| Exam Board | Edexcel |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Topic | Polar coordinates |
| Type | Region bounded by curve and tangent lines |
| Difficulty | Challenging +1.8 This is a multi-part polar coordinates question requiring area calculations using integration, finding tangent lines, and geometric reasoning. Part (a) is a standard polar area integral, but part (b) requires identifying tangent positions, computing rectangular area, and subtracting the curve area—demanding careful geometric visualization and extended calculation. The fractional power and two-lobed curve add complexity beyond routine FP2 exercises, though the techniques themselves are standard for Further Maths. |
| Spec | 4.09c Area enclosed: by polar curve |
\includegraphics{figure_1}
Figure 1 shows a closed curve $C$ with equation
$$r = 3(\cos 2\theta)^{\frac{1}{2}}, \quad \text{where } -\frac{\pi}{4} \leq \theta \leq \frac{\pi}{4}, \frac{3\pi}{4} \leq \theta \leq \frac{5\pi}{4}.$$
The lines $PQ$, $SR$, $PS$ and $QR$ are tangents to $C$, where $PQ$ and $SR$ are parallel to the initial line and $PS$ and $QR$ are perpendicular to the initial line. The point $O$ is the pole.
\begin{enumerate}[label=(\alph*)]
\item Find the total area enclosed by the curve $C$, shown unshaded inside the rectangle in Figure 1. [4]
\item Find the total area of the region bounded by the curve $C$ and the four tangents, shown shaded in Figure 1. [9]
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP2 Q8 [13]}}