Ok, to put it very simply, since vector A is along the z axis, its definition will be some magnitude in the z direction, with no magnitude in x or y. A = mk. Taking the cross product of this vector with any vector in the same direction will equal zero. Or alternatively, taking a cross product with any vector that is not in the same direction will result in a non-zero answer. So seeing that a) b) and c) all have some component of i or j, a cross product with any of them will result in a non-zero answer. A cross product with any vector that is a scalar multiple of the original vector (same direction) is zero, so if vector A was a little more complicated than just being along the z-axis, the problem would still be quite simple to quickly deduce the answer.
If you were a gambling person, and didn't know if they were referring to the cross product or the dot product in this question, or if you didn't know that the cross product is zero when parallel or perpendicular, you can deduce that by seeing more than one answer was perpendicular, one was neither perpendicular nor parallel, and only one was parallel, you can infer that d) would be your correct answer.
