Lemmara logoLemmaraWorked solution

4024/11 May/Jun 2023, Question 27: Expressing Vectors in Terms of a and b

Cambridge O Level Mathematics 4024 · Paper 11 (Non-calculator) · Non-calculator · 6 marks

اقرأ هذا الحل مشروحاً بالعربي ←

The question, in short

Parallelogram OABC has OA = 2a, OC = 3c; M is midpoint of BC and T divides OB with OT:TB = 2:1. Find vectors and show A, T, M are collinear.

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

Part (a) · 1 mark

Find vector OB in terms of a and c.

  1. 1→OB = →OA + →AB = 2a + 3c

Answer2a + 3c

Part (bi) · 1 mark

Express vector AM in terms of a and c in simplest form.

  1. 1→AM = →AB + →BM = 3c + (1/2)(−2a)
  2. 2= 3c − a

Answer3c − a

Part (bii) · 2 marks

Express vector AT in terms of a and c in simplest form.

  1. 1→OT = 2/3 × (2a + 3c) = (4/3)a + 2c
  2. 2→AT = −2a + (4/3)a + 2c
  3. 3= 2c − (2/3)a

Answer2c − 2/3 a or (6c − 2a)/3

Part (c) · 2 marks

Show that A, T and M lie on a straight line.

  1. 1(3/2) × →AT = (3/2)(2c − (2/3)a) = 3c − a = →AM
  2. 2So →AM = (3/2)→AT: AM is parallel to AT
  3. 3AM and AT both pass through A, so ATM is a straight line

Answer→AM = 3/2 →AT and AM is parallel to AT and both go through A so ATM is a straight line OR →TM = c − 1/3 a and →AT = 2→TM and AT is parallel to TM and both go through T so ATM is a straight line

Stuck on a step? Read the free lesson: Expressing Vectors in Terms of a and b

Practise more expressing vectors in terms of a and b questions

Real 4024 past-paper questions, marked on your working like the exam, with a worked solution for every part.

Practise free on Lemmara

More past-paper questions on expressing vectors in terms of a and b

Also searched as: 4024/11/M/J/23 · 4024_s23_qp_11 · 4024 s23 11 · question 27.
Lemmara is an independent study tool, not affiliated with or endorsed by Cambridge International Education. The worked solutions are Lemmara’s own.