Mathematics (AS & A Level) · Oct / Nov 2018 · Paper 12 · Question 7

O A B C P Q R 12 4 4 M N i k j 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 = 90Å. 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. [7] 11

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