Geometric properties in Argand diagram

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

1. Express \(z \omega\) in the form \(\mathrm { a } + \mathrm { bi }\), where \(a\) and \(b\) are real and in exact surd form.
2. Express \(z\) and \(\omega\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\). Give the exact values of \(r\) and \(\theta\) in each case.
3. 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.
4. Using your answers to part (b), prove that \(\tan \frac { 5 } { 12 } \pi = \frac { \sqrt { 3 } + 1 } { \sqrt { 3 } - 1 }\).

8 The complex numbers \(u\) and \(v\) are defined by \(u = - 4 + 2 \mathrm { i }\) and \(v = 3 + \mathrm { i }\).

1. Find \(\frac { u } { v }\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
2. Hence express \(\frac { u } { v }\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r\) and \(\theta\) are exact.
   In an Argand diagram, with origin \(O\), the points \(A , B\) and \(C\) represent the complex numbers \(u , v\) and \(2 u + v\) respectively.
3. State fully the geometrical relationship between \(O A\) and \(B C\).
4. Prove that angle \(A O B = \frac { 3 } { 4 } \pi\).

5

1. Solve the equation \(z ^ { 2 } - 6 \mathrm { i } z - 12 = 0\), giving the answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real and exact.
2. On a sketch of an Argand diagram with origin \(O\), show points \(A\) and \(B\) representing the roots of the equation in part (a).
3. Find the exact modulus and argument of each root.
4. Hence show that the triangle \(O A B\) is equilateral.

4 The cube roots of 1 are denoted by \(1 , \omega\) and \(\omega ^ { 2 }\), where the imaginary part of \(\omega\) is positive.

1. Show that \(1 + \omega + \omega ^ { 2 } = 0\).
   \includegraphics[max width=\textwidth, alt={}, center]{d12573dd-c0c2-4f0d-8e49-8fdf8d5864a5-2_616_748_1676_699} In the diagram, \(A B C\) is an equilateral triangle, labelled anticlockwise. The points \(A , B\) and \(C\) represent the complex numbers \(z _ { 1 } , z _ { 2 }\) and \(z _ { 3 }\) respectively.
2. State the geometrical effect of multiplication by \(\omega\) and hence explain why \(z _ { 1 } - z _ { 3 } = \omega \left( z _ { 3 } - z _ { 2 } \right)\).
3. Hence show that \(z _ { 1 } + \omega z _ { 2 } + \omega ^ { 2 } z _ { 3 } = 0\).

12 Fig. 12 shows a rhombus OACB in an Argand diagram. The points A and B represent the complex numbers \(z\) and \(w\) respectively. \begin{figure}[h] \end{figure} Prove that \(\arg ( z + w ) = \frac { 1 } { 2 } ( \arg z + \arg w )\).
[0pt] [A copy of Fig. 12 is provided in the Printed Answer Booklet.]