Solving equations involving complex fractions

Questions that require solving an equation where the unknown z appears in a fraction or quotient, necessitating algebraic manipulation before or after applying the conjugate method.

5 questions · Standard +0.1

4.02e Arithmetic of complex numbers: add, subtract, multiply, divide
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CAIE P3 2024 November Q4
5 marks Moderate -0.5
4 Find the complex number \(z\) satisfying the equation $$\frac { z - 3 \mathrm { i } } { z + 3 \mathrm { i } } = \frac { 2 - 9 \mathrm { i } } { 5 }$$ Give your answer in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
OCR MEI FP1 2016 June Q2
6 marks Standard +0.3
2 The complex number \(z _ { 1 }\) is \(2 - 5 \mathrm { j }\) and the complex number \(z _ { 2 }\) is \(( a - 1 ) + ( 2 - b ) \mathrm { j }\), where \(a\) and \(b\) are real.
  1. Express \(\frac { z _ { 1 } { } ^ { * } } { z _ { 1 } }\) in the form \(x + y \mathrm { j }\), giving \(x\) and \(y\) in exact form. You must show clearly how you obtain your
    answer.
  2. Given that \(\frac { z _ { 1 } { } ^ { * } } { z _ { 1 } } = z _ { 2 }\), find the exact values of \(a\) and \(b\).
CAIE P3 2014 November Q5
8 marks Standard +0.3
5 Throughout this question the use of a calculator is not permitted. The complex numbers \(w\) and \(z\) satisfy the relation $$w = \frac { z + \mathrm { i } } { \mathrm { i } z + 2 }$$
  1. Given that \(z = 1 + \mathrm { i }\), find \(w\), giving your answer in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
  2. Given instead that \(w = z\) and the real part of \(z\) is negative, find \(z\), giving your answer in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
WJEC Further Unit 1 Specimen Q2
11 marks Standard +0.3
Solve the equation \(2z + iz = \frac{-1 + 7i}{2 + i}\).
  1. Give your answer in Cartesian form [7]
  2. Give your answer in modulus-argument form. [4]
Pre-U Pre-U 9794/1 2010 June Q10
10 marks Standard +0.3
  1. Solve the equation \((2 + i)z = (4 + in)\). Give your answer in the form \(a + ib\), expressing \(a\) and \(b\) in terms of the real constant \(n\). [4]
  2. The roots of the equation \(z^2 + 8z + 25 = 0\) are denoted by \(z_1\) and \(z_2\).
    1. Find \(z_1\) and \(z_2\) and show these roots on an Argand diagram. [3]
    2. Find the modulus and argument in radians of each of \((z_1 + 1)\) and \((z_2 + 1)\). [3]