You sound like you almost got it... hmmm, did you screw up somewhere and position P and Q wrongly? Q should be on top while P below... i think.
EDIT: Hah... managed to solve it finally :P
This works :P
For my solution now:
Consider forces parallel to string for Q
CF1 sin(alpha) - T - 2mg cos(alpha) = 0 (forces balance)
2m(w^2)(QB sin(alpha)) sin (alpha) - T - 2mg cos(alpha) = 0
(2 QB) m(w^2) sin^2(alpha) - T - 2mg cos(alpha) = 0
Consider forces parallel to string for P
CF2 sin(alpha) + mg cos(alpha) - T = 0 (forces balance)
m(w^2)(BP sin(alpha)) sin (alpha) + mg cos(alpha) - T = 0
(BP) m(w^2) sin^2(alpha) + mg cos(alpha) - T = 0
Solving simultaneously
(BP) m(w^2) sin^2(alpha) + mg cos(alpha) = (2 QB) m(w^2) sin^2(alpha) - 2mg cos(alpha)
m(w^2)(2QB - BP) sin^2(alpha) = 3mg cos(alpha)
w^2 = {3g cos(alpha)} / {(2QB - BP) sin^2(alpha)}
Pretty close now, just need to figure out the inequality:
2QB - BP = 3QB - length of string ; given as 'a' in question
since QB < a
3QB < 3a
3QB - a <2a>
Therefore, {(3QB - a)^ -1} > {(2a)^ -1}
=> w^2 > {3g cos(alpha)} / {2a sin^2(alpha)}
For part (ii)
when (w^2) --> infinity,
{3g cos(alpha)} / {(2QB - BP) sin^2(alpha)} --> infinity
hence denominator --> zero
Therefore, 2QB - BP --> zero <=> BP/QB ~= 2
Suddenly got the inspiration to figure it out :P hope its understandable