Two lines have equations r = (-1 3 -4) + λ(2 3 -1) and r = (2 -3 -1) + μ(-1 -2 1).

Mark scheme

Marking points from the official mark scheme (Feb/Mar 2025). The mark or mark type is shown in brackets.

1. [B1] Express general point of a line in component form, e.g. (−1 + 2λ, 3 + 3λ, −4 −λ) or (2 – μ, −3 – 2μ, −1 + μ)
2. [M1] Equate at least two pairs of components and solve for λ or for μ
3. [A1] Obtain correct answer for λ or for μ Possible answers are 6, 12, 0 for λ and -9, -21, -3 for μ.
4. [A1] Verify that one component equation is not satisfied Can show by correctly obtaining 2 values of λ or 2 values of μ E.g. show 21 ≠ 15 for (11, 21, –10) and (11, 15, -10), or show -16 ≠ –22 for (23, 39, -16) and (23, 39, -22), or show -1 ≠ 5 for (−1, 3, −4) and (5, 3, -4).
5. [B1] Show that the lines are not parallel E.g. (2 3 -1) and (-1 -2 1) at least 2 components required. Just a statement that direction vectors are not scalar multiple of each other insufficient, if direction vectors have not been clearly identified. Also, told answer is skew.
6. [M1] Carry out correct process for evaluating the scalar product of (2 3 -1) and (-1 -2 1) E.g. (2 x −1) + (3 × −2) + (−1 × 1) or –2 – 6 – 1 or -9.
7. [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 Allow for any pair of vectors here but must be consistent between scalar product and magnitudes.
8. [A1] Obtain answer AWRT 169.1° or 2.95ᶜ Allow 169°.