| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2024 |
| Session | February |
| Marks | 11 |
| Topic | Polar coordinates |
| Type | Area between two polar curves |
| Difficulty | Challenging +1.8 This is a challenging Further Maths polar coordinates question requiring: (1) understanding of polar curve sketching for a four-petaled rose curve, (2) finding intersection points by solving r₁ = r₂ leading to a non-trivial trigonometric equation, (3) setting up and evaluating polar area integrals with limits involving the inverse sine, (4) extensive trigonometric manipulation to reach the specific form with the golden ratio appearing. The 9-mark allocation and algebraic complexity place it well above average difficulty, though it follows standard polar area techniques. |
| Spec | 4.09b Sketch polar curves: r = f(theta)4.09c Area enclosed: by polar curve |
The diagram shows the polar curve $C_1$ with equation $r = 2\sin\theta$
The diagram also shows part of the polar curve $C_2$ with equation $r = 1 + \cos 2\theta$
\includegraphics{figure_10}
\begin{enumerate}[label=(\alph*)]
\item On the diagram above, complete the sketch of $C_2$ [2 marks]
\item Show that the area of the region shaded in the diagram is equal to
$$k\pi + m\alpha - \sin 2\alpha + q\sin 4\alpha$$
where $\alpha = \sin^{-1}\left(\frac{\sqrt{5}-1}{2}\right)$, and $k$, $m$ and $q$ are rational numbers. [9 marks]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2024 Q10 [11]}}