Modulus-argument form conversions

9

1. 1. Without using a calculator, express the complex number \(\frac { 2 + 6 \mathrm { i } } { 1 - 2 \mathrm { i } }\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
   2. Hence, without using a calculator, express \(\frac { 2 + 6 \mathrm { i } } { 1 - 2 \mathrm { i } }\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\), giving the exact values of \(r\) and \(\theta\).
2. On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying both the inequalities \(| z - 3 \mathrm { i } | \leqslant 1\) and \(\operatorname { Re } z \leqslant 0\), where \(\operatorname { Re } z\) denotes the real part of \(z\). Find the greatest value of \(\arg z\) for points in this region, giving your answer in radians correct to 2 decimal places.

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

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

2 Find the roots of the quadratic equation \(x ^ { 2 } - 8 x + 17 = 0\) in the form \(a + b \mathrm { j }\).
Express these roots in modulus-argument form.

2 The complex number \(z\) has modulus \(2 \sqrt { 3 }\) and argument \(- \frac { 1 } { 3 } \pi\). Giving your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are exact real numbers, and showing clearly how you obtain them, find

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

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

1. Express \(\frac { z + w } { w }\) in the form \(a + b \mathrm { j }\).
2. Express \(w\) in modulus-argument form.
3. Show \(w\) on an Argand diagram, indicating its modulus and argument.

4 The complex number \(z _ { 1 }\) is \(3 - 2 \mathrm { j }\) and the complex number \(z _ { 2 }\) has modulus 5 and argument \(\frac { \pi } { 4 }\).

1. Express \(z _ { 2 }\) in the form \(a + b \mathrm { j }\), giving \(a\) and \(b\) in exact form.
2. Represent \(z _ { 1 } , z _ { 2 } , z _ { 1 } + z _ { 2 }\) and \(z _ { 1 } - z _ { 2 }\) on a single Argand diagram.

8 Let \(z = \cos \theta + \mathrm { i } \sin \theta\). Show that $$1 + z = 2 \cos \frac { 1 } { 2 } \theta \left( \cos \frac { 1 } { 2 } \theta + \mathrm { i } \sin \frac { 1 } { 2 } \theta \right)$$ By considering \(( 1 + z ) ^ { n }\), where \(n\) is a positive integer, deduce the sum of the series $$\binom { n } { 1 } \sin \theta + \binom { n } { 2 } \sin 2 \theta + \ldots + \binom { n } { n } \sin n \theta$$

1. The complex numbers \(z , v\) and \(w\) are related by the equation

$$z = \frac { v } { w }$$ Given that \(v = - 16 + 11 \mathrm { i }\) and \(w = 5 + 2 \mathrm { i }\), find \(z\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\).

5. In this question you must show detailed reasoning.

1. Given that $$z _ { 1 } = 6 \left( \cos \left( \frac { \pi } { 3 } \right) + i \sin \left( \frac { \pi } { 3 } \right) \right) \quad \text { and } \quad z _ { 2 } = 6 \sqrt { 3 } \left( \cos \left( \frac { 5 \pi } { 6 } \right) + i \sin \left( \frac { 5 \pi } { 6 } \right) \right)$$ show that $$z _ { 1 } + z _ { 2 } = 12 \left( \cos \left( \frac { 2 \pi } { 3 } \right) + i \sin \left( \frac { 2 \pi } { 3 } \right) \right)$$
2. Given that $$\arg ( z - 5 ) = \frac { 2 \pi } { 3 }$$ determine the least value of \(| \boldsymbol { z } |\) as \(Z\) varies.
   [0pt] [BLANK PAGE]

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

1. Express \(z _ { 1 }\) in modulus-argument form, giving the modulus in exact form and the argument correct to 3 significant figures.
2. Find \(z _ { 1 } z _ { 2 }\) in terms of \(a\), writing your answer in the form \(c + \mathrm { i } d\).
3. The real and imaginary parts of a complex number on an Argand diagram are \(x\) and \(y\) respectively. Given that the point representing \(z _ { 1 } z _ { 2 }\) lies on the line \(y = x\), find the value of \(a\).
4. Given instead that \(z _ { 1 } z _ { 2 } = \left( z _ { 1 } z _ { 2 } \right) ^ { * }\) find the value of \(a\). In this question you must show detailed reasoning.
5. Express \(( 2 + 3 \mathrm { i } ) ^ { 3 }\) in the form \(a + \mathrm { i } b\).
6. Hence verify that \(2 + 3 \mathrm { i }\) is a root of the equation \(3 z ^ { 3 } - 8 z ^ { 2 } + 23 z + 52 = 0\).
7. Express \(3 z ^ { 3 } - 8 z ^ { 2 } + 23 z + 52\) as the product of a linear factor and a quadratic factor with real coefficients.