O-LevelAdditional MathematicsCircular measure4037/21 · May/June 2025 · Q912 Marks

In this question, all lengths are in centimetres and all angles are in radians. [Figure showing a sector AOB of a circle and a cone formed from it] The diagram shows a sector AOB of a circle, centre O, radius 15. Angle AOB = 6π/5. The sector is made into a cone with points A and B touching, as shown.

(a)Find the curved surface area of the cone.[2]
(b)The top of the cone is a horizontal circle. Find the circumference of the circular top.[2]
(c)Hence find the radius of the circular top and the perpendicular height of the cone.[2]
(d)(i)When the depth of the water in the cone is h, the radius of the circular top of the water is r. Find an expression for r in terms of h.[1]
(d)(ii)The water is poured into the cone at a constant rate of 27 cm³ per second. Find the rate at which the depth of the water is rising when the depth of the water is 4.[5]

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Mark scheme

Marking points from the official mark scheme (May/June 2025). The mark or mark type is shown in brackets.

[M1] for 1/2 x 15² x 6π/5 oe
[A1] 135π isw or 424 or 424.1 to 424.2
[M1] for 15 x 6π/5 oe
[A1] 18π or 56.5 to 56.6
[B1] for radius = 9 OR
[M1] for a correct Pythagoras' statement e.g. their r² + height² = 15²
[B1] radius = 9, height = 12
[B1] r = 3/4 h oe
[B1] FT their r = kh where k is a positive constant e.g. V = 1/3 π (their kh)² x h
[B1] FT their k where k is a positive constant e.g. dV/dh = (theirk)² π x h²
[B1] Relevant, correct chain rule including dh/dt soi e.g. dh/dt = dV/dt ÷ dV/dh
[M1] 27 x 16 / (9π x 16) oe
[A1] 0.955 or 0.9549[29...]

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