O-LevelAdditional MathematicsVectors in two dimensions4037/12 · Oct/Nov 2014 · Q129 Marks
The position vectors of points A and B relative to an origin O are a and b respectively. The point P is such that OP = μOA. The point Q is such that OQ = λOB. The lines AQ and BP intersect at the point R. [Figure X.X]
- (i)Express AQ in terms of λ, a and b.[1]
- (ii)Express BP in terms of μ, a and b.[1]
- (iii)It is given that 3AR = AQ and 8BR = 7BP. Express OR in terms of λ, a and b.[2]
- (iv)Express OR in terms of μ, a and b.[2]
- (v)Hence find the value of μ and of λ.[3]
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Mark scheme
Marking points from the official mark scheme (Oct/Nov 2014). The mark or mark type is shown in brackets.
- [1] B1 AQ = λb - a
- [1] B1 BP = μa - b
- [1] M1 for a + 1/3 their (i)
- [1] A1 Allow unsimplified OR = 2/3 a + 1/3 λb
- [1] M1 for b + 7/8 their (ii)
- [1] A1 Allow unsimplified OR = 7/8 μa + 1/8 b
- [1] M1 for equating (iii) and (iv) and then equating like vectors
- [1] A1 μ = 16/21 Allow 0.762
- [1] A1 λ = 3/8 Allow 0.375
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