(a)[3]
Two convergent geometric progressions, $P$ and $Q$, share the same sum to infinity. The first and second terms of $P$ are $6$ and $6r$ respectively. The first and second terms of $Q$ are $12$ and $-12r$ respectively. Find the common sum to infinity.
(b)[5]
An arithmetic progression has first term $\cos \theta$ and second term $\cos \theta + \sin^2 \theta$, where $0 \leq \theta \leq \pi$. The sum of the first $13$ terms is $52$. Find the possible values of $\theta$.