Lemmara logoLemmaraWorked solution

WMA11/01 Oct 2023, Question 8: Simultaneous Equations with a Curve

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 1 (WMA11) · Calculator · 7 marks

The question, in short

Show algebraically that curves C₁ (xy = 15/2 − 5x) and C₂ (y = x³ − (7/2)x − 5) intersect where 2x⁴ − 7x² − 15 = 0, then find the exact distance between intersection points P and Q.

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

Part (a) · 2 marks

Show that substituting the equation of C₂ into C₁ leads to the quartic equation 2x⁴ − 7x² − 15 = 0.

  1. 1Substitute y from C₂ into C₁: x(x³ − (7/2)x − 5) = 15/2 − 5x
  2. 2x⁴ − (7/2)x² − 5x = 15/2 − 5x
  3. 3x⁴ − (7/2)x² − 15/2 = 0
  4. 4Multiply by 2: 2x⁴ − 7x² − 15 = 0

AnswerShown: x(x³ − (7/2)x − 5) = 15/2 − 5x ⇒ x⁴ − (7/2)x² = 15/2 ⇒ 2x⁴ − 7x² − 15 = 0

Part (b) · 5 marks

Using algebra, find the coordinates of P and Q and hence the exact distance PQ.

  1. 1Factorise as a quadratic in x²: (2x² + 3)(x² − 5) = 0
  2. 22x² = −3 has no real roots, so x² = 5 and x = ±√5
  3. 3y = 15/(2x) − 5: at x = √5, y = −5 + (3/2)√5; at x = −√5, y = −5 − (3/2)√5
  4. 4Differences: Δx = 2√5, Δy = 3√5
  5. 5PQ = √((2√5)² + (3√5)²) = √(20 + 45) = √65

AnswerPQ = √65

Stuck on a step? Read the free lesson: Simultaneous Equations with a Curve

Practise more simultaneous equations with a curve questions

Real XMA01 past-paper questions, marked on your working like the exam, with a worked solution for every part.

Practise free on Lemmara

More past-paper questions on simultaneous equations with a curve

Also searched as: WMA11 October 2023 · WMA11/01 · wma11 oct23 · IAS Maths Pure Mathematics 1 October 2023 · question 8.
Lemmara is an independent study tool, not affiliated with or endorsed by Pearson Edexcel. The worked solutions are Lemmara’s own.