Equations with conjugate of expressions

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 \(z\).

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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 \(z ^ { 2 } + 10 \mathrm { i } z - 29 = 0\).

8. (a) 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 \(( z - 2 \mathrm { i } ) ^ { * }\).
   (1 mark)
2. Solve the equation $$( z - 2 i ) ^ { * } = 4 i z + 3$$ giving your answer in the form \(a + b \mathrm { i }\).
   (b) 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 \(p - q \mathrm { i }\) is not a root of the equation \(z ^ { 2 } + 10 \mathrm { i } z - 29 = 0\).
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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 \(a + b \mathrm { i }\).

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1. Show that the equation $$\left( 2 z - z ^ { * } \right) ^ { * } = z ^ { 2 }$$ has exactly four solutions and state these solutions.
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2. 1. Plot the four solutions to the equation in part (a) on the Argand diagram below and join them together to form a quadrilateral with one line of symmetry.
      \includegraphics[max width=\textwidth, alt={}, center]{8f7a5fc0-6936-4aed-a173-e221bf86e4fd-09_842_860_406_589} 6
3. (ii) Show that the area of this quadrilateral is \(\frac { \sqrt { 15 } } { 2 }\) square units.