Question

Let $S_{1}$ is the complete solution set of the inequality $cos^{- 1}\left(x\right)>cos^{- 1}\left(x^{2}\right)$ and $S_{2}$ is the complete solution set of the inequality $\left(c o t^{- 1} x\right)^{2}-5cot^{- 1}x+6>0$ , then $S_{1}\cap S_{2}$ is

• A. $[- 1,0)$
• B. $[cot 3,0)$
• C. $[cot 2 , 0)$
• D. $\left[- 1 , c o t 2\right]$

Answer 1

Answer: (C)

$\cos ^{-1} x>\cos ^{-1} x^{2} \Rightarrow x $\left\{\because \cos ^{-1} x\right.$ is a decreasing function $\}$ $\Rightarrow x(x-1)>0 \& x \in[-1,1]$
$\Rightarrow x \in(-\infty, 0) \cup[1, \infty) \& x \in[-1,1]$
$\Rightarrow x \in[-1,0) \ldots(1)$
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 $x<\cot 3 \ldots$
$\left\{\because \cot ^{-1} x\right.$ is a decreasing function $\}$
$\left(1\right)\cap \left(2\right)\Rightarrow \left[c o t 2 , 0\right)$