| Exam Board | OCR |
|---|---|
| Module | Further Pure Core 2 (Further Pure Core 2) |
| Year | 2021 |
| Session | June |
| Marks | 9 |
| Topic | Complex Numbers Argand & Loci |
| Type | Square roots of complex numbers |
| Difficulty | Standard +0.8 This question requires finding cube roots using De Moivre's theorem (converting to modulus-argument form, dividing argument by 3, adding 2π/3 for each root), then analyzing the geometric symmetry of the resulting equilateral triangle. While the calculation is systematic, it demands multiple conversions between forms, careful angle arithmetic with multiple roots, and geometric insight about symmetry lines through the centroid—going beyond routine complex number exercises. |
| Spec | 4.02d Exponential form: re^(i*theta)4.02r nth roots: of complex numbers |
4 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.
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.
\item Write down the angles that any lines of symmetry of triangle $A B C$ make with the positive real axis, justifying your answer.
\end{enumerate}
\hfill \mbox{\textit{OCR Further Pure Core 2 2021 Q4 [9]}}