A-LevelMathematicsCircular measure9709/12 · May/June 2013 · Q47 Marks
The diagram shows a square ABCD of side 10 cm. The mid-point of AD is O and BXC is an arc of a circle with centre O. [Figure 4.1]
- (i)Show that angle BOC is 0.9273 radians, correct to 4 decimal places.[2]
- (ii)Find the perimeter of the shaded region.[3]
- (iii)Find the area of the shaded region.[2]
This question uses a diagram from the original paper, which isn't shown here. Open the full paper from the past papers archive to see it.
Mark scheme
Marking points from the official mark scheme (May/June 2013). The mark or mark type is shown in brackets.
- [1] M1 For (i): Correct trigonometry setup to find angle BOC (e.g., using tan(angle/2) = 5/10)
- [1] A1 For (i): Calculates angle BOC = 0.9273 radians (AG - answer given)
- [1] B1 For (ii): Calculates radius OB = √(10² + 5²) = √125 ≈ 11.18 cm
- [1] M1 For (ii): Uses arc length formula s = rθ with θ in radians (√125 × 0.9273)
- [1] A1 For (ii): Calculates perimeter = 20.4 cm
- [1] M1 For (iii): Uses sector area formula Area = ½r²θ with r = √125 and θ in radians
- [1] A1 For (iii): Calculates area = 7.96 cm² (allow 7.95 or 7.96)
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