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4024/22 Oct/Nov 2023, Question 8: Constructing Equations

Cambridge O Level Mathematics 4024 · Paper 22 (Calculator) · Calculator · 9 marks

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The question, in short

A large rectangle has height x cm and length 4 times its height; a 3 cm border surrounds an inner rectangle of area 80 cm². Form and solve an equation, then find the shaded area.

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

Part (a) · 4 marks

Using the dimensions of the inner rectangle, form an equation in x and show it rearranges to 2x² − 15x − 22 = 0.

  1. 1Small rectangle: height x − 6, length 4x − 6
  2. 2(4x − 6)(x − 6) = 80
  3. 34x² − 24x − 6x + 36 = 80
  4. 44x² − 30x − 44 = 0
  5. 52x² − 15x − 22 = 0

AnswerCorrect simplification to 2x²−15x−22=0

Part (b) · 3 marks

Solve 2x² − 15x − 22 = 0, showing working, giving both answers to 2 decimal places.

  1. 1x = (15 ± √((−15)² − 4 × 2 × (−22)))/(2 × 2)
  2. 2= (15 ± √401)/4
  3. 3x = −1.26 and 8.76

Answer−1.26 and 8.76

Part (c) · 2 marks

Using the valid value of x, calculate the area of the shaded border.

  1. 1x = 8.756…
  2. 2Shaded area = x × 4x − 80 = 4 × 8.756…² − 80
  3. 3= 226.6… ≈ 227 cm²

Answer227 or 226.6 to 226.9…

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Also searched as: 4024/22/O/N/23 · 4024_w23_qp_22 · 4024 w23 22 · question 8.
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