A-LevelMathematicsCircular measure9709/12 · Oct/Nov 2011 · Q68 Marks
The diagram shows a circle C₁ touching a circle C₂ at a point X. Circle C₁ has centre A and radius 6 cm, and circle C₂ has centre B and radius 10 cm. Points D and E lie on C₁ and C₂ respectively and DE is parallel to AB. Angle DAX = ¹/₃π radians and angle EBX = θ radians.
- (i)By considering the perpendicular distances of D and E from AB, show that the exact value of θ is sin⁻¹ (3√3 / 10).[3]
- (ii)Find the perimeter of the shaded region, correct to 4 significant figures.[5]
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Mark scheme
Marking points from the official mark scheme (Oct/Nov 2011). The mark or mark type is shown in brackets.
- [3] D to AX = 6sin(π/3) = 6√3/2
- [3] E to AX = 10sinθ
- [3] Equate these → θ = sin⁻¹(3√3 / 10)
- [5] Arc DX = 6×π/3 = 2π
- [5] Arc EX = 10×0.5464
- [5] Horizontal steps = 6cosπ/3 and 10cosθ
- [5] DE = 10+6 - 6cosπ/3 – 10cosθ
- [5] Perimeter = arc DX+ arc EX + DE
- [5] → 16.20
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