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4024/11 Oct/Nov 2025, Question 24: Constructing Equations

Cambridge O Level Mathematics 4024 · Paper 11 (Non-calculator) · Non-calculator · 10 marks

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The question, in short

Rectangle has length x cm and area 30 cm²; triangle sides are x − 1, x + 2 and 2x − 3 cm. Use these to find the rectangle width, form and solve an equation, then find the rectangle perimeter.

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

Part (a) · 1 mark

Write an expression in x for the width of the rectangle, using its length x and area 30 cm².

  1. 1Width = area ÷ length = 30/x

Answer30/x

Part (b) · 4 marks

Set the rectangle perimeter equal to the triangle perimeter, form an equation in x, and show it simplifies to x² − x − 30 = 0.

  1. 12(x + 30/x) = (x − 1) + (x + 2) + (2x − 3)
  2. 22x + 60/x = 4x − 2
  3. 32x² + 60 = 4x² − 2x
  4. 42x² − 2x − 60 = 0, divide by 2: x² − x − 30 = 0

Answer2(30/x + x) = x − 1 + x + 2 + 2x − 3, 60 + 2x² = 4x² − 2x, 4x² − 2x² − 2x − 60 = 0 leading to x² − x − 30 = 0

Part (c) · 3 marks

Solve the quadratic equation x² − x − 30 = 0 for x.

  1. 1x² − x − 30 = (x − 6)(x + 5) = 0
  2. 2x = 6 or x = −5

Answer6, −5

Part (d) · 2 marks

Using the valid value of x, calculate the perimeter of the rectangle.

  1. 1From part (c), x = 6 or x = −5; a length must be positive, so x = 6.
  2. 2Width = area ÷ length = 30 ÷ 6 = 5
  3. 3Perimeter = 2(6 + 5) = 22

Answer22

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Also searched as: 4024/11/O/N/25 · 4024_w25_qp_11 · 4024 w25 11 · question 24.
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