With respect to the origin O, the points A, B, C and D have position vectors given by OA = (1 5 3), OB = (0 4 1), OC = (1 -3 1) and OD = (3 -5 4). The line m passes through the points A and B.

Mark scheme

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

1. [M1] Carry out a correct method for finding a vector equation for m
2. [A1] Obtain r = (1 5 3) + λ( -1 -1 -2 )
3. [B1] Express general point of the second line in component form, i.e. (1 + 2μ, −3 – 2μ, 1 + 3μ)
4. [M1] Equate at least two pairs of corresponding components and solve for λ or for μ
5. [A1] Obtain λ = −4 or μ = −2
6. [A1] Obtain position vector of point of intersection is (-3 1 -5)
7. [B1] Find CP for a general point P on m, e.g. (0 8+λ 2+2λ)
8. [M1] Calculate scalar product of CP and a direction vector for m and equate the result to zero
9. [A1] Obtain λ = -2
10. [A1] Obtain answer (-1 3 -1)