Question

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

• A. $2$
• B. $4$
• C. $5$
• D. $6$

Answer 1

The correct option is D $6$

$a\sin x+\cos 2x=2a-7\Rightarrow a\sin x+1-2{\sin }^{2}x=2a-7\Rightarrow 2{\sin }^{2}x-a\sin x+2a-8=0\Rightarrow \sin x=\frac{a\pm \sqrt{a^2-8(2a-8)}}{4}=\frac{a\pm (a-8)}{4}\Rightarrow \sin x=2,\frac{a-4}{2}$
Since, $\sin x\ne 2$
If the equation has real solution(s), then
$\frac{a-4}{2}\in [-1,1]$
$\Rightarrow -1\le \frac{a-4}{2}\le 1\Rightarrow 2\le a\le 6$