4.02e Arithmetic of complex numbers: add, subtract, multiply, divide

1

1. The complex number 3-4i is denoted by \(z _ { 1 }\). Write \(z _ { 1 }\) in modulus-argument form, giving your angle in radians to 3 significant figures.
2. The complex number \(z _ { 2 }\) has modulus 6 and argument - 2.5 radians. Express \(z _ { 1 } z _ { 2 }\) in modulus-argument form with the angle in radians correct to 3 significant figures.

1 In this question you must show detailed reasoning.
For the complex number \(z\) it is given that \(| z | = 2\) and \(\arg z = \frac { 1 } { 6 } \pi\).
Find the following in the form \(a + \mathrm { i } b\), where \(a\) and \(b\) are exact numbers.

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

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 $$x + 3 \mathrm { i } y = z _ { 1 } + 4 \mathrm { i } z _ { 1 } *$$

8

1. 1. It is given that \(\alpha\) and \(\beta\) are the roots of the equation $$x ^ { 2 } - 2 x + 4 = 0$$ Without solving this equation, show that \(\alpha ^ { 3 }\) and \(\beta ^ { 3 }\) are the roots of the equation $$x ^ { 2 } + 16 x + 64 = 0$$ (6 marks)
   2. State, giving a reason, whether the roots of the equation $$x ^ { 2 } + 16 x + 64 = 0$$ are real and equal, real and distinct, or non-real.
2. Solve the equation $$x ^ { 2 } - 2 x + 4 = 0$$
3. Use your answers to parts (a) and (b) to show that $$( 1 + \mathrm { i } \sqrt { 3 } ) ^ { 3 } = ( 1 - \mathrm { i } \sqrt { 3 } ) ^ { 3 }$$

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 \(\left( z ^ { * } \right) ^ { 2 } = - z ^ { 2 }\).

6 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 $$( z + \mathrm { i } ) ^ { * }$$ where \(( z + \mathrm { i } ) ^ { * }\) denotes the complex conjugate of \(( z + \mathrm { i } )\).
2. Solve the equation $$( z + \mathrm { i } ) ^ { * } = 2 \mathrm { i } z + 1$$ giving your answer in the form \(a + b \mathrm { i }\).

7

1. Find the six roots of the equation \(z ^ { 6 } = 1\), giving your answers in the form \(\mathrm { e } ^ { \mathrm { i } \phi }\), where \(- \pi < \phi \leqslant \pi\).
2. It is given that \(w = \mathrm { e } ^ { \mathrm { i } \theta }\), where \(\theta \neq n \pi\).
   1. Show that \(\frac { w ^ { 2 } - 1 } { w } = 2 \mathrm { i } \sin \theta\).
   2. Show that \(\frac { w } { w ^ { 2 } - 1 } = - \frac { \mathrm { i } } { 2 \sin \theta }\).
   3. Show that \(\frac { 2 \mathrm { i } } { w ^ { 2 } - 1 } = \cot \theta - \mathrm { i }\).
   4. Given that \(z = \cot \theta - \mathrm { i }\), show that \(z + 2 \mathrm { i } = z w ^ { 2 }\).
3. 1. Explain why the equation $$( z + 2 \mathrm { i } ) ^ { 6 } = z ^ { 6 }$$ has five roots.
   2. Find the five roots of the equation $$( z + 2 \mathrm { i } ) ^ { 6 } = z ^ { 6 }$$ giving your answers in the form \(a + \mathrm { i } b\).

10

1. Show that \(\operatorname { det } \mathbf { A } = a + \mathrm { i }\) where \(a\) is an integer to be determined. 10 Matrix A is given by 10
2. Matrix B is given by $$\mathbf { B } = \left[ \begin{array} { c c } 14 - 2 \mathrm { i } & b \\ c & d \end{array} \right] \quad \text { and } \quad \mathbf { A B } = p$$ where \(b , c , d \in \mathbb { C }\) and \(p \in \mathbb { N }\) Find \(b , c , d\) and \(p\)

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]

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 \(w ^ { 2 } \left( w ^ { * } \right) ^ { 2 }\)

1 Find the imaginary part of $$\frac { 5 + \mathrm { i } } { 1 - \mathrm { i } }$$ Circle your answer.
-3
-2

4 In this question you must show detailed reasoning. You are given that \(\mathrm { f } ( \mathrm { z } ) = 4 \mathrm { z } ^ { 4 } - 12 \mathrm { z } ^ { 3 } + 41 \mathrm { z } ^ { 2 } - 128 \mathrm { z } + 185\) and that \(2 + \mathrm { i }\) is a root of the equation \(f ( z ) = 0\).

