O-LevelAdditional MathematicsCircular measure4037/21 · May/June 2023 · Q98 Marks
In this question all lengths are in centimetres and all angles are in radians.
- (a)The area of a sector of a circle of radius 24 is 432 cm². Find the length of the arc of the sector.[4]
- (b)(i)The diagram shows an isosceles triangle, OAB, with AO = AB = y and height AD. OCD is a sector of the circle with centre O. Angle AOB is α. Find an expression for OB in terms of y and α.[1]
- (b)(ii)Hence show that the area of the shaded region can be written as (y²/2)cosα(2sinα - αcosα).[3]
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 2023). The mark or mark type is shown in brackets.
- [M1] (1/2)×24² × θ = 432
- [A1] θ = 3/2 rads soi
- [M1] 24 × their θ
- [A1] 36 cao
- [B1] [OB =] 2y cos α oe
- [M2] (their 2y cos α)× y sin α / 2 - (1/2)×(their y cos α)² × α oe
- [A1] correct completion to y²/2 cos α (2 sin α - α cos α)
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