| Exam Board | OCR |
|---|---|
| Module | Further Pure Core 2 (Further Pure Core 2) |
| Year | 2018 |
| Session | December |
| Marks | 6 |
| Topic | Complex Numbers Argand & Loci |
| Type | Complex arithmetic operations |
| Difficulty | Easy -1.2 This question tests basic geometric interpretation of complex number operations (addition and multiplication) using Argand diagrams. Part (a) requires drawing a parallelogram for addition, while part (b) involves plotting points given modulus and argument, then using the rule that multiplication adds arguments and multiplies moduli. These are fundamental, routine applications of complex number geometry with no problem-solving or novel insight required—purely recall and direct application of standard results. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02k Argand diagrams: geometric interpretation4.02l Geometrical effects: conjugate, addition, subtraction4.02m Geometrical effects: multiplication and division |
| Answer | Marks | Guidance |
|---|---|---|
| (a) 2 lines drawn as shown to complete a parallelogram | B1 | Or 2 lines drawn to form a triangle which is either the upper or lower half of the parallelogram (split by the leading diagonal). eg Im axis with \(z_2\) and \(z_1\) marked on separate rays from O on Re axis |
| Cross (or \(z_1 + z_2\) unambiguously indicated) in the correct place | B1 | |
| (b)(i) \(z_3\) and \(z_4\) approximately correctly positioned and labelled. | B1 | If no labels shown then B1B1 can only follow if there is no ambiguity between points (eg magnitudes shown). |
| Approximate correct length (eg \(z_4\) length increased by 50%) and angle (about a quarter of the way round the 2nd quadrant). | B1 | \(r = 1.8, \theta = \frac{5}{8}\pi\) |
| (b)(ii) \(z_3\) and \(z_4\) approximately correctly positioned and labelled. | B1 | If no labels shown then B1B1 can only follow if there is no ambiguity between points (eg magnitudes shown). |
| Approximate correct length (eg \(z_3\) length halved) and either the same angle as part (b)(i) or about a quarter of the way round the 2nd quadrant. | B1 | \(r = 0.35, \theta = \frac{5}{8}\pi\) |
**(a)** 2 lines drawn as shown to complete a parallelogram | B1 | Or 2 lines drawn to form a triangle which is either the upper or lower half of the parallelogram (split by the leading diagonal). eg Im axis with $z_2$ and $z_1$ marked on separate rays from O on Re axis
Cross (or $z_1 + z_2$ unambiguously indicated) in the correct place | B1 |
**(b)(i)** $z_3$ and $z_4$ approximately correctly positioned and labelled. | B1 | If no labels shown then B1B1 can only follow if there is no ambiguity between points (eg magnitudes shown).
Approximate correct length (eg $z_4$ length increased by 50%) and angle (about a quarter of the way round the 2nd quadrant). | B1 | $r = 1.8, \theta = \frac{5}{8}\pi$
**(b)(ii)** $z_3$ and $z_4$ approximately correctly positioned and labelled. | B1 | If no labels shown then B1B1 can only follow if there is no ambiguity between points (eg magnitudes shown).
Approximate correct length (eg $z_3$ length halved) and either the same angle as part **(b)(i)** or about a quarter of the way round the 2nd quadrant. | B1 | $r = 0.35, \theta = \frac{5}{8}\pi$
---\begin{enumerate}[label=(\alph*)]
\item The Argand diagram below shows the two points which represent two complex numbers, $z_1$ and $z_2$.
\includegraphics{figure_1}
On the copy of the diagram in the Printed Answer Booklet
\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 $z_1 + z_2$.
\end{itemize} [2]
\item 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 $z_3z_4$ on the diagram in the Printed Answer Booklet. You should join each point to the origin with a straight line.
\begin{enumerate}[label=(\roman*)]
\item $|z_3| = 1.5$ and $|z_4| = 1.2$ [2]
\item $|z_3| = 0.7$ and $|z_4| = 0.5$ [2]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{OCR Further Pure Core 2 2018 Q1 [6]}}