# Question 1:

## Part (ii)

| Answer/Working | Marks | Guidance |
|---|---|---|
| $\mathbf{r} = \begin{pmatrix}1\\2\\4\end{pmatrix} + \lambda\begin{pmatrix}1\\-1\\2\end{pmatrix} = \begin{pmatrix}1+\lambda\\2-\lambda\\4+2\lambda\end{pmatrix}$ meets plane when: | | |
| $1+\lambda+2(2-\lambda)-3(4+2\lambda)=0$ | M1 | subst of **AB** in the plane |
| $\Rightarrow -7-7\lambda=0,\ \lambda=-1$ | A1 | |
| So B is $(0,3,2)$ | A1 | cao |
| $\overrightarrow{A'B} = \begin{pmatrix}0\\3\\2\end{pmatrix} - \begin{pmatrix}2\\4\\1\end{pmatrix} = \begin{pmatrix}-2\\-1\\1\end{pmatrix}$ | M1 | or $\overrightarrow{BA'}$, ft **only** on their B (condone $\overrightarrow{A'B}$ used as $\overrightarrow{BA'}$ or no label); can be implied by two correct coordinates |
| Eqn of line $A'B$ is $\mathbf{r} = \begin{pmatrix}2\\4\\1\end{pmatrix} + \lambda\begin{pmatrix}-2\\-1\\1\end{pmatrix}$ | B1 ft | $\begin{pmatrix}2\\4\\1\end{pmatrix}$ or their $B + \ldots$ |
| | A1 ft | $\ldots\lambda \times$ their $\overrightarrow{A'B}$ (or $\overrightarrow{BA'}$); ft only their B correctly |

## Part (iii)

| Answer/Working | Marks | Guidance |
|---|---|---|
| Angle between $\begin{pmatrix}1\\-1\\2\end{pmatrix}$ and $\begin{pmatrix}-2\\-1\\1\end{pmatrix}$ | M1 | correct vectors but ft their $\overrightarrow{A'B}$. Allow say $\begin{pmatrix}-1\\1\\-2\end{pmatrix}$ and/or $\begin{pmatrix}2\\1\\-1\end{pmatrix}$; condone a minor slip if intention is clear |
| $\cos\theta = \dfrac{1.(-2)+(-1).(-1)+2.1}{\sqrt{6}.\sqrt{6}}$ | M1 | correct formula (including $\cos\theta$) for their direction vectors from (ii); condone a minor slip if intention is clear |
| $= \dfrac{1}{6}$ | A1 | $\pm\frac{1}{6}$ or $99.6°$ from appropriate vectors only soi. **Do not allow either A mark if the correct B was found fortuitously in (ii)** |
| $\Rightarrow \theta = 80.4°$ | A1 | cao or better |

## Part (iv)

| Answer/Working | Marks | Guidance |
|---|---|---|
| Equation of BC is $\mathbf{r} = \begin{pmatrix}2\\4\\1\end{pmatrix} + \lambda\begin{pmatrix}-2\\-1\\1\end{pmatrix} = \begin{pmatrix}2-2\lambda\\4-\lambda\\1+\lambda\end{pmatrix}$ | | **NB this is not unique**, e.g. $\begin{pmatrix}0\\3\\2\end{pmatrix} + \mu\begin{pmatrix}2\\1\\-1\end{pmatrix}$ |
| Crosses $Oxz$ plane when $y=0$ | M1 | For putting $y=0$ in their line BC and solving for $\lambda$ |
| $\Rightarrow \lambda=4$ | A1 | Do not allow either A mark if B was found fortuitously in (ii); for A marks need fully correct work only. **NB this is not unique**, e.g. $\begin{pmatrix}0\\3\\2\end{pmatrix}+\mu\begin{pmatrix}2\\1\\-1\end{pmatrix}$ leads to $\mu=-3$ |
| $\Rightarrow \mathbf{r} = \begin{pmatrix}-6\\0\\5\end{pmatrix}$ so $(-6,0,5)$ | A1 | cao |

---