UPSC CSE Prelims
Data Sufficiency Previous Year Questions (PYQs)
Showing solved Previous Year Questions for Chapter: Data Sufficiency
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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.
For two distinct real numbers $x$ and $y$, which of them is bigger? Statement I : $x^2 < y < 1$ Statement II : $y < \sqrt{x} < 1$
Detailed Explanation:
Statement I alone: x² < y < 1 allows y to be anywhere between x² and 1. For x = 0.5, y = 0.3 gives x > y; for x = 0.5, y = 0.9 gives y > x. Insufficient.
Statement II alone: y < √x < 1 requires x ∈ (0,1) but doesn't fix the relationship. For x = 0.25, y = 0.4 gives y > x; for x = 0.81, y = 0.1 gives x > y. Insufficient.
Both together: x² < y < √x with x ∈ (0,1). Since x² < x < √x for x ∈ (0,1), y can still lie on either side of x. Examples: x = 0.25, y = 0.1 → x > y; x = 0.25, y = 0.4 → y > x. Both conditions are satisfied but the relationship remains ambiguous.
Therefore, the question cannot be answered even using both statements, making Option 4 the correct answer.
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The top of a table is rectangular and its dimensions are $6' \times 10'$. Two rectangular portions of the table top are painted in blue colour; both these portions have dimensions $2.5' \times 8'$ and each of them has exactly two sides common with two edges of the table top. If the table is fixed to the ground and the remaining portion of the table top is painted in white, how many different patterns are possible when observed from above?
Detailed Explanation:
The table is 6' × 10' (fixed orientation). Each blue rectangle is 2.5' × 8' with exactly two sides common with table edges—meaning each blue rectangle must occupy a corner. The 8' side must align with the 10' edge, and the 2.5' side with the 6' edge.
There are 4 corner positions. Since two blue rectangles are placed, we evaluate valid non-overlapping pairs: Adjacent corners on the same long/short side overlap (bottom-left + bottom-right, or top-left + top-right). Opposite or diagonal corner pairs do not overlap: (bottom-left + top-left), (bottom-right + top-right), (bottom-left + top-right), (bottom-right + top-left).
Therefore, 4 distinct non-overlapping patterns are possible, making Option 2 the correct answer.