AS June 2025 Paper 1 Q6
6 The complex numbers \(w\) and \(z\) are defined as follows:
\[w = 2(\cos 0.4 + \mathrm{i}\sin 0.4) \qquad \text{and} \qquad z = 6(\cos 1.2 + \mathrm{i}\sin 1.2)\](a) Express \(z^2\) in the form \(r(\cos\theta + \mathrm{i}\sin\theta)\) [2 marks]
(b) Express \(\dfrac{z}{w}\) in the form \(r(\cos\theta + \mathrm{i}\sin\theta)\) [2 marks]
| Scheme | Marks | AO |
|---|---|---|
| Correct method for the modulus. or Correct method for the argument. | M1 | 1.1a |
| Obtains \(36(\cos 2.4 + \mathrm{i}\sin 2.4)\) | A1 | 1.1b |
| (2) |
Typical solution
\[z^2 = 36(\cos 2.4 + \mathrm{i}\sin 2.4)\]| Scheme | Marks | AO |
|---|---|---|
| Correct method for the modulus. or Correct method for the argument. | M1 | 1.1a |
| Obtains \(3(\cos 0.8 + \mathrm{i}\sin 0.8)\) | A1 | 1.1b |
| (2) | ||
| (4 marks) |