A-LevelMathematicsCircular measure9709/12 · May/June 2020 · Q77 Marks
In the diagram, OAB is a sector of a circle with centre O and radius 2r, and angle AOB = π/6 radians. The point C is the midpoint of OA. [Figure 7.1]
- (a)Show that the exact length of BC is r√(5 – 2√3).[2]
- (b)Find the exact perimeter of the shaded region.[2]
- (c)Find the exact area of the shaded region.[3]
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Mark scheme
Marking points from the official mark scheme (May/June 2020). The mark or mark type is shown in brackets.
- [M1] BC² = r² + (2r)² - 2r(2r)cos(π/6) = 5r² - 2r²√3
- [A1] BC = r√(5 - 2√3)
- [M1 A1] Perimeter = (2πr/6) + r√(5 - 2√3)
- [M1] Sector area = (1/2)(2r)²(π/6) = πr²/3
- [M1] Triangle area = (1/2)r(2r)sin(π/6) = r²/2
- [A1] Shaded area = r²(π/3 - 1/2)
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