A circle, centre O and radius r cm, has a sector OAB of fixed area 10 cm². Angle AOB is θ radians and the perimeter of the sector is P cm.

Mark scheme

Marking points from the official mark scheme (May/June 2022). The mark or mark type is shown in brackets.

1. [3] B1 (for 1/2 r² θ = 10, θ = 20/r²)
2. M1 For substituting their θ in P (for [P=] 2r + rθ)
3. A1 (for [P=] 2r + 20/r)
4. [3] M1 For attempt to differentiate their answer to part (a) to obtain the form of dP/dr = 2 + a/r² (for dP/dr = 2 - 20/r²)
5. Dep M1 for equating their dP/dr to zero and attempt to solve
6. A1 cao (for When dP/dr = 0, r = √10)
7. [2] M1 for a complete method, allow valid alternatives, if differentiated, must be in the form of d²P/dr² = k/r^x (for d²P/dr² = 40/r³)
8. A1 for a correct conclusion (for As r is positive, d²P/dr² is also positive so minimum)
9. [1] B1 (for θ = 2)