4.02c Complex notation: z, z*, Re(z), Im(z), |z|, arg(z)

1

1. The complex number \(\mathrm { a } + \mathrm { ib }\) is denoted by \(z\).
   1. Write down \(z ^ { * }\).
   2. Find \(\operatorname { Re } ( \mathrm { iz } )\).
2. The complex number \(w\) is given by \(w = \frac { 5 + \mathrm { i } \sqrt { 3 } } { 2 - \mathrm { i } \sqrt { 3 } }\).
   1. In this question you must show detailed reasoning. Express \(w\) in the form \(\mathrm { x } + \mathrm { iy }\).
   2. Convert \(w\) to modulus-argument form.

2 Two complex numbers are given by \(u = - 1 + \mathrm { i }\) and \(v = - 2 - \mathrm { i }\).

1. 1. Find \(\mathrm { u } - \mathrm { v }\) in the form \(\mathrm { a } + \mathrm { bi }\), where \(a\) and \(b\) are real.
   2. In this question you must show detailed reasoning. Find \(\frac { \mathrm { u } } { \mathrm { v } }\) in the form \(\mathrm { a } + \mathrm { bi }\), where \(a\) and \(b\) are real.
2. Express \(u\) in exact modulus-argument form.

3 In this question you must show detailed reasoning.
The complex numbers \(z _ { 1 }\) and \(z _ { 2 }\) are given by \(z _ { 1 } = - 2 + 2 i\) and \(z _ { 2 } = 2 \left( \cos \frac { 1 } { 6 } \pi + i \sin \frac { 1 } { 6 } \pi \right)\).

1. Find the modulus and argument of \(z _ { 1 }\).
2. Hence express \(\frac { z _ { 1 } } { z _ { 2 } }\) in exact modulus-argument form.

5. \begin{figure}[h] \end{figure} Figure 1 shows an Argand diagram.
The set of points, \(A\), that lies within the shaded region, including its boundaries, is defined by $$A = \{ z : p \leqslant \arg ( z ) \leqslant q \} \cap \{ z : | z | \leqslant r \}$$ where \(p , q\) and \(r\) are positive constants.

1. Write down the values of \(p , q\) and \(r\). Given that \(w = - 2 \sqrt { 3 } + 2 \mathrm { i }\) and \(\mathrm { z } \in A\),
2. find the maximum value of \(| w - z | ^ { 2 }\) giving your answer in an exact simplified form.

8

1. The complex number \(z\) is given by \(z = x + i y\) where \(x , y \in \mathbb { R }\) 8
   1. (i) Write down the complex conjugate \(z ^ { * }\) in terms of \(x\) and \(y\) 8
   2. (ii) Hence prove that \(z z ^ { * }\) is real for all \(z \in \mathbb { C }\) 8
   3. The complex number \(w\) satisfies the equation $$3 w + 10 \mathrm { i } = 2 w ^ { \star } + 5$$ 8
   4. 1. Find \(w\) 8
   5. (ii) Calculate the value of \(w ^ { 2 } \left( w ^ { * } \right) ^ { 2 }\)

2 Find arg ( \(- 4 - 7 \mathrm { i }\) ) to the nearest degree.
Circle your answer.
[0pt] [1 mark] \(- 120 ^ { \circ }\) \(- 60 ^ { \circ }\) \(30 ^ { \circ }\) \(60 ^ { \circ }\)

7 Given that \(z\) is a complex number, prove that \(z z ^ { * } = | z | ^ { 2 }\).

8 The complex numbers \(w\) and \(z\) are given by \(w = 3 - \mathrm { i }\) and \(z = 1 + \mathrm { i }\).

1. Express \(\frac { z } { w }\) in the form \(p + \mathrm { i } q\) where \(p\) and \(q\) are real numbers.
2. On the same Argand diagram, mark the points representing \(z , w\) and \(\frac { z } { w }\).
3. Find the value in radians of \(\arg w\).
4. Show that \(z + \frac { 2 } { z }\) is a real number.

4 The complex number \(p\) satisfies the equation $$p + \mathrm { i } p ^ { * } = 2 \left( p - \mathrm { i } p ^ { * } \right) - 8$$ Determine the exact values of the modulus and argument of \(p\).

The complex numbers \(z_1\) and \(z_2\) are given by $$z_1 = 3 + 5\text{i} \quad \text{and} \quad z_2 = -2 + 6\text{i}$$

1. Show \(z_1\) and \(z_2\) on a single Argand diagram. [2]
2. Without using your calculator and showing all stages of your working,
   1. determine the value of \(|z_1|\) [1]
   2. express \(\frac{z_1}{z_2}\) in the form \(a + b\text{i}\), where \(a\) and \(b\) are fully simplified fractions. [3]
3. Hence determine the value of \(\arg \frac{z_1}{z_2}\) Give your answer in radians to 2 decimal places. [2]

\(z = 3 - i\) Determine the value of \(zz*\) Circle your answer. [1 mark] \(10\) \(\qquad\) \(\sqrt{10}\) \(\qquad\) \(10 - 2i\) \(\qquad\) \(10 + 2i\)

Given that \(z_1 = 2\left(\cos \frac{\pi}{6} + i \sin \frac{\pi}{6}\right)\) and \(z_2 = 2\left(\cos \frac{3\pi}{4} + i \sin \frac{3\pi}{4}\right)\)

1. Find the value of \(|z_1z_2|\) [1 mark]
2. Find the value of \(\arg\left(\frac{z_1}{z_2}\right)\) [1 mark]
3. Sketch \(z_1\) and \(z_2\) on the Argand diagram below, labelling the points as \(P\) and \(Q\) respectively. [2 marks]
4. A third complex number \(w\) satisfies both \(|w| = 2\) and \(-\pi < \arg w < 0\) Given that \(w\) is represented on the Argand diagram as the point \(R\), find the angle \(PRQ\). Fully justify your answer. [3 marks]

Given that \(z\) is a complex number, and that \(z^*\) is the complex conjugate of \(z\), which of the following statements is not always true? Circle your answer. [1 mark] \((z^*)^* = z\) \quad\quad \(zz^* = |z|^2\) \quad\quad \((-z)^* = -(z^*)\) \quad\quad \(z - z^* = z^* - z\)

Given that \(\arg(a + bi) = \varphi\), where \(a\) and \(b\) are positive real numbers and \(0 < \varphi < \frac{\pi}{2}\), three of the following four statements are correct. Which statement is not correct? Tick \((\checkmark)\) one box. [1 mark] \(\arg(-a - bi) = \pi - \varphi\) \(\arg(a - bi) = -\varphi\) \(\arg(b + ai) = \frac{\pi}{2} - \varphi\) \(\arg(b - ai) = \varphi - \frac{\pi}{2}\)

The complex numbers \(z_1\) and \(z_2\) are such that \(|z_1| = 2\), \(\arg(z_1) = \frac{7}{12}\pi\), \(|z_2| = \sqrt{2}\) and \(\arg(z_2) = -\frac{1}{8}\pi\).

1. Find, in exact form, the modulus and argument of \(\frac{z_1}{z_2}\). [3]
2. Let \(z_3 = \left(\frac{z_1}{z_2}\right)^n\). It is given that \(n\) is the least positive integer for which \(z_3\) is a positive real number. Find this value of \(n\) and the exact value of \(z_3\). [4]