range of ${\sin }^{-1}x$ is $[-\frac{\pi }{2},\frac{\pi }{2}]$

$\forall x\in [-1,1],{\sin }^{-1}x\in [-\frac{\pi }{2},\frac{\pi }{2}]$

$⟹3{\sin }^{-1}a\in [-\frac{\pi }{2},\frac{\pi }{2}]$

$⟹{\sin }^{-1}a\in [-\frac{\pi }{6},\frac{\pi }{6}]$

$⟹a\in [\sin (-\frac{\pi }{6}),\sin \left(\frac{\pi }{6}\right)]$

$⟹a\in [-\frac{1}{2},\frac{1}{2}]$

$⟹-\frac{1}{2}\le a\le \frac{1}{2}$