A2 June 2025 Paper 1 Q2
2 In this question you must show detailed reasoning.
Find the acute angle between the planes \(2x - y + 2z = 5\) and \(x + 2y + z = 8\). [4]
| Scheme | Marks | AO |
|---|---|---|
| DR Normal vectors are \(2\mathbf{i} - \mathbf{j} + 2\mathbf{k}\) and \(\mathbf{i} + 2\mathbf{j} + \mathbf{k}\) | B1 | 1.1 |
| \(\cos\theta = \dfrac{(2\mathbf{i} - \mathbf{j} + 2\mathbf{k}) \cdot (\mathbf{i} + 2\mathbf{j} + \mathbf{k})}{\sqrt{2^2 + (-1)^2 + 2^2}\sqrt{1^2 + 2^2 + 1^2}}\) | M1 | 1.1 |
| \(= \dfrac{2}{3\sqrt{6}}\) | A1 | 1.1 |
| \(\theta = 74.2^\circ\) or better | A1 | 1.1 |
| [4] |
Notes
B1: both soi
M1: scalar product formula with \(\cos\theta\), allow a slip
A1: with scalar product = 2 soi
A1: or 1.30 radians or better, www. Mark final answer. Answer with no supporting working scores 0 marks.
Alternative method 1
| Scheme | Marks |
|---|---|
| Normal vectors are \(2\mathbf{i} - \mathbf{j} + 2\mathbf{k}\) and \(\mathbf{i} + 2\mathbf{j} + \mathbf{k}\) | B1 |
| \(\sin\theta = \dfrac{|(2\mathbf{i} - \mathbf{j} + 2\mathbf{k}) \times (\mathbf{i} + 2\mathbf{j} + \mathbf{k})|}{\sqrt{2^2 + (-1)^2 + 2^2}\sqrt{1^2 + 2^2 + 1^2}}\) | M1 |
| \(= \dfrac{\sqrt{50}}{3\sqrt{6}}\) | A1 |
| \(\theta = 74.2^\circ\) or better | A1 |
B1: both soi
M1: vector product formula with \(\sin\theta\), allow a slip
A1: with |vector product| = \(\sqrt{50}\) soi
A1: or 1.30 radians or better, www. Mark final answer. Answer with no supporting working scores 0 marks.
Alternative method 2
| Scheme | Marks |
|---|---|
| Normal vectors are \(2\mathbf{i} - \mathbf{j} + 2\mathbf{k}\) and \(\mathbf{i} + 2\mathbf{j} + \mathbf{k}\) | B1 |
| \(\sin\alpha = \dfrac{(2\mathbf{i} - \mathbf{j} + 2\mathbf{k}) \cdot (\mathbf{i} + 2\mathbf{j} + \mathbf{k})}{\sqrt{2^2 + (-1)^2 + 2^2}\sqrt{1^2 + 2^2 + 1^2}}\) | M1 |
| \(= \dfrac{2}{3\sqrt{6}}\) | A1 |
| \(\theta = 90 - \alpha = 74.2^\circ\) or better | A1 |
B1: both soi
M1: scalar product formula with \(\sin\theta\), allow a slip
A1: with scalar product = 2 soi
A1: or 1.30 radians or better, www. Mark final answer. Answer with no supporting working scores 0 marks.