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4PM1/01R Jun 2024, Question 8: Vector Geometry

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1R · Calculator · 10 marks

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The question, in short

Trapezium ABCD has BC = 3a parallel to AD = 5a and CD = 2b; find vector AB, the fraction k with BX = k BD, and the area ratio of triangle CXD to the trapezium.

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

Part (a) · 1 mark

Express side AB as a simplified vector using a and b.

  1. 1AB = AD + DC + CB
  2. 2= 5a − 2b − 3a = 2a − 2b

AnswerAB = 2a − 2b

Part (b) · 5 marks

Given diagonals BD and AC meet at X with BX = k BD, use vectors to determine k.

  1. 1BD = BC + CD = 3a + 2b, so BX = k(3a + 2b)
  2. 2AC = AD + DC = 5a − 2b
  3. 3BX = BA + λAC = −2a + 2b + λ(5a − 2b)
  4. 4a: 3k = 5λ − 2; b: 2k = 2 − 2λ
  5. 5k = 3/8 (λ = 5/8)

Answerk = 3/8

Part (c) · 4 marks

Determine the ratio between the area of triangle CXD and the area of trapezium ABCD.

  1. 1Triangle CXD = (5/8) × triangle BCD (XD = (5/8)BD)
  2. 2BC : AD = 3 : 5 with the same height, so triangle BCD = (3/8) × ABCD
  3. 3CXD/ABCD = (5/8) × (3/8) = 15/64
  4. 4Ratio 15 : 64

Answer15 : 64

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