Multiplication and powers of complex numbers

1. $$z = 3 + 2 \mathrm { i } , \quad w = 1 - \mathrm { i }$$ Find in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are real constants,

1. \(z w\)
2. \(\frac { z } { w ^ { * } }\), showing clearly how you obtained your answer. Given that $$| z + k | = \sqrt { 53 } \text {, where } k \text { is a real constant }$$
3. find the possible values of \(k\).

1. $$z = 5 - 3 \mathrm { i } , \quad w = 2 + 2 \mathrm { i }$$ Express in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are real constants,

1. \(z ^ { 2 }\),
2. \(\frac { z } { w }\).

1. The complex numbers \(z\) and \(w\) are given by

$$z = 8 + 3 \mathrm { i } , \quad w = - 2 \mathrm { i }$$ Express in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are real constants,

1. \(z - w\),
2. \(z w\).

4 The complex number \(3 - 4 \mathrm { i }\) is denoted by \(z\). Giving your answers in the form \(x + \mathrm { i } y\), and showing clearly how you obtain them, find

1. \(2 z + 5 z ^ { * }\),
2. \(( z - \mathrm { i } ) ^ { 2 }\),
3. \(\frac { 3 } { z }\).

3 The complex numbers \(2 + 3 \mathrm { i }\) and \(4 - \mathrm { i }\) are denoted by \(z\) and \(w\) respectively. Express each of the following in the form \(x + \mathrm { i } y\), showing clearly how you obtain your answers.

1. \(z + 5 w\),
2. \(z ^ { * } w\), where \(z ^ { * }\) is the complex conjugate of \(z\),
3. \(\frac { 1 } { w }\).

5 The complex numbers \(3 - 2 \mathrm { i }\) and \(2 + \mathrm { i }\) are denoted by \(z\) and \(w\) respectively. Find, giving your answers in the form \(x + \mathrm { i } y\) and showing clearly how you obtain these answers,

1. \(2 z - 3 w\),
2. \(( \mathrm { i } z ) ^ { 2 }\),
3. \(\frac { z } { w }\).

3 The complex number \(2 + \mathrm { i }\) is denoted by \(z\), and the complex conjugate of \(z\) is denoted by \(z ^ { * }\).

1. Express \(z ^ { 2 }\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real, showing clearly how you obtain your answer.
2. Show that \(4 z - z ^ { 2 }\) simplifies to a real number, and verify that this real number is equal to \(z z ^ { * }\).
3. Express \(\frac { z + 1 } { z - 1 }\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real, showing clearly how you obtain your answer.

4 Two complex numbers, \(\alpha\) and \(\beta\), are given by \(\alpha = 1 - 2 \mathrm { j }\) and \(\beta = - 2 - \mathrm { j }\).

1. Represent \(\beta\) and its complex conjugate \(\beta ^ { * }\) on an Argand diagram.
2. Express \(\alpha \beta\) in the form \(a + b \mathrm { j }\).
3. Express \(\frac { \alpha + \beta } { \beta }\) in the form \(a + b \mathrm { j }\).

3 The complex numbers \(z\) and \(w\) are given by \(z = 5 - 2 \mathrm { i }\) and \(w = 3 + 7 \mathrm { i }\). Giving your answers in the form \(x + \mathrm { i } y\) and showing clearly how you obtain them, find

1. \(4 z - 3 w\),
2. \(z ^ { * } w\).

1 Two complex numbers are given by \(\alpha = - 3 + \mathrm { j }\) and \(\beta = 5 - 2 \mathrm { j }\).
Find \(\alpha \beta\) and \(\frac { \alpha } { \beta }\), giving your answers in the form \(a + b \mathrm { j }\), showing your working.

8 The complex number \(5 + 4 \mathrm { j }\) is denoted by \(\alpha\).

1. Find \(\alpha ^ { 2 }\) and \(\alpha ^ { 3 }\), showing your working.
2. The real numbers \(q\) and \(r\) are such that \(\alpha ^ { 3 } + \mathrm { q } \alpha ^ { 2 } + 11 \alpha + \mathrm { r } = 0\). Find \(q\) and \(r\). Let \(\mathrm { f } ( \mathrm { z } ) = \mathrm { z } ^ { 3 } + \mathrm { qz } ^ { 2 } + 11 \mathrm { z } + \mathrm { r }\), where \(q\) and \(r\) are as in part (ii).
3. Solve the equation \(\mathrm { f } ( z ) = 0\).
4. Solve the equation \(z ^ { 4 } + q z ^ { 3 } + 11 z ^ { 2 } + r z = z ^ { 3 } + q z ^ { 2 } + 11 z + r\).

2 In this question you must show detailed reasoning.
Given that \(z _ { 1 } = 3 + 2 \mathrm { i }\) and \(z _ { 2 } = - 1 - \mathrm { i }\), find the following, giving each in the form \(a + b \mathrm { i }\).

1. \(z _ { 1 } ^ { * } z _ { 2 }\)
2. \(\frac { z _ { 1 } + 2 z _ { 2 } } { z _ { 2 } }\)

3

1. Solve the following equations, giving each root in the form \(a + b \mathrm { i }\) :
   1. \(x ^ { 2 } + 9 = 0\);
   2. \(( x + 2 ) ^ { 2 } + 9 = 0\).
2. 1. Expand \(( 1 + x ) ^ { 3 }\).
   2. Express \(( 1 + 2 \mathrm { i } ) ^ { 3 }\) in the form \(a + b \mathrm { i }\).
   3. Given that \(z = 1 + 2 \mathrm { i }\), find the value of $$z ^ { * } - z ^ { 3 }$$

1. (i) (a) Show that

$$\frac { 2 + 3 \mathrm { i } } { 5 + \mathrm { i } } = k ( 1 + \mathrm { i } )$$ where \(k\) is a constant to be determined.
(Solutions relying on calculator technology are not acceptable.) Given that

• \(n\) is a positive integer
• \(\left( \frac { 2 + 3 \mathrm { i } } { 5 + \mathrm { i } } \right) ^ { n }\) is a real number
  (b) use the answer to part (a) to write down the smallest possible value of \(n\).
  (ii) The complex number \(z = a + b \mathrm { i }\) where \(a\) and \(b\) are real constants.

