Question

Assertion :The equation $a\sin x+\cos 2x=2a-7$ possesses a solution if $a\in [2,6]$. Reason: $-1\le \sin x\le 1$ $\forall x\in R.$

• 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

a) $a\sin x+\cos 2x=2a-7$
$\Rightarrow a\sin x+1-2{\sin }^{2}x=2a-7$
$\Rightarrow 2{\sin }^{2}x-a\sin x+2a-8=0$
$\Rightarrow \sin x=\frac{a\pm \sqrt{a^2-4\times 2\times 2(a-4)}}{4}$
$=\frac{a\pm \sqrt{a^2-16a+64}}{4}$
$=\frac{a-4}{2}$ or $2$ $\because -1\le \sin x\le 1$
$\Rightarrow -1\le \frac{a-4}{2}\le 1\Rightarrow 2\le a\le 6$
Hence, Assertion (A) and Reason (R) both are true& Reason (R) is correct explanation of Assertion (A).