Question

If the solutions for $\theta$ of $\cos$ of $\cos p\theta +\cos q\theta =0,p>q>0$ 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. $\frac{1}{p+q}$

Answer 1

Answer: (B)

$\frac{2\pi }{p+q}$

Given: $\cos p\theta +\cos q\theta =0$

$\Rightarrow \cos p\theta =-\cos q\theta$

$\Rightarrow \cos p\theta =\cos (\pi -q\theta )$

$\Rightarrow p\theta =2n\pi \pm \pi -q\theta$

$\Rightarrow \theta (p+q)=2n\pi \pm \pi$

$\Rightarrow \theta =\frac{2n\pi \pm \pi }{p+q}$

So, difference $=\frac{2\pi }{p+q}$

Hence, option $B.$