Question

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

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

Answer 1

Answer: (A)

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