Real and imaginary part expressions

1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

$$z _ { 1 } = 3 + 2 i \quad z _ { 2 } = 2 + 3 i \quad z _ { 3 } = a + b i \quad a , b \in \mathbb { R }$$

1. Determine the exact value of \(\left| z _ { 1 } + z _ { 2 } \right|\) Given that \(w = \frac { z _ { 2 } z _ { 3 } } { z _ { 1 } }\)
2. determine \(w\) in terms of \(a\) and \(b\), giving your answer in the form \(x + \mathrm { i } y\), where \(x , y \in \mathbb { R }\) Given also that \(w = \frac { 4 } { 13 } + \frac { 58 } { 13 } \mathrm { i }\)
3. determine the value of \(a\) and the value of \(b\)
4. determine arg \(w\), giving your answer in radians to 4 significant figures.

1 The complex number \(x + \mathrm { i } y\) is denoted by \(z\). Express \(3 z z ^ { * } - | z | ^ { 2 }\) in terms of \(x\) and \(y\).

2 Given that \(z = a + b \mathrm { j }\), find \(\operatorname { Re } \left( \frac { z } { z ^ { * } } \right)\) and \(\operatorname { Im } \left( \frac { z } { z ^ { * } } \right)\).

3 The complex number \(z\) is defined by $$z = x + 2 \mathrm { i }$$ where \(x\) is real.

1. Find, in terms of \(x\), the real and imaginary parts of:
   1. \(z ^ { 2 }\);
   2. \(z ^ { 2 } + 2 z ^ { * }\).
2. Show that there is exactly one value of \(x\) for which \(z ^ { 2 } + 2 z ^ { * }\) is real.

8. The function f is defined, for any complex number \(z\), by $$\mathrm { f } ( z ) = \frac { \mathrm { i } z - 1 } { \mathrm { i } z + 1 }$$ Suppose throughout that \(x\) is a real number.

1. Show that $$\operatorname { Re } f ( x ) = \frac { x ^ { 2 } - 1 } { x ^ { 2 } + 1 } \quad \text { and } \quad \operatorname { Im } f ( x ) = \frac { 2 x } { x ^ { 2 } + 1 }$$
2. Show that \(\mathrm { f } ( x ) \mathrm { f } ( x ) ^ { * } = 1\), where \(\mathrm { f } ( x ) ^ { * }\) is the complex conjugate of \(\mathrm { f } ( x )\).

8. The function f is defined, for any complex number \(z\), by $$\mathrm { f } ( z ) = \frac { \mathrm { i } z - 1 } { \mathrm { i } z + 1 }$$ Suppose throughout that \(x\) is a real number.

1. Show that $$\operatorname { Re } f ( x ) = \frac { x ^ { 2 } - 1 } { x ^ { 2 } + 1 } \quad \text { and } \quad \operatorname { Im } f ( x ) = \frac { 2 x } { x ^ { 2 } + 1 }$$
2. Show that \(\mathrm { f } ( x ) \mathrm { f } ( x ) ^ { * } = 1\), where \(\mathrm { f } ( x ) ^ { * }\) is the complex conjugate of \(\mathrm { f } ( x )\). Spare Paper

3 It is given that \(z = x + \mathrm { i } y\), where \(x\) and \(y\) are real numbers.

1. Write down, in terms of \(x\) and \(y\), an expression for \(z ^ { * }\), the complex conjugate of \(z\).
2. Find, in terms of \(x\) and \(y\), the real and imaginary parts of $$2 z - \mathrm { i } z ^ { * }$$
3. Find the complex number \(z\) such that $$2 z - \mathrm { i } z ^ { * } = 3 \mathrm { i }$$

1 It is given that \(z _ { 1 } = 2 + \mathrm { i }\) and that \(z _ { 1 } { } ^ { * }\) is the complex conjugate of \(z _ { 1 }\).
Find the real numbers \(x\) and \(y\) such that $$x + 3 \mathrm { i } y = z _ { 1 } + 4 \mathrm { i } z _ { 1 } *$$

9 The complex number \(z\) is such that $$z = \frac { 1 + \mathrm { i } } { 1 - k \mathrm { i } }$$ where \(k\) is a real number. 9

1. Find the real part of \(z\) and the imaginary part of \(z\), giving your answers in terms of \(k\)
   9
2. In the case where \(k = \sqrt { 3 }\), use part (a) to show that $$\cos \frac { 7 \pi } { 12 } = \frac { \sqrt { 2 } - \sqrt { 6 } } { 4 }$$ \(\_\_\_\_\) The region \(R\) on an Argand diagram satisfies both \(| z + 2 \mathrm { i } | \leq 3\) and \(- \frac { \pi } { 6 } \leq \arg ( z ) \leq \frac { \pi } { 2 }\)