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4024/12 Oct/Nov 2023, Question 24: Expressing Vectors in Terms of a and b

Cambridge O Level Mathematics 4024 · Paper 12 (Non-calculator) · Non-calculator · 5 marks

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The question, in short

Triangle OAB with P on AB dividing it 2:3; given vectors OA = 4a and OP = 3a + 2b, find AP, OB, and QP where Q on OA makes QP parallel to OB.

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

Part (ai) · 1 mark

Find vector AP in terms of a and b, in simplest form.

  1. 1→AP = →AO + →OP = −4a + 3a + 2b
  2. 2= 2b − a

Answer2b − a

Part (aii) · 3 marks

Find vector OB in terms of a and b, in simplest form.

  1. 1AP : PB = 2 : 3 so →AB = 5/2 →AP = 5/2(2b − a) = 5b − 5/2 a
  2. 2→OB = →OA + →AB = 4a + 5b − 5/2 a
  3. 3= 3/2 a + 5b

Answer3/2 a + 5b or (3a + 10b)/2

Part (b) · 1 mark

Q lies on OA with QP parallel to OB; find vector QP.

  1. 1QP ∥ OB so triangle AQP is similar to triangle AOB with scale factor AP/AB = 2/5
  2. 2→QP = 2/5 →OB = 2/5(3/2 a + 5b)
  3. 3= 3/5 a + 2b

Answer3/5 a + 2b

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Also searched as: 4024/12/O/N/23 · 4024_w23_qp_12 · 4024 w23 12 · question 24.
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