Question

The equation $\cos 2x+a\sin x=2a-7$ possesses a solution if

• A. $a<2$
• B. $2\le a\le 6$
• C. $a>6$
• D. $a$ is any integer

Answer 1

Answer: (B)

$2\le a\le 6$

$\cos 2x=1-2{\sin }^{2}x$
Hence
$1-2{\sin }^{2}x+a\sin \left(x\right)=2a-7$
Or
$2{\sin }^2\left(x\right)-a\sin \left(x\right)+(2a-8)=0$
Now
$\sin \left(x\right)=\frac{a\pm \sqrt{a^2-8(2a-8)}}{4}$
Or
$\sin \left(x\right)=\frac{a\pm \sqrt{a^2-16a+64}}{4}$
Hence
$\sin \left(x\right)=\frac{a\pm (a-8)}{4}$
Now
$|\sin \left(x\right)|<1$.
Hence
$\sin \left(x\right)=\frac{2a-8}{4}$

$=\frac{a-4}{2}$.
Now
$-1\le \sin \left(x\right)\le 1$.
Or
$-1\le \frac{a-4}{2}\le 1$
Or
$-2\le a-4\le 2$
Or
$2\le a\le 6$