WMA11/01 Jan 2024, Question 9: Cubic Equations and Curve Meetings

Part (a) · 2 marks

Sketch C₁, marking where it crosses the axes with coordinates.

1. 1Maximum at x = 2 and a root at x = 0, so by symmetry the other root is x = 4
2. 2Sketch an ∩ shape through (0, 0) and (4, 0) with vertex (2, 20)

Answer∩-shaped parabola through the origin (0, 0), rising to its maximum at (2, 20) and crossing the x-axis again at (4, 0)

Part (b) · 3 marks

Find the quadratic expression f(x) for C₁.

1. 1f(x) = Ax(x − 4) (roots 0 and 4)
2. 2Through (2, 20): 20 = A × 2 × (−2) ⇒ A = −5
3. 3f(x) = −5x(x − 4) = 20x − 5x²

Answerf(x) = 20x − 5x²

Part (c) · 5 marks

Solve algebraically for the coordinates of P, the negative-x intersection of C₁ and C₂, showing all steps.

1. 1x(x² − 4) = −5x(x − 4)
2. 2x³ − 4x = −5x² + 20x ⇒ x³ + 5x² − 24x = 0
3. 3x(x² + 5x − 24) = 0 ⇒ x(x + 8)(x − 3) = 0
4. 4x = 0, 3 or −8; P has negative x, so x = −8
5. 5y = −8((−8)² − 4) = −8 × 60 = −480
6. 6P(−8, −480)

AnswerP(−8, −480)

More past-paper questions on cubic equations and curve meetings

• WMA11/01 Jun 2025 Q7 · 10 marks
• WMA11/01 Oct 2023 Q4 · 7 marks
• WMA11/01 Oct 2022 Q4 · 6 marks
• WMA11/01 Jun 2021 Q8 · 9 marks
• WMA11/01 Jan 2020 Q5 · 7 marks