9.12. For each decision problem below, determine whether it is decidable or undecidable, and prove your answer.

a. Given a TM T, does it ever reach a state other than its initial state if it starts with a blank tape?

b. Given a TM T and a nonhalting state q of T, does T ever enter state q when it begins with a blank tape?

c. Given a TM T and a nonhalting state q of T, is there an input string x that would cause T eventually to enter state q?

d. Given a TM T, does it accept the string in an even number of moves?

e. Given a TM T, is there a string it accepts in an even number of moves?

f. Given a TM T and a string w, does T loop forever on input w?