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4MA1/2HR Jun 2022, 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

Parallelogram ABCD has AB→ = a, BC→ = b, DM→ = 1/2 b, with K on AB and L on CD. Given KLM straight and λ:μ = 1:2, use vectors to find λ and μ.

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

Solution · 5 marks

  1. 1AK = λa, so KB = (1 − λ)a
  2. 2CD = −a, and CL = μCD = -μa
  3. 3DL = DC + CL = a − μa = (1 − μ)a, so LD = (μ − 1)a
  4. 4DM = (1/2)b
  5. 5KM = KB + BC + CD + DM = (1 − λ)a + b − a + (1/2)b
  6. 6KM = -λa + (3/2)b
  7. 7LM = LD + DM = (μ − 1)a + (1/2)b
  8. 8KLM is a straight line, so KM = x LM for some scalar x
  9. 9-λa + (3/2)b = x[(μ − 1)a + (1/2)b]
  10. 10Compare the b terms: 3/2 = x × 1/2, so x = 3
  11. 11Compare the a terms: -λ = 3(μ − 1), so λ + 3μ = 3
  12. 12λ : μ = 1 : 2, so μ = 2λ
  13. 13λ + 3(2λ) = 3, so 7λ = 3
  14. 14λ = 3/7 and μ = 2 × 3/7 = 6/7

Answerλ = 3/7, μ = 6/7

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