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JEE Main 202430 JanuaryEvening Shift
Exam
JEE Main
Year
2024
Date
30 January
Shift
Evening Shift
Subject
Mathematics
Chapter
Trigonometric Ratios & Identities

JEE Main 2024 · 30 January · Evening Shift · Mathematics · Trigonometric Ratios & Identities

For α, β ∈ (0, (π)/(2)) let 3 ( α + β ) = 2 ( α - β ) and a real...

For α,β∈\alpha, \beta \in\left(0, \dfrac{\pi}{2}\right)letlet3\sin ( \alpha + \beta ) = 2\sin ( \alpha - \beta ) and a real number kk be such that tanα=tanβtan\alpha = tan\beta. Then the value ofkk is equal to

Options

  1. A−5- 5
  2. B55
  3. C23\dfrac{2}{3}
  4. D−23- \dfrac{2}{3}

Correct answer

A. −5- 5

Step-by-step solution

Given:

 3sin⁡(α+β)=2sin⁡(α−β)3\sin \left(\alpha + \beta\right) = 2\sin \left(\alpha - \beta\right)

⇒sin⁡(α+β)sin⁡(α−β)=23⇒ \dfrac{\sin \left(\alpha + \beta\right)}{\sin \left(\alpha - \beta\right)} = \dfrac{2}{3}

⇒sin⁡(α+β)+sin⁡(α−β)sin⁡(α+β)−sin⁡(α−β)=2+32−3⇒ \dfrac{\sin \left(\alpha + \beta\right) + \sin \left(\alpha - \beta\right)}{\sin \left(\alpha + \beta\right) - \sin \left(\alpha - \beta\right)} = \dfrac{2 + 3}{2 - 3}

⇒2sinαcosβ2cosαsinβ=−5⇒ \dfrac{2sin\alpha cos\beta}{2cos\alpha sin\beta} = - 5

⇒tanαtanβ=−5⇒ \dfrac{tan\alpha}{tan\beta} = - 5

⇒tanα=−5tanβ⇒ tan\alpha = - 5tan\beta

⇒k=−5⇒ k = - 5

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