A2 June 2025 Paper 1 Q1
1 The complex number \(z\) satisfies the equation \(z + 2\mathrm{i}z^* + 1 - 4\mathrm{i} = 0\).
You are given that \(z = x + \mathrm{i}y\), where \(x\) and \(y\) are real numbers.
Determine the values of \(x\) and \(y\). [4]
| Scheme | Marks | AO |
|---|---|---|
| \(z^* = x - \mathrm{i}y\) | B1 | 1.2 |
| \(x + \mathrm{i}y + 2\mathrm{i}x + 2y + 1 - 4\mathrm{i} \; [= 0]\) | M1 | 1.1 |
| \(\Rightarrow x + 2y + 1 = 0,\; y + 2x - 4 = 0\) | M1 | 3.1a |
| \(\Rightarrow x = 3,\; y = -2\) | A1 | 1.1 |
| [4] |
Notes
B1: seen or used
M1: \(\mathrm{i}^2 = -1\) used with their \(z^*\); soi by correct equating of real and imaginary parts
M1: equating their real and imaginary parts (imaginary component may still be in terms of i), allow a slip including a missing term
A1: www. Accept \([z =]\, 3 - 2\mathrm{i}\). An M1 step must be seen before final answer.