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0580/42 Feb/Mar 2022, Question 8: Equations That Become Quadratics

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

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The question, in short

Darpan runs 12 km at x km/h and cycles 26 km at x + 10 km/h, total time 2 hours 48 minutes. Form an equation, simplify to a quadratic, solve it, and find his running time in minutes.

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Part (a) · 3 marks

Write an equation in x for the total time in hours, using 12/x for the running time and the cycling time over 26 km.

  1. 1Cycling time = 26/(x + 10)
  2. 22 hours 48 minutes = 2 + 48/60 = 2.8 hours
  3. 312/x + 26/(x + 10) = 2.8

Answer12/x + 26/(x + 10) = 2.8

Part (b) · 4 marks

Show that the equation 12/x + 26/(x + 10) = 2.8 rearranges to 7x² − 25x − 300 = 0.

  1. 1Multiply by x(x + 10): 12(x + 10) + 26x = 2.8x(x + 10)
  2. 212x + 120 + 26x = 2.8x² + 28x
  3. 32.8x² − 10x − 120 = 0
  4. 4Multiply by 2.5: 7x² − 25x − 300 = 0

Answer12(x + 10) + 26x = 2.8x(x + 10), 2.8x² − 10x − 120 = 0 (or 30x + 300 + 65x = 7x² + 70x) leading to 7x² − 25x − 300 = 0

Part (c) · 4 marks

Solve 7x² − 25x − 300 = 0 using the quadratic formula, showing all working.

  1. 1x = (25 ± √((−25)² − 4 × 7 × (−300)))/(2 × 7)
  2. 2= (25 ± √9025)/14 = (25 ± 95)/14
  3. 3x = 120/14 = 8.571… or x = −70/14 = −5
  4. 4x = −5 and 8.57

Answer−5 and 8.57 or 8.571…

Part (d) · 2 marks

Using the valid value of x, calculate how many minutes Darpan takes to run the 12 km.

  1. 1Speed = 8.571… km/h
  2. 2Time = 12 ÷ 8.571… × 60
  3. 3= 84 minutes

Answer84 to 84.01…

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