UPSC Prelims 2026 CSAT Paper II · Q3 of 70 Comprehension medium

Passage

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 : x²y² is even. Statement II : 1 + x² + y² is odd.

  1. (a) Select this option if the question can be answered using one of these statements alone, but cannot be answered using other statement
  2. (b) Select this option if the question can be answered using either statement alone
  3. (c) Select this option if the question can be answered using both the statements together, but cannot be answered using either statement alone ✓ UPSC's answer
  4. (d) Select this option if the question cannot be answered even using any of the statements

Why the answer is (c)

• Statement I states that $x^2y^2$ is even, which implies that the product $xy$ is even, meaning at least one of $x$ or $y$ is even, but it does not specify which one, so $x$ could be odd or even.

• Statement II states that $1 + x^2 + y^2$ is odd, which implies that $x^2 + y^2$ is even, meaning $x$ and $y$ must have the same parity (both even or both odd), but it does not determine if they are specifically even or odd.

• Combining both statements: Since $x$ and $y$ have the same parity (from Statement II) and their product is even (from Statement I), they cannot both be odd (as the product of two odd numbers is odd).

• Therefore, $x$ and $y$ must both be even.

• Since $x$ is even, the question 'is x even?' can be answered affirmatively only when both statements are used together.

• Thus, the correct option is (c), as the question can be answered using both statements together but not either alone.

Why the other options are wrong

(a) Select this option if the question can be answered using one of these statements alone…
Statement I alone is insufficient because $x$ could be odd if $y$ is even, and Statement II alone is insufficient because $x$ could be odd if $y$ is also odd.
(b) Select this option if the question can be answered using either statement alone
Neither statement alone provides enough information to determine the parity of $x$ definitively; Statement I allows $x$ to be odd, and Statement II allows $x$ to be odd.
(d) Select this option if the question cannot be answered even using any of the statements
The question can be answered by combining the information from both statements, which forces $x$ and $y$ to be even, so it is not unanswerable.

Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2026, held on 24 May 2026. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.

Reading the answer is not the same as getting it right under a clock. Practise this question with UPSC's negative marking, and anything you miss goes into an error notebook until you get it right twice.

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