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NTA Abhyas 2020
Exam
NTA Abhyas
Year
2020
Subject
Mathematics
Chapter
Inverse Trigonometric Functions

NTA Abhyas 2020 · Mathematics · Inverse Trigonometric Functions

Let S _ 1 is the complete solution set of the inequality cos^ - 1...

Let S1isthecompletesolutionsetoftheinequality{S}_{1} is the complete solution set of the inequalitycos^{- 1} \left(x\right) > cos^{- 1} \left(x^{2}\right) and {S}_{2}is the complete solution set of theinequality(cot−1x)the inequality {\left(cot^{- 1}x\right)}^{2} -5cot−1x+6>05cot^{- 1}x + 6 > 0, then S1∩S2{S}_{1} \cap {S}_{2} isLet S _ 1 is the complete solution set of the inequality cos^ - 1 (x) > cos^ - 1 (x^ 2 ) and S _ 2 is the complete

Options

  1. A[−1,0)\left[- 1,0\right)
  2. B[cot⁡3,0)\left[\cot 3,0\right)
  3. C[cot⁡2,0)\left[\cot 2, 0\right)
  4. D[−1,cot⁡2]\left[- 1, \cot 2\right]

Correct answer

C. [cot⁡2,0)\left[\cot 2, 0\right)

Step-by-step solution

cos−1x>cos−1x2⇒x<x2x∈[−1,1]<br/>∵ cos −1xisadecreasingfunctioncos^{- 1}x > cos^{- 1}x^{2}⇒ x < x^{2} x \in \left[-1, 1\right]<br />{ ∵\text{ cos }^{- 1}x is a decreasing function}
⇒x(x–1)>0x∈[−1,1]<br/>⇒x \left(x – 1\right) > 0 x \in \left[-1,1\right]<br />⇒x \in \left(- \infty,0\right) \cup \left[1, \infty\right)x∈[−1,1]x \in \left[-1, 1\right]
⇒x∈[−1,0)…(1)<br/>Now⇒ x \in \left[-1, 0\right) … \left(1\right)<br />Now{ \left(cot^{- 1}x\right)}^{2} - 5cot^{- 1}x + 6 > 0
⇒(cot−1x−2)(cot−1x−3)⇒ \left(cot^{- 1}x - 2\right) \left(cot^{- 1}x - 3\right) > 0<br/><br />⇒ cot^{- 1}x < 2 or x < \cot 3…(2)<br/>∵ cot −1xisadecreasingfunction… \left(2\right)<br />{ ∵\text{ cot }^{- 1}x is a decreasing function}
Let S _ 1 is the complete solution set of the inequality cos^ - 1 (x) > cos^ - 1 (x^ 2 ) and S _ 2 is the complete
(1)\left(1\right)\cap \left(2\right)⇒[cot⁡2,0)⇒ \left[\cot 2, 0\right)
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