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**.