O-LevelMathematicsAlgebra and graphs4024/21 · May/June 2023 · Q19 Marks

A cuboid has dimensions x cm, (5-x) cm and 15 cm. [Figure 1.1]

(a)Show that the equation for the volume of the cuboid, ycm³, is y = 75x-15x².[1]
(b)Complete the table of values for y = 75x-15x². x | 1 | 1.5 | 2 | 2.5 | 3 | 3.5 | 4 y | 60 | 78.75 | 90 | | 90 | 78.75 | 60[1]
(c)Draw the graph of y = 75x-15x² for 1≤x≤ 4.[3]
(d)(i)[Volume of pyramid = 1/3 × base area × height] The diagram shows a pyramid with a square base of side length 9 cm. The pyramid has height x cm and volume ycm³. [Figure 1.2] Show that the equation for the volume of the pyramid is y = 27x.[1]
(d)(ii)By drawing a suitable straight line on the grid on page 2, find the height of the pyramid when the pyramid and the cuboid have the same volume.[3]

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Mark scheme

Marking points from the official mark scheme (May/June 2023). The mark or mark type is shown in brackets.

[1] (a) y = x × (5 – x) × 15 = 75x² – 15x
[1] (b) 93.75
[3] (c) Correct smooth curve B2FT for 6 or 7 points correctly plotted or B1FT for 4 or 5 points correctly plotted
[1] (d)(i) y = (1/3) × 9 × 9 × x = 27x or y = (1/3) × 9² × x = 27x
[1] (d)(ii) Correct ruled line crossing their curve B2 B1 for short or unruled line or for two correct coordinates soi
[1] (d)(ii) x-coordinate where ruled line y = 27x crosses y = 75x – 15x² B1 FT x-coordinate where their ruled line and their curve cross After 0 scored overall, SC1 for 3.2

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