A-LevelMathematicsCircular measure9709/12 · May/June 2017 · Q47 Marks
The diagram shows a circle with radius r cm and centre O. Points A and B lie on the circle and ABCD is a rectangle. Angle AOB = 2θ radians and AD = r cm. [Figure 4.1]
- (i)Express the perimeter of the shaded region in terms of r and θ.[3]
- (ii)In the case where r = 5 and θ = 1/6 π, find the area of the shaded region.[4]
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Mark scheme
Marking points from the official mark scheme (May/June 2017). The mark or mark type is shown in brackets.
- (AB) = 2rsinθ (or r√2-2cos2θ or rsin2θ/sin(π/2-θ) )
- [B1] Allow unsimplifed throughout eg r + r, 2θ/2 etc
- (Arc AB) = 2rθ
- [B1]
- (P =) 2r + 2rθ + 2rsinθ (or r√2-2cos2θ or rsin2θ/sin(π/2-θ) )
- [B1]
- Area sector AOB = ( ½ r² 2θ) = 25π/6 or 13.1
- [B1] Use of segment formula gives 2.26 B1B1
- Area triangle AOB = (½×2rsinθ×rcos0 or ½ × r² sin20)
- [B1]
- 25√3/4 -or10.8
- Area rectangle ABCD = (r × 2rsine) 25
- [B1]
- (Area =) Either 25 – (25π/6 – 25√3/4) or 22.7
- [B1] Correct final answer gets B4.
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