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

NTA Abhyas 2020 · Mathematics · Trigonometric Equations

The sum of the roots of the equation | 3 cos x - sin x| = 2 in...

The sum of the roots of the equation ∣3cosx−sinx∣=2\left|\sqrt{3}cos x - sin x\right| = 2 in [0,4π]\left[0,4\pi\right] is kπk\pi, then the value of 6k6k is

Correct answer

56

Step-by-step solution

3cosx−sinx=±2\sqrt{3}cos x - sin x = \pm 2
32cosx−12sinx=±1\dfrac{\sqrt{3}}{2}cos x - \dfrac{1}{2}sin x = \pm 1
cos⁡(x+π6)=±1\cos \left(x + \dfrac{\pi}{6}\right) = \pm 1
⇒x+π6=nπ⇒ x + \dfrac{\pi}{6} = n\pi
⇒x=nπ−π6⇒ x = n\pi - \dfrac{\pi}{6}
⇒x=π−π6,2π−π6,3π−π6,4π−π6⇒ x = \pi - \dfrac{\pi}{6}, 2\pi - \dfrac{\pi}{6}, 3\pi - \dfrac{\pi}{6}, 4\pi - \dfrac{\pi}{6}
⇒⇒ The sum of roots is 10π−2π3=283π10\pi - \dfrac{2\pi}{3} = \dfrac{28}{3}\pi
⇒6k=6×283=56⇒ 6k = 6 \times \dfrac{28}{3} = 56
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