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4PM1/01R Jun 2020, Question 8: Roots α and β of a Quadratic

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1R · Calculator · 7 marks

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The question, in short

Given f(x) = 3x² − x + 4 with roots α and β, and g(x) = x² − px + q with roots α + 1/α and β + 1/β, find p and q without solving f(x) = 0.

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Part (a) · 3 marks

Using relationships between coefficients and roots of f(x), show that p = 7/12.

  1. 1α + β = 1/3, αβ = 4/3
  2. 2p = sum of roots = α + β + (α + β)/(αβ)
  3. 3= 1/3 + (1/3)/(4/3) = 1/3 + 1/4 = 7/12

Answerp = 7/12 shown

Part (b) · 4 marks

Find the value of q from the roots of g(x).

  1. 1q = (α + 1/α)(β + 1/β) = αβ + 1/(αβ) + (α² + β²)/(αβ)
  2. 2α² + β² = (α + β)² − 2αβ = 1/9 − 8/3 = −23/9
  3. 3q = 4/3 + 3/4 + (−23/9)/(4/3)
  4. 4= 4/3 + 3/4 − 23/12 = 1/6

Answerq = 1/6

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