Division plus other arithmetic operations

Multi-part questions where division of complex numbers is one part, and other parts involve addition, subtraction, multiplication, powers, or conjugates, without requiring modulus or argument.

15 questions · Moderate -0.9

4.02e Arithmetic of complex numbers: add, subtract, multiply, divide
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OCR FP1 2006 January Q1
5 marks Moderate -0.8
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 \(x + i y\).
OCR FP1 2011 January Q2
6 marks Moderate -0.8
2 The complex numbers \(z\) and \(w\) are given by \(z = 4 + 3 \mathrm { i }\) and \(w = 6 - \mathrm { i }\). Giving your answers in the form \(x + \mathrm { i } y\) and showing clearly how you obtain them, find
  1. \(3 z - 4 w\),
  2. \(\frac { z ^ { * } } { w }\).
OCR FP1 2012 June Q1
5 marks Moderate -0.8
1 The complex numbers \(z\) and \(w\) are given by \(z = 6 - \mathrm { i }\) and \(w = 5 + 4 \mathrm { i }\). Giving your answers in the form \(x + \mathrm { i } y\) and showing clearly how you obtain them, find
  1. \(z + 3 w\),
  2. \(\frac { Z } { W }\).
OCR Further Pure Core AS 2024 June Q2
4 marks Moderate -0.3
2 In this question you must show detailed reasoning.
  1. Express \(\frac { 8 + \mathrm { i } } { 2 - \mathrm { i } }\) in the form \(\mathrm { a } + \mathrm { bi }\) where \(a\) and \(b\) are real.
  2. Solve the equation \(4 x ^ { 2 } - 8 x + 5 = 0\). Give your answer(s) in the form \(\mathrm { c } + \mathrm { di }\) where \(c\) and \(d\) are real.
Pre-U Pre-U 9794/2 2012 Specimen Q6
8 marks Moderate -0.8
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. \(\frac { 5 } { z }\).
Pre-U Pre-U 9794/1 2015 June Q8
11 marks Moderate -0.3
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.
Pre-U Pre-U 9794/1 2016 Specimen Q9
6 marks Easy -1.3
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.
Pre-U Pre-U 9794/1 2019 Specimen Q9
2 marks Easy -1.2
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.
Pre-U Pre-U 9794/1 2020 Specimen Q9
2 marks Easy -1.2
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.
Edexcel FP1 2013 June Q1
3 marks Easy -1.2
The complex numbers \(z\) and \(w\) are given by $$z = 8 + 3\text{i}, \quad w = -2\text{i}$$ Express in the form \(a + b\text{i}\), where \(a\) and \(b\) are real constants,
  1. \(z - w\), [1]
  2. \(zw\). [2]
OCR FP1 Q3
7 marks Moderate -0.8
The complex numbers \(2 + 3i\) and \(4 - i\) are denoted by \(z\) and \(w\) respectively. Express each of the following in the form \(x + iy\), showing clearly how you obtain your answers.
  1. \(z + 5w\), [2]
  2. \(z*w\), where \(z*\) is the complex conjugate of \(z\), [3]
  3. \(\frac{1}{w}\). [2]
OCR FP1 2005 June Q3
7 marks Easy -1.2
The complex numbers \(2 + 3\text{i}\) and \(4 - \text{i}\) are denoted by \(z\) and \(w\) respectively. Express each of the following in the form \(x + \text{i}y\), showing clearly how you obtain your answers.
  1. \(z + 5w\), [2]
  2. \(z^*w\), where \(z^*\) is the complex conjugate of \(z\), [3]
  3. \(\frac{1}{w}\). [2]
OCR MEI FP1 2007 June Q4
7 marks Moderate -0.8
Two complex numbers, \(\alpha\) and \(\beta\), are given by \(\alpha = 1 - 2\mathrm{j}\) and \(\beta = -2 - \mathrm{j}\).
  1. Represent \(\beta\) and its complex conjugate \(\beta^*\) on an Argand diagram. [2]
  2. Express \(\alpha\beta\) in the form \(a + b\mathrm{j}\). [2]
  3. Express \(\frac{\alpha + \beta}{\beta}\) in the form \(a + b\mathrm{j}\). [3]
SPS SPS FM 2022 February Q2
8 marks Moderate -0.8
The complex numbers \(3 - 2i\) and \(2 + i\) are denoted by \(z\) and \(w\) respectively. Find, giving your answers in the form \(x + iy\) and showing clearly how you obtain these answers,
  1. \(2z - 3w\), [2]
  2. \((iz)^2\), [3]
  3. \(\frac{z}{w}\). [3]
OCR FP1 AS 2017 Specimen Q2
4 marks Moderate -0.8
**In this question you must show detailed reasoning.** Given that \(z_1 = 3 + 2i\) and \(z_2 = -1 - i\), find the following, giving each in the form \(a + bi\).
  1. \(z_1^* z_2\) [2]
  2. \(\frac{z_1 + 2z_2}{z_2}\) [2]