| Exam Board | AQA |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2015 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex numbers 2 |
| Type | Roots of unity with trigonometric identities |
| Difficulty | Challenging +1.2 This is a structured Further Maths question on roots of unity that guides students through standard techniques: verifying a root, using geometric series sum, manipulating conjugate pairs, and deriving a trigonometric identity. While it requires multiple steps and some algebraic manipulation, each part follows predictable FP2 methods with clear signposting, making it moderately above average difficulty but not requiring novel insight. |
| Spec | 4.02n Euler's formula: e^(i*theta) = cos(theta) + i*sin(theta)4.02q De Moivre's theorem: multiple angle formulae4.05a Roots and coefficients: symmetric functions |
The complex number $\omega$ is given by $\omega = \cos\frac{2\pi}{5} + i\sin\frac{2\pi}{5}$.
**(a) [2 marks total]**
**(i) [1 mark]**
Verify that $\omega$ is a root of the equation $z^5 = 1$.
**(ii) [1 mark]**
Write down the three other non-real roots of $z^5 = 1$, in terms of $\omega$.
**(b) [3 marks total]**
**(i) [1 mark]**
Show that $1 + \omega + \omega^2 + \omega^3 + \omega^4 = 0$.
**(ii) [2 marks]**
Hence show that $\left(\omega + \frac{1}{\omega}\right)^2 + \left(\omega + \frac{1}{\omega}\right) - 1 = 0$.
**(c) [4 marks]**
Hence show that $\cos\frac{2\pi}{5} = \frac{\sqrt{5} - 1}{4}$.8 The complex number $\omega$ is given by $\omega = \cos \frac { 2 \pi } { 5 } + \mathrm { i } \sin \frac { 2 \pi } { 5 }$.
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Verify that $\omega$ is a root of the equation $z ^ { 5 } = 1$.
\item Write down the three other non-real roots of $z ^ { 5 } = 1$, in terms of $\omega$.
\end{enumerate}\item \begin{enumerate}[label=(\roman*)]
\item Show that $1 + \omega + \omega ^ { 2 } + \omega ^ { 3 } + \omega ^ { 4 } = 0$.
\item Hence show that $\left( \omega + \frac { 1 } { \omega } \right) ^ { 2 } + \left( \omega + \frac { 1 } { \omega } \right) - 1 = 0$.
\end{enumerate}\item Hence show that $\cos \frac { 2 \pi } { 5 } = \frac { \sqrt { 5 } - 1 } { 4 }$.
\end{enumerate}
\hfill \mbox{\textit{AQA FP2 2015 Q8 [9]}}