A-LevelMathematicsCircular measure9709/12 · Oct/Nov 2024 · Q66 Marks

The diagram shows a metal plate OABCDEF consisting of sectors of two circles, each with centre O. The radii of sectors AOB and EOF are r cm and the radius of sector COD is 2r cm. Angle AOB = angle EOF = θ radians and angle COD = 2θ radians. It is given that the perimeter of the plate is 14 cm and the area of the plate is 10 cm². Given that r > 3/2 and θ < 3/4, find the values of r and θ. [Figure X.X]

(a)The diagram shows a metal plate OABCDEF consisting of sectors of two circles, each with centre O. The radii of sectors AOB and EOF are r cm and the radius of sector COD is 2r cm. Angle AOB = angle EOF = θ radians and angle COD = 2θ radians. It is given that the perimeter of the plate is 14 cm and the area of the plate is 10 cm². Given that r > 3/2 and θ < 3/4, find the values of r and θ.[6]

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Mark scheme

Marking points from the official mark scheme (Oct/Nov 2024). The mark or mark type is shown in brackets.

[B1] [Perimeter =] r+rθ+r+2r×2θ+r+rθ+r [= 4r+6rθ]
[B1] [Area =] 1/2 r²θ + 1/2 (2r)²×2θ + 1/2 r²θ [= 5r²θ]
[M1*] 4r+6rθ=14 and 5r²θ=10 where a, b and c are constants ≠ 0. Terms may be uncollected.
[DM1] Eliminate θ to get an equation in r.
[DM1] Factorise or other accepted method for solving their 3-term quadratic.
[B1] r=2 and θ=0.5 Condone extra answers r = 3/2 and θ = 8/9.

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