These were my answers:
1. Show that...
2.(a) 2g/sqrt(3) = 11.3 (3 s.f.)
2.(b) omega = 2.51 rad s^-1 (3 s.f.)
3. Distance = R/5
4. k = 1.72 (3 s.f.)
EDIT: I made a silly mistake - the answer was actually k = 1.86.
5.(a) Prove centre of mass = 3r/8 from O (I couldn't remember how to do it, FFS!)
5.(b) Distance from O = Modulus of {3a(16-k)}/{16(8+k)}
5.(c) k = 16
6.(a) Show that...
6.(b) Minimum u = sqrt(3ag)
6.(c) u = sqrt(5ag)
7.(a) Show that l = 1.2
7.(b) Prove SHM (acceleration = -49x)
7.(c) v = 2.42 ms^-1 (3 s.f.)
7.(d) Time taken = 0.547 seconds (3 s.f.)