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4PM1/02 Jun 2024, Question 1: Solving Cubic Equations

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 2 · Calculator · 6 marks

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The question, in short

For f(x) = 6x³ − 13x² + ax − 10 with (3x − 2) a factor, show a = 21, then prove the curve y = f(x) meets the x-axis only once.

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Part (a) · 2 marks

Use the factor (3x − 2) to show that the constant a equals 21.

  1. 1f(2/3) = 0
  2. 26(2/3)³ − 13(2/3)² + (2/3)a − 10 = 0
  3. 316/9 − 52/9 + 2a/3 − 10 = 0, so 2a/3 = 14
  4. 4a = 21

Answera = 21 shown

Part (b) · 4 marks

Using a = 21, show algebraically that y = f(x) crosses the x-axis at only one point.

  1. 1Divide f(x) by (3x − 2): quotient 2x² − 3x + 5
  2. 2Discriminant of 2x² − 3x + 5: (−3)² − 4 × 2 × 5 = −31
  3. 3−31 < 0, so the quadratic has no real roots
  4. 4The only intersection with the x-axis is at x = 2/3

Answerf(x) = (3x − 2)(2x² − 3x + 5); 2x² − 3x + 5 has discriminant −31 < 0, so the only intersection is at x = 2/3

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