The number of solutions of the equation 2 θ (θ)/(2)+ (5 θ)/(2)=2 ^3 (5 θ)/(2) in [-(π)/(2), (π)/(2)] is :

Correct answer: A. 7 aligned & 2 θ (θ)/(2)+ (5 θ)/(2)=2 ^3 (5 θ)/(2) & (1)/(2)(2 2 θ (θ)/(2))+ (50)/(2) & =(1)/(2)( (15 θ)/(2)+3 (5 θ)/(2)) aligned or solving aligned & (3 θ)/(2)= (15 θ)/(2) & (15 θ)/(2)- (3 θ)/(2)=0 & 2 30 (9 θ)/(2)=0 aligned 3 θ=n π or (9 θ)/(2)=m π aligned & θ= n π 3 θ= 2 m π 9 & θ=\ -(π)/(2), (π)/(3), 0\ & θ=\ -(4 π)/(9), (-2 π)/(9), (4 π)/(9), (2 π)/(9)\ aligned

JEE Main 20257 AprilEvening Shift

Exam: JEE Main
Year: 2025
Date: 7 April
Shift: Evening Shift
Subject: Mathematics
Chapter: Trigonometric Equations

JEE Main 2025 · 7 April · Evening Shift · Mathematics · Trigonometric Equations

The number of solutions of the equation 2 θ (θ)/(2)+ (5 θ)/(2)=2 ^3...

The number of solutions of the equation

\cos 2 \theta \cos \frac{\theta}{2}+\cos \frac{5 \theta}{2}=2 \cos ^3 \frac{5 \theta}{2}

\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]

is :

Options

Correct answer

A. 7

Step-by-step solution

\begin{aligned}& \cos 2 \theta \cos \frac{\theta}{2}+\cos \frac{5 \theta}{2}=2 \cos ^3 \frac{5 \theta}{2} \\ & \frac{1}{2}\left(2 \cos 2 \theta \cos \frac{\theta}{2}\right)+\cos \frac{50}{2} \\ & =\frac{1}{2}\left(\cos \frac{15 \theta}{2}+3 \cos \frac{5 \theta}{2}\right)\end{aligned}

or solving

\begin{aligned}& \cos \frac{3 \theta}{2}=\cos \frac{15 \theta}{2} \\ & \cos \frac{15 \theta}{2}-\cos \frac{3 \theta}{2}=0 \\ & 2 \sin 30 \sin \frac{9 \theta}{2}=0\end{aligned}

3 \theta=\mathrm{n} \pi \text{\text{ or } } \frac{9 \theta}{2}=\mathrm{m} \pi

\begin{aligned}& \theta=\frac{\mathrm{n} \pi}{3} \quad \theta=\frac{2 \mathrm{ m} \pi}{9} \\ & \theta=\left\{-\frac{\pi}{2}, \frac{\pi}{3}, 0\right\} \\ & \theta=\left\{-\frac{4 \pi}{9}, \frac{-2 \pi}{9}, \frac{4 \pi}{9}, \frac{2 \pi}{9}\right\}\end{aligned}

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