| Exam Board | AQA |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2014 |
| Session | June |
| Marks | 7 |
| 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 standard Further Maths FP2 question on converting to exponential form and finding fourth roots. Part (a) is routine conversion of a pure imaginary number. Part (b) requires applying De Moivre's theorem systematically to find all four roots, which is a well-practiced technique at this level. The question is slightly above average difficulty due to being Further Maths content, but it's a textbook application with no novel problem-solving required. |
| Spec | 4.02d Exponential form: re^(i*theta)4.02r nth roots: of complex numbers |
(a) Express $9i$ in the form $re^{iy}$, where $r > 0$ and $-\pi < y \leq \pi$.
[2 marks]
(b) Solve the equation $z^4 + 9i = 0$, giving your answers in the form $re^{iy}$, where $r > 0$ and $-\pi < y \leq \pi$.
[5 marks]1
\begin{enumerate}[label=(\alph*)]
\item Express - 9 i in the form $r \mathrm { e } ^ { \mathrm { i } \theta }$, where $r > 0$ and $- \pi < \theta \leqslant \pi$.\\[0pt]
[2 marks]
\item Solve the equation $z ^ { 4 } + 9 \mathrm { i } = 0$, giving your answers in the form $r \mathrm { e } ^ { \mathrm { i } \theta }$, where $r > 0$ and $- \pi < \theta \leqslant \pi$.\\[0pt]
[5 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA FP2 2014 Q1 [7]}}