Question

The set of positive real values of the parameter 'a' for which the equation |sin2x|-|x|-a=0 does not have any real solution is

• A. $(\frac{3\sqrt{3}+\pi }{6},\infty )$
  
• B. $(\frac{3\sqrt{3}-\pi }{6},\infty )$
  
• C. $(\frac{3\sqrt{3}-\pi }{12},\infty )$
  
• D. $(\frac{3\sqrt{3}+\pi }{12},\infty )$

Answer 1

The correct option is B

$(\frac{3\sqrt{3}-\pi }{6},\infty )$

Let us first consider the case when y=|sin2x|, y=|x|+a touch each other.



If P is the point of contact, then at point P,
we have 1= 2cos2x (i.e slopes are equal)
$cos2x=\frac{1}{2}\Rightarrow x=\frac{\pi }{6}\therefore P=(\frac{\pi }{6},\frac{\sqrt{3}}{2})$
Now P lies on y=x+a
$\Rightarrow \frac{\sqrt{3}}{2}=\frac{\pi }{6}+a\Rightarrow a=\frac{3\sqrt{3}-\pi }{6}$
$\therefore$ For not to touch $a\in (\frac{3\sqrt{3}-\pi }{6},\infty )$