Given that

• \(\left| z ^ { 10 } \right| = 59049\)
• \(\arg \left( z ^ { 10 } \right) = - \frac { 5 \pi } { 3 }\)
  determine the value of \(a\) and the value of \(b\).

2 In this question you must show detailed reasoning. The complex numbers \(z _ { 1 }\) and \(z _ { 2 }\) are given by \(z _ { 1 } = 3 - 7 \mathrm { i }\) and \(z _ { 2 } = 2 + 4 \mathrm { i }\).

1. Express each of the following as exact numbers in the form \(a + b \mathrm { i }\).
   1. \(3 z _ { 1 } + 4 z _ { 2 }\)
   2. \(z _ { 1 } z _ { 2 }\)
   3. \(\frac { Z _ { 1 } } { Z _ { 2 } }\)
2. Write \(z _ { 1 }\) in modulus-argument form giving the modulus in exact form and the argument correct to \(\mathbf { 3 }\) significant figures.

2. The complex numbers \(3 - 2 \mathrm { i }\) and \(2 + \mathrm { i }\) are denoted by \(z\) and \(w\) respectively. Find, giving your answers in the form \(x + \mathrm { i } y\) and showing clearly how you obtain these answers,

1. \(2 z - 3 w\),
2. \(( \mathrm { i } z ) ^ { 2 }\),
3. \(\frac { z } { w }\).

8 In this question you must show detailed reasoning.

1. Express \(( 6 + 5 \mathrm { i } ) ( 7 + 5 \mathrm { i } )\) in the form \(a + b \mathrm { i }\).
2. You are given that \(17 ^ { 2 } + 65 ^ { 2 } = 4514\). Using the result in part (i) and by considering (6-5i)(7-5i) express 4514 as a product of its prime factors.

10. Given that \(z = 3 - 3 \mathrm { i }\) express, in the form \(a + \mathrm { i } b\), where \(a\) and \(b\) are real numbers,

1. \(z ^ { 2 }\),
2. \(\frac { 1 } { z }\).
3. Find the exact value of each of \(| z | , \left| z ^ { 2 } \right|\) and \(\left| \frac { 1 } { z } \right|\). The complex numbers \(z , z ^ { 2 }\) and \(\frac { 1 } { z }\) are represented by the points \(A , B\) and \(C\) respectively on an Argand diagram. The real number 1 is represented by the point \(D\), and \(O\) is the origin.
4. Show the points \(A\), \(B\), \(C\) and \(D\) on an Argand diagram.
5. Prove that \(\triangle O A B\) is similar to \(\triangle O C D\).

2 The complex number \(z\) is defined by $$z = 1 + \mathrm { i }$$

1. Find the value of \(z ^ { 2 }\), giving your answer in its simplest form.
2. Hence show that \(z ^ { 8 } = 16\).
3. Show that \(\left( z ^ { * } \right) ^ { 2 } = - z ^ { 2 }\).

4 The complex numbers \(w\) and \(z\) are defined as $$\begin{aligned} w & = 2 ( \cos \alpha + \mathrm { i } \sin \alpha )
z & = 3 ( \cos \beta + \mathrm { i } \sin \beta ) \end{aligned}$$ Find the product \(w z\)
Tick \(( \checkmark )\) one box. $$\begin{aligned} & 5 ( \cos ( \alpha \beta ) + \mathrm { i } \sin ( \alpha \beta ) )
& 6 ( \cos ( \alpha \beta ) + \mathrm { i } \sin ( \alpha \beta ) )
& 5 ( \cos ( \alpha + \beta ) + \mathrm { i } \sin ( \alpha + \beta ) )
& 6 ( \cos ( \alpha + \beta ) + \mathrm { i } \sin ( \alpha + \beta ) ) \end{aligned}$$ \includegraphics[max width=\textwidth, alt={}, center]{fd9715c4-9ce1-4608-aed6-f3d4f71208b5-03_113_113_762_1206}
\includegraphics[max width=\textwidth, alt={}, center]{fd9715c4-9ce1-4608-aed6-f3d4f71208b5-03_108_108_900_1206}
\includegraphics[max width=\textwidth, alt={}, center]{fd9715c4-9ce1-4608-aed6-f3d4f71208b5-03_113_113_1032_1206}
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5 Show that \(( 2 + \mathrm { i } ) ^ { 3 }\) is \(2 + 11 \mathrm { i }\)
[0pt] [3 marks]

3 In this question you must show detailed reasoning. In this question the principal argument of a complex number lies in the interval \([ 0,2 \pi )\).
Complex numbers \(z _ { 1 }\) and \(z _ { 2 }\) are defined by \(z _ { 1 } = 3 + 4 \mathrm { i }\) and \(z _ { 2 } = - 5 + 12 \mathrm { i }\).

1. Determine \(z _ { 1 } z _ { 2 }\), giving your answer in the form \(a + b \mathrm { i }\).
2. Express \(z _ { 2 }\) in modulus-argument form.
3. Verify, by direct calculation, that \(\arg \left( z _ { 1 } z _ { 2 } \right) = \arg \left( z _ { 1 } \right) + \arg \left( z _ { 2 } \right)\).