**(i)** Carry out a correct method for finding a vector equation for $AB$ | M1 |
Obtain $r = 2i - j + 3k + \lambda(1 + 2j + 2k)$, or equivalent | A1 |
Equate at least two pairs of components of general points on $AB$ and $l$ and solve for $\lambda$ or for $\mu$ | M1 |
Obtain correct answer for $\lambda$ or $\mu$, e.g. $\lambda = 1$ or $\mu = 0$; $\lambda = -\frac{4}{5}$ or $\mu = \frac{3}{5}$; or $\lambda = \frac{1}{4}$ or $\mu = -\frac{3}{2}$ | A1 |
Verify that not all three pairs of equations are satisfied and that the lines fail to intersect | A1 | [5]

**(ii)** **EITHER:** Obtain a vector parallel to the plane and not parallel to $l$, e.g. $i - 2j + k$ | B1 |
Use scalar product to obtain an equation in $a$, $b$ and $c$, e.g. $3a + b - c = 0$ | B1 |
Form a second relevant equation, e.g. $a - 2b + c = 0$ and solve for one ratio, e.g. $a : b$ | M1 |
Obtain final answer $a : b : c = 1 : 4 : 7$ | A1 |
Use coordinates of a relevant point and values of $a$, $b$ and $c$ in general equation and find $d$ | M1 |
Obtain answer $x + 4y + 7z = 19$, or equivalent | A1 |

**OR1:** Obtain a vector parallel to the plane and not parallel to $l$, e.g. $i - 2j + k$ | B1 |
Obtain a second relevant vector parallel to the plane and attempt to calculate their vector product, e.g. $(i - 2j + k) \times (3i + j - k)$ | M1 |
Obtain two correct components | A1 |
Obtain correct answer, e.g. $i + 4j + 7k$ | A1 |
Substitute coordinates of a relevant point in $x + 4y + 7z = d$, or equivalent, and find $d$ | M1 |
Obtain answer $x + 4y + 7z = 19$, or equivalent | A1 |

**OR2:** Obtain a vector parallel to the plane and not parallel to $l$, e.g. $i - 2j + k$ | B1 |
Using a relevant point and second relevant vector, form a 2-parameter equation for the plane | M1 |
State a correct equation, e.g. $r = 2i - j + 3k + s(i - 2j + k) + t(3i + j - k)$ | A1 |
State 3 correct equations in $x$, $y$, $z$ and $t$ | A1 |
Eliminate $s$ and $t$ | M1 |
Obtain answer $x + 4y + 7z = 19$, or equivalent | A1 |

**OR3:** Using the coordinates of $A$ and two points on $l$, state three simultaneous equations in $a$, $b$, $c$ and $d$, e.g. $a + b + 2c = d$, $2a - b + 3c = d$ and $4a + 2b + c = d$ | B1 |
Solve and find one ratio, e.g. $a : b$ | M1 |
State one correct ratio | A1 |
Obtain a correct ratio of three of the unknowns, e.g. $a : b : c = 1 : 4 : 7$, or equivalent | A1 |
Either use coordinates of a relevant point and the found ratio to find the fourth unknown, e.g. $d$, or find the ratio $a : b : c : d$ | M1 |
Obtain answer $x + 4y + 7z = 19$, or equivalent | A1 |

**OR4:** Obtain a vector parallel to the plane and not parallel to $l$, e.g. $i - 2j + k$ | B1 |
Using a relevant point and second relevant vector, form a determinant equation for the plane | M1 |
State a correct equation, e.g. $\begin{vmatrix}x-2 & y+1 & z-3\\ 1 & -2 & 1 \\ 3 & 1 & -1\end{vmatrix} = 0$ | A1 |
Attempt to expand the determinant | M1 |
Obtain or imply two correct cofactors | A1 |
Obtain answer $x + 4y + 7z = 19$, or equivalent | A1 | [6]