| Exam Board | AQA |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2009 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex numbers 2 |
| Type | Direct nth roots: roots with geometric or algebraic follow-up |
| Difficulty | Standard +0.8 This is a Further Maths FP2 question requiring students to verify a given root satisfies an equation (finding parameter a), then find remaining roots using de Moivre's theorem and argument spacing. While systematic, it requires confident manipulation of exponential form, argument arithmetic with fractions of π, and understanding of root distribution—moderately challenging for Further Maths students. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02d Exponential form: re^(i*theta)4.02q De Moivre's theorem: multiple angle formulae4.02r nth roots: of complex numbers |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(z^4 = 16e^{i\frac{4\pi}{12}}\) | M1 | Allow M1 if \(z^4 = 2e^{i\frac{\pi}{12}}\) |
| \(= 16\left(\cos\frac{\pi}{3} + i\sin\frac{\pi}{3}\right)\) | A1 | OE could be \(2ae^{i\frac{\pi}{3}}\) or \(2a\left(\cos\frac{\pi}{3} + i\sin\frac{\pi}{3}\right)\) |
| \(= 8 + 8\sqrt{3}i; \quad a = 8\) | A1F | 3 marks |
| (b) For other roots, \(r = 2\) | B1 | for realising roots are of form \(2 \times e^{i\theta}\) M1 for strictly correct \(\theta\) |
| \(\theta = \frac{\pi}{12} + \frac{2k\pi}{4}\) | M1A1 | i.e. must be \(\left(\frac{\pi}{3} + 2k\pi\right) \times \frac{1}{4}\), ft error in \(\frac{\pi}{12}\) or \(r\) |
| Roots are \(2e^{i\frac{7\pi}{12}}\), \(2e^{-i\frac{5\pi}{12}}\), \(2e^{-i\frac{11\pi}{12}}\) | A2,1, 0 F | 5 marks |
**(a)** $z^4 = 16e^{i\frac{4\pi}{12}}$ | M1 | Allow M1 if $z^4 = 2e^{i\frac{\pi}{12}}$
$= 16\left(\cos\frac{\pi}{3} + i\sin\frac{\pi}{3}\right)$ | A1 | OE could be $2ae^{i\frac{\pi}{3}}$ or $2a\left(\cos\frac{\pi}{3} + i\sin\frac{\pi}{3}\right)$
$= 8 + 8\sqrt{3}i; \quad a = 8$ | A1F | 3 marks | ft errors in $2^4$
**(b)** For other roots, $r = 2$ | B1 | for realising roots are of form $2 \times e^{i\theta}$ M1 for strictly correct $\theta$
$\theta = \frac{\pi}{12} + \frac{2k\pi}{4}$ | M1A1 | i.e. must be $\left(\frac{\pi}{3} + 2k\pi\right) \times \frac{1}{4}$, ft error in $\frac{\pi}{12}$ or $r$
Roots are $2e^{i\frac{7\pi}{12}}$, $2e^{-i\frac{5\pi}{12}}$, $2e^{-i\frac{11\pi}{12}}$ | A2,1, 0 F | 5 marks | accept $2e^{i\left(\frac{\pi}{12}+\frac{2k\pi}{4}\right)}$ for $k = -1, -2, 1$
**Total: 8 marks**1 Given that $z = 2 \mathrm { e } ^ { \frac { \pi \mathrm { i } } { 12 } }$ satisfies the equation
$$z ^ { 4 } = a ( 1 + \sqrt { 3 } i )$$
where $a$ is real:
\begin{enumerate}[label=(\alph*)]
\item find the value of $a$;
\item find the other three roots of this equation, 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 2009 Q1 [8]}}