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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 solution of the equation ∑ _ r = 1 ^ 5 (rx) = 0 lying...

The number of solution of the equation ∑r=15cos⁡(rx)=0{∑}_{r = 1}^{5}\cos \left(rx\right) = 0 lying in \left(0, \pi\right)$ is

Correct answer

5

Step-by-step solution

cosx+cos⁡2x+cos⁡3x+cos⁡4x+cos⁡5x=0cos x + \cos 2x + \cos 3x + \cos 4x + \cos 5x = 0
⇒sin5x2sinx2cos⁡3x=0⇒ \dfrac{sin \dfrac{5x}{2}}{sin \dfrac{x}{2}}\cos 3x = 0
⇒sin5x2=0⇒ sin \dfrac{5x}{2} = 0 or cos⁡3x=0\cos 3x = 0
but sinx2≠0sin \dfrac{x}{2} \ne 0
If x∈(0,π)⇒5x2∈(0,5π2)x \in \left(0, \pi\right) ⇒ \dfrac{5x}{2} \in \left(0, \dfrac{5\pi}{2}\right)
⇒⇒ Two solutions \left(\dfrac{5x}{2} = \pi, 2\pi\right)<br/>If<br />Ifx \in \left(0, \pi\right) ⇒ 3x \in \left(0, 3\pi\right)<br/><br />⇒ 3solutions(3x=π2,3π2,5π2)solutions \left(3x = \dfrac{\pi}{2}, \dfrac{3\pi}{2}, \dfrac{5\pi}{2}\right)
So, total five solutions
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