Reasoning : Logical Reasoning & Analytical Ability

Q 8 / 325

UPSC CSE Prelims 2026

Directions: 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. (a) Select this option if the question can be answered using one of these statements alone, but cannot be answered using other statement (b) Select this option if the question can be answered using either statement alone (c) Select this option if the question can be answered using both the statements together, but cannot be answered using either statement alone (d) Select this option if the question cannot be answered even using any of the statements
Question: If x and y are integers, then is x even?
  1. Statement I: x2y2 is even.
  2. Statement II: 1+x2+y2 is odd.

EXPLANATION

The correct option is C - Select this option if the question can be answered using both the statements together, but cannot be answered using either statement alone.

[as per provisional answerkey]

Analysis

The question asks for a definitive "Yes" or "No" to the question: Is x even?

Statement I alone: x2y2 is even.
For the product of two squares to be even, at least one of the base integers must be even. This leads to three possibilities:
1. x is even and y is odd.
2. x is odd and y is even.
3. Both x and y are even.
Since x could be even (Case 1 and 3) or odd (Case 2), Statement I alone is not sufficient to determine if x is even.

Statement II alone: 1+x2+y2 is odd.
If (1+x2+y2) is odd, then (x2+y2) must be even (since Odd + Even = Odd).
For the sum of two squares to be even, both x and y must have the same parity. This leads to two possibilities:
1. Both x and y are even (Even^2 + Even^2 = Even).
2. Both x and y are odd (Odd^2 + Odd^2 = Even).
Since x could be even or odd, Statement II alone is not sufficient.

Both statements together:
From Statement II, we know x and y must have the same parity (both even or both odd).
From Statement I, we know at least one of them must be even.
Combining these: If at least one is even AND both must have the same parity, then both x and y must be even.
This provides a unique "Yes" to the question. Therefore, both statements together are sufficient.

Why the other options are incorrect

  • Option (a) - Answered by one statement alone: As shown above, Statement I allows x to be odd if y is even, and Statement II allows x to be odd if y is also odd. Neither provides a unique answer.
  • Option (b) - Answered by either statement alone: This is incorrect because both statements individually result in multiple possibilities for the parity of x.
  • Option (d) - Cannot be answered even using both: This is incorrect because the intersection of the conditions (same parity + at least one even) leaves only one logical possibility: x is even.

Key Concept

This question tests the parity rules of integers: the product of integers is even if at least one factor is even, and the sum of integers is even if they share the same parity (both even or both odd).