Modulus and argument with operations

Questions that require finding modulus and/or argument after performing complex number operations (multiplication, division, addition) or for expressions involving multiple complex numbers.

8 questions · Moderate -0.1

4.02a Complex numbers: real/imaginary parts, modulus, argument4.02b Express complex numbers: cartesian and modulus-argument forms
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Edexcel F1 2024 June Q4
8 marks Standard +0.3
  1. In this question you must show all stages of your working.
\section*{Solutions relying entirely on calculator technology are not acceptable.} The complex number \(z\) is defined by $$2 = - 3 + 4 i$$
  1. Determine \(\left| z ^ { 2 } - 3 \right|\)
  2. Express \(\frac { 50 } { z ^ { * } }\) in the form \(k z\), where \(k\) is a positive integer.
  3. Hence find the value of \(\arg \left( \frac { 50 } { z ^ { * } } \right)\) Give your answer in radians to 3 significant figures.
Edexcel FP1 2012 January Q1
7 marks Moderate -0.5
  1. Given that \(z _ { 1 } = 1 - \mathrm { i }\),
    1. find \(\arg \left( z _ { 1 } \right)\).
    Given also that \(z _ { 2 } = 3 + 4 \mathrm { i }\), find, in the form \(a + \mathrm { i } b , a , b \in \mathbb { R }\),
  2. \(z _ { 1 } z _ { 2 }\),
  3. \(\frac { z _ { 2 } } { z _ { 1 } }\). In part (b) and part (c) you must show all your working clearly.
OCR MEI FP1 2010 June Q8
10 marks Moderate -0.3
8 Two complex numbers, \(\alpha\) and \(\beta\), are given by \(\alpha = \sqrt { 3 } + \mathrm { j }\) and \(\beta = 3 \mathrm { j }\).
  1. Find the modulus and argument of \(\alpha\) and \(\beta\).
  2. Find \(\alpha \beta\) and \(\frac { \beta } { \alpha }\), giving your answers in the form \(a + b \mathrm { j }\), showing your working.
  3. Plot \(\alpha , \beta , \alpha \beta\) and \(\frac { \beta } { \alpha }\) on a single Argand diagram.
OCR MEI FP1 2012 June Q2
7 marks Standard +0.3
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.
WJEC Further Unit 1 2022 June Q1
12 marks Standard +0.3
  1. The complex numbers \(z , w\) are given by \(z = 3 - 4 \mathrm { i } , w = 2 - \mathrm { i }\).
    1. (i) Find the modulus and argument of \(z w\).
      (ii) Express \(z w\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\).
    2. The complex numbers \(v , w , z\) satisfy the equation \(\frac { 1 } { v } = \frac { 1 } { w } - \frac { 1 } { z }\). Find \(v\) in the form \(a + \mathrm { i } b\), where \(a , b\) are real.
    3. The complex conjugate of \(v\) is denoted by \(\bar { v }\).
    Show that \(v \bar { v } = k\), where \(k\) is a real number whose value is to be determined.
CAIE P3 2016 November Q7
9 marks Standard +0.3
  1. Find the modulus and argument of \(z\).
  2. Express each of the following in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real and exact:
    1. \(z + 2 z ^ { * }\);
    2. \(\frac { z ^ { * } } { \mathrm { i } z }\).
    3. On a sketch of an Argand diagram with origin \(O\), show the points \(A\) and \(B\) representing the complex numbers \(z ^ { * }\) and \(\mathrm { i } z\) respectively. Prove that angle \(A O B\) is equal to \(\frac { 1 } { 6 } \pi\).
OCR FP1 AS 2021 June Q2
13 marks Moderate -0.3
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.
Edexcel FP1 Q44
10 marks Moderate -0.8
$$z = -2 + i.$$
  1. Express in the form \(a + ib\)
    1. \(\frac{1}{z}\)
    2. \(z^2\). [4]
  2. Show that \(|z^2 - z| = 5\sqrt{2}\). [2]
  3. Find \(\arg (z^2 - z)\). [2]
  4. Display \(z\) and \(z^2 - z\) on a single Argand diagram. [2]