| Exam Board | Edexcel |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Topic | Polar coordinates |
| Type | Area between two polar curves |
| Difficulty | Challenging +1.2 This is a structured polar coordinates question with clear steps: finding intersection points requires basic substitution and solving cos θ = 1/2; the area calculation uses the standard polar area formula ∫½r²dθ with symmetry. While it involves multiple steps and careful integration, the techniques are standard FP2 material with no novel insights required. The 'show that' format provides a target to verify, reducing difficulty slightly. |
| 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}
A logo is designed which consists of two overlapping closed curves.
The polar equations of these curves are
$$r = a(3 + 2\cos \theta) \quad \text{and}$$
$$r = a(5 - 2 \cos \theta), \quad 0 \leq \theta < 2\pi.$$
Figure 1 is a sketch (not to scale) of these two curves.
\begin{enumerate}[label=(\alph*)]
\item Write down the polar coordinates of the points $A$ and $B$ where the curves meet the initial line. [2]
\item Find the polar coordinates of the points $C$ and $D$ where the two curves meet. [4]
\item Show that the area of the overlapping region, which is shaded in the figure, is
$$\frac{a^2}{3}(49\pi - 48\sqrt{3}).$$ [8]
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP2 Q20 [14]}}