Question

If the equation $a\sin x+\cos 2x=2a-7$ possesses a solution, then find the value of $a$.

• A. The value of a is $3\le a\le 6$
• B. The value of a is $4\le a\le 6$
• C. The value of a is $1\le a\le 6$
• D. The value of a is $2\le a\le 6$

Answer 1

The correct option is D The value of a is $2\le a\le 6$

The given equation can be written as

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

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

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

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

Equation has solution only if $-1\le (a-4)/2\le 1$.

Thus,$-2\le (a-4)\le 2$

$\Rightarrow 2\le a\le 6$