0606/12 Oct/Nov 2025, Question 1: Sketching Cubics and Cubic Inequalities
Cambridge IGCSE Additional Mathematics 0606 · Paper 12 (Non-calculator) · Non-calculator · 5 marks
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The question, in short
Sketch the cubic y = (1 − x)(x − 5)(2x − 5) on the given axes, marking where it crosses the axes, then use the sketch to solve the inequality.
Summarised in our own words. Use the official paper for the exact wording and any diagrams.
Part (a) · 3 marks
On the blank axes, draw the curve y = (1 − x)(x − 5)(2x − 5) and state the coordinates of its intercepts with the x-axis and y-axis.
- 1Leading term (−x)(x)(2x) = −2x³, so the curve falls to the right and rises to the left
- 2x-intercepts: 1 − x = 0, x − 5 = 0, 2x − 5 = 0 ⇒ x = 1, 2.5, 5
- 3y-intercept: (1)(−5)(−5) = 25
- 4Minimum between 1 and 2.5 (below the axis), maximum between 2.5 and 5 (above the axis)
AnswerNegative cubic: comes down from top left through (0, 25), crosses the x-axis at x = 1, minimum in the 4th quadrant, crosses at x = 2.5, maximum in the 1st quadrant, crosses at x = 5 and goes down to the bottom right
Part (b) · 2 marks
Using your sketch, find the set of x values for which (1 − x)(x − 5)(2x − 5) ≤ 0.
- 1Read where the curve is on or below the x-axis
- 2Between x = 1 and x = 2.5 the curve is below the axis
- 3For x ≥ 5 the curve is below the axis
- 41 ≤ x ≤ 2.5 or x ≥ 5
Answer1 ≤ x ≤ 2.5, x ≥ 5
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Also searched as: 0606/12/O/N/25 · 0606_w25_qp_12 · 0606 w25 12 · question 1.
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