Question

Let $S_1$ is the complete solution set of the inequality $co{s}^{-1}\left(x\right)>co{s}^{-1}\left(x^2\right)$ and $S_2$ is the complete solution set of the inequality ${\left(co{t}^{-1}x\right)}^2-5co{t}^{-1}x+6>0$ , then $S_1\cap S_2$ is

• A. $[-1,0)$
• B. $[cot3,0)$
• C. $[cot2,0)$
• D. $\left[-1,cot2\right]$

Answer 1

Answer: (C)

${\cos }^{-1}x>{\cos }^{-1}{x}^2\Rightarrow x<x^2$ \& $x\in [-1,1]$
$\{\because {\cos }^{-1}x$ is a decreasing function $\}$ $\Rightarrow x(x-1)>0\&x\in [-1,1]$
$\Rightarrow x\in (-\infty ,0)\cup [1,\infty )\&x\in [-1,1]$
$\Rightarrow x\in [-1,0)\dots (1)$
Now ${\left({\cot }^{-1}x\right)}^2-5{\cot }^{-1}x+6>0$
$\Rightarrow \left({\cot }^{-1}x-2\right)\left({\cot }^{-1}x-3\right)>0$
$\Rightarrow {\cot }^{-1}x<2$ or $x<\cot 3\dots$
$\{\because {\cot }^{-1}x$ is a decreasing function $\}$
$\left(1\right)\cap \left(2\right)\Rightarrow \left[cot2,0\right)$