Question

Match the column

	List I		List II
A.	Range of $f\left(x\right)={\sin }^{-1}x+{\cos }^{-1}x+{\cot }^{-1}x$ is	1.	$[0,\frac{\pi }{2})\cup (\frac{\pi }{2},\pi ]$
B.	Range of $f\left(x\right)=\cot -1x+{\tan }^{-1}x+cose{c}^{-1}x$ is	2.	$[\frac{\pi }{2},\frac{3\pi }{2}]$
C.	Range of $f\left(x\right)={\cot }^{-1}x+{\tan }^{-1}x+{\cos }^{-1}x$ is	3.	$\{0,\pi \}$
D.	Range of $f\left(x\right)={\sec }^{-1}x+cose{c}^{-1}x+{\sin }^{-1}x$ is	4.	$(\frac{\pi }{2},\frac{3\pi }{2})$

• A. A- 1, B- 4, C- 2, D- 3
• B. A- 4, B- 1, C- 3, D- 2
• C. A- 4, B- 2, C- 1, D- 3
• D. A- 4, B- 1, C- 2, D- 3

Answer 1

Answer: (D)

A- 4, B- 1, C- 2, D- 3

A) $f\left(x\right)={\sin }^{-1}x+{\cos }^{-1}x+{\cot }^{-1}x=\frac{\pi }{2}+{\cot }^{-1}x$
${\cot }^{-1}xϵ(0,\pi )\Rightarrow f\left(x\right)ϵ(\frac{\pi }{2},\frac{3\pi }{2})$
B) $f\left(x\right)={\cot }^{-1}x+{\tan }^{-1}x+{\csc }^{-1}x=\frac{\pi }{2}+{\csc }^{-1}x$
As ${\csc }^{-1}xϵ[-\frac{\pi }{2},0)\cup \left(0\frac{\pi }{2}\right]\Rightarrow f\left(x\right)ϵ[0,\frac{\pi }{2})\cup (\frac{\pi }{2},\pi ]$
C) $f\left(x\right)={\cot }^{-1}x+{\tan }^{-1}x+{\cos }^{-1}x=\frac{\pi }{2}+{\cos }^{-1}x$
As ${\cos }^{-1}xϵ[0,\pi ]\Rightarrow f\left(x\right)=[\frac{\pi }{2},\frac{3\pi }{2}]$
D) $f\left(x\right)={\sec }^{-1}x+{\csc }^{-1}x+{\sin }^{-1}x$
As ${\sec }^{-1}xϵ[0,\frac{\pi }{2})\cup (\frac{\pi }{2},\pi ]$, ${\csc }^{-1}xϵ[-\frac{\pi }{2},0)\cup \left(0\frac{\pi }{2}\right]$ and ${\sin }^{-1}xϵ[-\frac{\pi }{2},\frac{\pi }{2}]$
$\Rightarrow f\left(x\right)ϵ\{0,\pi \}$