In the diagram, OABCD is a pyramid with vertex D. The horizontal base OABC is a square of side 4 units. The edge OD is vertical and OD = 4 units. The unit vectors i, j and k are parallel to OA, OC and OD respectively. The midpoint of AB is M and the point N on CD is such that DN = 3NC.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2021). The mark or mark type is shown in brackets.

1. [B1] State OM = 4i + 2j
2. [M1] Use a correct method to find ON
3. [A1] Obtain answer 3j + k
4. [M1] Use a correct method to find a line equation for MN
5. [A1] Obtain answer r = 3j + k + λ(4i – j – k), or equivalent
6. [M1] Taking a general point P on MN, form an equation in λ by either equating a relevant scalar product to zero or equating the derivative of OP² to zero or using Pythagoras in triangle OPM or OPN
7. [A1 OF] Obtain λ = 2/9
8. [M1] Use correct method to find OP
9. [A1] Obtain the given answer correctly
10. [M1] Use a scalar product to find the projection of OM (or ON) on MN
11. [A1] Obtain answer 14/√18 or 4/√18
12. [M1] Use Pythagoras to obtain the perpendicular
13. [A1] Obtain the given answer correctly