UPSC Prelims 2026 CSAT, question 3

Topic: Data sufficiency

Directions · Q1 to Q5

Directions for the following 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.
(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?

Statement I : $x^2y^2$ is even.

Statement II : $1+x^2+y^2$ is odd.

(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 aloneCorrect answer
(d) Select this option if the question cannot be answered even using any of the statements

Solution.

Answer: (c) Select this option if the question can be answered using both the statements together, but cannot be answered using either statement alone

Statement I alone is not sufficient: $x^2 y^2$ is even whenever at least one of $x$ or $y$ is even, so $(x, y) = (1, 2)$ gives odd $x$ while $(x, y) = (2, 1)$ gives even $x$ .
Statement II alone is not sufficient: $1 + x^2 + y^2$ is odd when $x^2 + y^2$ is even, meaning $x$ and $y$ have the same parity (both odd or both even).
Combining Statements I and II, $x$ and $y$ have the same parity and at least one of them is even, which forces both $x$ and $y$ to be even.