**(i)**

**EITHER:**

Express general point of $l$ or $m$ in component form, e.g. $(2 + \lambda, -\lambda, 1 + 2\lambda)$ or $(\mu, 2 + 2\mu, 6 - 2\mu)$ | B1 |
Equate at least two pairs of components and solve for $\lambda$ or for $\mu$ | M1 |
Obtain correct answer for $\lambda$ or $\mu$ (possible answers for $\lambda$ are $-2$, $\frac{1}{4}$, $7$ and for $\mu$ are $0, 2\frac{1}{4}, -4\frac{1}{2}$) | A1 |
Verify that all three component equations are not satisfied | A1 |

**OR:**

State a relevant scalar triple product, e.g. $(2\mathbf{i} - 2\mathbf{j} - 5\mathbf{k}) \cdot ((\mathbf{i} + \mathbf{j} + 2\mathbf{k}) \times (\mathbf{i} + 2\mathbf{j} - 2\mathbf{k}))$ | B1 |
Attempt to use the correct method of evaluation | M1 |
Obtain at least two correct simplified terms of the expansion of the triple product or of the corresponding determinant, e.g. $-4$, $-8$, $-15$ | A1 |
Obtain correct non-zero value, e.g. $-27$, and state that the lines do not intersect | A1 | [4]

**(ii)**

Carry out the correct process for evaluating scalar product of direction vectors for $l$ and $m$ | M1 |
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 | M1 |
Obtain answer $47.1°$ or 0.822 radians | A1 | [3]

**(iii)**

**EITHER:**

Use scalar product to obtain $a - b + 2c = 0$ | B1 |
Obtain $a + 2b - 2c = 0$, or equivalent, from a scalar product, or by subtracting two point equations obtained from points on $m$, and solve for one ratio, e.g. $a : b$ | M1* |
Obtain $a : b : c = -2 : 4 : 3$, or equivalent | A1 |
Substitute coordinates of a point on $m$ and values for $a$, $b$ and $c$ in general equation and evaluate $d$ | M1(dep*) |
Obtain answer $-2x + 4y + 3z = 26$, or equivalent | A1 |

**OR1:**

Attempt to calculate vector product of direction vectors of $l$ and $m$ | M1* |
Obtain two correct component | A1 |
Obtain $-2\mathbf{i} + 4\mathbf{j} + 3\mathbf{k}$, or equivalent | A1 |
Form a plane equation and use coordinates of a relevant point to evaluate $d$ | M1(dep*) |
Obtain answer $-2x + 4y + 3z = 26$, or equivalent | A1 |

**OR2:**

Form a two-parameter plane equation using relevant vectors | M1* |
State a correct equation e.g. $\mathbf{r} = 2\mathbf{j} + 6\mathbf{k} + s(\mathbf{i} - \mathbf{j} + 2\mathbf{k}) + t(\mathbf{i} + 2\mathbf{j} - 2\mathbf{k})$ | A1 |
State three correct equations in $x$, $y$, $z$, $s$ and $t$ | A1 |
Eliminate $s$ and $t$ | M1(dep*) |
Obtain answer $-2x + 4y + 3z = 26$, or equivalent | A1 | [5]