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BITSAT 2020
Exam
BITSAT
Year
2020
Subject
Mathematics
Chapter
Application of Derivatives

BITSAT 2020 · Mathematics · Application of Derivatives

Let x+y=3- 4 θ and x-y=4 2 θ then the greatest of x y is

Let x+y=3−cos⁡4θx+y=3-\cos 4 \theta and x−y=4sin⁡2θx-y=4 \sin 2 \theta then the greatest of xyx y is

Options

  1. A34\frac{3}{4}
  2. B1
  3. C12\frac{1}{2}
  4. D2

Correct answer

B. 1

Step-by-step solution

x=3−cos⁡4θ+4sin⁡2θ2=3−(1−sin⁡22θ)+4sin⁡2θ2=(1+sin⁡2θ)2y=3−cos⁡4θ−4sin⁡2θ2=3−(1−sin⁡22θ)+4sin⁡2θ2=(1−sin⁡2θ)2∴xy=(1−sin⁡22θ)2=cos⁡42θ≤1x=\frac{3-\cos 4 \theta+4 \sin 2 \theta}{2}=\frac{3-\left(1-\sin ^{2} 2 \theta\right)+4 \sin 2 \theta}{2}=(1+\sin 2 \theta)^{2}y=\frac{3-\cos 4 \theta-4 \sin 2 \theta}{2}=\frac{3-\left(1-\sin ^{2} 2 \theta\right)+4 \sin 2 \theta}{2}=(1-\sin 2 \theta)^{2}\therefore x y=\left(1-\sin ^{2} 2 \theta\right)^{2}=\cos ^{4} 2 \theta \leq 1
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