A Question is given followed by two Statements I and II. Consider the Question and the Statements.

Question: Is (x + y) an integer?

Statement-I: (2x + y) is an integer.

Statement-II: (x + 2y) is an integer.

Which one of the following is correct in respect of the above Question and the Statements?

Correct Option: (d) The Question cannot be answered even by using both the Statements together

We need to determine if (x + y) is an integer. From Statement-I: (2x + y) is an integer. This is not sufficient. For example, if x = 0.5 and y = 0, then 2x + y = 1 (an integer), but x + y = 0.5 (not an integer). Alternatively, if x = 1 and y = 1, then 2x + y = 3 (an integer) and x + y = 2 (an integer). Thus, Statement-I alone is insufficient. From Statement-II: (x + 2y) is an integer. Similarly, if x = 0 and y = 0.5, then x + 2y = 1 (an integer), but x + y = 0.5 (not an integer). Thus, Statement-II alone is insufficient. Combining both statements: Let 2x + y = I_1 and x + 2y = I_2, where I_1 and I_2 are integers. Adding these equations gives 3x + 3y = I_1 + I_2, which simplifies to x + y = (I_1 + I_2) / 3. Since the sum of two integers (I_1 + I_2) is not necessarily a multiple of 3, (x + y) can be a fraction (e.g., if I_1 = 1, I_2 = 1, then x + y = 2/3, which is not an integer). Therefore, even with both statements, we cannot determine if (x + y) is an integer. Hence, the correct option is (d).