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 : X is a collection of certain odd numbers whereas Y is a collection of certain even numbers. T consists of the numbers all of which are either from X or from Y. Is every number of T from Y? Statement I : The sum of any two numbers belonging to T is even. Statement II : If both p and q are picked from T, then (p - 1)q is even.

(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 ✓ UPSC's answer

Why the answer is (d)

• Statement I: the sum of any two numbers of T is even, so all numbers of T have the same parity. T could be all odd (from X) or all even (from Y). Not sufficient.

• Statement II: (p − 1)q is even for all p, q in T. If T had both an even p and an odd q, (p − 1)q would be odd × odd = odd, which is not allowed. So again T is all odd or all even, and both kinds satisfy the condition. Not sufficient.

• Together the two statements say the same thing (one parity throughout) and still allow an all-odd T.

• So the question cannot be answered even using both statements: (d).

Why the other options are wrong

(a) Select this option if the question can be answered using one of these statements alone…: Neither statement alone rules out an all-odd T.
(b) Select this option if the question can be answered using either statement alone: Neither statement alone is sufficient.
(c) Select this option if the question can be answered using both the statements together…: Together they still allow T to be entirely odd, so the answer stays open.

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.