Question

The equation $\cos 2x+a\sin x=2a-7$ has a solution for

• A. all $a$
• B. $a>6$
• C. $a<2$
• D. $a\in [2,6]$

Answer 1

Answer: (D)

$a\in [2,6]$

Given equation is $\cos 2x+a\sin x=2a-7$

$\Rightarrow 1-2{\sin }^{2}x+a\sin x=2a-7$

$\Rightarrow 2{\sin }^{2}x-a\sin x+2a-8=0$

We have a formula for solving quadratic equation $a{x}^2+bx+c=0$ is

$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

$\Rightarrow \sin x=\frac{a\pm \sqrt{a^2-16(a-4)}}{4}=\frac{a\pm (a-8)}{4}$

$\Rightarrow \sin x=\frac{a-4}{2}$ or $\sin x=2$ (not possible)

$\therefore \sin x=\frac{a-4}{2}$

$\because -1\le \sin x\le 1$

$\therefore -1\le \frac{a-4}{2}\le 1$

$\Rightarrow 2\le a\le 6⟹a\in [2,6]$