Question

Let $f(x)={\sin }^{-1}(2x-1)+{\cos }^{-1}\left(2\sqrt{x-x^2}\right)+{\tan }^{-1}\left(\frac{1}{1+\left[x^2\right]}\right)$ where $[k]$ denotes greatest integer less than or equal to $k$.

Column I		Column II
P	$f\left(\frac{1}{6}\right)$ is equal to	1	$\frac{\pi }{6}$
Q	$f\left(\frac{3}{4}\right)$ is equal to	2	$\frac{\pi }{4}$
R	${\sin }^{-1}(\tan (f(1)))$ is equal to	3	$\frac{\pi }{3}$
S	$\sum _{r=1}^{10}f\left(\frac{r}{20}\right)$ is equal to	4	$\frac{7\pi }{12}$
		5	$\frac{5\pi }{2}$

• A. $P\to 2$; Q $\to 4$; R $\to 1;S\to 3$
• B. $P\to 2;Q\to 4;R\to 1;S\to 5$
• C. $P\to 5;Q\to 4;R\to 1$; $S\to 3$
• D. $P\to 5;Q\to 4;R\to 1;S\to 5$

Answer 1

Answer: (B)

$(x)={\sin }^{-1}(2x-1)+{\cos }^{-1}\left(2\sqrt{x-x^2}\right)+{\tan }^{-1}\left(\frac{1}{1+\left[x^2\right]}\right)$
Domain of $f(x)=[0,1]$
$f(x)=\{\begin{array}{ll}0+\frac{\pi }{4}=\frac{\pi }{4},& x\in \left[0,\frac{1}{2}\right]\\ 2{\sin }^{-1}(2x-1)+\frac{\pi }{4},& x\in \left(\frac{1}{2},1\right)\\ \pi +{\tan }^{-1}\left(\frac{1}{2}\right),& x=1\end{array}$
Now, verify the options.