Question:
If the domain of the function $f(x)=\frac{\cos ^{-1} \sqrt{x^{2}-x+1}}{\sqrt{\sin ^{-1}\left(\frac{2 x-1}{2}\right)}}$ is the interval $(\alpha, \beta]$, then $\alpha+\beta$ is equal to :
Correct Option: 1
Solution:
$O \leq x^{2}-x+1 \leq 1$
$\Rightarrow x^{2}-x \leq 0$
$\Rightarrow x \in[0,1]$
Also, $0<\sin ^{-1}\left(\frac{2 x-1}{2}\right) \leq \frac{\pi}{2}$
$\Rightarrow 0<\frac{2 x-1}{2} \leq 1$
$\Rightarrow 0<2 \mathrm{x}-1 \leq 2$
$1<2 x \leq 3$
$\frac{1}{2} Taking intersection $x \in\left(\frac{1}{2}, 1\right)$ $\Rightarrow \alpha=\frac{1}{2}, \beta=1$ $\Rightarrow \alpha+\beta=\frac{3}{2}$