| Exam Board | OCR |
|---|---|
| Module | H240/03 (Pure Mathematics and Mechanics) |
| Year | 2023 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Topic | Arithmetic Sequences and Series |
| Type | Trigonometric arithmetic progression |
| Difficulty | Standard +0.8 This question requires students to set up equations using the arithmetic progression property (u₃ - u₁ = 2d and u₄ - u₃ = d), manipulate trigonometric expressions to find θ in a restricted domain, then apply the sum formula for an arithmetic series. While it combines multiple A-level topics (sequences, trigonometry), the steps are relatively standard once the AP property is recognized, making it moderately challenging but not requiring exceptional insight. |
| Spec | 1.04h Arithmetic sequences: nth term and sum formulae1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=1 |
The first, third and fourth terms of an arithmetic progression are $u_1$, $u_3$ and $u_4$ respectively, where
$$u_1 = 2 \sin \theta, \quad u_3 = -\sqrt{3} \cos \theta, \quad u_4 = \frac{7}{3} \sin \theta,$$
and $\frac{1}{2}\pi < \theta < \pi$.
\begin{enumerate}[label=(\alph*)]
\item Determine the exact value of $\theta$. [3]
\item Hence determine the value of $\sum_{r=1}^{100} u_r$. [3]
\end{enumerate}
\hfill \mbox{\textit{OCR H240/03 2023 Q6 [6]}}