Introduction:
The integral ∫ E.dl is known as the line integral of the electric field E along a closed path, and it represents the work done by the electric field in moving a unit positive charge around the closed path.

Answer:

Case a) Electric field caused by a static charge:
When the electric field is caused by a static charge, the line integral ∫ E.dl is not zero. This is because the electric field lines originate from the positive charge and terminate at the negative charge. When a positive charge is moved along a closed path, the electric field does work on it, resulting in a non-zero line integral. Therefore, the answer is not (a).

Case b) Electric field caused by a time-varying magnetic field:
When the electric field is caused by a time-varying magnetic field, the line integral ∫ E.dl is not zero. This is because a time-varying magnetic field induces an electric field according to Faraday's law of electromagnetic induction. This induced electric field can do work on a charge moving along a closed path, resulting in a non-zero line integral. Therefore, the answer is not (b).

Case c) Electric field caused by both a static charge and a time-varying magnetic field:
If the electric field is caused by both a static charge and a time-varying magnetic field, the line integral ∫ E.dl can be non-zero or zero depending on the specific scenario. If the contributions from the static charge and the time-varying magnetic field cancel each other out along the closed path, then the line integral can be zero. However, in most cases, the line integral will not be zero as either the static charge or the time-varying magnetic field will dominate. Therefore, the answer is not (c).

Case d) Electric field caused by none of the above:
If the electric field is not caused by a static charge or a time-varying magnetic field, then it is possible for the line integral ∫ E.dl to be zero. This could occur if the electric field is conservative, which means that the work done by the electric field in moving a charge around a closed path is path-independent. In such cases, the line integral will be zero. Therefore, the answer can be (d) if the electric field is conservative and not caused by a static charge or a time-varying magnetic field.

Conclusion:
In conclusion, the integral ∫ E.dl can be zero if the electric field is conservative and not caused by a static charge or a time-varying magnetic field. In all other cases, the line integral will be non-zero.