O-LevelMathematicsMensuration4024/22 · Oct/Nov 2023 · Q1012 Marks
[Volume of a cone = 1/3 πr²h] [Curved surface area of a cone = πrl] A solid is formed from a large cone with a small cone removed from the centre. The small cone is mathematically similar to the large cone. The vertex of the large cone is vertically below the vertex of the small cone. The height of the large cone is 21 cm and the diameter of the top is 18 cm. The height of the small cone is 14 cm. [Figure Cone with Conical Hole]
- (a)(i)Show that the volume of the solid is 399π cm³.[3]
- (a)(ii)Calculate the total surface area of the solid.[6]
- (b)The height of a cylinder is 13 cm, correct to the nearest centimetre. The radius of the base of the cylinder is 4.5 cm, correct to the nearest 0.1 centimetre. Calculate the upper bound of the volume, in cm³, of the cylinder.[3]
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Mark scheme
Marking points from the official mark scheme (Oct/Nov 2023). The mark or mark type is shown in brackets.
- [3] M1 18×14/21 or 18/21 = x/14 oe leading to x = 12
- or 9×14/21 = y/21 or 9/21 = y/14 oe leading to y = 6
- or (14/21)^3 oe
- A2 No errors between volume calculations and 399π
- M1 for 1/3 π×9^2×21 or 1/3 π×(their r)^2×14
- [6] M2 for √21² + 9² and √14² + their 6² or 2/3 √21² + 9² oe
- or M1 for 21² + 9² or 14² + their 6²
- AND
- M2 for π×9×their√522 + π×their 6×their√232 oe
- or (1 - (14/21)^2) π×9×their√522 oe
- or M1 for π×9×their√522 oe
- or π×their 6×their√232 oe
- or (14/21)^2 π×9×their√522 oe
- AND
- M1 for π(9² – (their 6)²) oe
- [3] B1 for 13.5 seen or 4.55 seen
- M1 for π×their 4.55^2 ×their 13.5
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