The GMAT almost never even mentions 'mode', and in the very rare questions where it does, all you need to know is its definition. As this question is designed, we need to be concerned about all kinds of potential technicalities -- how do we answer if the list might have no mode (which would be true if the set were 1, 2, 3, 4, say)? What if it has two modes?
Here, from Statement 1, if the product of two elements in the list is never positive, we can't possibly have two (or more) positive numbers in the list. Nor can we have two (or more) negative numbers in the list. So we have at most one positive, and at most one negative, and since the rest of the numbers can only be zero, we must have two, three, or four 0's in the list, and 0 must be the mode.
From Statement 2, we know every number in the list is the same, so a (or b, or c, or d) is the mode, but we don't have a numerical value for a, so we can't answer the question, and the answer is A.
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