| Exam Board | Edexcel |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2005 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Complex transformations and mappings |
| Difficulty | Challenging +1.2 This is a multi-part FP2 question involving standard locus identification (circle), finding a point from modulus-argument conditions, and showing a transformation maps a circle to a line. Part (c) requires algebraic manipulation but follows a predictable method (substitute z=2i/w, use |z-3i|=3). While it's Further Maths content and has multiple steps, each component uses routine techniques without requiring novel geometric insight or particularly sophisticated reasoning. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02l Geometrical effects: conjugate, addition, subtraction4.02m Geometrical effects: multiplication and division4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Circle drawn | M1 | |
| Correct circle, centre \((0, 3)\), radius \(3\) | A1 | 2 marks total |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Drawing correct half-line passing as shown | B1 | |
| Find either \(x\) or \(y\) coordinate of \(A\) | M1A1 | |
| \(z = -\dfrac{3\sqrt{2}}{2} + \left(3 + \dfrac{3\sqrt{2}}{2}\right)i\) | A1 | 4 marks total; Algebraic approach using \(y = 3-x\) and equation of circle gains M1A1 only unless second solution ruled out; B1 by implication if correct |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\ | z - 3i\ | = 3 \rightarrow \left\ |
| \(\Rightarrow \dfrac{\ | 2i - 3i\omega\ | }{\ |
| \(\Rightarrow \ | \omega - \tfrac{2}{3}\ | = \ |
| Line with equation \(u = \tfrac{1}{3}\) \((x = \tfrac{1}{3})\) | A1 | 5 marks total [11] |
# Question 9:
## Part (a)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Circle drawn | M1 | |
| Correct circle, centre $(0, 3)$, radius $3$ | A1 | 2 marks total |
## Part (b)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Drawing correct **half**-line passing as shown | B1 | |
| Find either $x$ or $y$ coordinate of $A$ | M1A1 | |
| $z = -\dfrac{3\sqrt{2}}{2} + \left(3 + \dfrac{3\sqrt{2}}{2}\right)i$ | A1 | 4 marks total; Algebraic approach using $y = 3-x$ and equation of circle gains M1A1 only unless second solution ruled out; B1 by implication if correct |
## Part (c)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\|z - 3i\| = 3 \rightarrow \left\|\dfrac{2i}{\omega} - 3i\right\| = 3$ | M1 | |
| $\Rightarrow \dfrac{\|2i - 3i\omega\|}{\|\omega\|} = 3$ | A1 | |
| $\Rightarrow \|\omega - \tfrac{2}{3}\| = \|\omega\|$ | M1A1 | |
| Line with equation $u = \tfrac{1}{3}$ $(x = \tfrac{1}{3})$ | A1 | 5 marks total **[11]** |
---9. A complex number $z$ is represented by the point $P$ in the Argand diagram. Given that
$$| z - 3 \mathrm { i } | = 3$$
\begin{enumerate}[label=(\alph*)]
\item sketch the locus of $P$.
\item Find the complex number $z$ which satisfies both $| z - 3 i | = 3$ and $\arg ( z - 3 i ) = \frac { 3 } { 4 } \pi$.
The transformation $T$ from the $z$-plane to the $w$-plane is given by
$$w = \frac { 2 \mathrm { i } } { z }$$
\item Show that $T$ maps $| z - 3 i | = 3$ to a line in the $w$-plane, and give the cartesian equation of this line.\\
(5)(Total 11 marks)
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP2 2005 Q9 [11]}}