Complex Numbers Arithmetic

1

1. Express \(( 1 + 8 i ) ( 2 - i )\) in the form \(x + i y\), showing clearly how you obtain your answer.
2. Hence express \(\frac { 1 + 8 i } { 2 + i }\) in the form \(x + i y\).

5 Show that \(( 2 + \mathrm { i } ) ^ { 3 }\) is \(2 + 11 \mathrm { i }\)
[0pt] [3 marks]

6. Given that \(z\) is the complex number \(x + i y\) and satisfies $$| z | + z = 6 - 2 i$$ find the value of \(x\) and the value of \(y\).

1. Solve the equation \(2 z - 5 \mathrm { i } z ^ { * } = 12\)
[0pt] [4 marks]

2 In this question you must show detailed reasoning. You are given that \(x = 2 + 5 \mathrm { i }\) is a root of the equation \(x ^ { 3 } - 2 x ^ { 2 } + 21 x + 58 = 0\).
Solve the equation.

2 You are given that \(\alpha = - 3 + 4 \mathrm { j }\).

1. Calculate \(\alpha ^ { 2 }\).
2. Express \(\alpha\) in modulus-argument form.

4 Use an algebraic method to find the square roots of the complex number 21-20i.

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

9

1. Use an algebraic method to find the square roots of the complex number \(5 + 12 \mathrm { i }\).
2. Find \(( 3 - 2 \mathrm { i } ) ^ { 2 }\).
3. Hence solve the quartic equation \(x ^ { 4 } - 10 x ^ { 2 } + 169 = 0\).

14 Show that the two values of \(b\) given on line 36 are equivalent.

7 In an Argand diagram the points representing the numbers \(2 + 3 \mathrm { i }\) and \(1 - \mathrm { i }\) are two adjacent vertices of a square, \(S\).

1. Find the area of \(S\).
2. Find all the possible pairs of numbers represented by the other two vertices of \(S\).

2 Find the roots of the quadratic equation \(z ^ { 2 } - 4 z + 13 = 0\).
Find the modulus and argument of each root.

3 One root of the quadratic equation \(x ^ { 2 } + a x + b = 0\), where \(a\) and \(b\) are real, is the complex number \(4 - 3 \mathrm { i }\). Find the values of \(a\) and \(b\).

3. $$\mathrm { f } ( z ) = z ^ { 4 } + a z ^ { 3 } + 6 z ^ { 2 } + b z + 65$$ where \(a\) and \(b\) are real constants.
Given that \(z = 3 + 2 \mathbf { i }\) is a root of the equation \(\mathrm { f } ( z ) = 0\), show the roots of \(\mathrm { f } ( z ) = 0\) on a single Argand diagram.

3. $$f ( x ) = \left( x ^ { 2 } + 4 \right) \left( x ^ { 2 } + 8 x + 25 \right)$$

1. Find the four roots of \(\mathrm { f } ( x ) = 0\).
2. Find the sum of these four roots.

2 In this question you must show detailed reasoning.

1. Write the complex number \(- 24 + 7 \mathrm { i }\) in modulus-argument form.
2. Solve the simultaneous equations given below, giving your answers in cartesian form. $$\begin{aligned} i z + 3 w & = - 7 i
   - 6 z + 5 i w & = 3 + 13 i \end{aligned}$$

5 In this question you must show detailed reasoning.

1. Express \(( 2 + 3 \mathrm { i } ) ^ { 3 }\) in the form \(a + \mathrm { i } b\).
2. Hence verify that \(2 + 3\) i is a root of the equation \(3 z ^ { 3 } - 8 z ^ { 2 } + 23 z + 52 = 0\).
3. Express \(3 z ^ { 3 } - 8 z ^ { 2 } + 23 z + 52\) as the product of a linear factor and a quadratic factor with real coefficients.

2

1. Given that \(z = a + b \mathrm { j }\), express \(| z |\) and \(z ^ { * }\) in terms of \(a\) and \(b\).
2. Prove that \(z z ^ { * } - | z | ^ { 2 } = 0\).

