Question

All $x$ satisfying the inequality $({\cot }^{-1}x{)}^2-7({\cot }^{-1}x)+10>0$, lie in the interval :

• A. $(\cot 5,\cot 4)$
• B. $(\cot 2,\infty )$
• C. $(-\infty ,\cot 5)\cup (\cot 4,\cot 2)$
• D. $(-\infty ,\cot 5)\cup (\cot 2,\infty )$

Answer 1

The correct option is B $(\cot 2,\infty )$

$({\cot }^{-1}x{)}^2-7({\cot }^{-1}x)+10>0\Rightarrow ({\cot }^{-1}x{)}^2-5\left({\cot }^{-1}x\right)-2\left({\cot }^{-1}x\right)+10>0\Rightarrow \left({\cot }^{-1}x\right)\left[\right({\cot }^{-1}x)-5]-2\left[\right({\cot }^{-1}x)-5]>0\Rightarrow ({\cot }^{-1}x-2)({\cot }^{-1}x-5)>0\Rightarrow {\cot }^{-1}x\in (-\infty ,2)\cup (5,\infty )\cdots \left(1\right)$

We know that ${\cot }^{-1}x\in (0,\pi )\cdots \left(2\right)$
So, from $\left(1\right)$ and $\left(2\right)$
$0<{\cot }^{-1}x<2$
As, ${\cot }^{-1}x$ is decreasing function,
$\Rightarrow x\in (\cot 2,\infty )$