Directions for the next 5 (five) items : Each item in this section contains a question followed by two statements. Answer each item using the following instructions and mark your response on the Answer Sheet accordingly. Question : If x and y are integers, then is x even? Statement I : x²y² is even. Statement II : 1 + x² + y² is odd.

Why the correct option is correct: According to the fundamental rules of parity in number theory, the parity of an integer (whether it is even or odd) determines the parity of its square (x is even if and only if x² is even).

• Evaluating Statement I: The expression x²y² is even. For a product to be even, at least one of its factors must be an even integer. Thus, either x is even, y is even, or both are even. We cannot definitively say if x is even (e.g., if x=3 and y=2, x²y² = 36 which is even, but x is undeniably odd). Thus, Statement I alone is insufficient.
• Evaluating Statement II: The expression 1 + x² + y² is odd. Subtracting the odd number 1 from this expression dictates that the sum x² + y² must be even. The sum of two integers is even if and only if both share the exact same parity (either both are even, or both are odd). Thus, x could still be either even or odd. Statement II alone is insufficient.
• Combining Both Statements: Statement II requires x and y to have identical parity. Statement I mandates that at least one of them must be even. The only mathematical way to satisfy both conditions simultaneously is if both x and y are even integers. Therefore, we can definitively answer the question: "Yes, x is even." Since both statements are required to deduce this, Option C is the correct answer.

Why the wrong options are incorrect:

• Option A is incorrect: Statement I alone cannot determine if x is the even variable; y could be the sole even integer.
• Option B is incorrect: Neither statement independently answers the question, as Statement II alone leaves the possibility that both integers are odd.
• Option D is incorrect: Combining the two statements successfully eliminates all alternative possibilities except x being even, meaning the question can indeed be answered.

Concluding Takeaway: Parity Rules: For UPSC CSAT Data Sufficiency, internalize standard parity rules: Odd + Odd = Even and Even × Odd = Even. Always check if a mathematical operation allows one variable to mask the parity of the other!