Up to (d) I think no explanation is required. They are simple to understand that they are non regular.
e) a^n ; n is the product of two prime numbers.
OK, every non prime number can be represented as product of prime numbers you know.
For example 62=2*31, 52=2*2*13 etc... So you saw not necesserily only two prime numbers, In 52 you saw 3 prime numbers required.
So we can cover some numbers except prime numbers using product of prime numbers only.
Prime numbers & some non prime numbers can not be covered using any two prime numbers.
The language is Sigma*- {a^prime nums,a^(some non prime numbers) , a,epsilon} I cant see any pattern in them. Not regular.
(1 is a special num so I will not talk about it, you can subtract as per need)
PS: If it would be n is the product of any number of prime numbers, then also it is not regular...Though numbers like 52 can be covered using 3 prime numbers....but how to cover prime numbers itself ? They do not have any divisor)
f) a^n ; n is either prime or the product of two or more prime numbers.
Product of two or more prime numbers can cover all non prime numbers. Again we have prime numbers too. So all numbers are covered.
So language is Sigma*-{a,epsilon} = a^n, n>=2 Regular.
(PS: If you will just edit it & remove "or more" from the statement, then it will not be regular)
g) L* where L is qsn a) i.e a^n, n>=2 is a prime number.
(a^n, n>=2 is a prime number)*
n={2,3,5......}
(a^n)={ a^2,a^3, a^5......}
So (a^n)*={ epsilon,a^2,a^3, (a^2)(a^2),(a^5),(a^3)(a^3)......} So u can see it is clearly Sigma*-{a} = a^n, n!=1 Hence Regular.