| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2024 |
| Session | February |
| Marks | 6 |
| Topic | Geometric Sequences and Series |
| Type | GP with trigonometric terms |
| Difficulty | Standard +0.3 Part (a) is a straightforward infinite GP sum with r = sin 30° = 1/2, requiring only the formula S = a/(1-r). Part (b) requires setting up 1/(1-cos θ) = 2-√2, rearranging to find cos θ, then solving for θ in radians. Both parts are standard applications of the infinite GP formula with minimal algebraic manipulation, making this easier than average despite the trigonometric context. |
| Spec | 1.04j Sum to infinity: convergent geometric series |r|<11.05g Exact trigonometric values: for standard angles1.05o Trigonometric equations: solve in given intervals |
8.
\begin{enumerate}[label=(\alph*)]
\item Evaluate
$$\sum _ { n = 1 } ^ { \infty } \left( \sin 30 ^ { \circ } \right) ^ { n }$$
\item Find the smallest positive exact value of $\theta$, in radians, which satisfies the equation
$$\sum _ { n = 0 } ^ { \infty } ( \cos \theta ) ^ { n } = 2 - \sqrt { 2 }$$
[4 marks]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM Pure 2024 Q8 [6]}}