So the answer to the first problem is actually quite simple, but you have to be careful that you aren't tricked! Many of you probably thought out this variation at first, which looks completely right:

However, the key lies in white's throw in at 2 preventing black's atari at so it turns out to be ko if black plays at 1 below.

So that must mean our answer is the hane on the other side, which is absolutely correct.

The next problem is a bit more interesting, with this move below probably being the first instinct to recconect the black stones.

It even seems that the first instict may actually work with the throw in at 1:

But all is not what is seems as white is able to just capture all the stones with no problem at all:

So after looking over this problem again, this problem just seems to be a simple case of re-ordering. So the solution is to play at black 1, in the diagram below instead:

This move prevents white's 2 stones from connecting back up with his group without leaving a gap underneath. In the 2 variations played below, we can very clearly see that white cannot block black from connecting, and so we have our solution.

Hopefully this was helpful! Please correct me if you spot mistakes!

Thank you!