| Exam Board | Edexcel |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Marks | 16 |
| Paper | Download PDF ↗ |
| Topic | Polar coordinates |
| Type | Area between two polar curves |
| Difficulty | Challenging +1.2 This is a standard Further Pure 2 polar coordinates question covering intersection points, chord length, and area between curves. Parts (a)-(c) follow routine procedures (equating equations, using distance formula, integrating ½r² with correct limits), though the area calculation requires careful setup with multiple integrals. Part (d) is straightforward substitution. The multi-step nature and integration complexity elevate it slightly above average FP2 difficulty, but it remains a textbook-style question without requiring novel insight. |
| Spec | 1.10f Distance between points: using position vectors4.09b Sketch polar curves: r = f(theta)4.09c Area enclosed: by polar curve |
\includegraphics{figure_1}
Figure 1 is a sketch of the two curves $C_1$ and $C_2$ with polar equations
$$C_1 : r = 3a(1 - \cos \theta), \quad -\pi \leq \theta < \pi$$
and
$$C_2 : r = a(1 + \cos \theta), \quad -\pi \leq \theta < \pi.$$
The curves meet at the pole $O$, and at the points $A$ and $B$.
\begin{enumerate}[label=(\alph*)]
\item Find, in terms of $a$, the polar coordinates of the points $A$ and $B$. [4]
\item Show that the length of the line $AB$ is $\frac{3\sqrt{3}}{2}a$. [2]
The region inside $C_2$ and outside $C_1$ is shown shaded in Fig. 1.
\item Find, in terms of $a$, the area of this region. [7]
A badge is designed which has the shape of the shaded region.
Given that the length of the line $AB$ is $4.5$ cm,
\item calculate the area of this badge, giving your answer to three significant figures. [3]
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP2 Q32 [16]}}