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NTA Abhyas 2020
Exam
NTA Abhyas
Year
2020
Subject
Mathematics
Chapter
Trigonometric Equations

NTA Abhyas 2020 · Mathematics · Trigonometric Equations

The number of solutions, the equation ^ 4 x + ^ 4 x = x x has, in [...

The number of solutions, the equation sin⁡4x+cos⁡4x=sin⁡xcos⁡x\sin^{4}x + \cos^{4}x = \sin x \cos x has, in [π,5π][ \pi, 5\pi ] is/are

Correct answer

4

Step-by-step solution

sin⁡4x+cos⁡4x=sin⁡xcos⁡x\sin^{4}x + \cos^{4}x = \sin x \cos x
⇒(sin⁡2x+cos⁡2x)2−2sin⁡2xcos⁡2x=sin⁡xcos⁡x⇒ {( \sin^{2}x + \cos^{2}x )}^{2} - 2\sin^{2}x\cos^{2}x = \sin x \cos x
⇒2sin⁡2xcos⁡2x+sin⁡xcos⁡x−1=0⇒ 2\sin^{2}x\cos^{2}x + \sin x \cos x - 1 = 0
⇒(2sin⁡xcos⁡x−1)(sin⁡xcos⁡x+1)=0⇒ \left(2\sin x \cos x - 1\right)\left(\sin x \cos x + 1\right) = 0
⇒sin⁡2x=1⇒ \sin 2x = 1 …(i)
x∈[π,5π]⇒2x∈[2π,10π]x \in \left[\pi, 5\pi\right] ⇒ 2x \in \left[2\pi, 10\pi\right]
from (i), 2x=5π2,9π2,13π2,17π22x = \dfrac{5\pi}{2}, \dfrac{9\pi}{2}, \dfrac{13\pi}{2}, \dfrac{17\pi}{2}
⇒x=5π4,9π4,13π4,17π4⇒ x = \dfrac{5\pi}{4}, \dfrac{9\pi}{4}, \dfrac{13\pi}{4}, \dfrac{17\pi}{4}
⇒⇒ The number of solutions =4= 4
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