The diagram shows a solid cylinder standing on a horizontal circular base with centre O and radius 4 units. Points A, B and C lie on the circumference of the base such that AB is a diameter and angle BOC = 90°. Points P, Q and R lie on the upper surface of the cylinder vertically above A, B and C respectively. The height of the cylinder is 12 units. The mid-point of CR is M and N lies on BQ with BN = 4 units. Unit vectors i and j are parallel to OB and OC respectively and the unit vector k is vertically upwards. Evaluate PN.PM and hence find angle MPN.

1. (a)Evaluate PN ⋅ PM and hence find angle MPN.[7]

This question uses a diagram from the original paper, which isn't shown here. Open the full paper from the past papers archive to see it.