Question

Let $f(x)={\sin }^{-1}x+{\cos }^{-1}x+\pi (|x|-2)$. The true set of values of $k$ for which the equation $f(x)=k\pi$ possesses real solution is $[a,b]$ then the value of $2|a|+4|b|$ is equal to

• A. 2
• B. 3
• C. 5
• D. 7

Answer 1

Answer: (C)

$f(x)=\frac{\pi }{2}+\pi (|x|-2)x\in [-1,1]$
Equation $f(x)=k\pi \Rightarrow \frac{1}{2}+|x|-2=k$
$\text{ or }|x|=k+\frac{3}{2}\in [0,1]$
$\Rightarrow 0\le k+\frac{3}{2}\le 1\Rightarrow -\frac{3}{2}\le k\le -\frac{1}{2}$
$\Rightarrow a=\frac{-3}{2}\text{ and }b=\frac{-1}{2}$
$\therefore 2|a|+4|b|=5$