| Exam Board | AQA |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2008 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex numbers 2 |
| Type | Direct nth roots: find and express roots |
| Difficulty | Standard +0.3 This is a standard Further Maths FP2 question on finding nth roots of complex numbers. Part (a) requires routine conversion to exponential form (finding modulus and argument), and part (b) applies the standard formula for fifth roots by dividing the argument by 5 and adding multiples of 2π/5. While this is Further Maths content, it's a textbook application of a well-practiced technique with no novel problem-solving required. |
| Spec | 4.02d Exponential form: re^(i*theta)4.02r nth roots: of complex numbers |
| Answer | Marks | Guidance |
|---|---|---|
| (a) Any method for finding \(r\) or \(\theta\) | M1 | |
| \(r = 4\sqrt{2}\), \(\theta = \frac{\pi}{4}\) | A1A1 | 3 marks total |
| (b) \(z^3 = 4\sqrt{2}e^{i\frac{\pi}{4}}\) | M1 | M1 needs some reference to \(a + 2k\pi i\); A1 for \(r\); incorrect \(r, \theta\) in part (a) |
| \(z = \sqrt{2}e^{i\frac{\pi + 2k\pi i}{20}}\) | A1F, A1F | |
| \(z = \sqrt{2}e^{i\frac{\pi}{20}}\), \(\sqrt{2}e^{i\frac{9\pi}{20}}\), \(\sqrt{2}e^{i\frac{17\pi}{20}}\), \(\sqrt{2}e^{-i\frac{7\pi}{20}}\), \(\sqrt{2}e^{-i\frac{15\pi}{20}}\) | A2,1,0, F | 5 marks total; Accept \(r\) in any form eg \(32^{\frac{1}{10}}\); Correct but some answers outside range allow A1; it incorrect \(r, \theta\) in part (a) |
**(a)** Any method for finding $r$ or $\theta$ | M1 |
$r = 4\sqrt{2}$, $\theta = \frac{\pi}{4}$ | A1A1 | 3 marks total
**(b)** $z^3 = 4\sqrt{2}e^{i\frac{\pi}{4}}$ | M1 | M1 needs some reference to $a + 2k\pi i$; A1 for $r$; incorrect $r, \theta$ in part (a)
$z = \sqrt{2}e^{i\frac{\pi + 2k\pi i}{20}}$ | A1F, A1F |
$z = \sqrt{2}e^{i\frac{\pi}{20}}$, $\sqrt{2}e^{i\frac{9\pi}{20}}$, $\sqrt{2}e^{i\frac{17\pi}{20}}$, $\sqrt{2}e^{-i\frac{7\pi}{20}}$, $\sqrt{2}e^{-i\frac{15\pi}{20}}$ | A2,1,0, F | 5 marks total; Accept $r$ in any form eg $32^{\frac{1}{10}}$; Correct but some answers outside range allow A1; it incorrect $r, \theta$ in part (a)
**Question 1 Total: 8 marks**
---1
\begin{enumerate}[label=(\alph*)]
\item Express $4 + 4 \mathrm { i }$ in the form $r \mathrm { e } ^ { \mathrm { i } \theta }$, where $r > 0$ and $- \pi < \theta \leqslant \pi$.
\item Solve the equation
$$z ^ { 5 } = 4 + 4 i$$
giving your answers in the form $r \mathrm { e } ^ { \mathrm { i } \theta }$, where $r > 0$ and $- \pi < \theta \leqslant \pi$.
\end{enumerate}
\hfill \mbox{\textit{AQA FP2 2008 Q1 [8]}}