O-LevelAdditional MathematicsCircular measure4037/11 · May/June 2016 · Q1110 Marks
The diagram shows a circle, centre O, radius r cm. The points A and B lie on the circle such that angle AOB = 2θ radians. [Figure 11.1]
- (i)Find, in terms of r and θ, an expression for the length of the chord AB.[1]
- (ii)Given that the perimeter of the shaded region is 20 cm, show that r = 10 / (θ + sinθ).[2]
- (iii)Given that r and θ can vary, find the value of dr/dθ when θ = π/6.[4]
- (iv)Given that r is increasing at the rate of 15 cms⁻¹, find the corresponding rate of change of θ when θ = π/6.[3]
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Mark scheme
Marking points from the official mark scheme (May/June 2016). The mark or mark type is shown in brackets.
- [1] AB = 2r sin θ or √(r² + r² - 2r² cos 2θ) or r sin 2θ / sin(π/2 - θ) or r sin 2θ / cos θ
- [1] 2r sin θ + 2rθ = 20
- [1] r = 10 / (θ + sinθ)
- [1] dr/dθ calculation attempt
- [2] dr/dθ = (10(-(1 + cosθ))) / (θ + sinθ)²
- [1] dr/dθ when θ = π/6 = -17.8
- [1] dr/dt = 15
- [1] dθ/dt = dr/dt / (dr/dθ)
- [1] dθ/dt = 15 / (-17.8)
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