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4PM1/02 Jun 2024, Question 10: Vector Geometry

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 2 · Calculator · 11 marks

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The question, in short

Quadrilateral ABCD has AB = 3a + 4b, AC = 7a + 9b and AD = 4a + 5b. Prove ABCD is a parallelogram, then with BC extended to E and F on CD with CF : FD = 1 : 2, find vector AE as Xa + Yb.

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Part (a) · 3 marks

Use the given vectors AB, AC and AD to show that ABCD is a parallelogram.

  1. 1BC = −AB + AC = 4a + 5b = AD
  2. 2DC = −AD + AC = 3a + 4b = AB
  3. 3AB ∥ DC and BC ∥ AD, so ABCD is a parallelogram

AnswerBC = AD = 4a + 5b and DC = AB = 3a + 4b, so opposite sides are parallel and ABCD is a parallelogram

Part (b) · 8 marks

With BCE a straight line and CF : FD = 1 : 2, and A, F, E collinear, express AE as Xa + Yb with rational X and Y.

  1. 1CF = −(1/3)(3a + 4b)
  2. 2AF = AC + CF = 6a + (23/3)b
  3. 3AE = μAF = μ(6a + (23/3)b)
  4. 4AE = AB + λBC = (3 + 4λ)a + (4 + 5λ)b
  5. 56μ = 3 + 4λ and (23/3)μ = 4 + 5λ give λ = 3/2, μ = 3/2
  6. 6AE = 9a + (23/2)b

AnswerAE = 9a + (23/2)b

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