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4024/11 Oct/Nov 2025, Question 26: Vector Geometry

Cambridge O Level Mathematics 4024 · Paper 11 (Non-calculator) · Non-calculator · 7 marks

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The question, in short

In triangle OAB with vectors OA = a and OB = 4b, point T divides AB in ratio 3:2; work out vector expressions and a length ratio involving point Q on OB.

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

Part (a) · 1 mark

Express the vector from A to B using a and b.

  1. 1→AB = →AO + →OB = −a + 4b = 4b − a

Answer4b − a

Part (b) · 3 marks

Using the given ratio on AB, prove that T's position vector equals 2/5(a + 6b).

  1. 1→AT = 3/5 →AB = 3/5(4b − a)
  2. 2→OT = a + 3/5(4b − a) = a + 12/5 b − 3/5 a
  3. 3= 2/5 a + 12/5 b = 2/5(a + 6b)

Answera + 3/5(4b − a) or 4b − 2/5(4b − a), 2/5 a + 12/5 b leading to 2/5(a + 6b)

Part (c) · 3 marks

Given QT = 1/5(2a − 3b) with Q on OB, determine the ratio OQ : QB.

  1. 1→OQ = →OT + →TQ = 2/5 a + 12/5 b − 1/5(2a − 3b)
  2. 2= 2/5 a + 12/5 b − 2/5 a + 3/5 b = 3b
  3. 3→QB = 4b − 3b = b
  4. 4OQ : QB = 3 : 1

Answer3 : 1

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