Question

The equation $\cos 2x+a\sin x=2a-7$ has a solution, if :

• A. $a>6$
• B. $2\le a\le 6$
• C. $a>2$
• D. None of these

Answer 1

The correct option is B $2\le a\le 6$

Rewrite the given equation as
$a\sin x+1-2{\sin }^{2}x=2a-7$
or $2{\sin }^{2}x-a\sin x+2a-8=0$
$\therefore \sin x=\frac{a\pm (a-8)}{4}$
Hence $\sin x=(a-4)/2,2$ but $\sin x=\frac{a-4}{2}\Rightarrow -1\le \frac{a-4}{2}\le 1$
$ \Rightarrow -2 \le a - 4 \le 2 \Rightarrow 2 \le a
\le 6 $