UPSC Prelims 2026 CSAT, question 4

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 : $X$ 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 statementsCorrect answer

Solution.

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

Statement I alone is not sufficient: the sum of any two numbers in $T$ is even if all numbers in $T$ are odd (from $X$ , e.g. $\{3, 5\}$ ) or if all numbers in $T$ are even (from $Y$ , e.g. $\{2, 4\}$ ).
Statement II alone is not sufficient: $(p - 1)q$ is even for all $p, q \in T$ if every number in $T$ is odd (making $p - 1$ even) and also if every number in $T$ is even (making $q$ even).
Even using both statements together, $T$ can consist entirely of odd numbers from $X$ or entirely of even numbers from $Y$ , so neither statement nor both together can answer the question.

Trap: When every element of $T$ is odd, $p - 1$ is always even, so $(p - 1)q$ is automatically even.