O-LevelAdditional MathematicsCircular measure4037/11 · Oct/Nov 2011 · Q1012 Marks

Answer only one of the following two alternatives. [Figure 10.2 OR]

OR (i)The figure shows a sector OAB of a circle, centre O, radius 10 cm. Angle AOB = 2θ radians where 0 < θ < π/2. A circle centre C, radius r cm, touches the arc AB at the point D. The lines OA and OB are tangents to the circle at the points E and F respectively. Write down, in terms of r, the length of OC.[1]
OR (ii)Hence show that r = 10 sin θ / (1 + sin θ).[2]
OR (iii)Given that r can vary, find dr/dθ when r = 10/3.[6]
OR (iv)Given that r is increasing at 2 cms⁻¹, find the rate at which θ is increasing when θ = π/6.[3]

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Mark scheme

Marking points from the official mark scheme (Oct/Nov 2011). The mark or mark type is shown in brackets.

[1] (i) OC = 10 - r
[1] (ii) M1 for attempt to use sine sin θ = r/(10-r)
[1] (ii) A1 for correct attempt to simplify to given answer leading to r = 10 sin θ / (1 + sin θ)
[2] (iii) M1 for correct attempt to differentiate a quotient dr/dθ = 10 cos θ / (1 + sin θ)²
[1] (iii) A2, 1, 0 - 1 each error
[1] (iii) M1 for attempt to find sin or cos when r = 10/3, sin θ = 1/2, cos θ = √3/2
[1] (iii) M1 for substitution
[1] (iii) A1 dr/dθ = 20√3 / 9 (3.85)
[1] (iv) B1 dr/dt = 2
[1] (iv) M1 for correct use of rates of change dθ/dt = (dr/dt) * (dθ/dr)
[1] (iv) A1 leading to dθ/dt = 3√3 / 10 (0.520)

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