1. Another agreement
(a)to sign
(b)to be signed
will concern technical co-operation.
**Both are correct. The meaning is very close. The first one is active voice, and the second one is passive voice. Active voice brings more attention to the one doing the action (the one signing the agreement), and passive voice brings more attention to the agreement being signed. I believe 'b' sounds a little more natural and is more common in this case.
2. The greatest thing
(a) to remember.
(b)to be remembered
is this.
**This one is very similar to #1. 'a' is active voice, and 'b' is passive voice. In this case, we want to bring more attention to the 'thing' being remembered, so I believe 'b' is better.
3. The action
(a)to take
(b) be taken
is correct.
**Same as #2. 'b' is better because it brings attention to the 'action.' Just make sure to include 'to' before 'be taken.' "The action to be taken is correct."
4. That is a nice place
(a) to visit.
(b) to be visited.
**Both of these are technically correct, and the meaning is very close. However, I recommend 'a' because it sounds more natural than 'b.'
5. These are things
(a) to wear.
(b) to be worn.
**Again, both are technically correct, and the meaning is very close. I believe 'a' sounds better. One would usually say, "I have a lot of clothes to wear" rather than "I have a lot of clothes to be worn."
5. He is not a man
(a) to trust.
(b) to be trusted.
**Very similar. Both are correct, but 'b' sounds more natural. This is because 'b' brings more attention to the fact that we should not trust this man.
6. She was the first woman
(a) to have elected
(b) to have been elected
to such a new leading post.
**'a' is incorrect. The woman did not do the electing. She was elected by the people. 'b' is passive voice showing that the woman was elected to the post.