Question

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

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

Answer 1

The correct option is D $(-\infty ,\cot 2)\cup (\cot 5,\infty )$

let ${\cot }^{-1}x=t$

$t^2-7t+10>0$

on factorizing,

$(t-2)(t-5)>0$

${\cot }^{-1}x=2,5$

$x=\cot 2,\cot 5$

on plotting the $x$ on wavy curve

we get $xliein(-\infty ,\cot 2)\cup (\cot 5,\infty )$