| Exam Board | SPS |
|---|---|
| Module | SPS FM (SPS FM) |
| Year | 2024 |
| Session | November |
| Marks | 10 |
| Topic | Polar coordinates |
| Type | Maximum/minimum distance from pole or line |
| Difficulty | Standard +0.8 This is a Further Maths polar coordinates question requiring optimization (finding maximum r by differentiation or using trigonometric identities), proving a curve is a circle (completing the square in Cartesian form or using geometric properties), and finding its centre. While the techniques are standard for FM students, the multi-part nature and need to connect polar and Cartesian representations makes it moderately challenging but not exceptional. |
| Spec | 4.09a Polar coordinates: convert to/from cartesian4.09b Sketch polar curves: r = f(theta) |
6. A curve has polar equation $r = a ( \cos \theta + 2 \sin \theta )$, where $a$ is a positive constant and $0 \leq \theta \leq \pi$.
\begin{enumerate}[label=(\alph*)]
\item Determine the polar coordinates of the point on the curve which is furthest from the pole.
\item \begin{enumerate}[label=(\roman*)]
\item Show that the curve is a circle whose radius should be specified.
\item Write down the polar coordinates of the centre of the circle.\\[0pt]
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM 2024 Q6 [10]}}