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WMA11/01 Jun 2025, Question 10: The Discriminant and Nature of Roots

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 1 (WMA11) · Calculator · 7 marks

The question, in short

Given (k − 1)x⁶ + 4x³ + (k − 4) = 0 with constant k, find exact solutions when k = 4.5, then find all k giving no real roots, showing full working without calculator.

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Part (a) · 3 marks

Substitute k = 4.5 into the equation and solve exactly for x, showing all stages without calculator.

  1. 1k = 4.5: 3.5x⁶ + 4x³ + 0.5 = 0, so 7x⁶ + 8x³ + 1 = 0
  2. 2(x³ + 1)(7x³ + 1) = 0
  3. 3x³ = −1 or x³ = −1/7
  4. 4x = −1 or x = −∛(1/7)

Answerx = −1 and x = −∛(1/7)

Part (b) · 4 marks

Determine the set of values of k for which the equation has no real roots.

  1. 1Treat as a quadratic in x³: a = k − 1, b = 4, c = k − 4
  2. 2b² − 4ac = 16 − 4(k − 1)(k − 4) = 16 − 4(k² − 5k + 4) = 20k − 4k²
  3. 3No real roots when 20k − 4k² < 0, i.e. 4k(5 − k) < 0
  4. 4k < 0 or k > 5

Answerk < 0 or k > 5

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Also searched as: WMA11 June 2025 · WMA11/01 · wma11 jun25 · IAS Maths Pure Mathematics 1 June 2025 · question 10.
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