The points A and B have position vectors given by OA = i − 2j + 2k and OB = 3i + j + k. The line l has equation r = 2i + j + mk + µ(i – 2j – 4k), where m is a constant.

Mark scheme

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

1. [M1] Carry out a correct method for finding a vector equation for AB
2. [A1] Obtain r = i - 2j + 2k + λ(2i + 3j - k), or equivalent
3. [M1] Equate two pairs of components of general points on AB and l and solve for λ or for µ
4. [A1] Obtain correct answer for λ or µ e.g. λ= 1/2 or µ = 1/2
5. [A1] Obtain m = 3
6. [B1] EITHER: Use scalar product to obtain an equation in a, b and c, e.g. a – 2b – 4c = 0. OR 1: Attempt to calculate the vector product of relevant vectors, e.g. (i – 2j – 4k) × (2i + 3j – k). OR 2: Using a relevant point and relevant vectors, form a 2-parameter equation for the plane. OR 3: Using a relevant point and relevant vectors, form a determinant equation for the plane.
7. [M1] Form a second relevant equation and solve for one ratio (if using scalar product method), OR obtain two correct components (if using vector product method), OR state a correct 2-parameter equation (if using 2-parameter method), OR state a correct determinant equation (if using determinant method).
8. [A1] Obtain final answer for ratio a:b:c = 14:-7:7 (if scalar product method), OR obtain correct vector (e.g., 14i-7j+7k) (if vector product method), OR state 3 correct equations in x,y,z,s,t (if 2-parameter method), OR attempt to expand determinant (if determinant method).
9. [M1] Use coordinates of a relevant point and values of a, b and c to find d (if scalar product or vector product method), OR eliminate s and t (if 2-parameter method), OR obtain or imply two correct cofactors (if determinant method).
10. [A1] Obtain answer 14x – 7y + 7z = 42, or equivalent