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WMA11/01 Oct 2024, Question 4: Simultaneous Equations with a Curve

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

The question, in short

From a sketch of quadratic C₁ and cubic C₂, derive each equation from the given intercepts and point, then find where the two curves intersect in the first quadrant.

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

Part (a) · 3 marks

Find the quadratic f(x) for C₁ using its x-intercepts at 0 and 4 and minimum turning point (2, −4.8).

  1. 1Roots 0 and 4: f(x) = kx(x − 4)
  2. 2Minimum (2, −4.8): k × 2 × (2 − 4) = −4.8, so −4k = −4.8
  3. 3k = 1.2, so f(x) = 1.2x(x − 4) = 1.2x² − 4.8x

Answerf(x) = 1.2x(x − 4) = 1.2x² − 4.8x

Part (b) · 3 marks

Find the cubic g(x) for C₂ using its x-intercepts at 0 and 4 and the point (6, 7.2).

  1. 1C₂ crosses at 0 and touches at 4 (repeated root): g(x) = λx(x − 4)²
  2. 2Through (6, 7.2): λ × 6 × (6 − 4)² = 7.2, so 24λ = 7.2
  3. 3λ = 0.3, so g(x) = 0.3x(x − 4)²

Answerg(x) = 0.3x(x − 4)² = 0.3x³ − 2.4x² + 4.8x

Part (c) · 4 marks

Solve algebraically for the coordinates of the first-quadrant intersection point P of C₁ and C₂.

  1. 1Set 1.2x(x − 4) = 0.3x(x − 4)²
  2. 2For P, x ≠ 0 and x ≠ 4, so divide by x(x − 4): 1.2 = 0.3(x − 4)
  3. 3x − 4 = 4, so x = 8
  4. 4y = 1.2 × 8 × 4 = 38.4, so P = (8, 38.4)

AnswerP = (8, 38.4)

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Also searched as: WMA11 October 2024 · WMA11/01 · wma11 oct24 · IAS Maths Pure Mathematics 1 October 2024 · question 4.
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