Question

Assertion :The number of solutions ofthe equation ${\sin }^{-1}x+{\sin }^{-1}2x=\frac{\pi }{3}$ is only one. Reason: The sum of two positive angles cannot be negative.

• 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. Assertion false but Reason is true

Answer 1

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

Given, ${\sin }^{-1}x+{\sin }^{-1}2x=\frac{\pi }{3}$

$\Rightarrow {\sin }^{-1}2x=\frac{\pi }{3}-{\sin }^{-1}x$

$\Rightarrow 2x=\sin (\frac{\pi }{3}-{\sin }^{-1}x)$

$\Rightarrow 2x=\sin \frac{\pi }{3}\cos \left({\sin }^{-1}x\right)-\cos \frac{\pi }{3}\sin \left({\sin }^{-1}x\right)$

$\Rightarrow 2x=\frac{\sqrt{3}}{2}\sqrt{1-x^2}-\frac{1}{2}x$

$\Rightarrow {\left(5x\right)}^2=3(1-x^2)$

$\Rightarrow 28x^2=3$

$\therefore x=\pm \frac{\sqrt{3}}{2\sqrt{7}}=\frac{1}{2}\sqrt{\frac{3}{7}}$
Hence, there is only one solution of the given equation.
The sum of two positive angles can not be negative.

Reason is also true and is the correct explanation of assertion