| Exam Board | AQA |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2013 |
| Session | January |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Topic | Complex numbers 2 |
| Type | Direct nth roots: roots with geometric or algebraic follow-up |
| Difficulty | Challenging +1.2 This is a multi-part Further Maths question requiring conversion to exponential form, finding cube roots using De Moivre's theorem, calculating triangle area on an Argand diagram, and proving a trigonometric identity using roots of unity. While it involves several FP2 techniques, each part follows standard procedures without requiring novel insight—the methods are direct applications of syllabus content. The final part requires recognizing that real parts of roots sum to the cosine expression, which is a typical FP2 exam technique. Slightly above average difficulty due to the multi-step nature and Further Maths content. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02b Express complex numbers: cartesian and modulus-argument forms4.02d Exponential form: re^(i*theta)4.02k Argand diagrams: geometric interpretation4.02n Euler's formula: e^(i*theta) = cos(theta) + i*sin(theta)4.02r nth roots: of complex numbers |
\begin{enumerate}[label=(\alph*)]
\item Express $-4 + 4\sqrt{3}\text{i}$ in the form $r\text{e}^{\text{i}\theta}$, where $r > 0$ and $-\pi < \theta \leqslant \pi$. [3 marks]
\item
\begin{enumerate}[label=(\roman*)]
\item Solve the equation $z^3 = -4 + 4\sqrt{3}\text{i}$, giving your answers in the form $r\text{e}^{\text{i}\theta}$, where $r > 0$ and $-\pi < \theta \leqslant \pi$. [4 marks]
\item The roots of the equation $z^3 = -4 + 4\sqrt{3}\text{i}$ are represented by the points $P$, $Q$ and $R$ on an Argand diagram.
Find the area of the triangle $PQR$, giving your answer in the form $k\sqrt{3}$, where $k$ is an integer. [3 marks]
\end{enumerate}
\item By considering the roots of the equation $z^3 = -4 + 4\sqrt{3}\text{i}$, show that
$$\cos\frac{2\pi}{9} + \cos\frac{4\pi}{9} + \cos\frac{8\pi}{9} = 0$$ [4 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA FP2 2013 Q8 [14]}}