**(i)** State or imply a direction vector for $AB$ is $-\mathbf{i} + 2\mathbf{j} + 2\mathbf{k}$, or equivalent | B1 |

**Either:** State equation of $AB$ is $\mathbf{r} = 2\mathbf{i} + 2\mathbf{j} + \mathbf{k} + t(-\mathbf{i} + 2\mathbf{j} + 2\mathbf{k})$, or equivalent | B1√ |
Equate at least two pairs of components of $AB$ and $l$ and solve for $s$ or for $t$ | M1 |
Obtain correct answer for $s$ or for $t$, e.g. $s = 0$ or $t = -2; s = -\frac{3}{5}$ or $t = -\frac{1}{3}$ or $s = 5$ or $t = 3$ | A1 |
Verify that all three pairs of equations are not satisfied and that the lines fail to intersect | A1 | 5

**Or:** State a Cartesian equation for $AB$, e.g. $\frac{x-2}{-1} = \frac{y-2}{2} = \frac{z-1}{2}$, and for $l$, e.g. $\frac{x-4}{-1} = \frac{y+2}{2} = \frac{z-2}{2}$ | B1√ |
Solve a pair of equations, e.g. in $x$ and $y$, for one unknown | M1 |
Obtain one unknown, e.g. $x = 4$ or $y = -2$ | A1 |
Obtain corresponding remaining values, e.g. of $z$, and show lines do not intersect | A1 | 5

**Or:** Form a relevant triple scalar product, e.g. $(2\mathbf{i} - 4\mathbf{j} + \mathbf{k}) \cdot ((-\mathbf{i} + 2\mathbf{j} + 2\mathbf{k}) \times (\mathbf{i} + 2\mathbf{j} + \mathbf{k}))$ | B1√ |
Attempt to use correct method of evaluation | M1 |
Obtain at least two correct simplified terms of the three terms of the complete expansion of the triple product or of the corresponding determinant | A1 |
Obtain correct non-zero value, e.g. $-20$, and state that the lines do not intersect | A1 | 5

**(ii)** **Either:** Obtain a vector parallel to the plane and not parallel to $l$, e.g. $2\mathbf{i} - 4\mathbf{j} + \mathbf{k}$ | B1 |
Use scalar product to obtain an equation in $a, b$ and $c$, e.g. $a + 2b + c = 0$ | B1 |
Form a second relevant equation, e.g. $2a - 4b + c = 0$ and solve for one ratio, e.g. $a : b$ | M1 |
Obtain final answer $a : b : c = 6 : 1 : -8$ | A1 |
Use coordinates of a relevant point and values of $a, b$ and $c$ in general equation and find $d$ | M1 |
Obtain answer $6x + y - 8z = 6$, or equivalent | A1 | 6

**Or:** Obtain a vector parallel to the plane and not parallel to $l$, e.g. $2\mathbf{i} - 4\mathbf{j} + \mathbf{k}$ | B1 |
Obtain a second relevant vector parallel to the plane and attempt to calculate their vector product, e.g. $(\mathbf{i} + 2\mathbf{j} + \mathbf{k}) \times (2\mathbf{i} - 4\mathbf{j} + \mathbf{k})$ | M1 |
Obtain two correct components of the product | A1 |
Obtain correct answer, e.g. $6\mathbf{i} + \mathbf{j} - 8\mathbf{k}$ | A1 |
Substitute coordinates of a relevant point in $6x + y - 8z = d$, or equivalent, to find $d$ | M1 |
Obtain answer $6x + y - 8z = 6$, or equivalent | A1 | 6

**Or:** Obtain a vector parallel to the plane and not parallel to $l$, e.g. $2\mathbf{i} - 4\mathbf{j} + \mathbf{k}$ | B1 |
Obtain a second relevant vector parallel to the plane and correctly form a 2-parameter equation for the plane, e.g. $\mathbf{r} = 2\mathbf{i} + 2\mathbf{j} + \mathbf{k} + \lambda(2\mathbf{i} - 4\mathbf{j} + \mathbf{k}) + \mu(\mathbf{i} + 2\mathbf{j} + \mathbf{k})$ | M1 |
State 3 correct equations in $x, y, z, \lambda$ and $\mu$ | A1 |
Eliminate $\lambda$ and $\mu$ | M1 |
Obtain equation in any correct form | A1 |
Obtain answer $6x + y - 8z = 6$, or equivalent | A1 | 6