Question

The range of the function $y=2sin^{- 1} \left[x^{2} + \frac{1}{2}\right]+cos^{- 1}⁡\left[x^{2} - \frac{1}{2}\right]$ is (where, $\left[\cdot \right]$ denotes the greatest integer function)

• A. $\left(0 , \pi \right)$
• B. $\left[\pi , \frac{3 \pi }{2}\right]$
• C. $\left\{\pi \right\}$
• D. $\left\{\pi , \frac{3 \pi }{2}\right\}$

Answer 1

Answer: (D)

Let, $\left[x^{2} - \frac{1}{2}\right]=\left[x^{2} + \frac{1}{2} - 1\right]=\alpha \in I$
$\Rightarrow \left[x^{2} + \frac{1}{2}\right]-1=\alpha $
$\therefore y=2\left(sin\right)^{- 1} \left(\alpha + 1\right)+\left(cos\right)^{- 1}⁡\alpha , \, \alpha =\left\{- 1,0\right\}$
At $\alpha =-1$
$y=2\left(sin\right)^{- 1} \left(0\right)+\left(cos\right)^{- 1} ⁡ \left(- 1\right)=\pi $
At $\alpha =0$
$y=2\left(sin\right)^{- 1} \left(1\right)+\left(cos\right)^{- 1} ⁡ \left(0\right)=\frac{3 \pi }{2}$