| Exam Board | AQA |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2009 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Region shading with multiple inequalities |
| Difficulty | Standard +0.8 This is a Further Maths question requiring visualization of a locus (circle centered at 4i with radius 2) and then finding extreme argument values by identifying tangent lines from the origin. While the geometric insight needed is non-trivial, the calculation itself is straightforward trigonometry once the setup is understood—moderately challenging but not exceptional for FP2. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks | Guidance |
|---|---|---|
| - Circle | B1 | |
| - Correct centre | B1 | |
| - Correct radius | B1 | |
| - Inside shading | B1F | 4 marks |
| Answer | Marks | Guidance |
|---|---|---|
| - Correct points \(P_1\) and \(P_2\) indicated | B1F | Possibly by tangents drawn ft mirror image of circle in x-axis |
| - \(\sin\alpha = \frac{2}{4}\) | M1 | |
| - \(\alpha = \frac{\pi}{6}\) | A1 | |
| - Range is \(\frac{\pi}{3} \le \arg z \le \frac{2\pi}{3}\) | A1 | 4 marks; Deduct 1 for angles in degrees |
**(a)**
- Circle | B1 |
- Correct centre | B1 |
- Correct radius | B1 |
- Inside shading | B1F | 4 marks
- **Total: 4 marks**
**(b)**
- Correct points $P_1$ and $P_2$ indicated | B1F | Possibly by tangents drawn ft mirror image of circle in x-axis
- $\sin\alpha = \frac{2}{4}$ | M1 |
- $\alpha = \frac{\pi}{6}$ | A1 |
- Range is $\frac{\pi}{3} \le \arg z \le \frac{2\pi}{3}$ | A1 | 4 marks; Deduct 1 for angles in degrees
- **Total: 4 marks**
**Question 2 Total: 8 marks**
---2
\begin{enumerate}[label=(\alph*)]
\item Indicate on an Argand diagram the region for which $| z - 4 \mathrm { i } | \leqslant 2$.
\item The complex number $z$ satisfies $| z - 4 \mathrm { i } | \leqslant 2$. Find the range of possible values of $\arg z$.
\end{enumerate}
\hfill \mbox{\textit{AQA FP2 2009 Q2 [8]}}