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WMA12/01 Oct 2024, Question 8: Simultaneous Equations with a Curve

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 2 (WMA12) · Calculator · 8 marks

The question, in short

Two curves y = x² + 3 and y = 13 − 9/x² for x > 0 meet at P and Q; find the x coordinates of P and Q, then the exact area of the region R between the curves.

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

Part (a) · 4 marks

Solve algebraically for the x coordinates where the two curve equations are equal, giving the x values of P and Q.

  1. 1x² + 3 = 13 − 9/x²
  2. 2Multiply by x²: x⁴ + 3x² = 13x² − 9, so x⁴ − 10x² + 9 = 0
  3. 3(x² − 1)(x² − 9) = 0, so x² = 1 or x² = 9
  4. 4x > 0, so x = 1 (P) and x = 3 (Q)

AnswerP: x = 1, Q: x = 3

Part (b) · 4 marks

Set up and evaluate an algebraic definite integral between the two x values to find the exact area of R.

  1. 1Area = ∫ from 1 to 3 of [(13 − 9/x²) − (x² + 3)] dx = ∫ from 1 to 3 of (10 − 9x⁻² − x²) dx
  2. 2= [10x + 9/x − x³/3] from 1 to 3
  3. 3At x = 3: 30 + 3 − 9 = 24
  4. 4At x = 1: 10 + 9 − 1/3 = 56/3
  5. 5Area = 24 − 56/3 = 16/3

Answer16/3

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