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4024/22 Oct/Nov 2021, Question 7: Expressing Vectors in Terms of a and b

Cambridge O Level Mathematics 4024 · Paper 22 (Calculator) · Calculator · 8 marks

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The question, in short

Given points P(−5, 2) and Q(3, 7) with vector QR = (−4, 6), find midpoint, point R and length QR; then use quadrilateral OACB with vectors a and b to find AB and NC.

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

Part (ai) · 1 mark

Find the coordinates of the midpoint of the line segment PQ.

  1. 1Midpoint = ((−5 + 3)/2, (2 + 7)/2) = (−1, 4.5)

Answer(−1, 4.5)

Part (aii) · 1 mark

Find the coordinates of point R using the given vector from Q to R.

  1. 1R = (3 + (−4), 7 + 6) = (−1, 13)

Answer(−1, 13)

Part (aiii) · 2 marks

Calculate the length of the vector from Q to R.

  1. 1|QR| = √((−4)² + 6²) = √52
  2. 2= 7.211… ≈ 7.21

Answer7.21 or 7.211…

Part (bi) · 1 mark

Express the vector from A to B in terms of a and b.

  1. 1AB = AO + OB = −a + b = b − a

Answerb − a

Part (bii) · 3 marks

Using BN : NA = 1 : 3 and BC parallel to OA, express vector NC in terms of a and b in simplest form.

  1. 1BC = 1/2 a
  2. 2NB = 1/4 AB = 1/4(b − a)
  3. 3NC = NB + BC = 1/4 b − 1/4 a + 1/2 a
  4. 4= 1/4 a + 1/4 b

Answer(1/4)a + (1/4)b or (1/4)(a + b)

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Also searched as: 4024/22/O/N/21 · 4024_w21_qp_22 · 4024 w21 22 · question 7.
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