Equations with conjugate of expressions

Equations involving (z + a)* or similar conjugates of expressions, requiring first expanding the conjugate using (w)* = w* before substituting z = x + iy.

4 questions · Standard +0.3

4.02e Arithmetic of complex numbers: add, subtract, multiply, divide
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Edexcel F1 2015 January Q3
6 marks Standard +0.8
3. Given that \(z = x + \mathrm { i } y\), where \(x\) and \(y\) are real numbers, solve the equation $$( z - 2 i ) \left( z ^ { * } - 2 i \right) = 21 - 12 i$$ where \(z ^ { * }\) is the complex conjugate of \(z\).
Edexcel FP1 2018 June Q9
12 marks Standard +0.3
    1. Given that
$$\frac { 3 w + 7 } { 5 } = \frac { p - 4 \mathrm { i } } { 3 - \mathrm { i } } \quad \text { where } p \text { is a real constant }$$
  1. express \(w\) in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are real constants. Give your answer in its simplest form in terms of \(p\). Given that arg \(w = - \frac { \pi } { 2 }\)
  2. find the value of \(p\).
    (ii) Given that $$( z + 1 - 2 i ) ^ { * } = 4 i z$$ find \(z\), giving your answer in the form \(z = x + i y\), where \(x\) and \(y\) are real constants. \includegraphics[max width=\textwidth, alt={}, center]{89f82cd3-9afa-4431-bc74-a073909c903f-36_106_129_2469_1816}
AQA FP1 2013 June Q4
7 marks Standard +0.3
4
  1. It is given that \(z = x + y \mathrm { i }\), where \(x\) and \(y\) are real numbers.
    1. Write down, in terms of \(x\) and \(y\), an expression for \(( z - 2 \mathrm { i } ) ^ { * }\).
    2. Solve the equation $$( z - 2 \mathrm { i } ) ^ { * } = 4 \mathrm { i } z + 3$$ giving your answer in the form \(a + b \mathrm { i }\).
  2. It is given that \(p + q \mathrm { i }\), where \(p\) and \(q\) are real numbers, is a root of the equation \(z ^ { 2 } + 10 \mathrm { i } z - 29 = 0\). Without finding the values of \(p\) and \(q\), state why \(p - q\) i is not a root of the equation \(z ^ { 2 } + 10 \mathrm { i } z - 29 = 0\).
AQA FP1 2006 June Q6
7 marks Moderate -0.3
6 It is given that \(z = x + \mathrm { i } y\), where \(x\) and \(y\) are real numbers.
  1. Write down, in terms of \(x\) and \(y\), an expression for $$( z + \mathrm { i } ) ^ { * }$$ where \(( z + \mathrm { i } ) ^ { * }\) denotes the complex conjugate of \(( z + \mathrm { i } )\).
  2. Solve the equation $$( z + \mathrm { i } ) ^ { * } = 2 \mathrm { i } z + 1$$ giving your answer in the form \(a + b \mathrm { i }\).