CS 250 Homework  - Undecidable Problems

Show each of the following problems is undecidable:

1.  For an arbitrary Turing machine M and input w, does M revisit the starting cell during the course of the computation?

2. For an arbitrary Turing machine M and input w, does M use a finite amount of tape on input w?

3. For an arbitrary Turing machine M, does L(M) contain any strings of even length?