| Exam Board | Edexcel |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2003 |
| Session | June |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polar coordinates |
| Type | Area between two polar curves |
| Difficulty | Challenging +1.2 This is a standard FP2 polar coordinates question requiring: (a) direct substitution of θ=0, (b) solving r₁=r₂ to find intersection points, and (c) computing area between curves using the polar area formula ½∫r²dθ. While it involves multiple steps and careful setup of integration limits, the techniques are routine for Further Maths students and the question provides clear guidance through its parts. The algebraic manipulation is moderate but straightforward. |
| Spec | 4.09a Polar coordinates: convert to/from cartesian4.09b Sketch polar curves: r = f(theta)4.09c Area enclosed: by polar curve |
7.
\begin{figure}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\includegraphics[alt={},max width=\textwidth]{141c7b1b-4236-4433-84af-04fa9baa3d96-2_568_1431_1637_258}
\end{center}
\end{figure}
A logo is designed which consists of two overlapping closed curves.
The polar equations of these curves are $r = \boldsymbol { a } ( \mathbf { 3 } + \mathbf { 2 } \cos \boldsymbol { \theta } )$ 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 corrdinates of the points $A$ and $B$ where the curves meet the initial line.(2)
\item Find the polar coordinates of the points $\boldsymbol { C }$ and $\boldsymbol { D }$ where the two curves meet. (4)
\item Show that the area of the overlapping region, which is shaded in the figure, is
$$\frac { a ^ { 2 } } { 3 } ( 49 \pi - 48 \sqrt { } 3 )$$
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP2 2003 Q7 [14]}}