| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2016 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Geometric Sequences and Series |
| Type | Relationship between two GPs |
| Difficulty | Standard +0.3 Part (a) is a straightforward application of the sum to infinity formula for GPs, requiring students to set up two equations and solve simultaneously—standard bookwork with minimal problem-solving. Part (b) involves using the arithmetic series formula with trigonometric terms and solving a quadratic in sin²θ, which adds modest complexity but remains a routine multi-step question. Overall slightly easier than average due to direct application of formulas. |
| Spec | 1.04h Arithmetic sequences: nth term and sum formulae1.04j Sum to infinity: convergent geometric series |r|<1 |
\begin{enumerate}[label=(\alph*)]
\item Two convergent geometric progressions, $P$ and $Q$, have 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 value of the common sum to infinity. [3]
\item The first term of an arithmetic progression is $\cos\theta$ and the second term is $\cos\theta + \sin^2\theta$, where $0 \leq \theta \leq \pi$. The sum of the first $13$ terms is $52$. Find the possible values of $\theta$. [5]
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2016 Q9 [8]}}