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4024/21 May/Jun 2023, Question 1: Quadratic Graphs and the Parabola

Cambridge O Level Mathematics 4024 · Paper 21 (Calculator) · Calculator · 9 marks

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The question, in short

Cuboid with sides x, 5 − x and 15 cm; pyramid with 9 cm square base and height x cm. Derive volume formulas, tabulate, graph, and compare volumes.

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

Part (a) · 1 mark

Using the cuboid's three side lengths, derive the stated volume equation y = 75x − 15x².

  1. 1Volume = length × width × height
  2. 2y = x × (5 − x) × 15
  3. 3y = 15x(5 − x) = 75x − 15x²

Answery = x × (5 − x) × 15 = 75x − 15x²

Part (b) · 1 mark

Work out the missing y value in the table for y = 75x − 15x² at x = 2.5.

  1. 1y = 75 × 2.5 − 15 × 2.5²
  2. 2= 187.5 − 93.75 = 93.75

Answer93.75

Part (c) · 3 marks

Plot the curve y = 75x − 15x² on the grid for x from 1 to 4.

  1. 1Plot (1, 60), (1.5, 78.75), (2, 90), (2.5, 93.75), (3, 90), (3.5, 78.75), (4, 60)
  2. 2Join the points with a smooth curve (maximum at (2.5, 93.75))

AnswerCorrect smooth curve

Part (di) · 1 mark

Apply the pyramid volume formula to the 9 cm square base and height x to derive y = 27x.

  1. 1Base area = 9 × 9 = 81 cm²
  2. 2y = 1/3 × 81 × x = 27x

Answery = 1/3 × 9 × 9 × x = 27x or y = 1/3 × 9² × x = 27x

Part (dii) · 3 marks

Add a straight line to the grid and use its intersection with the curve to find x when both volumes match.

  1. 1Draw the ruled line y = 27x, e.g. through (1, 27) and (4, 108)
  2. 2Read the x-coordinate where it crosses the curve
  3. 3Check: 27x = 75x − 15x² gives 15x² = 48x, so x = 48/15
  4. 4Height = 3.2 cm

Answer3.2 (accept 3.1 to 3.3)

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Also searched as: 4024/21/M/J/23 · 4024_s23_qp_21 · 4024 s23 21 · question 1.
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