UPSC CSE Prelims
Numerical Problems Previous Year Questions (PYQs)
Practice solved questions for Numerical Problems with detailed step-by-step solutions, key insights, and trend analysis for UPSC CSE PRELIMS.
Solved Previous Year Questions
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$X$ 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.
Detailed Explanation:
Statement I: Sum of any two numbers in T is even → all numbers have same parity (all even OR all odd). Example: T = {2,4} (all from Y) or T = {1,3} (all from X). Insufficient alone.
Statement II: (p-1)q is even for any p,q from T → satisfied when all are even (q even) OR all are odd (p-1 even). Example: T = {2,4} or T = {1,3}. Insufficient alone.
Both together: Both statements allow T to be entirely even (every number from Y) OR entirely odd (no number from Y). Cannot determine definitively.
Therefore, the question cannot be answered even using both statements, making Option 4 the correct answer.
$X$ 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 $5$.
Detailed Explanation:
List all possible combinations of three different coins: 1+2+5=8, 1+2+10=13, 1+2+20=23, 1+5+10=16, 1+5+20=26, 1+10+20=31, 2+5+10=17, 2+5+20=27, 2+10+20=32, 5+10+20=35.
Statement I: Non-prime values are 8, 16, 26, 27, 32, 35—includes 32 (2+10+20) which lacks denomination 5, so insufficient. Statement II: Values with digit sum >5 are 8, 16, 17, 26, 27, 35—all include denomination 5, so sufficient.
Therefore, Statement II alone answers the question but Statement I does not, making Option 1 the correct answer.
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Common questions about Numerical Problems in UPSC CSE PRELIMS