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4PM1/01R Jun 2023, Question 10: Vector Geometry

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1R · Calculator · 11 marks

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The question, in short

Vector geometry on triangle OAB with OA = 2a, OB = 4b, M midpoint of OA, Y on AB with AY:YB = 3:1, X on OB extended. Find vector expressions, a ratio OB:OX, and an area ratio.

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

Part (a) · 3 marks

Find simplified vector expressions in terms of a and b for (i) AB and (ii) MY.

  1. 1AB = −OA + OB = −2a + 4b
  2. 2MY = MA + (3/4)AB
  3. 3= a + (3/4)(−2a + 4b)
  4. 4= −(1/2)a + 3b

Answer(i) AB = −2a + 4b (ii) MY = −(1/2)a + 3b

Part (b) · 5 marks

Given M, Y and X are collinear, find the ratio OB : OX.

  1. 1OX = μ × 4b
  2. 2Also OX = OM + λMY = a + λ(−a/2 + 3b) = a(1 − λ/2) + 3λb
  3. 3Compare a: 1 − λ/2 = 0, so λ = 2
  4. 4Compare b: 4μ = 3λ = 6, so μ = 3/2
  5. 5OB : OX = 2 : 3

AnswerOB : OX = 2 : 3

Part (c) · 3 marks

Find the ratio of the area of triangle YBX to the area of triangle OAX.

  1. 1Triangles YBX and ABX share the height from X; YB = (1/4)AB, so ΔYBX = (1/4)ΔABX
  2. 2Triangles ABX and OAX share the height from A; BX = (1/3)OX, so ΔABX = (1/3)ΔOAX
  3. 3ΔYBX/ΔOAX = (1/4) × (1/3) = 1/12
  4. 4Ratio 1 : 12

Answer1 : 12

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