| Exam Board | Edexcel |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Roots of polynomial equations |
| Difficulty | Moderate -0.8 This is a straightforward FP1 question testing basic complex number skills: using the quadratic formula, converting to modulus-argument form, plotting on an Argand diagram, and finding distance between roots. All parts are routine applications of standard techniques with no problem-solving or insight required, making it easier than average even for Further Maths. |
| Spec | 4.02b Express complex numbers: cartesian and modulus-argument forms4.02i Quadratic equations: with complex roots4.02k Argand diagrams: geometric interpretation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Solve quadratic to obtain \(z = -5 \pm 12\text{i}\) | M1 A1 A1 | (3 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \( | z_1 | = |
| \(\arg z_1 = 1.97\) and \(\arg z_2 = -1.97\) | M1 A1 A1 | (5 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Graph showing two conjugate points plotted and line through them | B1 B1 | (2 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \( | \pm 24\text{i} | = 24\) |
| (12 marks total) |
# Question 7:
## Part (a)
| Answer/Working | Marks | Guidance |
|---|---|---|
| Solve quadratic to obtain $z = -5 \pm 12\text{i}$ | M1 A1 A1 | (3 marks) |
## Part (b)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $|z_1| = |z_2| = 13$ | B1, B1 | |
| $\arg z_1 = 1.97$ and $\arg z_2 = -1.97$ | M1 A1 A1 | (5 marks) |
## Part (c)
| Answer/Working | Marks | Guidance |
|---|---|---|
| Graph showing two conjugate points plotted and line through them | B1 B1 | (2 marks) |
## Part (d)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $|\pm 24\text{i}| = 24$ | M1 A1 | (2 marks) |
| **(12 marks total)** | | |
---7. The quadratic equation
$$z ^ { 2 } + 10 z + 169 = 0$$
has complex roots $z _ { 1 }$ and $z _ { 2 }$.
\begin{enumerate}[label=(\alph*)]
\item Find each of these roots in the form $a + b \mathrm { i }$.
\item Find the modulus and argument of $z _ { 1 }$ and of $z _ { 2 }$.
Give the arguments in radians to 3 significant figures.
\item Illustrate the two roots on a single Argand diagram.
\item Find the value of $\left| z _ { 1 } - z _ { 2 } \right|$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP1 Q7 [12]}}