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4024/12 Oct/Nov 2024, Question 26: Vector Proofs

Cambridge O Level Mathematics 4024 · Paper 12 (Non-calculator) · Non-calculator · 4 marks

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The question, in short

Using vectors a and b for OA and OB with AB:BC = 1:3 and OBD straight, find AB, show OC, and determine k where AD = kOC.

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

Part (a) · 1 mark

Write vector AB using a and b.

  1. 1→AB = →AO + →OB = −a + 2b = 2b − a

Answer2b − a

Part (b) · 2 marks

Prove that vector OC equals 8b − 3a.

  1. 1AB : BC = 1 : 3, so →AC = 4→AB = 4(2b − a)
  2. 2→OC = →OA + →AC = a + 4(2b − a)
  3. 3= a + 8b − 4a = 8b − 3a

Answer→OC = a + 4(2b − a) or →OC = 2b + 3(2b − a), with correct working leading to →OC = 8b − 3a

Part (c) · 1 mark

Given AD = kOC, work out the value of k.

  1. 1D is on OB extended, so →OD is a multiple of b and →AD = →OD − a
  2. 2→AD = k(8b − 3a) = 8kb − 3ka, so the a part gives −3k = −1
  3. 3k = 1/3

Answer1/3

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Also searched as: 4024/12/O/N/24 · 4024_w24_qp_12 · 4024 w24 12 · question 26.
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