| Exam Board | AQA |
|---|---|
| Module | Further AS Paper 1 (Further AS Paper 1) |
| Session | Specimen |
| Marks | 3 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Single locus sketching |
| Difficulty | Moderate -0.5 Part (a) is a standard circle locus |z-2|=2 requiring only recognition and sketching. Part (b) combines modulus and argument to find a specific complex number using routine conversion from polar form—straightforward application of basic Further Maths techniques with no problem-solving insight required. Easier than average A-level but typical for introductory Further Maths loci questions. |
| Spec | 4.02b Express complex numbers: cartesian and modulus-argument forms4.02k Argand diagrams: geometric interpretation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Circle with centre \(2 + 0i\) | M1 | Ignore other features |
| Circle passing through \((0,0)\), \((4,0)\), close to \((2,2)\) and \((2,-2)\), with imaginary axis tangential | A1 | All features required |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(z - 2 = 2\left[\cos\left(-\frac{\pi}{3}\right) + i\sin\left(-\frac{\pi}{3}\right)\right]\) | M1 | Uses mod/arg forms |
| \(= 2\left[\frac{1}{2} + i\left(-\frac{\sqrt{3}}{2}\right)\right]\) | M1 | Substitutes exact values for cos and sin; allow one slip |
| \(z = 3 - \sqrt{3}\,i\) | A1 | Exact form required |
## Question 9(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Circle with centre $2 + 0i$ | M1 | Ignore other features |
| Circle passing through $(0,0)$, $(4,0)$, close to $(2,2)$ and $(2,-2)$, with imaginary axis tangential | A1 | All features required |
## Question 9(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| $z - 2 = 2\left[\cos\left(-\frac{\pi}{3}\right) + i\sin\left(-\frac{\pi}{3}\right)\right]$ | M1 | Uses mod/arg forms |
| $= 2\left[\frac{1}{2} + i\left(-\frac{\sqrt{3}}{2}\right)\right]$ | M1 | Substitutes exact values for cos and sin; allow one slip |
| $z = 3 - \sqrt{3}\,i$ | A1 | Exact form required |
---9
\begin{enumerate}[label=(\alph*)]
\item Sketch on the Argand diagram below, the locus of points satisfying the equation $| z - 2 | = 2$\\
\includegraphics[max width=\textwidth, alt={}, center]{e61d0202-49c9-4ed9-9fa3-f10734e17463-14_1417_1475_790_399}
9
\item Given that $| z - 2 | = 2$ and $\arg ( z - 2 ) = - \frac { \pi } { 3 }$, express $z$ in the form $a + b \mathrm { i }$, where $a$ and $b$ are real numbers.\\[0pt]
[3 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA Further AS Paper 1 Q9 [3]}}