CAIE A-Level Mathematics A2 3.9.7 Complex Numbers Questions

Practise using complex-number geometry on Argand diagrams to identify transformations, lines and geometric relationships.

Exam points

interpret multiplication and conjugation geometrically on Argand diagrams
identify geometric relationships between points and line segments
use complex representations to prove geometric properties

[Maximum number: 2]

The complex numbers z and $\omega$ are defined by z=1-i and $\omega=-3+3 \sqrt{3} \mathrm{i}$ .

On an Argand diagram, the points representing $\omega$ and $z \omega$ are A and B respectively.

Prove that O A B is an isosceles right-angled triangle, where O is the origin.

Figure for Answer to Question CAIE A-Level Mathematics A2 3.9.7 Complex Numbers Questions question 1 — CAIE A-Level Mathematics A2

Note: The question does not require the diagram.
If they use $\frac{5 \pi}{12}$ they need to demonstrate where it comes from.
Complex number equivalent to A B is $3 \sqrt{3}+3 \mathrm{i}$ .

Show |O A|=|A B|=6, hence isosceles
One mark for 'isosceles' and one mark for 'right angle'.
There will be alternatives e.g. use of Pythagoras (ratio of lengths is

$\angle A O B=\arg \omega-\arg z \omega=-\arg z=\frac{\pi}{4}$ hence third angle is a right
angle
$1: 1: \sqrt{2}$ ), expressing each number in "vector" form and using
scalar product or explaining the effect of multiplying by 1-i.

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CAIE A-Level Mathematics A2 3.9.7 Complex Numbers Questions question 1