A-LevelMathematicsCircular measure9709/12 · May/June 2015 · Q24 Marks
In the diagram, AYB is a semicircle with AB as diameter and OAXB is a sector of a circle with centre O and radius r. Angle AOB = 2θ radians. Find an expression, in terms of r and θ, for the area of the shaded region. [Diagram shown]
- In the diagram, AYB is a semicircle with AB as diameter and OAXB is a sector of a circle with centre O and radius r. Angle AOB = 2θ radians. Find an expression, in terms of r and θ, for the area of the shaded region.[4]
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 2015). The mark or mark type is shown in brackets.
- [B1] Radius of semicircle = ½AB = rsinθ aef
- [B1] Area of semicircle = ½πr²sin²θ Uses ½πr² with r = f(θ)
- [B1B1] Shaded area = semicircle – segment = A₁ – ½r²(2θ) + ½r²sin(2θ) B1 (-sector), B1 for + (triangle)
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