## Question 6:

### Part (a):

| Answer/Working | Mark | Guidance |
|---|---|---|
| Finds any two vectors $\pm\overrightarrow{PQ}$, $\pm\overrightarrow{PR}$ or $\pm\overrightarrow{QR}$: $\pm\begin{pmatrix}2\\3\\-9\end{pmatrix}$ or $\pm\begin{pmatrix}1\\2\\-1\end{pmatrix}$ or $\pm\begin{pmatrix}-1\\-1\\8\end{pmatrix}$ | M1 | Two out of three values correct is sufficient |
| A correct equation for the plane $\mathbf{r} = \mathbf{a} + \lambda\mathbf{b} + \mu\mathbf{c}$, with $\mathbf{a} = \begin{pmatrix}1\\-2\\4\end{pmatrix}$ or $\begin{pmatrix}3\\1\\-5\end{pmatrix}$ or $\begin{pmatrix}2\\0\\3\end{pmatrix}$, and **b**, **c** any two vectors from $\pm\begin{pmatrix}2\\3\\-9\end{pmatrix}$ or $\pm\begin{pmatrix}1\\2\\-1\end{pmatrix}$ or $\pm\begin{pmatrix}-1\\-1\\8\end{pmatrix}$ | A1 | Must start with $\mathbf{r} =\ldots$ |

### Part (b):

| Answer/Working | Mark | Guidance |
|---|---|---|
| Forms two simultaneous equations by setting $y=0$ and $z=0$: $-2+3\lambda+2\mu=0$ and $4-9\lambda-\mu=0$ | M1 | Uses their plane equation |
| Solves simultaneous equations: $\lambda=0.4$, $\mu=0.4$ | dM1 | Dependent on previous M1 |
| Uses values of $\mu$ and $\lambda$ to find $x$: $x=1+2\lambda+\mu=1+2(0.4)+(0.4)=\ldots$ | ddM1 | Depends on both method marks |
| $(2.2,\ 0,\ 0)$ | A1 | Accept $\left(\frac{11}{5},0,0\right)$ or $\left(\frac{33}{15},0,0\right)$ |

**Alternative:**

| Answer/Working | Mark | Guidance |
|---|---|---|
| $\begin{vmatrix}2&3&-9\\1&2&-1\end{vmatrix} = (-3+18)\mathbf{i}-(-2+9)\mathbf{j}+(4-3)\mathbf{k}$ | M1 | Allow one slip in expansion |
| $\begin{pmatrix}15\\-7\\1\end{pmatrix}\cdot\begin{pmatrix}1\\-2\\4\end{pmatrix}=15+14+4=33$ leading to $15x-7y+z=33$ | dM1 | Finds Cartesian equation |
| $15x-7(0)+(0)=33 \Rightarrow x=\ldots$ | ddM1 | Sets $y=0$ and $z=0$ |
| $(2.2,\ 0,\ 0)$ | A1 | Accept equivalent fractions |

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