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MCQEasyJEE 2024Trigonometric Ratios & Identities

JEE Mathematics 2024 Question with Solution

For α,β∈(0,π/2)\alpha, \beta \in (0, \pi/2), let 3sin⁡(α+β)=2sin⁡(α−β)3\sin(\alpha + \beta) = 2\sin(\alpha - \beta), and a real number kk be such that tan⁡α=ktan⁡β\tan\alpha = k\tan\beta. Then the value of kk is equal to:

  • A

    23\frac{2}{3}

  • B

    −5-5

  • C

    32\frac{3}{2}

  • D

    55

Answer

Correct answer:B

Step-by-step solution

Standard Method

Given: 3sin⁡(α+β)=2sin⁡(α−β)3\sin(\alpha + \beta) = 2\sin(\alpha - \beta) and tan⁡α=ktan⁡β\tan\alpha = k\tan\beta, where α,β∈(0,π/2)\alpha, \beta \in (0, \pi/2).

Find: The value of kk.

Use the identities

sin⁡(α+β)=sin⁡αcos⁡β+cos⁡αsin⁡β\sin(\alpha + \beta) = \sin\alpha\cos\beta + \cos\alpha\sin\beta

and

sin⁡(α−β)=sin⁡αcos⁡β−cos⁡αsin⁡β.\sin(\alpha - \beta) = \sin\alpha\cos\beta - \cos\alpha\sin\beta.

Substituting into the given equation,

3(sin⁡αcos⁡β+cos⁡αsin⁡β)=2(sin⁡αcos⁡β−cos⁡αsin⁡β)3(\sin\alpha\cos\beta + \cos\alpha\sin\beta) = 2(\sin\alpha\cos\beta - \cos\alpha\sin\beta)

Expanding,

3sin⁡αcos⁡β+3cos⁡αsin⁡β=2sin⁡αcos⁡β−2cos⁡αsin⁡β3\sin\alpha\cos\beta + 3\cos\alpha\sin\beta = 2\sin\alpha\cos\beta - 2\cos\alpha\sin\beta

Rearranging terms,

3sin⁡αcos⁡β−2sin⁡αcos⁡β=−3cos⁡αsin⁡β−2cos⁡αsin⁡β3\sin\alpha\cos\beta - 2\sin\alpha\cos\beta = -3\cos\alpha\sin\beta - 2\cos\alpha\sin\beta

So,

sin⁡αcos⁡β=−5cos⁡αsin⁡β\sin\alpha\cos\beta = -5\cos\alpha\sin\beta

Dividing both sides by cos⁡αcos⁡β\cos\alpha\cos\beta,

tan⁡α=−5tan⁡β\tan\alpha = -5\tan\beta

Comparing with tan⁡α=ktan⁡β\tan\alpha = k\tan\beta, we get

k=−5k = -5

Therefore, the correct option is B.

Direct Comparison After Expansion

Given: 3sin⁡(α+β)=2sin⁡(α−β)3\sin(\alpha + \beta) = 2\sin(\alpha - \beta).

Find: The value of kk in tan⁡α=ktan⁡β\tan\alpha = k\tan\beta.

Expand both sine terms and immediately collect like terms:

3(sin⁡αcos⁡β+cos⁡αsin⁡β)=2(sin⁡αcos⁡β−cos⁡αsin⁡β)3(\sin\alpha\cos\beta + \cos\alpha\sin\beta) = 2(\sin\alpha\cos\beta - \cos\alpha\sin\beta)

This gives

sin⁡αcos⁡β=−5cos⁡αsin⁡β\sin\alpha\cos\beta = -5\cos\alpha\sin\beta

Now divide by cos⁡αcos⁡β\cos\alpha\cos\beta to convert the expression into tangents:

tan⁡α=−5tan⁡β\tan\alpha = -5\tan\beta

Hence,

k=−5k = -5

Therefore, the correct option is B.

Common mistakes

  • A common mistake is using the identity for sin⁡(α−β)\sin(\alpha - \beta) with a plus sign. That changes the equation completely and gives a wrong value of kk. Use sin⁡(α−β)=sin⁡αcos⁡β−cos⁡αsin⁡β\sin(\alpha - \beta) = \sin\alpha\cos\beta - \cos\alpha\sin\beta carefully.

  • Some students move terms across the equation incorrectly and obtain +5+5 instead of −5-5. The sign changes during rearrangement are crucial. Collect the cos⁡αsin⁡β\cos\alpha\sin\beta terms with attention to the negative sign.

  • Another mistake is comparing the result before dividing by cos⁡αcos⁡β\cos\alpha\cos\beta. The given relation involves tan⁡α\tan\alpha and tan⁡β\tan\beta, so you must divide by cos⁡αcos⁡β\cos\alpha\cos\beta to convert the equation into tangent form.

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