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JEE Main 201610 AprilOnline
Exam
JEE Main
Year
2016
Date
10 April
Mode
Online
Subject
Mathematics
Chapter
Trigonometric Equations

JEE Main 2016 · 10 April · Online · Mathematics · Trigonometric Equations

Let P = \ θ: sinθ - cosθ = 2 cosθ\ and Q = \ θ: sinθ + cosθ = 2...

Let P={θ:sinθ−cosθ=2cosθ}P = \left\{\theta: sin\theta - cos\theta = \sqrt{2}cos\theta\right\} and Q={θ:sinθ+cosθ=2sinθ},Q = \left\{\theta: sin\theta + cos\theta = \sqrt{2}sin\theta\right\}, be two sets. Then

Options

  1. AP⊂QP ⊂ Q and Q−P≠ϕQ - P \ne ϕ
  2. BQ⊄PQ ⊄ P
  3. CP=QP = Q
  4. DP⊄QP ⊄ Q

Correct answer

C. P=QP = Q

Step-by-step solution

P:sinθ−cosθ=2cosθP: sin\theta - cos\theta = \sqrt{2}cos\theta

⇒sinθ=cosθ+2cosθ⇒ sin\theta = cos\theta + \sqrt{2}cos\theta

tanθ=2+1tan\theta = \sqrt{2} + 1

Q:sinθ+cosθ=2sinθ⇒(2−1)sinθ=cosθQ: sin\theta + cos\theta = \sqrt{2}sin\theta ⇒ \left(\sqrt{2} - 1\right)sin\theta = cos\theta

=tanθ=12−1=2+1= tan\theta = \dfrac{1}{\sqrt{2} - 1} = \sqrt{2} + 1

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