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4MA1/2HR Jun 2020, Question 25: Vector Geometry

Pearson Edexcel International GCSE Mathematics A 4MA1 · Paper 2HR (Higher) · Calculator · 5 marks

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The question, in short

Using vectors with OA→ = 2a and OB→ = 2b, given OA : AN = 1 : 4 and OM : MB = 1 : 1, find the ratio AP : PB in simplest form.

Summarised in our own words. Use the official paper for the exact wording and the diagram.

Solution · 5 marks

  1. 1OA : AN = 1 : 4, so ON = 5 × OA = 10a
  2. 2OM : MB = 1 : 1, so OM = ½ × OB = b
  3. 3AB = OB − OA = 2b − 2a
  4. 4MN = ON − OM = 10a − b
  5. 5P is on AB, so let AP = k(2b − 2a), giving OP = 2a + k(2b − 2a)
  6. 6P is on MN, so let MP = λ(10a − b), giving OP = b + λ(10a − b)
  7. 72a + k(2b − 2a) = b + λ(10a − b)
  8. 8Comparing a: 2 − 2k = 10λ
  9. 9Comparing b: 2k = 1 − λ, so λ = 1 − 2k
  10. 102 − 2k = 10(1 − 2k)
  11. 112 − 2k = 10 − 20k, so 18k = 8 and k = 4/9
  12. 12AP = 4/9 of AB, so PB = 5/9 of AB
  13. 13AP : PB = 4/9 : 5/9 = 4 : 5

Answer4 : 5

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