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 states that the sum of any two numbers in T is even, which implies all numbers in T have the same parity (all odd or all even), but does not specify which parity, so it cannot determine if T consists solely of even numbers (Y).",
"- Statement II states that for any p, q in T, (p - 1)q is even; if p is odd, (p-1) is even, making the product even regardless of q, but if p is even, q must be even, leading to ambiguous cases where T could contain odd numbers or be all even, thus failing to provide a definitive answer.",
"- Since neither statement alone nor together can definitively prove that every number in T is from Y (even numbers), the question cannot be answered with the given information.",
"- Therefore, the correct option is (d), as the data is insufficient to answer the question.
Why the other options are wrong
- (a) Select this option if the question can be answered using one of these statements alone…
- Option (a) is incorrect because neither statement alone is sufficient to answer the question definitively.
- (b) Select this option if the question can be answered using either statement alone
- Option (b) is incorrect because neither statement alone provides enough information to determine the parity of all elements in T.
- (c) Select this option if the question can be answered using both the statements together…
- Option (c) is incorrect because even combining both statements does not eliminate the possibility of T containing odd numbers, so the question remains 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.