| Exam Board | Edexcel |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Area calculations in complex plane |
| Difficulty | Standard +0.3 This is a straightforward FP1 question testing standard complex number operations: finding modulus and argument of a quotient (using |z/w| = |z|/|w| and arg(z/w) = arg(z) - arg(w)), plotting points on an Argand diagram, and basic geometric calculations. All parts follow routine procedures with no novel insight required, making it slightly easier than average even for Further Maths. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02b Express complex numbers: cartesian and modulus-argument forms4.02e Arithmetic of complex numbers: add, subtract, multiply, divide4.02f Convert between forms: cartesian and modulus-argument4.02k Argand diagrams: geometric interpretation4.02m Geometrical effects: multiplication and division |
Given that $z = -2\sqrt{2} + 2\sqrt{2}i$ and $w = 1 - i\sqrt{3}$, find
\begin{enumerate}[label=(\alph*)]
\item $\left|\frac{z}{w}\right|$, [3]
\item $\arg \left( \frac{z}{w} \right)$. [3]
\end{enumerate}
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item On an Argand diagram, plot points $A$, $B$, $C$ and $D$ representing the complex numbers $z$, $w$, $\left( \frac{z}{w} \right)$ and 4, respectively. [3]
\item Show that $\angle AOC = \angle DOB$. [2]
\item Find the area of triangle $AOC$. [2]
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP1 Q21 [13]}}