| Exam Board | Edexcel |
|---|---|
| Module | F2 (Further Pure Mathematics 2) |
| Year | 2017 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex numbers 2 |
| Type | Direct nth roots: purely real or imaginary RHS |
| Difficulty | Moderate -0.8 This is a straightforward application of de Moivre's theorem to find fifth roots of a real number. Students need to express 32 in polar form, apply the nth root formula, and write out five solutions—purely procedural with no problem-solving required. While it's Further Maths content, it's a standard textbook exercise testing basic technique. |
| Spec | 4.02r nth roots: of complex numbers |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance Notes |
| \(2(\cos 0 + i\sin 0)\) or \(2\) | B1 | Allow \(z=2\) or \(z=2(\cos 0 + i\sin 0)\) or \(2\cos0 + i\sin0\) or \(2+0i\) or \(2(\cos 0\pi + i\sin 0\pi)\) |
| \(2\left(\cos\frac{2\pi}{5} + i\sin\frac{2\pi}{5}\right)\) | B1 | This answer in this form. Do not allow e.g. \(2e^{\frac{2\pi i}{5}}\) but allow \(2\cos\frac{2\pi}{5} + 2i\sin\frac{2\pi}{5}\) |
| \(2\left(\cos\frac{2k\pi}{5} + i\sin\frac{2k\pi}{5}\right), (k=2,3,4)\) | M1 | Attempts at least 2 more solutions whose arguments differ by \(\frac{2\pi}{5}\). Allow if arguments are out of range |
| \(2\left(\cos\frac{4\pi}{5} + i\sin\frac{4\pi}{5}\right)\) | A1 | One further correct answer, allow brackets to be expanded |
| \(2\left(\cos\frac{6\pi}{5} + i\sin\frac{6\pi}{5}\right)\) and \(2\left(\cos\frac{8\pi}{5} + i\sin\frac{8\pi}{5}\right)\) | A1 | All correct, allow brackets to be expanded |
# Question 1:
| Answer/Working | Mark | Guidance Notes |
|---|---|---|
| $2(\cos 0 + i\sin 0)$ or $2$ | B1 | Allow $z=2$ or $z=2(\cos 0 + i\sin 0)$ or $2\cos0 + i\sin0$ or $2+0i$ or $2(\cos 0\pi + i\sin 0\pi)$ |
| $2\left(\cos\frac{2\pi}{5} + i\sin\frac{2\pi}{5}\right)$ | B1 | This answer in this form. Do not allow e.g. $2e^{\frac{2\pi i}{5}}$ but allow $2\cos\frac{2\pi}{5} + 2i\sin\frac{2\pi}{5}$ |
| $2\left(\cos\frac{2k\pi}{5} + i\sin\frac{2k\pi}{5}\right), (k=2,3,4)$ | M1 | Attempts at least 2 more solutions whose arguments differ by $\frac{2\pi}{5}$. Allow if arguments are out of range |
| $2\left(\cos\frac{4\pi}{5} + i\sin\frac{4\pi}{5}\right)$ | A1 | One further correct answer, allow brackets to be expanded |
| $2\left(\cos\frac{6\pi}{5} + i\sin\frac{6\pi}{5}\right)$ and $2\left(\cos\frac{8\pi}{5} + i\sin\frac{8\pi}{5}\right)$ | A1 | All correct, allow brackets to be expanded |
**Do not allow:** $2\left(\cos\frac{4\pi}{5} - i\sin\frac{4\pi}{5}\right)$ or $2\left(\cos\left(-\frac{4\pi}{5}\right) + i\sin\left(-\frac{4\pi}{5}\right)\right)$ for $2\left(\cos\frac{6\pi}{5} + i\sin\frac{6\pi}{5}\right)$
**Do not allow:** $2\left(\cos\frac{2\pi}{5} - i\sin\frac{2\pi}{5}\right)$ or $2\left(\cos\left(-\frac{2\pi}{5}\right) + i\sin\left(-\frac{2\pi}{5}\right)\right)$ for $2\left(\cos\frac{8\pi}{5} + i\sin\frac{8\pi}{5}\right)$
Answers in degrees: $0, 72, 144, 216, 288$ — penalise once on first occurrence. **[Total: 5]**
---\begin{enumerate}
\item Solve the equation
\end{enumerate}
$$z ^ { 5 } = 32$$
Give your answers in the form $r ( \cos \theta + \mathrm { i } \sin \theta )$, where $r > 0$ and $0 \leqslant \theta < 2 \pi$\\
\hfill \mbox{\textit{Edexcel F2 2017 Q1 [5]}}