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4024/21 Oct/Nov 2022, Question 8: Equations That Become Quadratics

Cambridge O Level Mathematics 4024 · Paper 21 (Calculator) · Calculator · 10 marks

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The question, in short

Two cyclists cover 50 km at speeds x and x − 3 km/h; form and solve a quadratic from their time difference, then find Marco's time.

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

Part (a) · 1 mark

Write an expression, in hours, for the time Lara takes to ride 50 km at x km/h.

  1. 1Time = distance ÷ speed
  2. 2Lara's time = 50/x hours

Answer50/x

Part (b) · 4 marks

Using Marco's 15-minute longer time, form an equation and show it rearranges to x² − 3x − 600 = 0.

  1. 1Marco's time = 50/(x − 3) hours; 15 minutes = 15/60 hour
  2. 250/(x − 3) − 50/x = 15/60
  3. 3Multiply by 60x(x − 3): 3000x − 3000(x − 3) = 15x(x − 3)
  4. 49000 = 15x² − 45x
  5. 5Divide by 15: 600 = x² − 3x
  6. 6x² − 3x − 600 = 0

Answer50/(x − 3) − 50/x = 15/60 leading to x² − 3x − 600 = 0

Part (c) · 3 marks

Solve x² − 3x − 600 = 0, showing working, giving both answers to 2 decimal places.

  1. 1x = (3 ± √(9 + 2400))/2
  2. 2= (3 ± √2409)/2
  3. 3√2409 = 49.08…
  4. 4x = 26.04 or x = −23.04

Answer26.04 and −23.04

Part (d) · 2 marks

Given x = 26.04, find Marco's time for the trail in hours and minutes, to the nearest minute.

  1. 1Marco's speed = 26.04 − 3 = 23.04 km/h
  2. 2Time = 50 ÷ 23.04 = 2.170 hours
  3. 30.170 × 60 = 10.2 minutes
  4. 42 hours 10 minutes

Answer2 hours 10 minutes

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Also searched as: 4024/21/O/N/22 · 4024_w22_qp_21 · 4024 w22 21 · question 8.
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