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WMA11/01 Jun 2021, Question 8: Cubic Equations and Curve Meetings

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 1 (WMA11) · Calculator · 9 marks

The question, in short

Complete the square for the quadratic 3x² + 6x + 9, find the minimum point P of C₁, then determine A, B, C, D for a cubic through P with x-intercepts −4, −2, 3.

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

Part (a) · 3 marks

Express 3x² + 6x + 9 as a(x + b)² + c and state the constants a, b and c.

  1. 13x² + 6x + 9 = 3(x² + 2x) + 9
  2. 2= 3[(x + 1)² − 1] + 9
  3. 3= 3(x + 1)² + 6

Answer3(x + 1)² + 6 (a = 3, b = 1, c = 6)

Part (b) · 1 mark

Use the completed-square form to write down the coordinates of the minimum point P.

  1. 1Minimum when x + 1 = 0
  2. 2P = (−1, 6)

AnswerP(−1, 6)

Part (c) · 5 marks

Find A, B, C and D for the cubic passing through P with x-axis crossings at −4, −2 and 3, showing your method.

  1. 1y = k(x + 4)(x + 2)(x − 3)
  2. 2Through P(−1, 6): 6 = k(3)(1)(−4) = −12k
  3. 3k = −1/2
  4. 4y = −½(x + 4)(x + 2)(x − 3) = −½(x³ + 3x² − 10x − 24)
  5. 5y = −½x³ − (3/2)x² + 5x + 12

AnswerA = −1/2, B = −3/2, C = 5, D = 12

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