| Exam Board | AQA |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Marks | 18 |
| 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 FP3 question requiring understanding of loci (circle with center -4+4i, radius 4), optimization using geometric reasoning (maximum |z| occurs when z lies on the line from origin through center), and argument calculations with exact trigonometry. Part (b) requires insight that max distance is center-to-origin plus radius. Multi-step with moderate conceptual demand, typical of Further Maths but not requiring exceptional creativity. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
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Please provide the extracted mark scheme content that needs to be cleaned, and I'll convert the unicode symbols to LaTeX notation and format it as requested.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
$$\arg ( z + 4 - 4 i ) = \frac { 1 } { 6 } \pi$$
Give your answer in the form $a + \mathrm { i } b$.
\end{enumerate}
\hfill \mbox{\textit{AQA FP3 Q5 [18]}}