Question

The set of values of $x$ satisfying the inequation $(2-\frac{\pi }{2})({\cot }^{-1}x-3)+{\tan }^{-1}x({\cot }^{-1}x-3)>0$ is

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

Answer 1

The correct option is C $(\cot 3,\cot 2)$

Given : $(2-\frac{\pi }{2})({\cot }^{-1}x-3)+{\tan }^{-1}x({\cot }^{-1}x-3)>0$
$\Rightarrow ({\cot }^{-1}x-3)[2-(\frac{\pi }{2}-{\tan }^{-1}x)]>0\Rightarrow ({\cot }^{-1}x-3)(2-{\cot }^{-1}x)>0(\because {\tan }^{-1}x+{\cot }^{-1}x=\frac{\pi }{2})\Rightarrow ({\cot }^{-1}x-3)({\cot }^{-1}x-2)<0\Rightarrow {\cot }^{-1}x\in (2,3)$
Since ${\cot }^{-1}x$ is a decreasing function, so
$x\in (\cot 3,\cot 2)$