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0580/21 May/Jun 2025, Question 21: Vector Proofs

Cambridge IGCSE Mathematics 0580 · Paper 21 (Extended) · Non-calculator · 5 marks

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The question, in short

Rhombus OABC has centre P with OP = 2m and AP = n; express two side vectors in m and n, then use AD = 10m − 3n to prove OADC is a trapezium.

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

Part (ai) · 1 mark

Write vector OA using m and n in simplest form.

  1. 1→OA = →OP + →PA = 2m − n

Answer2m − n

Part (aii) · 1 mark

Write vector OC using m and n in simplest form.

  1. 1Diagonals of a rhombus bisect each other, so →PC = →AP = n
  2. 2→OC = →OP + →PC = 2m + n

Answer2m + n

Part (b) · 3 marks

Given AD = 10m − 3n, show that quadrilateral OADC has one pair of parallel sides.

  1. 1→CD = →CO + →OA + →AD = −(2m + n) + (2m − n) + 10m − 3n
  2. 2→CD = 10m − 5n = 5(2m − n) = 5→OA
  3. 3So CD is parallel to OA (and 5 times as long), therefore OADC is a trapezium

Answer→CD = 10m − 5n = 5→OA, so CD is parallel to OA, therefore OADC is a trapezium

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Also searched as: 0580/21/M/J/25 · 0580_s25_qp_21 · 0580 s25 21 · question 21.
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