| Exam Board | OCR |
|---|---|
| Module | Further Pure Core 1 (Further Pure Core 1) |
| Year | 2021 |
| Session | June |
| Marks | 12 |
| 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 De Moivre's theorem, sum of roots, and deriving a trigonometric value). While it requires multiple steps and some algebraic manipulation, each part builds on the previous one with clear signposting, making it more accessible than questions requiring independent problem-solving insight. The techniques are standard for FP1 level. |
| Spec | 4.02n Euler's formula: e^(i*theta) = cos(theta) + i*sin(theta)4.02q De Moivre's theorem: multiple angle formulae |
6 In this question you must show detailed reasoning.\\
You are given the complex number $\omega = \cos \frac { 2 } { 5 } \pi + i \sin \frac { 2 } { 5 } \pi$ and the equation $z ^ { 5 } = 1$.
\begin{enumerate}[label=(\alph*)]
\item Show that $\omega$ is a root of the equation.
\item Write down the other four roots of the equation.
\item Show that $\omega + \omega ^ { 2 } + \omega ^ { 3 } + \omega ^ { 4 } = - 1$.
\item Hence show that $\left( \omega + \frac { 1 } { \omega } \right) ^ { 2 } + \left( \omega + \frac { 1 } { \omega } \right) - 1 = 0$.
\item Hence determine the value of $\cos \frac { 2 } { 5 } \pi$ in the form $a + b \sqrt { c }$ where $a , b$ and $c$ are rational numbers to be found.
Total Marks for Question Set 4: 38
\end{enumerate}
\hfill \mbox{\textit{OCR Further Pure Core 1 2021 Q6 [12]}}