Trigonometrical Equations 2 Question 1
1. All $x$ satisfying the inequality $\left(\cot ^{-1} x\right)^{2}-7\left(\cot ^{-1} x\right)+10>0$, lie in the interval
(2019 Main, 11 Jan II)
(a) $(-\infty, \cot 5) \cup(\cot 2, \infty)$
(b) $(\cot 5, \cot 4)$
(c) $(\cot 2, \infty)$
(d) $(-\infty, \cot 5) \cup(\cot 4, \cot 2)$
Show Answer
Answer:
Correct Answer: 1. (c)
Solution:
- Given, $\left(\cot ^{-1} x\right)^{2}-7\left(\cot ^{-1} x\right)+10>0$ $\Rightarrow\left(\cot ^{-1} x-2\right)\left(\cot ^{-1} x-5\right)>0$ $\Rightarrow \cot ^{-1} x<2$ or $\cot ^{-1} x>5$
(by factorisation)
By wavy curve method,
$\therefore \cot ^{-1} x \in(-\infty, 2) \cup(5, \infty)$ $\cot ^{-1} x \in(0,2) \quad\left[\because\right.$ Range of $\cot ^{-1} x$ is $\left.(0, \pi)\right]$
$\therefore x \in(\cot 2, \infty)$
