| Exam Board | Edexcel |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Division plus modulus/argument |
| Difficulty | Moderate -0.3 This is a structured multi-part question testing standard complex number operations (conjugate, division, modulus, argument) with a specific value given. Parts (a)-(d) involve routine calculations and verification of known properties, while part (e) uses standard sum/product of roots. Slightly easier than average due to the guided structure and straightforward arithmetic, though it requires familiarity with multiple FP1 techniques. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02e Arithmetic of complex numbers: add, subtract, multiply, divide4.02g Conjugate pairs: real coefficient polynomials4.02k Argand diagrams: geometric interpretation |
$$z = \sqrt{3} - i.$$
$z^*$ is the complex conjugate of $z$.
\begin{enumerate}[label=(\alph*)]
\item Show that $\frac{z}{z^*} = \frac{1}{2} - \frac{\sqrt{3}}{2} i$. [3]
\item Find the value of $\left| \frac{z}{z^*} \right|$. [2]
\item Verify, for $z = \sqrt{3} - i$, that $\arg \frac{z}{z^*} = \arg z - \arg z^*$. [4]
\item Display $z$, $z^*$ and $\frac{z}{z^*}$ on a single Argand diagram. [2]
\item Find a quadratic equation with roots $z$ and $z^*$ in the form $ax^2 + bx + c = 0$, where $a$, $b$ and $c$ are real constants to be found. [2]
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP1 Q38 [13]}}