The main difficulty with such questions is applying correct logic. The options could be true for some cases, but we must decide if they are necessarily true for all cases.
The line integral $W=\int_O^C F.ds$ is the work done on the particle by the external force. By the Work-Energy Theorem this equals the change in kinetic energy of the particle. We are told that there is no change in the KE of the particle for any of the 3 paths OAC, OBC, OPC. So for each of these paths the work done is $W=0$.
Therefore option 1 is correct : at least 3 such paths exist along which the line integral of force is zero.
A conservative force is one for which the work done between two points such as O and C is independent of the path taken. This is true for the 3 paths identified, because the work done is zero for each. But we cannot assume that it is true for all other paths between O and C. Even if it is true for all paths between O and C we do not know if it is true for all paths between any other pairs of points. So option 2 is not necessarily true.
Conversely, it is possible that the work done could be zero for all paths between all pairs of point. We don't have enough evidence to decide this issue. So option 3 is not necessarily true either.
If we were told that C coincides with O then we would know that option 4 would be false, because we would know there are at least 3 closed paths along which the line integral is zero. But we don't know if C coincides with O, so we don't know whether the line integral is zero for any closed path. We cannot be sure whether option 4 is true or false. All we can conclude is that option 4 is not necessarily true.