Standard +0.3 This is a straightforward Further Maths complex numbers question requiring systematic application of standard techniques: using conditions to set up simultaneous equations (part i), sketching circles on an Argand diagram (part ii-a), and finding min/max distances between two circles (part ii-b). All steps are routine with no novel insight required, making it slightly easier than average.
\begin{enumerate}[label=(\roman*)]
\item $z_1 = a + bi$ and $z_2 = c + di$
where $a$, $b$, $c$ and $d$ are real constants.
Given that
\begin{itemize}
\item $b > d$
\item $z_1 + z_2$ is real
\item $|z_1| = \sqrt{13}$
\item $|z_2| = 5$
\item $\text{Re}(z_2 - z_1) = 2$
\end{itemize}
show that $a = 2$ and determine the value of each of $b$, $c$ and $d$ [5]
\item \begin{enumerate}[label=(\alph*)]
\item On the same Argand diagram
\begin{itemize}
\item sketch the locus of points $z$ which satisfy $|z - 12| = 7$
\item sketch the locus of points $w$ which satisfy $|w - 5i| = 4$
\end{itemize}
showing the coordinates of any points of intersection with the axes. [2]
\item Determine the range of possible values of $|z - w|$ [3]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2026 Q6 [10]}}