There Exists

Which one is correct?

1. (a) There exist multiple methods...
1. (b) There exists multiple methods...

2. (a) There exists a solution...
2. (b) There exist a solution...

For 1 and 2, does the form of exist/exists depend on the singular/plural thingy that follows?

3. (a) There are many answers, where apple is one of them.
3. (b) There are many answers, where an apple is one of them.

4. (a) The machine is capable of performing....
4. (b) The machine is capable in performing...

5. (a) The paper is organized as follows.
5. (b) The paper is organized as follow.

Thanks.