| Exam Board | AQA |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2010 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Intersection of two loci |
| Difficulty | Standard +0.3 This is a standard Further Maths loci question requiring identification of a circle and perpendicular bisector, then shading their intersection. While it involves multiple steps and geometric interpretation, these are routine FP2 techniques with no novel problem-solving required, making it slightly easier than average. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks | Guidance |
|---|---|---|
| Circle | B1 | |
| Correct centre | B1 | |
| Touching y-axis | B1 | 3 marks total |
| Answer | Marks | Guidance |
|---|---|---|
| Straight line parallel to x-axis | B1, B1 | |
| through (0, 1) | B1 | 3 marks total |
| Answer | Marks | Guidance |
|---|---|---|
| Shading: inside circle above line | B1F, B1F | 2 marks |
### Part (a)(i)
Circle | B1 |
Correct centre | B1 |
Touching y-axis | B1 | 3 marks total | x-coordinate = $-2 \times$ y-coordinate in correct quadrant; condone (4, −2i)
### Part (a)(ii)
Straight line parallel to x-axis | B1, B1 |
through (0, 1) | B1 | 3 marks total | Assume (0,1) if distance up y-axis is half distance to top of circle; no other shading outside circle
### Part (b)
Shading: inside circle above line | B1F, B1F | 2 marks | Whole question reflected in x-axis loses 2 marks2 (a) On the same Argand diagram, draw:\\
(i) the locus of points satisfying $| z - 4 + 2 \mathrm { i } | = 4$;\\
(ii) the locus of points satisfying $| z | = | z - 2 \mathrm { i } |$.\\
(b) Indicate on your sketch the set of points satisfying both
$$| z - 4 + 2 i | \leqslant 4$$
and
$$| z | \geqslant | z - 2 \mathrm { i } |$$
\hfill \mbox{\textit{AQA FP2 2010 Q2 [8]}}