Directions for the next 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. 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.

To determine whether every number in set T belongs to set Y (which contains exclusively even numbers), we must establish if T is composed entirely of even numbers. According to fundamental rules of parity in number theory, we evaluate both statements:

Statement I: This states that the sum of any two numbers in T is even. Mathematically, the sum of two even integers is even (e.g., 2 + 4 = 6), and the sum of two odd integers is also even (e.g., 3 + 5 = 8). However, a mixed set would yield an odd sum (e.g., 2 + 3 = 5). Therefore, set T must be purely all-even (a subset of Y) or purely all-odd (a subset of X). Because it could be either, we cannot definitively answer the core question. Statement I is insufficient.

Statement II: This states that for any p and q picked from T, the product (p-1)q is even. If T consists solely of even numbers, q is even, making the product even. Conversely, if T consists solely of odd numbers, p is odd; thus, (p-1) is an even number, which again makes the entire product even. Therefore, Statement II alone is also insufficient to distinguish between an all-odd and all-even set.

Combining Both Statements: Even when taken together, both the 'all-even' and 'all-odd' scenarios perfectly satisfy all given conditions. Because the ambiguity remains, the question cannot be answered even using both statements together.

Why other options are incorrect:

• Option A: Incorrect because Statement I fails to eliminate the possibility of T being an all-odd set.
• Option B: Incorrect because Statement II similarly fails to definitively isolate T as an all-even set.
• Option C: Incorrect because combining the statements does not provide new limiting parameters to resolve the ambiguity.

Takeaway: In Data Sufficiency questions involving number systems and parity, always test opposite extremes (e.g., purely odd vs. purely even sets). If both scenarios satisfy the given conditions, the data is definitively insufficient.