| Exam Board | AQA |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2006 |
| Session | January |
| Marks | 9 |
| 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 FP2 question requiring understanding of loci as circles in the Argand diagram, optimization using geometric reasoning (finding maximum modulus on a circle), and combining modulus/argument conditions. Part (b) requires insight that maximum |z| occurs when z lies on the line from origin through center, and part (c) combines argument and modulus conditions requiring trigonometry. Multi-step with moderate conceptual demand, typical of harder FP2 questions but follows standard patterns. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Circle drawn | B1 | Circle |
| Correct centre | B1 | Correct centre |
| Touching both axes | B1 | Touching both axes |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \( | z | _{\max} = OK\) |
| \(= \sqrt{4^2+4^2}+4\) | A1F | Follow through circle in incorrect position |
| \(= 4(\sqrt{2}+1)\) | A1F | AG |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Correct position of \(z\), ie \(L\) | M1 | |
| \(a = -(4 - 4\cos\frac{1}{6}\pi) = -(4-2\sqrt{3})\) | A1F | Follow through circle in incorrect position |
| \(b = 4 + 4\sin\frac{1}{6}\pi = 6\) | A1F |
# Question 5:
## Part (a)
| Answer/Working | Marks | Guidance |
|---|---|---|
| Circle drawn | B1 | Circle |
| Correct centre | B1 | Correct centre |
| Touching both axes | B1 | Touching both axes |
## Part (b)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $|z|_{\max} = OK$ | M1 | Accept $\sqrt{4^2+4^2}+4$ as a method |
| $= \sqrt{4^2+4^2}+4$ | A1F | Follow through circle in incorrect position |
| $= 4(\sqrt{2}+1)$ | A1F | AG |
## Part (c)
| Answer/Working | Marks | Guidance |
|---|---|---|
| Correct position of $z$, ie $L$ | M1 | |
| $a = -(4 - 4\cos\frac{1}{6}\pi) = -(4-2\sqrt{3})$ | A1F | Follow through circle in incorrect position |
| $b = 4 + 4\sin\frac{1}{6}\pi = 6$ | A1F | |5 The complex number $z$ satisfies the relation
$$| z + 4 - 4 i | = 4$$
\begin{enumerate}[label=(\alph*)]
\item Sketch, on an Argand diagram, the locus of $z$.
\item Show that the greatest value of $| z |$ is $4 ( \sqrt { 2 } + 1 )$.
\item Find the value of $z$ for which
$$\arg ( z + 4 - 4 \mathrm { i } ) = \frac { 1 } { 6 } \pi$$
Give your answer in the form $a + \mathrm { i } b$.
\end{enumerate}
\hfill \mbox{\textit{AQA FP2 2006 Q5 [9]}}