## Question 9(i):

| Answer | Marks | Guidance |
|--------|-------|----------|
| Carry out correct method for finding a vector equation for $AB$ | M1 | |
| Obtain $\mathbf{r} = \mathbf{i}+2\mathbf{j}-\mathbf{k}+\lambda(2\mathbf{i}-\mathbf{j}+2\mathbf{k})$, or equivalent | A1 | |
| Equate two pairs of components of general points on *their* $AB$ and $l$ and solve for $\lambda$ or $\mu$ | M1 | $\begin{pmatrix}1+2\lambda\\2-\lambda\\-1+2\lambda\end{pmatrix}=\begin{pmatrix}2+\mu\\1+\mu\\1+2\mu\end{pmatrix}$ |
| Obtain correct answer for $\lambda$ or $\mu$, e.g. $\lambda=0,\,\mu=-1$ | A1 | |
| Verify all three equations are not satisfied and lines fail to intersect ($\neq$ is sufficient justification e.g. $2\neq 0$); conclusion needs to follow correct values | A1 | See table of alternatives in mark scheme |

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## Question 9(ii):

| Answer | Marks | Guidance |
|--------|-------|----------|
| State or imply midpoint has position vector $2\mathbf{i}+\dfrac{3}{2}\mathbf{j}$ | B1 | |
| Substitute in $2x-y+2z=d$ and find $d$ | M1 | Correct use of *their* direction for $AB$ and *their* midpoint |
| Obtain plane equation $4x-2y+4z=5$ | A1 | or equivalent e.g. $\mathbf{r}\cdot\begin{pmatrix}2\\-1\\2\end{pmatrix}=\dfrac{5}{2}$ |
| Substitute components of $l$ in plane equation and solve for $\mu$ | M1 | Correct use of their plane equation |
| Obtain $\mu=-\dfrac{1}{2}$ and position vector $\dfrac{3}{2}\mathbf{i}+\dfrac{1}{2}\mathbf{j}$ for point $P$ | A1 | Final answer; accept coordinates in place of position vector |