Question

Let $S$ be the set of all $\alpha \in R$ such that the equation, $\cos 2x+\alpha \sin x=2\alpha -7$ has a solution. Then $S$ is equal to:

• A. $R$
• B. $[2,6]$
• C. $[1,4]$
• D. $[3,7]$

Answer 1

The correct option is B $[2,6]$

$\cos 2x+\alpha \sin x=2\alpha -7$
$\Rightarrow 1-2{\sin }^{2}x+\alpha \sin x=2\alpha -7\Rightarrow 2{\sin }^{2}x-\alpha \sin x+2\alpha -8=0\Rightarrow \sin x=\frac{\alpha \pm \sqrt{{\alpha }^2-8(2\alpha -8)}}{4}\Rightarrow \sin x=\frac{\alpha -4}{2},2$
As $\sin x=2$ is not possible so,
$\sin x=\frac{\alpha -4}{2}\Rightarrow -1\le \frac{\alpha -4}{2}\le 1\Rightarrow 2\le \alpha \le 6$