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0606/22 Feb/Mar 2025, Question 2: Tangents, Normals and Stationary Points (Further)

Cambridge IGCSE Additional Mathematics 0606 · Paper 22 (Calculator) · Calculator · 9 marks

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The question, in short

For the curve y = (1/2)(3 − 2x)(x + 2)², locate stationary points, draw its shape with axis crossings, and determine k giving three distinct real roots.

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

Part (a) · 4 marks

Differentiate the given cubic and solve for the x-values where the gradient is zero.

  1. 1Product rule: dy/dx = (1/2)[2(x + 2)(3 − 2x) − 2(x + 2)²]
  2. 2= (1/2)(−6x² − 10x + 4) = −(x + 2)(3x − 1)
  3. 3dy/dx = 0: (x + 2)(3x − 1) = 0
  4. 4x = −2, x = 1/3

Answerx = −2, x = 1/3

Part (b) · 3 marks

Draw the cubic on the provided axes, marking where it meets the x-axis and y-axis.

  1. 1y = 0 when 3 − 2x = 0 or x + 2 = 0: x = 1.5 and x = −2 (repeated root, so the curve touches)
  2. 2x = 0: y = (1/2)(3)(4) = 6
  3. 3Leading term −x³, so y → −∞ as x → ∞ and y → +∞ as x → −∞
  4. 4Minimum at (−2, 0), maximum at x = 1/3 in the first quadrant

AnswerNegative cubic: touches the x-axis at x = −2 (minimum), crosses the y-axis at 6, has a maximum in the first quadrant, and crosses the x-axis at x = 1.5 continuing into the fourth quadrant

Part (c) · 2 marks

Use your sketch to state the range of k for which the horizontal line y = k cuts the curve three times.

  1. 1Maximum value: y at x = 1/3 is (1/2)(7/3)(7/3)² = 343/54
  2. 2Minimum value: y at x = −2 is 0
  3. 3The line y = k meets the curve three times when 0 < k < 343/54

Answer0 < k < 343/54 (or 0 < k < 6.35)

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Also searched as: 0606/22/F/M/25 · 0606_m25_qp_22 · 0606 m25 22 · question 2.
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