| Exam Board | Edexcel |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Topic | Polar coordinates |
| Type | Area between two polar curves |
| Difficulty | Challenging +1.3 This is a standard Further Maths polar coordinates question requiring intersection finding, distance calculation, and area integration. Part (a) involves solving a trigonometric equation and using the distance formula, while part (b) requires setting up and evaluating a polar area integral. The techniques are well-practiced in FP2, though the algebra and integration require care. It's moderately harder than average A-level due to being Further Maths content, but follows predictable patterns for this topic. |
| Spec | 4.09a Polar coordinates: convert to/from cartesian4.09b Sketch polar curves: r = f(theta)4.09c Area enclosed: by polar curve |
\includegraphics{figure_1}
The curve $C$ which passes through $O$ has polar equation
$$r = 4a(1 + \cos \theta), \quad -\pi < \theta \leq \pi$$
The line $l$ has polar equation
$$r = 3a \sec \theta, \quad -\frac{\pi}{2} < \theta < \frac{\pi}{2}.$$
The line $l$ cuts $C$ at the points $P$ and $Q$, as shown in Figure 1.
\begin{enumerate}[label=(\alph*)]
\item Prove that $PQ = 6\sqrt{3}a$. [6]
The region $R$, shown shaded in Figure 1, is bounded by $l$ and $C$.
\item Use calculus to find the exact area of $R$. [7]
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP2 Q45 [13]}}