Question

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

• A. $- 1 , 0$
• B. $\cot 3,0$
• C. $\cot 2 , 0$
• D. $(- 1 , \cot 2)$

Answer 1

Answer: (C)

$\cos ^{-1} x >\cos ^{-1} x^{2} $
$\Rightarrow x< x^{2} \, \&\, x \in-1,1$
$\left\{\because \cos ^{-1} x\right.$ 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 $\cot ^{-1} x^{2}-5 \cot ^{-1} x+6>0$
$\Rightarrow \cot ^{-1} x-2 \cot ^{-1} x-3>0 $
$\Rightarrow \cot ^{-1} x< 2 \text { or }^{-1} \cot ^{-1} x >3$
$x >\cot 2$ or $x< \cot 3 \ldots 2$
$\left\{\because \cot ^{-1} x\right.$ is a decreasing function $\}$

$1 \cap 2 \Rightarrow x \in \cot 2,0$