The lines l and m have vector equations r = −i + 3j + 4k + λ(2i – j – k) and r = 5i + 4j + 3k + µ(ai + bj + k) respectively, where a and b are constants.

Mark scheme

Marking points from the official mark scheme (May/June 2022). The mark or mark type is shown in brackets.

1. B1 Express general point of l or m in component form, i.e. (−1+2λ, 3−λ, 4-λ) or (5+aμ, 4+bμ, 3+μ)
2. M1 Equate components and eliminate either λ or μ e.g. μ = (2λ-6)/(a), λ = (1-b)/μ, μ = (λ-4)/1, λ = (a+6)/(2+a)
3. M1 Eliminate the other parameter or obtain a second expression in the first λ and μ are not required to be the subject of the equations.
4. A1 Show intermediate steps to obtain 2b - a = 4 AG
5. *M1 Using the correct process equate the scalar product of the direction vectors to zero (2i-j-k).(ai+bj+k) = 0 SOI.
6. A1 Obtain 2a-b-1=0 OE e.g. 2(2b-4)-b-1=0
7. DM1 Solve simultaneous equations for a or for b
8. A1 Obtain a = 2, b = 3
9. M1 Substitute found values in component equations and solve for λ or for μ
10. A1 Obtain answer 3i + j + 2k from either λ = 2 or μ = -1 Accept as coordinates or equivalent.