| Exam Board | Edexcel |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Region shading with multiple inequalities |
| Difficulty | Standard +0.8 This is a multi-part Further Maths question requiring understanding of loci (circle with center 5+12i, radius 3), then optimization using geometric reasoning. Part (a) is routine, but parts (b) and (c) require visualizing that max/min |z| occur on the line through origin and center, and max/min arg(z) occur at tangent points. While systematic, it demands geometric insight beyond standard techniques, placing it moderately above average difficulty. |
| Spec | 4.02c Complex notation: z, z*, Re(z), Im(z), |z|, arg(z)4.02k Argand diagrams: geometric interpretation |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Notes |
| Locus is a circle | B1 | |
| Centre is at \((5, 12)\) | B1 | |
| Radius is \(3\) | B1 (3) |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Notes |
| Finds distance from centre to origin is \(13\) | M1 | |
| Maximum modulus is \(13 + 3 = 16\) | M1 A1 | |
| Minimum modulus is \(13 - 3 = 10\) | A1 (4) |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Notes |
| Finds \(\arctan\frac{12}{5}\) | M1 | |
| Uses \(\arctan\frac{12}{5} \pm \arcsin\frac{3}{13}\) | M1 | |
| Obtains \(0.94\) and \(1.41\) radians | A1 A1 (4) |
# Question 8:
## Part (a):
| Working | Marks | Notes |
|---------|-------|-------|
| Locus is a circle | B1 | |
| Centre is at $(5, 12)$ | B1 | |
| Radius is $3$ | B1 (3) | |
## Part (b):
| Working | Marks | Notes |
|---------|-------|-------|
| Finds distance from centre to origin is $13$ | M1 | |
| Maximum modulus is $13 + 3 = 16$ | M1 A1 | |
| Minimum modulus is $13 - 3 = 10$ | A1 (4) | |
## Part (c):
| Working | Marks | Notes |
|---------|-------|-------|
| Finds $\arctan\frac{12}{5}$ | M1 | |
| Uses $\arctan\frac{12}{5} \pm \arcsin\frac{3}{13}$ | M1 | |
| Obtains $0.94$ and $1.41$ radians | A1 A1 (4) | |
---8. A complex number $z$ satisfies the equation
$$| z - 5 - 12 i | = 3$$
\begin{enumerate}[label=(\alph*)]
\item Describe in geometrical terms with the aid of a sketch, the locus of the point which represents $z$ in the A rgand diagram.
For points on this locus, find
\item the maximum and minimum values for $| z |$,
\item the maximum and minimum values for arg $z$, giving your answers in radians to 2 decimal places.
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP2 Q8 [11]}}