| Exam Board | Edexcel |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Marks | 18 |
| Paper | Download PDF ↗ |
| Topic | Polar coordinates |
| Type | Area between two polar curves |
| Difficulty | Challenging +1.2 This is a multi-part polar coordinates question requiring sketching curves, finding intersections, and computing areas using integration. While it involves several steps and the final part requires combining given information, the techniques are standard for FP2: recognizing standard polar curves (circle and cardioid), solving simultaneous polar equations, and applying the polar area formula. The final part is somewhat mechanical given the provided area. More challenging than average A-level but routine for Further Maths students who have practiced polar coordinate problems. |
| Spec | 4.09a Polar coordinates: convert to/from cartesian4.09b Sketch polar curves: r = f(theta)4.09c Area enclosed: by polar curve |
The curve $C$ has polar equation $r = 3a \cos \theta$, $-\frac{\pi}{2} \leq \frac{\pi}{2}$. The curve $D$ has polar equation $r = a(1 + \cos \theta)$, $-\pi \leq \theta < \pi$. Given that $a$ is a positive constant,
\begin{enumerate}[label=(\alph*)]
\item sketch, on the same diagram, the graphs of $C$ and $D$, indicating where each curve cuts the initial line. [4]
The graphs of $C$ intersect at the pole $O$ and at the points $P$ and $Q$.
\item Find the polar coordinates of $P$ and $Q$. [3]
\item Use integration to find the exact value of the area enclosed by the curve $D$ and the lines $\theta = 0$ and $\theta = \frac{\pi}{3}$. [7]
The region $R$ contains all points which lie outside $D$ and inside $C$.
Given that the value of the smaller area enclosed by the curve $C$ and the line $\theta = \frac{\pi}{3}$ is
$$\frac{3a^2}{16}(2\pi - 3\sqrt{3}),$$
\item show that the area of $R$ is $\pi a^2$. [4]
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP2 Q4 [18]}}