Q. If the solutions for $\theta,\cos{p\theta + \cos{q\theta = 0,\ p > 0,\ q > 0}}$ are in AP, then the numerically smallest common difference of AP is
A) $\frac{\pi}{p + q}$
B) $\frac{2\pi}{p + q}$
C) $\frac{\pi}{2(p + q)}$
D) $\frac{1}{p + q}$
Explanation
(b)
Given, $\cos{p\theta = - \cos{q\theta = \cos{(\pi + q\theta)}}}$
$\Rightarrow \ p\theta = 2n\pi \pm (\pi + q\theta),\ n \in I$
$\Rightarrow \ \theta = \frac{(2n + 1)\pi}{p - q}\ or\ \frac{(2n - 1)\pi}{p + q},\ n \in I$
Angle $\theta = \frac{(2n + 1)\pi}{p - q}$ gives an AP with common difference $\frac{2\pi}{p - q}$ and $\theta = \frac{(2n - 1)\pi}{p - q}$ gives also an AP with common difference $\frac{2\pi}{p + q}$
Certainly, $\frac{2\pi}{p + q} < \left| \frac{2\pi}{p - q} \right|$
$\therefore$ The smallest common difference is $\frac{2\pi}{p + q}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.