| Exam Board | Edexcel |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Geometric properties using complex numbers |
| Difficulty | Moderate -0.3 This is a routine FP1 question testing standard complex number operations (squaring, reciprocal, modulus) and basic Argand diagram geometry. Parts (a)-(c) are straightforward algebraic manipulation, part (d) is plotting, and part (e) requires showing similarity using scaling properties of complex multiplication—all standard techniques with no novel insight required. Slightly easier than average A-level due to being mostly procedural. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02e Arithmetic of complex numbers: add, subtract, multiply, divide4.02k Argand diagrams: geometric interpretation |
Given that $z = 3 - 3i$ express, in the form $a + ib$, where $a$ and $b$ are real numbers,
\begin{enumerate}[label=(\alph*)]
\item $z^2$, [2]
\item $\frac{1}{z}$, [2]
\item Find the exact value of each of $|z|$, $|z^2|$ and $\left|\frac{1}{z}\right|$. [2]
\end{enumerate}
The complex numbers $z$, $z^2$ and $\frac{1}{z}$ are represented by the points $A$, $B$ and $C$ respectively on an Argand diagram. The real number 1 is represented by the point $D$, and $O$ is the origin.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{3}
\item Show the points $A$, $B$, $C$ and $D$ on an Argand diagram. [2]
\item Prove that $\triangle OAB$ is similar to $\triangle OCD$. [3]
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP1 Q10 [11]}}