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JEE Advanced

Exam: JEE Advanced
Subject: Mathematics
Chapter: Trigonometry

JEE Advanced · Mathematics · Trigonometry

Which of the following sets can be the subset of the general...

Which of the following sets can be the subset of the general solution of the equation

1+\cos 3 x=2 \cos 2 x ?

Options

A $n \pi+\frac{\pi}{3}$
B $\mathrm{n} \pi+\frac{\pi}{6}$
C $n \pi-\frac{\pi}{6}$
D $2 n \pi$

Correct answer

D. $2 n \pi$

Step-by-step solution

\begin{aligned} & 1+4 \cos ^3 x-3 \cos x=2\left(2 \cos ^2 x-1\right) \\ & 4 \cos ^3 x-4 \cos ^2 x-3 \cos x+3=0 \\ & \text { let } \cos x=t\end{aligned}

\begin{aligned} & 4 t^3-4 t^2-3 t+3=0 \\ & \Rightarrow 4 t^2(t-1)-3(t-1)=0 \\ & \Rightarrow \quad(t-1)\left(4 t^2-3\right)=0\end{aligned}

t=1

or

t^2=\frac{3}{4}\cos x=1

or

\cos ^2 x=\frac{3}{4}\cos x=1

or

\cos ^2 x=\cos ^2 \frac{\pi}{6}x=2 n \pi

or

x=n \pi \pm \frac{\pi}{6}$

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