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4024/23 May/Jun 2025, Question 18: Equations That Become Quadratics

Cambridge O Level Mathematics 4024 · Paper 23 (Calculator) · Calculator · 10 marks

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The question, in short

Cuboid with length x cm; height is three times the length, width is 4 cm under the length. Work with expressions, a surface area equation, its solution, and the height.

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

Part (a) · 2 marks

Give expressions in x for the cuboid's height and its width.

  1. 1Height is 3 times the length: 3x
  2. 2Width is 4 cm less than the length: x − 4

Answerheight = 3x, width = x − 4

Part (b) · 4 marks

Using a surface area of 200 cm², write an equation in x and demonstrate it reduces to 7x² − 16x − 100 = 0.

  1. 1Faces: x × 3x, x(x − 4), 3x(x − 4), each twice
  2. 22x(3x) + 2x(x − 4) + 2(3x)(x − 4) = 200
  3. 36x² + 2x² − 8x + 6x² − 24x = 200
  4. 414x² − 32x − 200 = 0
  5. 5Divide by 2: 7x² − 16x − 100 = 0

Answer7x² − 16x − 100 = 0 shown

Part (c) · 3 marks

Find both roots of 7x² − 16x − 100 = 0, showing working, rounded to 2 decimal places.

  1. 1a = 7, b = −16, c = −100
  2. 2x = (16 ± √((−16)² − 4 × 7 × (−100)))/(2 × 7)
  3. 3= (16 ± √3056)/14
  4. 4x = 5.09 or x = −2.81

Answer5.09 and −2.81

Part (d) · 1 mark

Given the roots x = 5.09 and x = −2.81, work out the cuboid's height.

  1. 1Length must be positive, x = 5.091…
  2. 2Height = 3x = 3 × 5.091… = 15.27 cm

Answer15.27 to 15.3

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Also searched as: 4024/23/M/J/25 · 4024_s25_qp_23 · 4024 s25 23 · question 18.
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