The volume of a solid circular cylinder of radius r cm is 250π cm³.

Mark scheme

Marking points from the official mark scheme (May/June 2013). The mark or mark type is shown in brackets.

1. [1] M1 For (i): Makes h the subject from the volume formula (πr²h = 250π → h = 250/r²)
2. [1] M1 For (i): Substitutes h into the surface area formula (S = 2πrh + 2πr²) to obtain the given expression (AG - check all formulae)
3. [1] B1 For (ii): Correct derivative of the first term of S (dS/dr = 4πr)
4. [1] B1 For (ii): Correct derivative of the second term of S (-500π/r²)
5. [1] M1 For (ii): Sets dS/dr = 0 and attempts to solve for r (4πr - 500π/r² = 0 → r³ = 125 → r = 5)
6. [1] A1 For (ii): Correct stationary value of S = 150π (at r=5)
7. [1] M1 For (iii): Finds the second derivative d²S/dr² (d²S/dr² = 4π + 1000π/r³)
8. [1] A1 For (iii): Determines the nature as Minimum (since d²S/dr² is positive at r=5)