O-LevelAdditional MathematicsVectors in two dimensions4037/12 · May/June 2020 · Q810 Marks

The diagram shows a triangle OAB such that OA = a and OB = b. The point P lies on OA such that OP = 3/4 OA. The point Q is the mid-point of AB. The lines OB and PQ are extended to meet at the point R. Find, in terms of a and b,

(a)AB,[1]
(b)PQ. Give your answer in its simplest form.[3]
(c)It is given that nPQ = QR and BR = kb, where n and k are positive constants. Find QR in terms of n, a and b.[1]
(d)Find QR in terms of k, a and b.[2]
(e)Hence find the value of n and of k.[3]

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Mark scheme

Marking points from the official mark scheme (May/June 2020). The mark or mark type is shown in brackets.

[1] b-a
[1] For 1/4 a or -3/4 a
[1] For 1/2 (b-a) or 1/2 (a+b)
[1] Correct and simplified
[1] FT on their answer to (b)
[1] For use of their (a) and kb
[1]
[1] For equating their (c) and (d) and then equating like vectors to obtain 2 equations
[1] n=2
[1] k=1/2

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Topic: Vectors in two dimensions, O-Level Additional Mathematics (syllabus 4037). Structured question, 10 marks.
Source: O-Level Additional Mathematics, May/June 2020 Paper 1 Variant 2, question 8.

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