Question

The set of values of a for which the equation $\sin x(\sin x+\cos x)=a$ has real solutions is

• A. $[1-\sqrt{2},1+\sqrt{2}]$
• B. $[2-\sqrt{3},2+\sqrt{3}]$
• C. $[0,2+\sqrt{3}]$
• D. $\left[\frac{1-\sqrt{2}}{2},\frac{1+\sqrt{2}}{2}\right]$

Answer 1

The correct option is D $\left[\frac{1-\sqrt{2}}{2},\frac{1+\sqrt{2}}{2}\right]$

We have,
$\sin x(\sin x+\cos x)=a$
$\Rightarrow 2{\sin }^{2}x+2\sin x\cos x=2a$
$\Rightarrow 1-\cos 2x+\sin 2x=2a$
$\Rightarrow \sin 2x-\cos 2x=2a-1$
This equation will have real solutions, if
$|2a-1|\le \sqrt{1+1}$
$\Rightarrow 1-\sqrt{2}\le 2a\le 1+\sqrt{2}$
$\Rightarrow aϵ\left[\frac{1-\sqrt{2}}{2},\frac{1+\sqrt{2}}{2}\right]$