3. Given that \(z = x + \mathrm { i } y\), where \(x\) and \(y\) are real numbers, solve the equation $$( z - 2 i ) \left( z ^ { * } - 2 i \right) = 21 - 12 i$$ where \(z ^ { * }\) is the complex conjugate of \(z\).

9 In this question you must show detailed reasoning. You are given that \(a\) is a real root of the equation \(x ^ { 4 } + x ^ { 3 } + 3 x ^ { 2 } - 5 x = 0\).
You are also given that \(a + 2 + 3 \mathrm { i }\) is one root of the equation
\(z ^ { 4 } - 2 ( 1 + a ) z ^ { 3 } + ( 21 a - 10 ) z ^ { 2 } + ( 86 - 80 a ) z + ( 285 a - 195 ) = 0\). Determine all possible values of \(z\).

6 Find the complex numbers \(w\) which satisfy the equation \(w ^ { 2 } + 2 \mathrm { i } w ^ { * } = 1\) and are such that \(\operatorname { Re } w \leqslant 0\). Give your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.

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

1. Find, in terms of \(x\) and \(y\), the real and imaginary parts of $$( z - \mathrm { i } ) \left( z ^ { * } - \mathrm { i } \right)$$
2. Given that $$( z - \mathrm { i } ) \left( z ^ { * } - \mathrm { i } \right) = 24 - 8 \mathrm { i }$$ find the two possible values of \(z\).

1 Show that \(\frac { 5 } { 2 - 4 \mathrm { i } } = \frac { 1 } { 2 } + \mathrm { i }\).

4 Solve the quadratic equation \(( 3 + \mathrm { i } ) w ^ { 2 } - 2 w + 3 - \mathrm { i } = 0\), giving your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.

4 Find the complex number \(z\) satisfying the equation $$\frac { z - 3 \mathrm { i } } { z + 3 \mathrm { i } } = \frac { 2 - 9 \mathrm { i } } { 5 }$$ Give your answer in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.

4 Find a cubic equation with real coefficients, two of whose roots are \(2 - \mathrm { i }\) and 3.

9

1. It is given that \(( 1 + 3 \mathrm { i } ) w = 2 + 4 \mathrm { i }\). Showing all necessary working, prove that the exact value of \(\left| w ^ { 2 } \right|\) is 2 and find \(\arg \left( w ^ { 2 } \right)\) correct to 3 significant figures.
2. On a single Argand diagram sketch the loci \(| z | = 5\) and \(| z - 5 | = | z |\). Hence determine the complex numbers represented by points common to both loci, giving each answer in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\).

1. (a) Determine the roots of the equation

$$z ^ { 6 } = 1$$ giving your answers in the form \(\mathrm { e } ^ { \mathrm { i } \theta }\) where \(0 \leqslant \theta < 2 \pi\)
(b) Show the roots of the equation in part (a) on a single Argand diagram.
(c) Show that $$( \sqrt { 3 } + i ) ^ { 6 } = - 64$$ (d) Hence, or otherwise, solve the equation $$z ^ { 6 } + 64 = 0$$ giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\) where \(0 \leqslant \theta < 2 \pi\)

2 You are given that \(z _ { 1 }\) and \(z _ { 2 }\) are complex numbers.
\(z _ { 1 } = 3 + 3 \sqrt { 3 } \mathrm { j }\), and \(z _ { 2 }\) has modulus 5 and argument \(\frac { \pi } { 3 }\).

1. Find the modulus and argument of \(z _ { 1 }\), giving your answers exactly.
2. Express \(z _ { 2 }\) in the form \(a + b \mathrm { j }\), where \(a\) and \(b\) are to be given exactly.
3. Explain why, when plotted on an Argand diagram, \(z _ { 1 } , z _ { 2 }\) and the origin lie on a straight line.

3 The complex number \(2 - \mathrm { i }\) is denoted by \(z\).

1. Find \(| z |\) and \(\arg z\).
2. Given that \(a z + b z ^ { * } = 4 - 8 \mathrm { i }\), find the values of the real constants \(a\) and \(b\).