Moderate -0.8 This is a straightforward Further Maths question testing basic geometric interpretation of complex number operations (addition via parallelogram rule, multiplication via adding arguments and multiplying moduli). All parts require only direct application of standard rules with no problem-solving or novel insight—essentially bookwork with diagram drawing. While it's Further Maths content, the execution is purely procedural and simpler than typical A-level problem-solving questions.
The Argand diagram below shows the two points which represent two complex numbers, \(z _ { 1 }\) and \(z _ { 2 }\).
\includegraphics[max width=\textwidth, alt={}, center]{20816f61-154d-4491-9d2d-4c62687bf81e-02_321_592_276_347}
On the copy of the diagram in the Resource Materials.
draw an appropriate shape to illustrate the geometrical effect of adding \(z _ { 1 }\) and \(z _ { 2 }\),
indicate with a cross \(( \times )\) the location of the point representing the complex number \(z _ { 1 } + z _ { 2 }\).
You are given that \(\arg z _ { 3 } = \frac { 1 } { 4 } \pi\) and \(\arg z _ { 4 } = \frac { 3 } { 8 } \pi\).
In each part, sketch and label the points representing the numbers \(z _ { 3 } , z _ { 4 }\) and \(z _ { 3 } z _ { 4 }\) on the diagram in the Resource Materials. You should join each point to the origin with a straight line.
\(\left| z _ { 3 } \right| = 1.5\) and \(\left| z _ { 4 } \right| = 1.2\)
\(\left| z _ { 3 } \right| = 0.7\) and \(\left| z _ { 4 } \right| = 0.5\)
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\begin{enumerate}[label=(\alph*)]
\item The Argand diagram below shows the two points which represent two complex numbers, $z _ { 1 }$ and $z _ { 2 }$.\\
\includegraphics[max width=\textwidth, alt={}, center]{20816f61-154d-4491-9d2d-4c62687bf81e-02_321_592_276_347}
On the copy of the diagram in the Resource Materials.
\begin{itemize}
\item draw an appropriate shape to illustrate the geometrical effect of adding $z _ { 1 }$ and $z _ { 2 }$,
\item indicate with a cross $( \times )$ the location of the point representing the complex number $z _ { 1 } + z _ { 2 }$.
\item You are given that $\arg z _ { 3 } = \frac { 1 } { 4 } \pi$ and $\arg z _ { 4 } = \frac { 3 } { 8 } \pi$.
\end{itemize}
In each part, sketch and label the points representing the numbers $z _ { 3 } , z _ { 4 }$ and $z _ { 3 } z _ { 4 }$ on the diagram in the Resource Materials. You should join each point to the origin with a straight line.
\begin{enumerate}[label=(\roman*)]
\item $\left| z _ { 3 } \right| = 1.5$ and $\left| z _ { 4 } \right| = 1.2$
\item $\left| z _ { 3 } \right| = 0.7$ and $\left| z _ { 4 } \right| = 0.5$
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{OCR Further Pure Core 2 2021 Q1 [6]}}