| Exam Board | SPS |
|---|---|
| Module | SPS FM (SPS FM) |
| Year | 2021 |
| Session | November |
| Marks | 13 |
| Topic | Polar coordinates |
| Type | Region bounded by curve and tangent lines |
| Difficulty | Challenging +1.8 This is a challenging Further Maths polar coordinates question requiring: (a) integration of a non-standard polar curve with careful attention to the domain split, and (b) finding tangent lines to a polar curve and computing areas between curve and tangents. The polar area formula application is routine, but identifying tangent positions and computing the composite shaded region requires solid geometric understanding and multi-step calculation. The curve equation r = 3√cos(2θ) with its restricted domain adds complexity beyond standard polar area questions. |
| Spec | 4.09c Area enclosed: by polar curve |
\includegraphics{figure_1}
Figure 1 shows a closed curve $C$ with equation
$$r = 3\sqrt{\cos(2\theta)}, \quad \text{where } -\frac{\pi}{4} < \theta \leq \frac{\pi}{4}, \quad \frac{3\pi}{4} < \theta \leq \frac{5\pi}{4}$$
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 marks]
\item Find the total area of the region bounded by the curve $C$ and the four tangents, shown shaded in Figure 1.
[9 marks]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM 2021 Q10 [13]}}