O-LevelAdditional MathematicsVectors in two dimensions4037/21 · May/June 2024 · Q129 Marks

The diagram shows a triangle OBC. OA : OB = 4 : 7 and OD : OC = 4 : 7. OB = b and OC = c. The point P is the point of intersection of AC and BD such that AP = λAC and BP = μBD where λ and μ are scalars.

(a)Find two expressions for OP, each in terms of b, c and a scalar, and hence show that P divides both AC and DB in the ratio 4: 7.[7]
(b)The point Q is such that OQ = 2/7 b + 2/7 c. Use a vector method to show that O, Q and P are collinear. Justify your answer.[2]

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Mark scheme

Marking points from the official mark scheme (May/June 2024). The mark or mark type is shown in brackets.

[B3] for OP = b + λ(c – (4/7)b) and OP = (4/7)c + μ(b – (4/7)c) or OP = (1 – 4λ/7)b + λc and OP = μb + (4/7)(1 – μ)c oe
[M1] Equates components e.g.: (1 – 4λ/7) = μ and λ = (4/7)(1 – μ) (FT providing at least B1 awarded and two expressions for OP in terms of b, c, λ and μ found)
[A1] λ = 4/11
[A1] μ = 7/11
[A1] and conclusion AP : AC = 4 : 11 therefore AP : PC = 4 : 7 BP : BD = 7 : 11 therefore DP : PB = 4 : 7 oe
[M1] for OP = (4/11)(b+c) or OP = (4/11)b + (4/11)c
[A1] OP and OQ are scalar multiples of each other and have a point in common oe

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