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4024/22 Oct/Nov 2024, Question 10: Equations That Become Quadratics

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

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The question, in short

Using apple cost x cents per kg and $9 spent, derive and solve a quadratic, then find Narinder's total cost for 1.5 kg apples and 0.8 kg pears.

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

Part (a) · 1 mark

Write an expression, in kilograms, for the mass of apples bought for $9 at x cents per kilogram.

  1. 1$9 = 900 cents
  2. 2Mass = 900/x kg

Answer900/x

Part (b) · 4 marks

With pears costing 40 cents/kg more, form an equation from the 0.75 kg mass difference and show it simplifies to x² + 40x − 48 000 = 0.

  1. 1Mass of pears = 900/(x + 40)
  2. 2900/x − 900/(x + 40) = 0.75
  3. 3900(x + 40) − 900x = 0.75x(x + 40)
  4. 436 000 = 0.75x² + 30x
  5. 5Multiply by 4/3: 48 000 = x² + 40x, so x² + 40x − 48 000 = 0

Answer900/x − 900/(x + 40) = 0.75, 900(x + 40) − 900x = 0.75x(x + 40), correct expansion of brackets and rearrangement to x² + 40x − 48 000 = 0

Part (c) · 3 marks

Solve x² + 40x − 48 000 = 0, showing your working.

  1. 1Find two numbers that multiply to −48 000 and add to 40
  2. 2−200 × 240 = −48 000 and −200 + 240 = 40
  3. 3So (x − 200)(x + 240) = 0
  4. 4x = 200 or x = −240

Answer200 and −240

Part (d) · 2 marks

Using 200 and 240 cents per kg, work out the total dollars Narinder pays for 1.5 kg apples and 0.8 kg pears.

  1. 1From part (c), x² + 40x − 48 000 = 0 gives x = 200 (x must be positive)
  2. 2Apples cost 200 cents/kg and pears cost 200 + 40 = 240 cents/kg
  3. 31.5 × 200 + 0.8 × 240 = 300 + 192 = 492 cents
  4. 4492 cents = $4.92

Answer4.92

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Also searched as: 4024/22/O/N/24 · 4024_w24_qp_22 · 4024 w24 22 · question 10.
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