A2 October 2020 Paper 1 Q8
8
(a) Given that the lines \(\mathbf{r} = \begin{pmatrix} 0 \\ 2 \\ 2 \end{pmatrix} + \lambda\begin{pmatrix} -1 \\ 1 \\ 3 \end{pmatrix}\) and \(\mathbf{r} = \begin{pmatrix} -1 \\ 2 \\ k \end{pmatrix} + \mu\begin{pmatrix} 2 \\ 3 \\ 4 \end{pmatrix}\) meet, determine \(k\). [5]
(b) In this question you must show detailed reasoning.
Find the acute angle between the two lines. [4]
Find the acute angle between the two lines. [4]
| Scheme | Marks | AO |
|---|---|---|
| \(-\lambda = -1 + 2\mu,\ 2 + \lambda = 2 + 3\mu,\ 2 + 3\lambda = k + 4\mu\) | M1 | 3.1a |
| \(\lambda = 3\mu,\ -3\mu = -1 + 2\mu\) | M1 | 1.1 |
| \(\Rightarrow \mu = 1/5,\ \lambda = 3/5\) \(2 + 9/5 = k + 4/5 \Rightarrow k = 3\) | A1A1 A1 | 1.1,1.1 1.1 |
| [5] |
| Scheme | Marks | AO |
|---|---|---|
| DR \(\cos\theta = \dfrac{(-\mathbf{i} + \mathbf{j} + 3\mathbf{k}).(2\mathbf{i} + 3\mathbf{j} + 4\mathbf{k})}{\sqrt{(-1)^2 + 1^2 + 3^2}\sqrt{2^2 + 3^2 + 4^2}}\) | M1A1 | 1.1,1.1 |
| \(= \dfrac{13}{\sqrt{11}\sqrt{29}}\) | B1 | 1.1 |
| \(\Rightarrow \theta = 43.3^\circ\) | A1 | 1.1 |
| [4] |
Notes
B1: soi
A1: accept 0.756 rad