| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2023 |
| Session | November |
| Topic | Polar coordinates |
| Type | Area of region with line boundary |
| Difficulty | Challenging +1.8 This is a challenging Further Maths question requiring conversion between Cartesian and polar forms of a circle, geometric visualization of set intersections, and area calculation using polar integration. The algebraic manipulation to derive the polar equation and the exact area calculation (likely involving sector minus triangle or polar integration) require strong technical skills and multi-step reasoning beyond standard A-level, though the individual techniques are within FM syllabus. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.09a Polar coordinates: convert to/from cartesian4.09c Area enclosed: by polar curve |
(a) (i) Show on an Argand diagram the locus of points given by the values of $z$ satisfying
$$|z - 4 - 3i| = 5$$
Taking the initial line as the positive real axis with the pole at the origin and given that
$$\theta \in [\alpha, \alpha + \pi], \text{ where } \alpha = -\arctan\left(\frac{4}{3}\right),$$
(ii) show that this locus of points can be represented by the polar curve with equation
$$r = 8\cos\theta + 6\sin\theta$$ (6)
The set of points $A$ is defined by
$$A = \left\{z : 0 \leqslant \arg z \leqslant \frac{\pi}{3}\right\} \cap \{z : |z - 4 - 3i| \leqslant 5\}$$
(b) (i) Show, by shading on your Argand diagram, the set of points $A$.
(ii) Find the exact area of the region defined by $A$, giving your answer in simplest form. (7)
\hfill \mbox{\textit{SPS SPS FM Pure 2023 Q7}}