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 :

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

Answer 1

Answer: (A)

$0 \leq x^{2}-x+1 \leq 1$
$\Rightarrow x^{2}-x \leq 0$
$\Rightarrow x \in[0,1]$
Also, $0 < \sin ^{-1}\left(\frac{2 x-1}{2}\right) \leq \frac{\pi}{2}$
$\Rightarrow 0 < \frac{2 x-1}{2} \leq 1$
$\Rightarrow 0 < 2 x-1 \leq 2$
$1 < 2 x \leq 3$
$\frac{1}{2} < x \leq \frac{3}{2}$
Taking intersection $x \in\left(\frac{1}{2}, 1\right]$
$\Rightarrow \alpha=\frac{1}{2}, \beta=1 $
$\Rightarrow \alpha+\beta=\frac{3}{2}$