Data Sufficiency

If the question is asking "is sqrt(x) an integer?", what it's really asking is - is x a perfect square? If it is a square, we'll get an integer for a root. If not, we won't.

Statement (1) tells us that sqrt(4x) is an integer. In other words, 4x is a perfect square. Since 4 is already a perfect square, that means that x would also have to be a perfect square to get a product that is also a perfect square. For example, (4)(9) = 36. Square * square = square. If x is any non-square integer, then sqrt(4x) would not be an integer. Sufficient.

If, as you suggest, x = 1, that would satisfy the statement. 1 is a perfect square, as is (4)(1). If x = 1, then sqrt(1) is definitely an integer, so this is sufficient.

Statement (2) tells us that sqrt(3x) is not an integer. This means that 3x is not a perfect square. But does that necessarily mean that x is not a perfect square? x could be a non-square, such as 5: sqrt(3*5) = sqrt(15), which is not an integer. This satisfies the statement, and gives us a "no" answer - sqrt(5) is not an integer.

But could x be a square? Try x = 1. sqrt(3*1) = sqrt(3), which is not an integer. This satisfies the statement, but gives us a "yes" answer to the question - sqrt(1) IS an integer. Insufficient.

Choosing x=1 does not change the fact that statement (1) is sufficient and (2) is not. The answer is A.