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0580/42 May/Jun 2023, Question 8: Equation of the Tangent Line

Cambridge IGCSE Mathematics 0580 · Paper 42 (Extended) · Calculator · 13 marks

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The question, in short

Expand a product of three linear factors to reach a given cubic, sketch that cubic showing axis crossings, then find two tangents with gradient 10 and give their equations as y = mx + c.

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

Part (ai) · 2 marks

Multiply out (x − 4)(x + 1)(x − 2) and show it simplifies to x³ − 5x² + 2x + 8.

  1. 1(x − 4)(x + 1) = x² + x − 4x − 4 = x² − 3x − 4
  2. 2(x² − 3x − 4)(x − 2) = x³ − 2x² − 3x² + 6x − 4x + 8
  3. 3y = x³ − 5x² + 2x + 8

Answerx³ − 4x² + x² − 4x − 2x² + 8x − 2x + 8 leading to y = x³ − 5x² + 2x + 8

Part (aii) · 4 marks

Sketch y = x³ − 5x² + 2x + 8 on the axes, marking where it crosses the x-axis and y-axis.

  1. 1y = 0 when x = 4, x = −1 or x = 2 (from the factors)
  2. 2x = 0 gives y = 8
  3. 3Sketch a positive cubic (rising at the right) crossing the x-axis at −1, 2 and 4 and the y-axis at 8

AnswerCorrect labelled sketch: positive cubic crossing the x-axis at −1, 2 and 4 only and crossing the y-axis at 8 only

Part (b) · 7 marks

Find the two points where the cubic's gradient is 10, then write the tangent equations in y = mx + c form, showing working.

  1. 1dy/dx = 3x² − 10x + 2
  2. 23x² − 10x + 2 = 10, so 3x² − 10x − 8 = 0
  3. 3(3x + 2)(x − 4) = 0, so x = 4 or x = −2/3
  4. 4x = 4: y = 0, so 0 = 10 × 4 + c, c = −40: y = 10x − 40
  5. 5x = −2/3: y = −8/27 − 20/9 − 4/3 + 8 = 112/27, so c = 112/27 + 20/3 = 292/27
  6. 6y = 10x − 40 and y = 10x + 292/27

Answer10x − 40 and 10x + 292/27

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Also searched as: 0580/42/M/J/23 · 0580_s23_qp_42 · 0580 s23 42 · question 8.
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