Question

The maximum integral value of a for which the equation $a\sin x+\cos 2x=2a-7$ has a solution is

Answer 1

$a\sin x+\cos 2x=2a-7$
$\Rightarrow 2{\sin }^{2}x-a\sin x+2a-8=0$
$\Rightarrow \sin x=\frac{a\pm (a-8)}{4}$
$\Rightarrow \sin x=2$(not possible), $\sin x=\frac{a-4}{2}$
Now, $-1\le \sin x\le 1$
$\Rightarrow -1\le \frac{a-4}{2}\le 1$
$\Rightarrow 2\le a\le 6$
So, the maximum integral value of $a$ is $6$.