Division plus modulus/argument

Multi-part questions where division of complex numbers is one part, and other parts ask for modulus and/or argument of a complex number.

10 questions · Moderate -0.5

4.02a Complex numbers: real/imaginary parts, modulus, argument4.02e Arithmetic of complex numbers: add, subtract, multiply, divide
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Edexcel F1 2017 January Q5
8 marks Moderate -0.3
  1. The complex number \(z\) is given by
$$z = - 7 + 3 i$$ Find
  1. \(| z |\)
  2. \(\arg z\), giving your answer in radians to 2 decimal places. Given that \(\frac { z } { 1 + \mathrm { i } } + w = 3 - 6 \mathrm { i }\)
  3. find the complex number \(w\), giving your answer in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are real numbers. You must show all your working.
  4. Show the points representing \(z\) and \(w\) on a single Argand diagram.
Edexcel FP1 2013 January Q2
8 marks Moderate -0.3
2. $$z = \frac { 50 } { 3 + 4 \mathrm { i } }$$ Find, in the form \(a + \mathrm { i } b\) where \(a , b \in \mathbb { R }\),
  1. \(z\),
  2. \(z ^ { 2 }\). Find
  3. \(| z |\),
  4. \(\arg z ^ { 2 }\), giving your answer in degrees to 1 decimal place.
Edexcel FP1 2013 June Q7
11 marks Moderate -0.3
7. $$z _ { 1 } = 2 + 3 \mathrm { i } , \quad z _ { 2 } = 3 + 2 \mathrm { i } , \quad z _ { 3 } = a + b \mathrm { i } , \quad a , b \in \mathbb { R }$$
  1. Find the exact value of \(\left| z _ { 1 } + z _ { 2 } \right|\). Given that \(w = \frac { z _ { 1 } z _ { 3 } } { z _ { 2 } }\),
  2. find \(w\) in terms of \(a\) and \(b\), giving your answer in the form \(x + \mathrm { i } y , \quad x , y \in \mathbb { R }\) Given also that \(w = \frac { 17 } { 13 } - \frac { 7 } { 13 } \mathrm { i }\),
  3. find the value of \(a\) and the value of \(b\),
  4. find \(\arg w\), giving your answer in radians to 3 decimal places.
CAIE P3 2020 Specimen Q6
8 marks Moderate -0.5
6 The complex numbers \(1 + 3 \mathrm { i }\) and \(4 + 2 \mathrm { i }\) are denoted by \(u\) and \(v\) respectively.
  1. Find \(\frac { u } { v }\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
  2. State the argument of \(\frac { u } { v }\).
    In an Argand diagram, with origin \(O\), the points \(A , B\) and \(C\) represent the complex numbers \(u , v\) and \(u - v\) respectively.
  3. State fully the geometrical relationship between \(O C\) and \(B A\).
  4. Show that angle \(A O B = \frac { 1 } { 4 } \pi\) radians.
OCR FP1 2014 June Q2
5 marks Moderate -0.8
2 The complex number \(7 + 3 \mathrm { i }\) is denoted by \(z\). Find
  1. \(| z |\) and \(\arg z\),
  2. \(\frac { z } { 4 - \mathrm { i } }\), showing clearly how you obtain your answer.
OCR MEI Further Pure Core 2024 June Q2
7 marks Moderate -0.8
2 Two complex numbers are given by \(u = - 1 + \mathrm { i }\) and \(v = - 2 - \mathrm { i }\).
    1. Find \(\mathrm { u } - \mathrm { v }\) in the form \(\mathrm { a } + \mathrm { bi }\), where \(a\) and \(b\) are real.
    2. In this question you must show detailed reasoning. Find \(\frac { \mathrm { u } } { \mathrm { v } }\) in the form \(\mathrm { a } + \mathrm { bi }\), where \(a\) and \(b\) are real.
  2. Express \(u\) in exact modulus-argument form.
Edexcel F1 2022 January Q2
8 marks Moderate -0.8
The complex numbers \(z_1\) and \(z_2\) are given by $$z_1 = 3 + 5\text{i} \quad \text{and} \quad z_2 = -2 + 6\text{i}$$
  1. Show \(z_1\) and \(z_2\) on a single Argand diagram. [2]
  2. Without using your calculator and showing all stages of your working,
    1. determine the value of \(|z_1|\) [1]
    2. express \(\frac{z_1}{z_2}\) in the form \(a + b\text{i}\), where \(a\) and \(b\) are fully simplified fractions. [3]
  3. Hence determine the value of \(\arg \frac{z_1}{z_2}\) Give your answer in radians to 2 decimal places. [2]
Edexcel FP1 Q1
5 marks Moderate -0.3
Given that \(z = 22 + 4i\) and \(\frac{z}{w} = 6 - 8i\), find
  1. \(w\) in the form \(a + bi\), where \(a\) and \(b\) are real, [3]
  2. the argument of \(z\), in radians to 2 decimal places. [2]
Edexcel FP1 Q38
13 marks Moderate -0.3
$$z = \sqrt{3} - i.$$ \(z^*\) is the complex conjugate of \(z\).
  1. Show that \(\frac{z}{z^*} = \frac{1}{2} - \frac{\sqrt{3}}{2} i\). [3]
  2. Find the value of \(\left| \frac{z}{z^*} \right|\). [2]
  3. Verify, for \(z = \sqrt{3} - i\), that \(\arg \frac{z}{z^*} = \arg z - \arg z^*\). [4]
  4. Display \(z\), \(z^*\) and \(\frac{z}{z^*}\) on a single Argand diagram. [2]
  5. Find a quadratic equation with roots \(z\) and \(z^*\) in the form \(ax^2 + bx + c = 0\), where \(a\), \(b\) and \(c\) are real constants to be found. [2]
OCR MEI Further Pure Core AS Specimen Q1
4 marks Moderate -0.8
The complex number \(z_1\) is \(1+ i\) and the complex number \(z_2\) has modulus 4 and argument \(\frac{\pi}{3}\).
  1. Express \(z_2\) in the form \(a + bi\), giving \(a\) and \(b\) in exact form. [2]
  2. Express \(\frac{z_2}{z_1}\) in the form \(c + di\), giving \(c\) and \(d\) in exact form. [2]