Modulus-argument form conversions

A question is this type if and only if it asks to convert between Cartesian form (a + bi) and modulus-argument form r(cos θ + i sin θ) or re^(iθ).

9 questions · Moderate -0.2

4.02b Express complex numbers: cartesian and modulus-argument forms
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OCR MEI FP1 2008 January Q2
5 marks Moderate -0.8
2 You are given that \(\alpha = - 3 + 4 \mathrm { j }\).
  1. Calculate \(\alpha ^ { 2 }\).
  2. Express \(\alpha\) in modulus-argument form.
OCR MEI FP1 2005 June Q2
5 marks Moderate -0.8
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.
OCR FP1 2016 June Q2
7 marks Standard +0.3
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. \(\frac { 1 } { \left( z ^ { * } - 5 \mathrm { i } \right) ^ { 2 } }\).
OCR MEI FP1 2011 June Q2
8 marks Moderate -0.8
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.
OCR MEI FP1 2013 June Q4
6 marks Moderate -0.8
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.
CAIE FP1 2012 November Q8
9 marks Challenging +1.2
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 $$\binom { n } { 1 } \sin \theta + \binom { n } { 2 } \sin 2 \theta + \ldots + \binom { n } { n } \sin n \theta$$
WJEC Further Unit 1 2024 June Q1
5 marks Moderate -0.5
  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 \(r ( \cos \theta + \mathrm { i } \sin \theta )\).
CAIE P3 2019 November Q6
7 marks Standard +0.3
  1. Express \(w\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real and exact.
    The complex number \(1 + 2 \mathrm { i }\) is denoted by \(u\). The complex number \(v\) is such that \(| v | = 2 | u |\) and \(\arg v = \arg u + \frac { 1 } { 3 } \pi\).
  2. Sketch an Argand diagram showing the points representing \(u\) and \(v\).
  3. Explain why \(v\) can be expressed as \(2 u w\). Hence find \(v\), giving your answer in the form \(a + \mathrm { i } b\), where \(a\) and \(b\) are real and exact.
Pre-U Pre-U 9794/1 Specimen Q5
10 marks Moderate -0.3
5 The complex number \(z\) satisfies the equation \(2 z - \mathrm { i } = \mathrm { i } z + 2\).
  1. Express \(z\) in the form \(a + \mathrm { i } b\) where \(a\) and \(b\) are rational numbers.
  2. Find the exact value of \(| z |\) and of \(\arg ( z )\).
  3. Express \(z ^ { 2 }\) in the form \(c + \mathrm { i } d\) where \(c\) and \(d\) are rational numbers.
  4. Verify that \(\tan ( 2 \arg ( z ) ) = \tan \left( \arg \left( z ^ { 2 } \right) \right)\) using an appropriate trigonometrical identity.