| Exam Board | Edexcel |
|---|---|
| Module | F2 (Further Pure Mathematics 2) |
| Session | Specimen |
| 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 Further Maths question combining standard loci (perpendicular bisector and circle intersection) with an inversion transformation. Part (a) is routine, part (b) requires solving simultaneous geometric equations, and part (c) involves algebraic manipulation of the transformation—all established F2 techniques without requiring novel insight. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Vertical straight line through 3 on real axis, \(\text{Re}(z) = 3\) | B1 B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Points where line \(x = 3\) meets circle centre \((3, 4)\) radius \(5\) | M1 | |
| Complex numbers are \(3 + 9i\) and \(3 - i\) | A1 A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\ | z - 6\ | = \ |
| \(\therefore \ | 30 - 6w\ | = \ |
| Circle with Cartesian equation \((u-5)^2 + v^2 = 25\) | M1 A1 |
## Question 6:
### Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Vertical straight line through 3 on real axis, $\text{Re}(z) = 3$ | B1 B1 | |
### Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Points where line $x = 3$ meets circle centre $(3, 4)$ radius $5$ | M1 | |
| Complex numbers are $3 + 9i$ and $3 - i$ | A1 A1 | |
### Part (c):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\|z - 6\| = \|z\| \Rightarrow \|\frac{30}{w} - 6\| = \|\frac{30}{w}\|$ | M1 | |
| $\therefore \|30 - 6w\| = \|30\| \Rightarrow \therefore \|5 - w\| = \|5\|$ | M1 A1 | |
| Circle with Cartesian equation $(u-5)^2 + v^2 = 25$ | M1 A1 | |\begin{enumerate}
\item A complex number $z$ is represented by the point $P$ in the Argand diagram.\\
(a) Given that $| z - 6 | = | z |$, sketch the locus of $P$.\\
(b) Find the complex numbers $z$ which satisfy both $| z - 6 | = | z |$ and $| z - 3 - 4 \mathrm { i } | = 5$.
\end{enumerate}
The transformation $T$ from the $z$-plane to the $w$-plane is given by $w = \frac { 30 } { z }$.\\
(c) Show that $T$ maps $| z - 6 | = | z |$ onto a circle in the $w$-plane and give the cartesian equation of this circle.
\hfill \mbox{\textit{Edexcel F2 Q6 [10]}}