The important idea is that more than one death can occur in a week. Hence, in order to tackle question (a) and (b), the probability of losing a particular life over the time period must be found.

(a) For the cat to die during the next 10 weeks, it must have lost all 9 lives by then. Hence, the probability of the cat not losing a particular life is $$(0.8)^{10}$$, and correspondingly, the probability that the cat loses a particular life is $$1-(0.8)^{10} = 0.89263$$

Hence, the probability of losing all 9 lives over the next 10 weeks is $$(0.89263)^9 = 0.360$$

(b) By using similar logic to above, we find the probability that the cat is dead at the end of the 5th week is $$(1-0.8^{5})^9 = 0.02807$$

and continuing, we find the probability that the cat is dead at the end of the 4th week is $$(1-0.8^{4})^9 = 0.00872$$

Hence, we find the probability that the cat dies during the 5th week is equal to $$P(\text{Dead at the end of week 5})-P(\text{Dead at the end of week 4}) = 0.02807-0.00872 = 0.019$$