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4PM1/02R Jun 2019, Question 11: Roots α and β of a Quadratic

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 2R · Calculator · 12 marks

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The question, in short

Quadratic x² − px + q = 0 with p > 0 has roots α, β; given 2αβ = 3 and 4(α² + β²) = k² − 6k − 3, k > 3, find q, p in terms of k, then k, then an integer-coefficient equation.

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

Part (a) · 5 marks

(i) State q using the product of roots. (ii) Express p in terms of k using the sum of roots and the given identity.

  1. 1q = αβ = 3/2
  2. 2α² + β² = (k² − 6k − 3)/4
  3. 3(α + β)² = α² + β² + 2αβ = (k² − 6k − 3)/4 + 3 = (k − 3)²/4
  4. 4p = α + β = (k − 3)/2 (positive root since p > 0, k > 3)

Answerq = 3/2; p = (k − 3)/2

Part (b) · 2 marks

Using 7αβ = 3(α + β) together with earlier results, determine the value of k.

  1. 17 × 3/2 = 3 × (k − 3)/2
  2. 27 = k − 3
  3. 3k = 10

Answerk = 10

Part (c) · 5 marks

Hence form a quadratic with integer coefficients whose roots are α/(α + β) and β/(α + β).

  1. 1α + β = (10 − 3)/2 = 7/2
  2. 2Sum of new roots = (α + β)/(α + β) = 1
  3. 3Product = αβ/(α + β)² = (3/2)/(7/2)² = 6/49
  4. 4x² − x + 6/49 = 0
  5. 549x² − 49x + 6 = 0

Answer49x² − 49x + 6 = 0

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