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) $(-\infty, \cot 5) \cup(\cot 4, \cot 2)$

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Answer:

Correct Answer: 1. (c)

Solution:

  1. 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 $\left.(0, \pi)\right]$

$\therefore x \in(\cot 2, \infty)$



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