Hm....
1. There are infinitely many primes whose last digit is 1.
I think this is true, but I don't remember how to prove it. IIRC there is a theorem that every arithmetic sequence where the starting value and the step size are relatively prime contains infinitely many primes.
2. There are infinitely many primes whose last two digits are 35.
False: every such number is divisible by 5.
3. There are infinitely many integers n such that n, n + 2, and n + 4 are all prime.
No, only n=3 has this property. Every such sequence includes a number divisible by 2 or 3.
4. For every positive integer n, the number n2 -79n +1601 is prime.
False: if n=80, then n^2 -79n +1601=1681=41^2
5. If the square root of 36n2 - 12n + 1 is an integer m, then m will be prime.
False: 36 n^2 - 12 n + 1=(6n-1)^2 is always a square. If n=6, then m=35 is not prime.