Question

If $\left(\cot ^{-1} x\right)^{2}-7\left(\cot ^{-1} x\right)+10>0$ , then the range of $x$ will be

• A. $\left(\right.-\in fty, \, cot 2\left.\right)$
• B. $\left(- \in fty , \, cot 5\right)$
• C. $\left(cot 2 , \, cot ⁡ 5\right)$
• D. $\left(cot 2 , \, \in fty\right)$

Answer 1

Answer: (D)

$\left(\left(cot\right)^{- 1} x - 5\right)\left(\left(cot\right)^{- 1} ⁡ x - 2\right)>0$
$\Rightarrow \left(cot\right)^{- 1} x \in \left(- \in fty , \, 2\right)\cup\left(5 , \, \in fty\right)$
$\Rightarrow \left(cot\right)^{- 1} x\in \left(0 , \, 2\right)$ (Taking intersection with range of $cot^{- 1} x$ )
$\Rightarrow n\in \left(\right.cot 2 , \, \in fty \left.\right)$