Hi Kunjesh,
Yes, you're right, with that formula, we must have lambda > 1/60, or mu could go negative.
We can see this make sense again when we answer part (ii)
The question isn't very good because it doesn't tell us whether it is an annual force of mortality or a daily one. (The English in it is pretty bad too, it's missing the word "the" in 2 places!) Let's assume the forces are daily...
We have:
90q0 = 0.95
90p0 = 0.05
30p0 *60p30 = 0.05
exp{-0.03*30}*exp{-integral from 0 to 60 of (-0.5 + lamda*x dx)} = 0.05
exp(-0.9)*exp{-(-0.5x + 0.5 lambda*x^2) evaluated between 0 and 60} = 0.05
exp(-0.9)*exp{30 - 1800 lambda} = 0.05
So 60p30 = exp{30 - 1800 lambda} would be negative if lambda < 1/60. I think the question is fine but it would have been helpful to state that lambda > 1/60 or make this a part of the question,
Good luck!
John