If the domain of the function


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 :

  1. $\frac{3}{2}$

  2. 2

  3. $\frac{1}{2}$

  4. 1

Correct Option: 1


$O \leq x^{2}-x+1 \leq 1$

$\Rightarrow x^{2}-x \leq 0$

$\Rightarrow x \in[0,1]$

Also, $0<\sin ^{-1}\left(\frac{2 x-1}{2}\right) \leq \frac{\pi}{2}$

$\Rightarrow 0<\frac{2 x-1}{2} \leq 1$

$\Rightarrow 0<2 \mathrm{x}-1 \leq 2$

$1<2 x \leq 3$


Taking intersection

$x \in\left(\frac{1}{2}, 1\right)$

$\Rightarrow \alpha=\frac{1}{2}, \beta=1$

$\Rightarrow \alpha+\beta=\frac{3}{2}$

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