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1.Relation and Function
hard
If the domain of the function $f(\mathrm{x})=\frac{\cos ^{-1} \sqrt{x^{2}-x+1}}{\sqrt{\sin ^{-1}\left(\frac{2 x-1}{2}\right)}}$ is the interval $(\alpha, \beta]$, then $\alpha+\beta$ is equal to:
A
$2$
B
$\frac{3}{2}$
C
$\frac{1}{2}$
D
$1$
(JEE MAIN-2021)
Solution
$0 \leq x^{2}-x+1 \leq 1$
$\Rightarrow x^{2}-x \leq 0$
$\Rightarrow x \in[0,1]$
$\text { Also, } 0\,<\,\sin ^{-1}\left(\frac{2 x-1}{2}\right) \leq \frac{\pi}{2}$
$\Rightarrow 0\,<\,\frac{2 x-1}{2} \leq 1$
$\Rightarrow 0\,<\,2 x-1 \leq 2$
$1\,<\,2 x \leq 3$
$\frac{1}{2}\,<\,x \leq \frac{3}{2}$
Taking intersection
$x \in\left(\frac{1}{2}, 1\right]$
$\Rightarrow \alpha=\frac{1}{2}, \beta=1$
$\Rightarrow \alpha=\frac{1}{2}, \beta=1$
Standard 12
Mathematics
