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4PM1/02 Jan 2020, Question 4: Vector Proofs

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 2 · Calculator · 7 marks

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The question, in short

Quadrilateral ABCD has AB = 7i + pj, AC = 11i − pj and AD = 4i − 2pj; show it is a parallelogram, then use |BD| = 3√10 to find p and a unit vector parallel to BD.

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

Part (a) · 3 marks

Show that ABCD is a parallelogram for every value of p.

  1. 1DC = AC − AD = (11i − pj) − (4i − 2pj) = 7i + pj
  2. 2So DC = AB for all p
  3. 3AB and DC are parallel and equal in length, so ABCD is a parallelogram

AnswerDC = 7i + pj = AB (and BC = 4i − 2pj = AD), so opposite sides are parallel and equal: ABCD is a parallelogram

Part (b) · 3 marks

Given |BD| = 3√10, find the possible values of p.

  1. 1BD = AD − AB = (4i − 2pj) − (7i + pj) = −3i − 3pj
  2. 2|BD|² = 9 + 9p² = (3√10)² = 90
  3. 3p² = 9
  4. 4p = ±3

Answerp = 3, p = −3

Part (c) · 1 mark

Given p > 0, find a unit vector parallel to BD.

  1. 1p = 3: BD = −3i − 9j
  2. 2|BD| = √90 = 3√10
  3. 3Unit vector = (−3i − 9j)/(3√10) = −(i + 3j)/√10

Answer±(−3i − 9j)/(3√10), i.e. ±(i + 3j)/√10

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