Modulus and argument calculations

8 The variable complex number \(z\) is given by $$z = 1 + \cos 2 \theta + i \sin 2 \theta$$ where \(\theta\) takes all values in the interval \(- \frac { 1 } { 2 } \pi < \theta < \frac { 1 } { 2 } \pi\).

1. Show that the modulus of \(z\) is \(2 \cos \theta\) and the argument of \(z\) is \(\theta\).
2. Prove that the real part of \(\frac { 1 } { z }\) is constant.

5 The complex number \(2 + y \mathrm { i }\) is denoted by \(a\), where \(y\) is a real number and \(y < 0\). It is given that \(\mathrm { f } ( a ) = a ^ { 3 } - a ^ { 2 } - 2 a\).

1. Find a simplified expression for \(\mathrm { f } ( a )\) in terms of \(y\).
2. Given that \(\operatorname { Re } ( \mathrm { f } ( a ) ) = - 20\), find \(\arg a\).

11 The complex number \(z\) is defined by \(z = \frac { 5 a - 2 \mathrm { i } } { 3 + a \mathrm { i } }\), where \(a\) is an integer. It is given that \(\arg z = - \frac { 1 } { 4 } \pi\).

1. Find the value of \(a\) and hence express \(z\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
2. Express \(z ^ { 3 }\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\). Give the simplified exact values of \(r\) and \(\theta\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

5

1. The complex number \(u\) is given by $$u = \frac { \left( \cos \frac { 1 } { 7 } \pi + i \sin \frac { 1 } { 7 } \pi \right) ^ { 4 } } { \cos \frac { 1 } { 7 } \pi - i \sin \frac { 1 } { 7 } \pi }$$ Find the exact value of \(\arg u\).
2. The complex numbers \(u\) and \(u ^ { * }\) are plotted on an Argand diagram. Describe the single geometrical transformation that maps \(u\) onto \(u ^ { * }\) and state the exact value of \(\arg u ^ { * }\).
   \includegraphics[max width=\textwidth, alt={}, center]{9da6c2ac-31aa-4063-88b9-e15e38bedd8a-06_2715_35_110_2012}
   \includegraphics[max width=\textwidth, alt={}, center]{9da6c2ac-31aa-4063-88b9-e15e38bedd8a-07_588_869_255_603} The variables \(x\) and \(y\) satisfy the equation \(a y = b ^ { x }\), where \(a\) and \(b\) are constants. The graph of \(\ln y\) against \(x\) is a straight line passing through the points ( \(0.50,2.24\) ) and ( \(3.40,8.27\) ), as shown in the diagram. Find the values of \(a\) and \(b\). Give each value correct to 1 significant figure.

2. $$z = 5 \sqrt { } 3 - 5 i$$ Find

1. \(| z |\),
2. \(\arg ( z )\), in terms of \(\pi\). $$w = 2 \left( \cos \frac { \pi } { 4 } + i \sin \frac { \pi } { 4 } \right)$$ Find
3. \(\left| \frac { w } { z } \right|\),
4. \(\quad \arg \left( \frac { w } { z } \right)\), in terms of \(\pi\).

2 You are given the complex numbers \(z = \sqrt { 32 } ( 1 + \mathrm { j } )\) and \(w = 8 \left( \cos \frac { 7 } { 12 } \pi + \mathrm { j } \sin \frac { 7 } { 12 } \pi \right)\).

1. Find the modulus and argument of each of the complex numbers \(z , z ^ { * } , z w\) and \(\frac { z } { w }\).
2. Express \(\frac { z } { w }\) in the form \(a + b \mathrm { j }\), giving the exact values of \(a\) and \(b\).
3. Find the cube roots of \(z\), in the form \(r \mathrm { e } ^ { \mathrm { j } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).
4. Show that the cube roots of \(z\) can be written as $$k _ { 1 } w ^ { * } , \quad k _ { 2 } z ^ { * } \quad \text { and } \quad k _ { 3 } \mathrm { j } w ,$$ where \(k _ { 1 } , k _ { 2 }\) and \(k _ { 3 }\) are real numbers. State the values of \(k _ { 1 } , k _ { 2 }\) and \(k _ { 3 }\).

1 Given that \(z _ { 1 } = 4 e ^ { \mathrm { i } \frac { \pi } { 3 } }\) and \(z _ { 2 } = 2 e ^ { \mathrm { i } \frac { \pi } { 4 } }\)
state the value of \(\arg \left( \frac { z _ { 1 } } { z _ { 2 } } \right)\)
Circle your answer.
[0pt] [1 mark]
\(\frac { \pi } { 12 }\)
\(\frac { 4 } { 3 }\)
\(\frac { 7 \pi } { 12 }\)
2

1. Given that

$$\begin{aligned} z _ { 1 } & = 3 \left( \cos \left( \frac { \pi } { 3 } \right) + \mathrm { i } \sin \left( \frac { \pi } { 3 } \right) \right) \\ z _ { 2 } & = \sqrt { 2 } \left( \cos \left( \frac { \pi } { 12 } \right) - \mathrm { i } \sin \left( \frac { \pi } { 12 } \right) \right) \end{aligned}$$

1. write down the exact value of
   1. \(\left| Z _ { 1 } Z _ { 2 } \right|\)
   2. \(\arg \left( \mathrm { z } _ { 1 } \mathrm { z } _ { 2 } \right)\) Given that \(w = z _ { 1 } z _ { 2 }\) and that \(\arg \left( w ^ { n } \right) = 0\), where \(n \in \mathbb { Z } ^ { + }\)
2. determine
   1. the smallest positive value of \(n\)
   2. the corresponding value of \(\left| w ^ { n } \right|\)