The equations of two lines l and m are r = 3i − j − 2k + λ(−i + j + 4k) and r = 4i + 4j – 3k + μ(2i + j − 2k) respectively.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2017). The mark or mark type is shown in brackets.

1. [M1] (i) Equate at least two pairs of components of general points on l and m and solve for λ or for μ
2. [A1] (i) Obtain correct answer for λ or μ, e.g. λ = 3 or μ = -2; λ = 0 or μ = -½; or λ = ⅔ or μ = -3/2
3. [A1] (i) Verify that not all three pairs of equations are satisfied and that the lines fail to intersect
4. [*M1] (ii) Carry out correct process for evaluating scalar product of direction vectors for l and m
5. [DM1] (ii) 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
6. [A1] (ii) Obtain answer 45° or ¼π (0.785) radians
7. [(M1)] (iii) Attempt to calculate vector product of relevant vectors, e.g (-i+j+4k)×(2i + j - 2k)
8. [A1] (iii) Obtain two correct components
9. [A1] (iii) Obtain correct answer, e.g. -6i+6j-3k
10. [M1] (iii) Substitute (3, -2, -1) in -6x+6y - 3z = d, or equivalent, and find d
11. [A1)] (iii) Obtain answer -2x + 2y - z = −9, or equivalent