| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2024 |
| Session | February |
| Marks | 9 |
| Topic | Complex numbers 2 |
| Type | Direct nth roots: roots with geometric or algebraic follow-up |
| Difficulty | Standard +0.8 This is a Further Maths question requiring conversion to modulus-argument form, application of de Moivre's theorem for cube roots, and geometric reasoning about symmetry in the Argand diagram. While the techniques are standard for FM students, the multi-step nature (simplify the complex number, find modulus/argument, apply cube root formula for three roots, then analyze geometric symmetry) and the need for careful angle calculations elevate this above average difficulty. |
| Spec | 4.02q De Moivre's theorem: multiple angle formulae4.02r nth roots: of complex numbers |
In this question you must show detailed reasoning.
The complex number $-4 + i\sqrt{48}$ is denoted by $z$.
\begin{enumerate}[label=(\alph*)]
\item Determine the cube roots of $z$, giving the roots in exponential form. [6]
\end{enumerate}
The points which represent the cube roots of $z$ are denoted by $A$, $B$ and $C$ and these form a triangle in an Argand diagram.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Write down the angles that any lines of symmetry of triangle $ABC$ make with the positive real axis, justifying your answer. [3]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2024 Q9 [9]}}