SATHEE: Trigonometrical Equations 2 Question 1

Trigonometrical Equations 2 Question 1

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

(2019 Main, 11 Jan II)

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

(b) $(\cot 5, \cot 4)$

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

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Answer:

Correct Answer: 1. (c)

Solution:

Given, ${(\cot^{- 1} x)}^{2} - 7 (\cot^{- 1} x) + 10 > 0$ $\Rightarrow (\cot^{- 1} x - 2) (\cot^{- 1} x - 5) > 0$ $\Rightarrow \cot^{- 1} x < 2$ or $\cot^{- 1} x > 5$

(by factorisation)

By wavy curve method,

$∴ \cot^{- 1} x \in (- \infty, 2) \cup (5, \infty)$ $\cot^{- 1} x \in (0, 2) [∵$ Range of $\cot^{- 1} x$ is $(0, π)]$

$∴ x \in (\cot 2, \infty)$

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Trigonometrical Equations 2 Question 1

1. All x satisfying the inequality (cot−1⁡x)2−7(cot−1⁡x)+10>0, lie in the interval

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