JEE 2023 — Mathematics PYQ

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Question

JEE 2023 — Mathematics PYQ

JEE | Mathematics | 2023

If $m$ and $n$ respectively are the numbers of positive and negative values of $θ$ in the interval $[- π, π]$ that satisfy the equation $cos 2 θ cos \frac{θ}{2} = cos 3 θ cos \frac{9 θ}{2}$ , then $mn$ is equal to

Choose the correct answer:

Explanation

Solving:

$2\cos 2\theta \cos \frac{\theta}{2} = 2\cos 3\theta \cos \frac{9\theta}{2}$

$\cos \frac{5\theta}{2} + \cos \frac{3\theta}{2} = \cos \frac{15\theta}{2} + \cos \frac{3\theta}{2}$

$\cos \frac{15\theta}{2} - \cos \frac{5\theta}{2} = 0 \implies -2\sin 5\theta \sin \frac{5\theta}{2} = 0$

$\sin 5\theta = 0 \implies 5\theta = k\pi \implies \theta = 0, \pm\frac{\pi}{5}, \pm\frac{2\pi}{5}, \pm\frac{3\pi}{5}, \pm\frac{4\pi}{5}, \pm\pi$

$\sin \frac{5\theta}{2} = 0 \implies \frac{5\theta}{2} = k\pi \implies \theta = 0, \pm\frac{2\pi}{5}, \pm\frac{4\pi}{5}$ (already included)

Positive values ( $m$ ): $\frac{\pi}{5}, \frac{2\pi}{5}, \frac{3\pi}{5}, \frac{4\pi}{5}, \pi$ (Note: $0$ is neither positive nor negative) $\implies m = 5$

Negative values ( $n$ ): $-\frac{\pi}{5}, -\frac{2\pi}{5}, -\frac{3\pi}{5}, -\frac{4\pi}{5}, -\pi \implies n = 5$

$mn = 5 \times 5 = 25$