Question

The sum of the solutions of the equation$2{\sin }^{-1}\sqrt{x^2+x+1}+{\cos }^{-1}\sqrt{x^2+x}=\frac{3\pi }{2}$ is

• A. 0
• B. -1
• C. 1
• D. 2

Answer 1

The correct option is B -1

Let $f\left(x\right)=2si{n}^{-1}\left(\sqrt{x^2+x+1}\right)+co{s}^{-1}\left(\sqrt{x^2+x}\right)=\frac{3\pi }{2}$
Hence
$f\left(0\right)=2si{n}^{-1}\left(1\right)+co{s}^{-1}\left(0\right)=\frac{3\pi }{2}$
$=0$
Hence $x=0$ is a root of $f\left(x\right)=0$
$f(-1)=2si{n}^{-1}\left(1\right)+co{s}^{-1}\left(0\right)=\frac{3\pi }{2}$
$=0$
Hence $x=-1$ is another root of $f\left(x\right)=0$
Therefore sum of the roots will be
$0+(-1)$
$=-1$