| Exam Board | Edexcel |
|---|---|
| Module | F2 (Further Pure Mathematics 2) |
| Session | Specimen |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex numbers 2 |
| Type | Complex number arithmetic and simplification |
| Difficulty | Standard +0.3 This is a standard Further Maths question testing routine application of modulus-argument form and de Moivre's theorem. Part (a) requires straightforward calculation, part (b) is direct application of de Moivre's theorem, and part (c) involves finding fourth roots using standard techniques. While it's Further Maths content (inherently harder), the question follows a predictable template with no novel problem-solving required. |
| Spec | 4.02b Express complex numbers: cartesian and modulus-argument forms4.02q De Moivre's theorem: multiple angle formulae4.02r nth roots: of complex numbers |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Modulus \(= 16\) | B1 | |
| Argument \(= \arctan(-\sqrt{3}) = \frac{2\pi}{3}\) | M1 A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(z^3 = 16^3(\cos(\frac{2\pi}{3}) + i\sin(\frac{2\pi}{3}))^3 = 16^3(\cos 2\pi + i\sin 2\pi) = 4096\) or \(16^3\) | M1 A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(w = 16^{\frac{1}{4}}(\cos(\frac{2\pi}{3}) + i\sin(\frac{2\pi}{3}))^{\frac{1}{4}} = 2(\cos(\frac{\pi}{6}) + i\sin(\frac{\pi}{6}))\) \((= \sqrt{3} + i)\) | M1 A1ft | |
| OR \(-1 + \sqrt{3}i\) OR \(-\sqrt{3} - i\) OR \(1 - \sqrt{3}i\) | M1A2 (1,0) |
## Question 4:
### Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Modulus $= 16$ | B1 | |
| Argument $= \arctan(-\sqrt{3}) = \frac{2\pi}{3}$ | M1 A1 | |
### Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $z^3 = 16^3(\cos(\frac{2\pi}{3}) + i\sin(\frac{2\pi}{3}))^3 = 16^3(\cos 2\pi + i\sin 2\pi) = 4096$ or $16^3$ | M1 A1 | |
### Part (c):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $w = 16^{\frac{1}{4}}(\cos(\frac{2\pi}{3}) + i\sin(\frac{2\pi}{3}))^{\frac{1}{4}} = 2(\cos(\frac{\pi}{6}) + i\sin(\frac{\pi}{6}))$ $(= \sqrt{3} + i)$ | M1 A1ft | |
| OR $-1 + \sqrt{3}i$ OR $-\sqrt{3} - i$ OR $1 - \sqrt{3}i$ | M1A2 (1,0) | |
---4.
$$z = - 8 + ( 8 \sqrt { } 3 ) \mathrm { i }$$
\begin{enumerate}[label=(\alph*)]
\item Find the modulus of $z$ and the argument of $z$.
Using de Moivre's theorem,
\item find $z ^ { 3 }$,
\item find the values of $w$ such that $w ^ { 4 } = z$, giving your answers in the form $a + \mathrm { i } b$, where $a , b \in \mathbb { R }$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel F2 Q4 [10]}}