A Question is given followed by two Statements I and II. Consider the Question and the Statements.
Question: Is (x + y) an integer?
Which one of the following is correct in respect of the above Question and the Statements?
The Question asks whether (x + y) is an integer.
Statement-I: (2x + y) is an integer.
Consider an example: If 2x + y = 3. If x = 1, then y = 1, making x + y = 2 (an integer). However, if x = 1.5, then y = 0, making x + y = 1.5 (not an integer). Therefore, Statement-I alone is insufficient to determine if (x + y) is an integer.
Statement-II: (x + 2y) is an integer.
Consider an example: If x + 2y = 3. If y = 1, then x = 1, making x + y = 2 (an integer). However, if y = 1.5, then x = 0, making x + y = 1.5 (not an integer). Therefore, Statement-II alone is insufficient to determine if (x + y) is an integer.
Combining Statement-I and Statement-II:
Adding these two expressions: (2x + y) + (x + 2y) = 3x + 3y = 3(x + y).
Since the sum of two integers is an integer, 3(x + y) must be an integer. Let 3(x + y) = K, where K is an integer. This implies .
The expression may or may not be an integer. For instance, if K = 3, then x + y = 1 (an integer). If K = 4, then (not an integer).
Thus, even by using both statements together, it cannot be definitively determined whether (x + y) is an integer.
Options 1, 2, and 3 are incorrect because, as demonstrated above, neither statement alone nor both statements combined are sufficient to answer the question.