Question

The set of all the values of $x$ satisfying the inequality $\left(\right. \left(c o t\right)^{- 1} x \left.\right)^{2}-7\left(\right.\left(c o t\right)^{- 1}x\left.\right)+10>0$ is

• A. $\left(\right.-\in fty, \, cot \, 5\left.\right)\cup\left(\right.cot \, 4,cot 2\left.\right)$
• B. $\left(\right.cot \, 5,cot \, 4$ )
• C. $\left(\right.-\in fty, \, cot \, 5\left.\right)\cup\left(\right.cot \, 2,\in fty\left.\right)$
• D. $\left(c o t \, 2 , \, \in fty\right)$

Answer 1

Answer: (D)

$\left(\cot ^{-1} x-5\right)\left(\cot ^{-1} x-2\right)>0$
$\Rightarrow \cot ^{-1} x \in(-\infty, 2) \cup(5, \infty) \ldots$...(i)
But, $\cot ^{-1} x \in(0, \pi)$...(ii)
Taking intersection of (I) and (ii), we get, $\Rightarrow \cot ^{-1} x \in(0,2)$
$\Rightarrow \quad x \in(\cot 2, \infty)$