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4024/22 Oct/Nov 2020, Question 10: Solving Quadratics by Factorising

Cambridge O Level Mathematics 4024 · Paper 22 (Calculator) · Calculator · 10 marks

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The question, in short

Amira drives 40 km to work in two stages; use given speed expressions to form and solve a quadratic, then find her whole-journey average speed.

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

Part (a) · 1 mark

Show that the average speed for the first 30 km, taking x minutes, equals 1800/x km/h.

  1. 1x minutes = x/60 hours
  2. 2Speed = 30 ÷ (x/60) = 30/x × 60 = 1800/x

Answer30/x × 60 = 1800/x

Part (b) · 3 marks

Using the two speed expressions differing by 8 km/h, form an equation in x and show it reduces to x² + 125x − 5625 = 0.

  1. 11800/x = 600/(x − 25) − 8
  2. 2Multiply by x(x − 25): 1800(x − 25) = 600x − 8x(x − 25)
  3. 31800x − 45 000 = 600x − 8x² + 200x
  4. 48x² + 1000x − 45 000 = 0
  5. 5x² + 125x − 5625 = 0

AnswerCorrect rearrangement to x² + 125x − 5625 = 0

Part (c) · 3 marks

Solve x² + 125x − 5625 = 0, showing working, giving both roots to 1 decimal place.

  1. 1x = (−125 ± √(125² − 4 × 1 × (−5625)))/2
  2. 2= (−125 ± √38 125)/2
  3. 3x = 35.1 or x = −160.1

Answer35.1, −160.1

Part (d) · 3 marks

Given the final 10 km takes 25 minutes less than the first 30 km, calculate the average speed in km/h for all 40 km.

  1. 1Time for first 30 km = 35.12… min, for last 10 km = 35.12… − 25 = 10.12… min
  2. 2Total time = 2 × 35.12… − 25 = 45.25… min
  3. 3Average speed = 40 ÷ (45.25…/60) = 53.03… ≈ 53.0 km/h

Answer53.0 to 53.1

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