O-LevelAdditional MathematicsCircular measure4037/22 · May/June 2022 · Q97 Marks
In this question all lengths are in centimetres. [Figure 9]
- (a)In the case when the area of the sector OPQ is equal to the area of the shaded region, show that tan φ = 2φ.[4]
- (b)In the case when the perimeter of the sector OPQ is equal to half the perimeter of the shaded region, find an expression for tanφ in terms of φ.[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 2022). The mark or mark type is shown in brackets.
- [B1] [area sector =] 1/2 a²(2φ) oe or 1/2 a²φ oe
- [B1] [shaded area = ] 2 × 1/2 × 1/2 a(a tan φ) oe or a(a tan φ) - 1/2 a²(2φ) oe soi or [area kite =] 2a² or [area OPT=] a² nfww and PT = a tan φ oe, nfww
- [M1] Correct equation using correct areas e.g. a²φ = 1/2 a(a tan φ) or a(a tan φ) - a²φ = a²φ soi
- [A1] Correct completion to given equation tan φ = 2φ
- [M2] M1 for arc length = 2aφ soi or for PT = a tan φ and M1 for perimeter of sector = 2a + 2aφ and perimeter of shaded region = 2a tanφ (or equivalent expressions involving a and φ)
- [A1] tan φ = 2 + 2φ
Write your answer and get it marked
On Oracle Prep you can write or photograph your own answer and have it marked by AI against this mark scheme, with feedback on what you missed.
Sign up free →