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0580/42 Oct/Nov 2021, Question 8: Equations That Become Quadratics

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

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The question, in short

Kaito runs a 12 km path at x km/h; Yuki walks it at (x − 4) km/h taking 1.5 hours more. Form and solve a quadratic, then find Yuki's time; also find a lower bound for a bus's speed.

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

Part (ai) · 1 mark

Give an expression in x for how many hours Kaito's run takes.

  1. 1Time = distance ÷ speed = 12/x hours

Answer12/x or 12 ÷ x

Part (aii) · 4 marks

Using Yuki's 1.5 hour longer time and speed x − 4, form an equation in x and show it reduces to x² − 4x − 32 = 0.

  1. 112/(x − 4) − 12/x = 1.5
  2. 212x − 12(x − 4) = 1.5x(x − 4)
  3. 348 = 1.5x² − 6x
  4. 41.5x² − 6x − 48 = 0
  5. 5Divide by 1.5: x² − 4x − 32 = 0

Answer1.5x² − 6x − 48 = 0 leading to x² − 4x − 32 = 0

Part (aiii) · 3 marks

Solve x² − 4x − 32 = 0 using factorisation.

  1. 1(x − 8)(x + 4) = 0
  2. 2x = 8 or x = −4

Answer−4 and 8

Part (aiv) · 2 marks

Work out how long Yuki's walk over the 12 km takes, in hours.

  1. 1Speed must be positive, so x = 8 and Yuki walks at 8 − 4 = 4 km/h
  2. 2Time = 12 ÷ 4 = 3 hours

Answer3

Part (b) · 3 marks

A 440 km journey (nearest 10 km) takes 6 hours (nearest half hour); find the lowest possible average speed.

  1. 1Lower bound distance = 435 km, upper bound time = 6.25 h
  2. 2Lower bound speed = 435 ÷ 6.25 = 69.6 km/h

Answer69.6

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Also searched as: 0580/42/O/N/21 · 0580_w21_qp_42 · 0580 w21 42 · question 8.
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