UPSC Prelims 2026 CSAT, question 1

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 : $X$ receives three coins of different denominations : $1,2,5,10$ and $20$ . If the total amount received by $X$ is $m$ , does $X$ receive a coin of denomination $5$ ?

Statement I : $m$ is not a prime number.

Statement II : The sum of the digits of $m$ is greater than $5$ .

(a) Select this option if the question can be answered using one of these statements alone, but cannot be answered using other statementCorrect answer
(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

Solution.

Answer: (a) Select this option if the question can be answered using one of these statements alone, but cannot be answered using other statement

The 10 possible three-coin sums $m$ from $\{1, 2, 5, 10, 20\}$ are $8, 16, 17, 26, 27, 35$ (which include coin $5$ ) and $13, 23, 31, 32$ (which exclude coin $5$ ).
Statement I alone is not sufficient: if $m$ is not prime, $m$ can be $8 = 1+2+5$ (includes $5$ ) or $32 = 2+10+20$ (excludes $5$ ).
Statement II alone is sufficient: the digit sums for the four combinations without coin $5$ are $4, 5, 4, 5$ (all at most $5$ ), whereas every combination with coin $5$ has digit sum $7, 8,$ or $9$ (strictly greater than $5$ ).

Trap: Check $2 + 10 + 20 = 32$ under Statement I: $32$ is composite but does not use coin $5$ , so Statement I alone fails.