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4PM1/02 Jun 2020, Question 6: Roots α and β of a Quadratic

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 2 · Calculator · 11 marks

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The question, in short

Given f(x) = 4x² − 3x − 5 with roots α and β, use root relationships without solving to build a new equation and find p and q in g(x) = 4x² + px + q.

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Part (a) · 6 marks

Using α + β and αβ from f(x), form an equation with integer coefficients whose roots are 2α/β and 2β/α.

  1. 1α + β = 3/4, αβ = −5/4
  2. 2Sum = 2α/β + 2β/α = 2[(α + β)² − 2αβ]/(αβ) = 2(9/16 + 5/2)/(−5/4) = −49/10
  3. 3Product = (2α/β)(2β/α) = 4
  4. 4x² + (49/10)x + 4 = 0
  5. 510x² + 49x + 40 = 0

Answer10x² + 49x + 40 = 0

Part (b) · 5 marks

Given g(x) = 4x² + px + q has roots 3α + β and α + 3β, find the constants p and q.

  1. 1Sum of roots = 4(α + β) = 3 = −p/4 ⇒ p = −12
  2. 2Product = (3α + β)(α + 3β) = 3(α² + β²) + 10αβ
  3. 3= 3(9/16 + 5/2) − 10 × 5/4 = −53/16
  4. 4q/4 = −53/16 ⇒ q = −53/4

Answerp = −12, q = −53/4

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