1. Express \(\mathrm { f } ( \mathrm { z } )\) as the product of two quadratic factors with integer coefficients.
2. Solve \(f ( z ) = 0\). Two loci on an Argand diagram are defined by \(C _ { 1 } = \left\{ z : | z | = r _ { 1 } \right\}\) and \(C _ { 2 } = \left\{ z : | z | = r _ { 2 } \right\}\) where \(r _ { 1 } > r _ { 2 }\). You are given that two of the points representing the roots of \(\mathrm { f } ( \mathrm { z } ) = 0\) are on \(C _ { 1 }\) and two are on \(C _ { 2 } . R\) is the region on the Argand diagram between \(C _ { 1 }\) and \(C _ { 2 }\).
3. Find the exact area of \(R\).
4. \(\omega\) is the sum of all the roots of \(\mathrm { f } ( \mathrm { z } ) = 0\). Determine whether or not the point on the Argand diagram which represents \(\omega\) lies in \(R\).

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 \(\arg \left( z _ { 1 } z _ { 2 } \right) = \arg \left( z _ { 1 } \right) + \arg \left( z _ { 2 } \right)\).

8

1. Solve the equation \(\omega + 2 + 7 \mathrm { i } = 3 \omega ^ { * } - \mathrm { i }\).
2. Prove algebraically that, for non-zero \(z , z = - z ^ { * }\) if and only if \(z\) is purely imaginary.
3. The complex numbers \(z\) and \(z ^ { * }\) are represented on an Argand diagram by the points \(A\) and \(B\) respectively.
   1. State, for any \(z\), the single transformation which transforms \(A\) to \(B\).
   2. Use a geometric argument to prove that \(z = z ^ { * }\) if and only if \(z\) is purely real.

2 In this question you must show detailed reasoning.
The complex number \(7 - 4 \mathrm { i }\) is denoted by \(z\).

1. Giving your answers in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are rational numbers, find the following.
   1. \(3 z - 4 z ^ { * }\)
   2. \(( z + 1 - 3 \mathrm { i } ) ^ { 2 }\)
   3. \(\frac { z + 1 } { z - 1 }\)
2. Express \(z\) in modulus-argument form giving the modulus exactly and the argument correct to 3 significant figures.
3. The complex number \(\omega\) is such that \(z \omega = \sqrt { 585 } ( \cos ( 0.5 ) + \mathrm { i } \sin ( 0.5 ) )\). Find the following.

6 The complex number \(5 - 3 \mathrm { i }\) is denoted by \(z\). Giving your answers in the form \(x + \mathrm { i } y\), and showing clearly how you obtain them, find

1. \(\quad 6 z - z ^ { * }\),
2. \(\quad ( z - \mathrm { i } ) ^ { 2 }\),
3. \(\frac { 5 } { z }\).

11

1. Determine \(p\) and \(q\) given that \(( p + \mathrm { i } q ) ^ { 2 } = 63 - 16 \mathrm { i }\) and that \(p\) and \(q\) are real.
2. Let \(\mathrm { f } ( z ) = z ^ { 3 } - A z ^ { 2 } + B z - C\) for complex numbers \(A , B\) and \(C\).
   1. Given that the cubic equation \(\mathrm { f } ( z ) = 0\) has roots \(\alpha = - 7 \mathrm { i } , \beta = 3 \mathrm { i }\) and \(\gamma = 4\), determine each of \(A , B\) and \(C\).
   2. Find the roots of the equation \(\mathrm { f } ^ { \prime } ( z ) = 0\).

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.

9 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. \(2 z + z ^ { * }\),
2. \(\frac { 5 } { z }\).
3. Show \(z\) and \(z ^ { * }\) on an Argand diagram.

1 Without using a calculator, determine the possible values of \(a\) and \(b\) for which \(( a + \mathrm { i } b ) ^ { 2 } = 21 - 20 \mathrm { i }\).

4 A sequence of complex numbers is defined by $$u _ { 1 } = 1 + \mathrm { i } \quad \text { and } \quad u _ { n + 1 } = \mathrm { i } u _ { n } ( n = 1,2,3 , \ldots )$$

1. Find \(u _ { 2 } , u _ { 3 } , u _ { 4 } , u _ { 5 }\) and \(u _ { 6 }\).
2. Describe the behaviour of the sequence.
3. Hence evaluate \(\sum _ { n = 1 } ^ { 73 } u _ { n }\).

5 The complex numbers \(u\) and \(v\) are given by \(u = 3 + 2 \mathrm { i }\) and \(v = 1 + 4 \mathrm { i }\).

1. Given that \(a u ^ { 2 } + b v ^ { * } = 7 + 36 \mathrm { i }\) find the values of the real constants \(a\) and \(b\).
2. Show the points representing \(u\) and \(v\) on an Argand diagram and hence sketch the locus given by \(| z - u | = | z - v |\). Find the point of intersection of this locus with the imaginary axis.

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

1. \(2 z + z ^ { * }\),
2. \(\frac { 5 } { z }\).
3. Show \(z\) and \(z ^ { * }\) on an Argand diagram.