| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2023 |
| Session | February |
| Marks | 9 |
| Topic | Complex numbers 2 |
| Type | Geometric properties in Argand diagram |
| Difficulty | Challenging +1.2 This question requires understanding rotation in the complex plane (multiplying by ω = e^(i2π/3)) to find B and C, then calculating midpoints and using the medial triangle area property (1/4 of original). While it involves multiple steps and exact arithmetic with surds, the techniques are standard for Further Maths complex numbers with no novel geometric insight required. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02m Geometrical effects: multiplication and division |
In an Argand diagram, the points $A$, $B$ and $C$ are the vertices of an equilateral triangle with its centre at the origin. The point $A$ represents the complex number $6 + 2i$.
\begin{enumerate}[label=(\alph*)]
\item Find the complex numbers represented by the points $B$ and $C$, giving your answers in the form $x + iy$, where $x$ and $y$ are real and exact.
[6]
\end{enumerate}
The points $D$, $E$ and $F$ are the midpoints of the sides of triangle $ABC$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Find the exact area of triangle $DEF$.
[3]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2023 Q11 [9]}}