WMA11/01 June 2024 Question 8: Sketching Cubics and Reciprocal Curves

Question 1

Sketch C₁, marking where it crosses the x-axis and y-axis with their coordinates.

Accepted Answer

y = x(2 − x)(2 + x): roots x = −2, 0, 2 Leading term −x³, so the curve starts top left and ends bottom right Minimum between −2 and 0, maximum between 0 and 2 Crosses the axes at (−2, 0), (0, 0) and (2, 0) Answer: Negative cubic: comes down from the top left, crosses the x-axis at (−2, 0), has a minimum, passes through the origin (0, 0), has a maximum, crosses at (2, 0), then goes down

Question 2

Show that intersection x-coordinates of C₁ and C₂ satisfy x⁴ − 4x² + A = 0.

Accepted Answer

x(4 − x²) = A/x Multiply by x: 4x² − x⁴ = A x⁴ − 4x² + A = 0 Answer: Shown: x(4 − x²) = A/x gives 4x² − x⁴ = A, so x⁴ − 4x² + A = 0

Question 3

Find the range of A for which C₁ and C₂ meet at 4 distinct points.

Accepted Answer

Let u = x²: u² − 4u + A = 0 needs two distinct positive roots Distinct roots: b² − 4ac > 0, so 16 − 4A > 0, giving A < 4 Both roots positive: product of roots A > 0 (their sum 4 is already positive) So 0 < A < 4 Answer: 0 < A < 4

WMA11/01 Jun 2024, Question 8: Sketching Cubics and Reciprocal Curves

The question, in short

Part (a) · 3 marks

Part (b) · 1 mark

Part (c) · 3 marks

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