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WMA11/01 Jun 2024, Question 7: The Quadratic Formula and Completing the Square

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 1 (WMA11) · Calculator · 8 marks

The question, in short

For f(x) = 2x³ − kx² + 14x + 24 with constant k, find derivatives, use intersection of f′(x) and f″(x) at A with x = 5 to find k, then coordinates of B.

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

Part (ai) · 2 marks

Find f′(x) in simplest form.

  1. 1Differentiate each term: 2x³ → 6x², −kx² → −2kx, 14x → 14, 24 → 0
  2. 2f′(x) = 6x² − 2kx + 14

Answerf′(x) = 6x² − 2kx + 14

Part (aii) · 1 mark

Find f″(x).

  1. 1Differentiate f′(x) = 6x² − 2kx + 14
  2. 2f″(x) = 12x − 2k

Answerf″(x) = 12x − 2k

Part (b) · 2 marks

Given the x coordinate of A is 5, find the value of k.

  1. 1At x = 5: f′(5) = f″(5)
  2. 2150 − 10k + 14 = 60 − 2k
  3. 3104 = 8k, so k = 13

Answerk = 13

Part (c) · 3 marks

Hence find the coordinates of B.

  1. 16x² − 26x + 14 = 12x − 26
  2. 26x² − 38x + 40 = 0, so 3x² − 19x + 20 = 0
  3. 3(3x − 4)(x − 5) = 0; x = 5 is A, so x = 4/3
  4. 4y = f″(4/3) = 12(4/3) − 26 = −10
  5. 5B = (4/3, −10)

AnswerB = (4/3, −10)

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