Answers

I.
For A->C to be false, A must be true and C false. Since A is true, for the first premise to be true, B must be true. Since C is false, the second premise is true independently of whether D is true or false.  Hence, here's an assignment showing invalidity:
A: true
B: true
C: false.

II.
For RvT to be false, both R and T must be false.  If R is false, then the first premise is true; and if T is false, the second premise is true. Hence here's an assignment showing invalidity:
R: false
T: false
Note that the assignment of s doesn't matter.

III.
A<->C can be false in two (mutually exclusive) ways:

  1. A is true and C is false
  2. A is false and C is true.
Let's try the first alternative.  If A is true, then for the first premise to be true, (BvC) must be true.  Since I notice that B is the antecedent in the second premise, I'll suppose that B is false and C is true (the reason for doing this is that this leaves open the possibility for (D&E) to be true or false as I might need later on.  In other words, generally it pays to give a truth assignement to a letter as late as possible).  Since B is false, the second premise is true.  Since the third premise has D as an antecedent, and I have not committed D yet, I'll say now that D is false.  Hence, the third premise is true.  Since A is true, the only way for the fourth premise to be true is for F to be false, and fortunately for me, I have not committed F yet.  So, I'll say that F is false.  Hence, here's an assignment:
A: true
B: false
C: true
D: false
F: false
Note that E's assignment doesn't matter.

IV.
H&J  can be false in more than one way.  Let's try H false.  The first premise also can be true in more than one way.  Let's say that G is true and F false.  Then, the second premise will be true.  Since G is true, for the third premise to be true, J must be true.  Here's the assignment:
F: false
G: true
H: false
J: true