| Exam Board | Edexcel |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2010 |
| Session | June |
| Marks | 10 |
| 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 Further Maths question involving standard loci (perpendicular bisector and circle intersection) and a transformation. Part (a) is routine, part (b) requires solving simultaneous equations but is mechanical, and part (c) involves algebraic manipulation to show the image is a circle. While it requires multiple techniques, each step follows standard FP2 procedures without requiring novel insight or particularly challenging problem-solving. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02k Argand diagrams: geometric interpretation4.02l Geometrical effects: conjugate, addition, subtraction |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Vertical straight line drawn | B1 | |
| Line passes through \(\text{Re}(z) = 3\) on real axis | B1 | (2 marks total) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Points where line \(x=3\) meets circle centre \((3,4)\) with radius \(5\) | M1 | |
| Complex numbers are \(3 + 9i\) and \(3 - i\) | A1 A1 | (3 marks total) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\ | z-6\ | = \ |
| \(\therefore \ | 30 - 6w\ | = \ |
| Circle with Cartesian equation \((u-5)^2 + v^2 = 25\) | M1 A1 | (5 marks total) |
## Question 6:
### Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Vertical straight line drawn | B1 | |
| Line passes through $\text{Re}(z) = 3$ on real axis | B1 | (2 marks total) |
### Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Points where line $x=3$ meets circle centre $(3,4)$ with radius $5$ | M1 | |
| Complex numbers are $3 + 9i$ and $3 - i$ | A1 A1 | (3 marks total) |
### Part (c):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\|z-6\| = \|z\| \Rightarrow \left\|\frac{30}{w} - 6\right\| = \left\|\frac{30}{w}\right\|$ | M1 | |
| $\therefore \|30 - 6w\| = \|30\| \Rightarrow \therefore \|5 - w\| = \|5\|$ | M1 A1 | |
| Circle with Cartesian equation $(u-5)^2 + v^2 = 25$ | M1 A1 | (5 marks total) |
---6. A complex number $z$ is represented by the point $P$ in the Argand diagram.
\begin{enumerate}[label=(\alph*)]
\item Given that $| z - 6 | = | z |$, sketch the locus of $P$.
\item Find the complex numbers $z$ which satisfy both $| z - 6 | = | z |$ and $| z - 3 - 4 \mathrm { i } | = 5$.
The transformation $T$ from the $z$-plane to the $w$-plane is given by $w = \frac { 30 } { z }$.
\item Show that $T$ maps $| z - 6 | = | z |$ onto a circle in the $w$-plane and give the cartesian equation of this circle.
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP2 2010 Q6 [10]}}