Question

The equation $2\sin x=|x|+a$ has no solution for $a\in$

• A. $(\frac{3\sqrt{3}-\pi }{3},\infty )$
• B. $(\frac{3\sqrt{3}+\pi }{3},\infty )$
• C. $(-\infty ,\frac{3\sqrt{3}-\pi }{3})$
• D. $(-\infty ,\frac{3\sqrt{3}+\pi }{3})$

Answer 1

The correct option is A $(\frac{3\sqrt{3}-\pi }{3},\infty )$

$2\sin x=|x|+a$
$|x|+a=\{\begin{array}{cc}x+a,& x\ge 0\\ -x+a,& x<0\end{array}$

$\text{Slope of}(x+a)=\frac{d}{dx}(x+a)=1$
$\frac{d}{dx}\left(2\sin x\right)=2\cos x$

Now, $2\cos x=1$
$\Rightarrow x=\frac{\pi }{3}$

$\text{I}:|x|+a;a<0$
$\text{II}:|x|+a;a=0$
$\text{III, IV}:|x|+a;a>0$

Therefore, for no solution
$a>2\sin \frac{\pi }{3}-\frac{\pi }{3}$
$\Rightarrow a>\frac{3\sqrt{3}-\pi }{3}$
$\Rightarrow a\in (\frac{3\sqrt{3}-\pi }{3},\infty )$