| Exam Board | Edexcel |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Roots of unity and special equations |
| Difficulty | Moderate -0.8 This is a straightforward Further Pure 1 question on cube roots of unity with standard techniques: factorising a cubic given one root, solving a quadratic formula, and plotting on an Argand diagram. While it's Further Maths content, the methods are routine and well-practiced, making it easier than average overall but not trivial due to the multi-step nature and complex number manipulation required. |
| Spec | 4.02i Quadratic equations: with complex roots4.02k Argand diagrams: geometric interpretation4.02r nth roots: of complex numbers |
\begin{enumerate}[label=(\alph*)]
\item Using that 3 is the real root of the cubic equation $x^3 - 27 = 0$, show that the complex roots of the cubic satisfy the quadratic equation $x^2 + 3x + 9 = 0$. [2]
\item Hence, or otherwise, find the three cube roots of 27, giving your answers in the form $a + ib$, where $a, b \in \mathbb{R}$. [3]
\item Show these roots on an Argand diagram. [2]
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP1 Q11 [7]}}