| Exam Board | OCR |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex numbers 2 |
| Type | Basic roots of unity properties |
| Difficulty | Standard +0.8 This is a multi-part FP3 question on roots of unity requiring algebraic manipulation, complex number representation, and factorization. Part (i) is routine expansion (1 mark). Part (ii) requires finding 7th roots of unity and plotting them (standard but requires care). Part (iii) is the challenging component: students must pair conjugate roots using the result from (i) to obtain real quadratic factors, requiring insight into how complex conjugates combine and systematic organization of multiple factors. While structured with guidance, the synthesis in part (iii) elevates this above routine Further Maths exercises. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02r nth roots: of complex numbers |
\begin{enumerate}[label=(\roman*)]
\item Show that $(z - e^{i\theta})(z - e^{-i\theta}) \equiv z^2 - (2\cos \theta)z + 1$. [1]
\item Write down the seven roots of the equation $z^7 = 1$ in the form $e^{i\theta}$ and show their positions in an Argand diagram. [4]
\item Hence express $z^7 - 1$ as the product of one real linear factor and three real quadratic factors. [5]
\end{enumerate}
\hfill \mbox{\textit{OCR FP3 Q7 [10]}}