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WMA11/01 Oct 2024, Question 9: Differentiating Polynomials

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 1 (WMA11) · Calculator · 10 marks

The question, in short

Use the cubic f(x) = (x + 5)(3x² − 4x + 20) and its sketch to find where f(x) ≥ 0, differentiate it, locate point P, and transform point R(−4, 84).

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

Part (a) · 1 mark

State the set of x values satisfying f(x) ≥ 0.

  1. 13x² − 4x + 20 has discriminant 16 − 240 < 0, so it is always positive
  2. 2So f(x) ≥ 0 exactly when x + 5 ≥ 0: x ≥ −5

Answerx ≥ −5

Part (b) · 3 marks

Differentiate f(x) and simplify the derivative.

  1. 1Expand: (x + 5)(3x² − 4x + 20) = 3x³ − 4x² + 20x + 15x² − 20x + 100
  2. 2f(x) = 3x³ + 11x² + 100
  3. 3f′(x) = 9x² + 22x

Answerf′(x) = 9x² + 22x

Part (c) · 4 marks

Given the tangent at P is parallel to that at R(−4, 84), find P's x coordinate.

  1. 1Gradient at R: f′(−4) = 9(16) + 22(−4) = 144 − 88 = 56
  2. 2Set 9x² + 22x = 56, so 9x² + 22x − 56 = 0
  3. 3(9x − 14)(x + 4) = 0
  4. 4x = −4 is R itself, so P has x = 14/9

Answerx = 14/9

Part (di) · 1 mark

Give the image of R(−4, 84) under y = f(x − 3).

  1. 1y = f(x − 3) is a translation 3 units in the positive x-direction
  2. 2R(−4, 84) moves to (−4 + 3, 84) = (−1, 84)

Answer(−1, 84)

Part (dii) · 1 mark

Give the image of R(−4, 84) under y = 4f(x).

  1. 1y = 4f(x) is a vertical stretch, scale factor 4
  2. 2R(−4, 84) moves to (−4, 4 × 84) = (−4, 336)

Answer(−4, 336)

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