| Exam Board | Edexcel |
|---|---|
| Module | F1 (Further Pure Mathematics 1) |
| Year | 2014 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Geometric relationships on Argand diagram |
| Difficulty | Standard +0.8 This is a Further Maths question requiring multiple geometric proofs on the Argand diagram. While plotting points (a,c) is routine, proving perpendicularity (b) requires showing the product of gradients equals -1 or using complex number arguments, and proving OPRQ is a square (d) demands showing equal side lengths and right angles using modulus and argument properties. The multi-step proof structure and geometric insight needed place this above average difficulty. |
| Spec | 1.10a Vectors in 2D: i,j notation and column vectors4.02a Complex numbers: real/imaginary parts, modulus, argument4.02k Argand diagrams: geometric interpretation |
\begin{enumerate}
\item Given that $z _ { 1 } = - 3 - 4 \mathrm { i }$ and $z _ { 2 } = 4 - 3 \mathrm { i }$\\
(a) show, on an Argand diagram, the point $P$ representing $z _ { 1 }$ and the point $Q$ representing $z _ { 2 }$\\
(b) Given that $O$ is the origin, show that $O P$ is perpendicular to $O Q$.\\
(c) Show the point $R$ on your diagram, where $R$ represents $z _ { 1 } + z _ { 2 }$\\
(d) Prove that $O P R Q$ is a square.\\
\end{enumerate}
\hfill \mbox{\textit{Edexcel F1 2014 Q5 [7]}}