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NDA 2 2024 Maths, question 33

Topic: Trigonometry

Q33

If ptan⁡(θ−30∘)=qtan⁡(θ+120∘)p\tan (\theta-30^{\circ})=q\tan (\theta+120^{\circ}), then what is (p+q)/(p−q)(p+q)/(p-q) equal to?

  1. (a) sin⁡2θ\sin 2\theta
  2. (b) cos⁡2θ\cos 2\theta
  3. (c) 2sin⁡2θ2\sin 2\theta
  4. (d) 2cos⁡2θ2\cos 2\thetaCorrect answer

Solution.

Answer: (d) 2cos⁡2θ2\cos 2\theta

  1. Set A=θ−30° and B=θ+120°.
  2. The given relation implies (p+q)/(p−q)=(tan B+tan A)/(tan B−tan A).
  3. Convert the ratio to sin(A+B)/sin(B−A)=sin(2θ+90°)/sin150°=2cos2θ, wherever the displayed expressions are defined.