| Exam Board | Edexcel |
|---|---|
| Module | F2 (Further Pure Mathematics 2) |
| Year | 2021 |
| Session | October |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex numbers 2 |
| Type | Direct nth roots: purely real or imaginary RHS |
| Difficulty | Standard +0.3 This is a straightforward application of De Moivre's theorem to find fifth roots of a complex number. Students need to express 32i in polar form, apply the nth root formula, and list all five roots with appropriate angles. While it requires knowledge of Further Maths content (complex numbers in exponential form), the procedure is mechanical and well-practiced, making it slightly easier than average. |
| Spec | 4.02d Exponential form: re^(i*theta)4.02r nth roots: of complex numbers |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Notes |
| \(z^5 - 32i = 0 \Rightarrow z^5 = 32 \Rightarrow r = 2\) | B1 | Correct value for \(r\). May be shown explicitly or used correctly. |
| \(5\theta = \frac{\pi}{2} + 2n\pi \Rightarrow \theta = \frac{\pi}{10} + \frac{2n\pi}{5}\) | M1 | Applies a correct strategy establishing at least 2 values of \(\theta\). Can be awarded if initial angle (\(\frac{\pi}{2}\) or \(\frac{\pi}{10}\)) is incorrect but strategy is otherwise correct. |
| \(z = 2e^{i\frac{\pi}{10}}, 2e^{i\frac{\pi}{2}}, 2e^{i\frac{9\pi}{10}}, 2e^{i\frac{13\pi}{10}}, 2e^{i\frac{17\pi}{10}}\) or \(z = 2e^{i\left(\frac{\pi}{10}+\frac{2n\pi}{5}\right)},\ n=0,1,2,3,4\) | A1ft | At least 2 correct, follow through their \(r\) |
| (all five roots as above) | A1 | All correct. Must have \(r = 2\) |
## Question 1:
| Working/Answer | Mark | Notes |
|---|---|---|
| $z^5 - 32i = 0 \Rightarrow z^5 = 32 \Rightarrow r = 2$ | B1 | Correct value for $r$. May be shown explicitly or used correctly. |
| $5\theta = \frac{\pi}{2} + 2n\pi \Rightarrow \theta = \frac{\pi}{10} + \frac{2n\pi}{5}$ | M1 | Applies a correct strategy establishing at least 2 values of $\theta$. Can be awarded if initial angle ($\frac{\pi}{2}$ or $\frac{\pi}{10}$) is incorrect but strategy is otherwise correct. |
| $z = 2e^{i\frac{\pi}{10}}, 2e^{i\frac{\pi}{2}}, 2e^{i\frac{9\pi}{10}}, 2e^{i\frac{13\pi}{10}}, 2e^{i\frac{17\pi}{10}}$ or $z = 2e^{i\left(\frac{\pi}{10}+\frac{2n\pi}{5}\right)},\ n=0,1,2,3,4$ | A1ft | At least 2 correct, follow through their $r$ |
| (all five roots as above) | A1 | All correct. Must have $r = 2$ |
---\begin{enumerate}
\item Solve the equation
\end{enumerate}
$$z ^ { 5 } - 32 i = 0$$
giving each answer in the form $r \mathrm { e } ^ { \mathrm { i } \theta }$ where $0 < \theta < 2 \pi$\\
\hfill \mbox{\textit{Edexcel F2 2021 Q1 [4]}}