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JEE Advanced 2015Paper 1
Exam
JEE Advanced
Year
2015
Paper
Paper 1
Subject
Mathematics
Chapter
Trigonometric Equations

JEE Advanced 2015 · Paper 1 · Mathematics · Trigonometric Equations

The number of distinct solutions of the equation (5)/(4) ^ 2 2x + ^...

The number of distinct solutions of the equation 54cos⁡22x+cos⁡4x+sin⁡4x+cos⁡6x+sin⁡6x=2\dfrac{5}{4}\cos^{2} 2x + \cos^{4} x + \sin^{4} x + \cos^{6} x + \sin^{6} x = 2 in the interval [0,2π][ 0, 2\pi ] is

Correct answer

8

Step-by-step solution

54cos⁡22x+(sin⁡2x+cos⁡2x)2+(sin⁡2x+cos⁡2x)3−5sin⁡2x.cos⁡2x=2\dfrac{5}{4}\cos^{2} 2x +{\left(\sin^{2} x + \cos^{2} x\right)}^{2} + {\left(\sin^{2} x + \cos^{2} x\right)}^{3} - 5\sin^{2} x . \cos^{2} x = 2
⇒54(1−sin⁡22x)−54sin⁡22x+2=2⇒ \dfrac{5}{4}\left(1 - \sin^{2} 2x\right) - \dfrac{5}{4}\sin^{2} 2x + 2 = 2
⇒1−2sin⁡22x=0⇒ 1 - 2\sin^{2} 2x = 0
⇒sin⁡22x=12⇒ \sin^{2} 2x =\dfrac{1}{2}
⇒2x=nπ±π4⇒ 2x = n\pi \pm \dfrac{\pi}{4}
⇒x=nπ2±π8⇒ x = \dfrac{n\pi}{2} \pm \dfrac{\pi}{8}
Number of solution = 8
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