A2 June 2024 Paper 1 Q7
7 The complex numbers \(z\) and \(w\) satisfy the simultaneous equations
\[z + w^* = 5\]\[3z^* - w = 6 + 4\mathrm{i}\]Find \(z\) and \(w\) [5 marks]
| Scheme | Marks | AO |
|---|---|---|
| Writes \(z = x + \mathrm{i}y \qquad z^* = x - \mathrm{i}y\) \(w = u + \mathrm{i}v \qquad w^* = u - \mathrm{i}v\) OE PI or Obtains the conjugate of one of the equations Eg \(z^* + w = 5\) | M1 | 1.1a |
| Forms two of \(x + u = 5\) \(y - v = 0\) \(3x - u = 6\) \(-3y - v = 4\) OE or eliminates one complex unknown | M1 | 1.1a |
| Obtains at least two correct values of \(x\), \(y\), \(u\) or \(v\) or obtains one of \(z^* = \dfrac{11}{4} + \mathrm{i}\), \(w^* = \dfrac{9}{4} + \mathrm{i}\) | M1 | 1.1a |
| Obtains the values \(\dfrac{11}{4}, -1, \dfrac{9}{4}, -1\) | A1 | 1.1b |
| Obtains \(z = \dfrac{11}{4} - \mathrm{i}\) \(w = \dfrac{9}{4} - \mathrm{i}\) | A1 | 1.1b |
| (5 marks) |
Typical solution
Let
\[z = x + \mathrm{i}y \qquad z^* = x - \mathrm{i}y\]\[w = u + \mathrm{i}v \qquad w^* = u - \mathrm{i}v\]where \(x, y, u, v \in \mathbb{R}\)
\[x + \mathrm{i}y + u - \mathrm{i}v = 5\]Re: \(x + u = 5 \quad \ldots(1)\)
Im: \(y - v = 0 \Rightarrow y = v \quad \ldots(2)\)
\[3(x - \mathrm{i}y) - (u + \mathrm{i}v) = 6 + 4\mathrm{i}\]Re: \(3x - u = 6 \quad \ldots(3)\)
Im: \(-3y - v = 4 \quad \ldots(4)\)
(2), (4) \(\Rightarrow y = -1,\ v = -1\)
(1), (3) \(\Rightarrow x = \dfrac{11}{4},\ u = \dfrac{9}{4}\)
\[z = \frac{11}{4} - \mathrm{i}\]\[w = \frac{9}{4} - \mathrm{i}\]