| Exam Board | AQA |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2009 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Optimization of modulus on loci |
| Difficulty | Standard +0.8 This is a multi-part Further Maths question requiring: (a) finding circle equation from diameter endpoints (routine), (b) sketching two circles (straightforward), and (c) optimization requiring geometric insight that maximum distance between points on two circles equals sum of radii plus distance between centers. Part (c) elevates this above standard exercises, requiring spatial reasoning and careful calculation with surds, but the geometric principle is a known technique in FP2. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks | Guidance |
|---|---|---|
| (a) Centre \(-1 - i\) or \((-1, -1)\) | B1 | |
| Radius 5 | M1 | |
| \( | z + 1 + i | = 5\) or \( |
| (b) \(C_1\) correct centre, correct radius | B1F | |
| \(C_2\) correct centre, correct radius. Touching \(x\)-axis | B1 | |
| B1F | 3 marks | ft errors in (a) but fit circles need to intersect and \(C_1\) enclose \((0,0)\); error in plotting centre |
| (c) \(O_1O_2 = 3\sqrt{5}\) | M1A1 | allow if circles misplaced but \(O_1O_2\) is still \(3\sqrt{5}\) |
| Correct length identified | m1 | |
| Length is \(9 + 3\sqrt{5}\) | M1, A1F | 5 marks |
**(a)** Centre $-1 - i$ or $(-1, -1)$ | B1 |
Radius 5 | M1 |
$|z + 1 + i| = 5$ or $|z - (-1 - i)| = 5$ | A1F | A1F | 4 marks | ft incorrect centre if used. ft $|z + 1 + i| = 10$ earns M0B1
**(b)** $C_1$ correct centre, correct radius | B1F |
$C_2$ correct centre, correct radius. Touching $x$-axis | B1 |
B1F | 3 marks | ft errors in (a) but fit circles need to intersect and $C_1$ enclose $(0,0)$; error in plotting centre
**(c)** $O_1O_2 = 3\sqrt{5}$ | M1A1 | allow if circles misplaced but $O_1O_2$ is still $3\sqrt{5}$
Correct length identified | m1 |
Length is $9 + 3\sqrt{5}$ | M1, A1F | 5 marks | ft if $r$ is taken as 10
**Total: 12 marks**6
\begin{enumerate}[label=(\alph*)]
\item Two points, $A$ and $B$, on an Argand diagram are represented by the complex numbers $2 + 3 \mathrm { i }$ and $- 4 - 5 \mathrm { i }$ respectively. Given that the points $A$ and $B$ are at the ends of a diameter of a circle $C _ { 1 }$, express the equation of $C _ { 1 }$ in the form $\left| z - z _ { 0 } \right| = k$.
\item A second circle, $C _ { 2 }$, is represented on the Argand diagram by the equation $| z - 5 + 4 \mathrm { i } | = 4$. Sketch on one Argand diagram both $C _ { 1 }$ and $C _ { 2 }$.
\item The points representing the complex numbers $z _ { 1 }$ and $z _ { 2 }$ lie on $C _ { 1 }$ and $C _ { 2 }$ respectively and are such that $\left| z _ { 1 } - z _ { 2 } \right|$ has its maximum value. Find this maximum value, giving your answer in the form $a + b \sqrt { 5 }$.
\end{enumerate}
\hfill \mbox{\textit{AQA FP2 2009 Q6 [12]}}