Question

Match the following by appropriately matching the lists based on the information given in $\text{Column I}$ and $\text{Column II}$.

$\begin{array}{cc}ColumnI& ColumnII\\ \text{a. Range of}f\left(x\right)={\sin }^{-1}x+{\cos }^{-1}x+{\cot }^{-1}x\text{is}& \text{p.}[0,\frac{\pi }{2})\cup (\frac{\pi }{2},\pi ]\\ \text{b. Range of}f\left(x\right)={\cot }^{-1}x+{\tan }^{-1}x+{\text{cosec}}^{-1}x\text{is}& \text{q.}[\frac{\pi }{2},\frac{3\pi }{2}]\\ \text{c. Range of}f\left(x\right)={\cot }^{-1}x+{\tan }^{-1}x+{\cos }^{-1}x\text{is}& \text{r.}\{0,\pi \}\\ \text{d. Range of}f\left(x\right)={\sec }^{-1}x+{\text{cosec}}^{-1}x+{\sin }^{-1}x\text{is}& \text{s.}[\frac{3\pi }{4},\frac{5\pi }{4}]\end{array}$

• A. $a-p;b-q;c-r;d-s$
• B. $a-s;b-r;c-q;d-p$
• C. $a-s;b-p;c-q;d-r$
• D. $a-q;b-s;c-p;d-r$

Answer 1

The correct option is C $a-s;b-p;c-q;d-r$

a. $f\left(x\right)={\sin }^{-1}x+{\cos }^{-1}x+{\cot }^{-1}x=\frac{\pi }{2}+{\cot }^{-1}x,x\in [-1,1]$
For $x\in [-1,1],{\cot }^{-1}x\in [\frac{\pi }{4},\frac{3\pi }{4}]$
$\Rightarrow \frac{\pi }{2}+{\cot }^{-1}x\in [\frac{3\pi }{4},\frac{5\pi }{4}]$

b. $f\left(x\right)={\cot }^{-1}x+{\tan }^{-1}x+{cosec}^{-1}x$
$=\frac{\pi }{2}+{cosec}^{-1}x,\text{where}x\in (-\infty ,-1]\cup [1,\infty )$
Now ${cosec}^{-1}x\in [-\frac{\pi }{2},0)\cup (0,\frac{\pi }{2}]$
$\Rightarrow \frac{\pi }{2}+{cosec}^{-1}x\in [0,\frac{\pi }{2})\cup (\frac{\pi }{2},\pi ]$

c. $f\left(x\right)={\cot }^{-1}x+{\tan }^{-1}x+{\cos }^{-1}x$
$=\frac{\pi }{2}+{\cos }^{-1}x,$ where $x\in [-1,1]$
$\Rightarrow \frac{\pi }{2}+{\cos }^{-1}x\in [\frac{\pi }{2},\frac{3\pi }{2}]$

d. ${\sec }^{-1}x+{cosec}^{-1}x+{\sin }^{-1}x,$ where $x\in \{-1,1\}$
$=\frac{\pi }{2}+{\sin }^{-1}x,$ where $x\in \{-1,1\}$
Hence, $f\left(x\right)\in \{0,\pi \}$