| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2023 |
| Session | November |
| Marks | 8 |
| Topic | Polar coordinates |
| Type | Area enclosed by polar curve |
| Difficulty | Standard +0.3 This is a straightforward polar curves question requiring standard techniques: identifying key points from the equation, recognizing symmetry properties, and applying the polar area formula with a simple trigonometric integral. The integration of cos²θ is routine A-level content, and the symmetry allows halving the work. While it requires multiple steps, each is standard procedure with no novel insight needed. |
| Spec | 4.09a Polar coordinates: convert to/from cartesian4.09b Sketch polar curves: r = f(theta)4.09c Area enclosed: by polar curve |
Fig. 5 shows the curve with polar equation $r = a(3 + 2\cos\theta)$ for $-\pi \leqslant \theta \leqslant \pi$, where $a$ is a constant.
\includegraphics{figure_2}
(a) Write down the polar coordinates of the points A and B. [2]
(b) Explain why the curve is symmetrical about the initial line. [2]
(c) In this question you must show detailed reasoning.
Find in terms of $a$ the exact area of the region enclosed by the curve. [4]
\hfill \mbox{\textit{SPS SPS FM Pure 2023 Q2 [8]}}