a) EITHER: Express general point of $l$ or $m$ in component form, e.g. $\left( {2 + \lambda ,{\text{ }} - \lambda ,{\text{ }}1 + 2\lambda } \right)$ or $\left( {\mu ,{\text{ }}2 + 2\mu ,{\text{ }}6 - 2\mu } \right)$

Equate at least two pairs of components and solve for $\lambda $ or for $\mu $

Obtain correct answer for $\lambda $ or $\mu $ (possible answers for $\lambda $ are ${ - 2}$, $\frac{1}{4}$, $7$ and for $\mu $ are $0$, $2\frac{1}{4}$, $ - 4\frac{1}{2}$)

Verify that all three component equations are not satisfied

OR: State a relevant scalar triple product, e.g.

$\left( {2i - 2j - 5k} \right).\,\left( {\left( {i - j + 2k} \right) \times \left( {i + 2j - 2k} \right)} \right){\text{ }}$

Attempt to use the correct method of evaluation

Obtain at least two correct simplified terms of the three terms of the expansion of the triple product or of the corresponding determinant, e.g. $ - 4,{\text{ }} - 8,{\text{ }} - 15$

Obtain correct non-zero value, e.g. $ - 27$, and state that the lines do not intersect

b) Carry out the correct process for evaluating scalar product of direction vectors for $l$ and $m$

Using the correct process for the moduli, divide the scalar product by the product of the moduli and evaluate the inverse cosine of the result

Obtain answer ${47.1^ \circ }$ or $0.822$ radians

c) EITHER: Use scalar product to obtain $\alpha  - b + 2c = 0$

Obtain $\alpha  + 2b - 2c = 0$, or equivalent, from a scalar product, or by ubtracting two point equations obtained from points on $m$, and solve for one ratio, e.g. $\alpha {\text{ }}:{\text{ }}b$

Obtain $\alpha :b:c =  - 2:4:3$, or equivalent

Substitute coordinates of a point on $m$ and values for $\alpha $, $b$ and $c$ in general equation and evaluate $d$

OR1: Attempt to calculate vector product of direction vectors of $l$ and $m$ 

Obtain two correct components

Obtain $ - 2i + 4j + 3k$, or equivalent

Form a plane equation and use coordinates of a relevant point to evaluate $d$

Obtain answer $ - 2x + 4y + 3z = 26$, or equivalent

OR2: Form a two-parameter plane equation using relevant vectors

State a correct equation e.g. $r = 2j + 6k + s\left( {i - j + 2k} \right) + t\left( {i + 2j - 2k} \right)$

State three correct equations in $x,{\text{ }}y,{\text{ }}z,{\text{ }}s$ and $t$

Eliminate $s$ and $t$

Obtain answer $ - 2x + 4y + 3z = 26$, or equivalent