Question

If the domain of the function $f\left(x\right)=\frac{{\cos }^{-1}\sqrt{x^2-x+1}}{\sqrt{{\sin }^{-1}\left(\frac{2x-1}{2}\right)}}$ is the interval $(\alpha ,\beta ],$ then $\alpha +\beta$ is equal to:

• A. $1$
• B. $\frac{3}{2}$
• C. $\frac{1}{2}$
• D. $2$

Answer 1

The correct option is B $\frac{3}{2}$

$f\left(x\right)=\frac{{\cos }^{-1}\left(\sqrt{x^2-x+1}\right)}{\sqrt{{\sin }^{-1}\left(\frac{2x-1}{2}\right)}}$

$0<\frac{2x-1}{2}\le 1$

$\Rightarrow 0<2x-1\le 2$

$\Rightarrow 1<2x\le 3$

$\Rightarrow \frac{1}{2}<x\le \frac{3}{2}\cdots \left(i\right)$

and $0\le x^2-x+1\le 1$

$x^2-x\le 0$

$x(x-1)\le 0$

$0\le x\le 1\cdots \left(ii\right)$

from $\left(i\right),\left(ii\right)$

$\therefore$ Domain is $x\in (\frac{1}{2},1]=(\alpha ,\beta ]$
$\Rightarrow \alpha +\beta =\frac{3}{2}$