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4PM1/01 Jan 2020, Question 10: Vector Geometry

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1 · Calculator · 13 marks

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The question, in short

Parallelogram OABC has OA = a, OC = c, with M and N midpoints of AB and BC; line OB meets MN at X. Find vectors, parameters and an area ratio.

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

Part (a) · 2 marks

Find OB and MN as expressions in a and c.

  1. 1OB = OA + AB = a + c
  2. 2MN = MB + BN = ½c − ½a = ½(c − a)

AnswerOB = a + c; MN = ½(c − a)

Part (b) · 8 marks

Using MX = λMN and OX = μOB, apply a vector method to determine λ and μ.

  1. 1OX = OA + AM + λMN = a + ½c + ½λ(c − a)
  2. 2OX = μ(a + c)
  3. 3Equate a: 1 − λ/2 = μ
  4. 4Equate c: ½ + λ/2 = μ
  5. 5So 1 − λ/2 = ½ + λ/2, giving λ = 1/2
  6. 6μ = ½ + ¼ = 3/4

Answerλ = 1/2, μ = 3/4

Part (c) · 3 marks

Hence find the simplest form of the ratio of the area of quadrilateral OXNC to that of parallelogram OABC.

  1. 1XB = ¼OB and BN = ½BC, so area XBN = ¼ × ½ × area OBC = ⅛ area OBC
  2. 2Area OBC = ½ area OABC, so XBN = 1/16 of the parallelogram
  3. 3OXNC = OBC − XBN = 8/16 − 1/16 = 7/16 of the parallelogram
  4. 4Ratio 7 : 16

Answer7 : 16

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