Cambridge A Level Mathematics 9709 — 2005 Oct/Nov Paper 1 · Variant 1

2 In the diagram, OAB and OCD are radii of a circle, centre O and radius 16 cm. Angle AOC = α radians. AC and BD are arcs of circles, centre O and radii 10 cm and 16 cm respectively. (i) In the case where α = 0.8, find the area of the shaded region. [2] (ii) Find the value of α for which the perimeter of the shaded region is 28.9 cm. [3]

Mark scheme: 21 (i) 3 sin θ − 2 cos θ − 3 = 0 Use of s² + c² = 1 M1 Use of s²+c²=1 to eliminate sine. 3cos²θ + 2cosθ = 0 A1 Correct equation cosθ = 0, θ = 90° B1 Co. or cosθ = −2/3, θ = 131.8 ° A1 Co. ( to 1 d.p or more – there must be only this answer in the range 0 to 180)) [4] 2 α = 0.8, radii 10cm and 16 cm (i) Area = ½.16².0.8 − ½.10².0.8 M1 Use of ½r²θ once.. → 62.4 cm² A1 Co. [2] (ii) Arcs are 10α and 16α M1 Use of s=rθ once. 12 + 10α + 16α = 28.9 DM1 Forming an eqn, including the 6 + 6. → α = 0.65 A1 co [3] 3 (i) 2dsin30 + 2d√3sin60 M1 For one - allow even if decimals used = 2d.½ + 2d√3.√3/2 = 4d A1 Co – exact answer only. [2] ans to (i) (ii) tan θ = 2 d cos 30 + 2 3d cos 60 M1 Use of tan = opp/adj in correct triangle DM1 For horizontal step attempted 2 tan θ =

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