Q. A question is given followed by two Statements I and II. Consider the Question and the Statements and mark the correct option.
Question: Let P, Q, R, S be distinct non-zero digits. If PP × PQ = RRSS, where P ≤ 3 and Q ≤ 4, then what is Q equal to?
Statement I: R = 1.
Statement II: S = 2.
Which one of the following is correct in respect of the above Question and the Statements?
(a) The Question can be answered by using one of the Statements alone, but cannot be answered using the other statement alone.
(b) The Question can be answered by using either Statement alone.
(c) The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone.
(d) The Question can be answered even without using any of the Statements.
Correct Answer : d)
UPSC Prelims 2025 CSAT
Explanation :
We need to find digits P,Q,R,S (all nonzero and distinct), with P≤3, Q≤4, such that
PP × PQ = RRSS
(where “PP” means the two-digit number with both digits P, etc.).
Step 1. List possible values of PP (since P≤3 and nonzero):
P=1 → PP=11
P=2 → PP=22
P=3 → PP=33
Step 2. For each P, try Q (1 through 4, but Q≠P) and compute PP×PQ. We need a 4-digit product of the form RRSS.
1) P=1 → PP=11, PQ=10+Q gives 11×11=121 up to 11×14=154, all 3-digit. No good.
2) P=2 → PP=22, PQ=20+Q gives 22×21=462, 22×23=506, 22×24=528, all 3-digit. No good.
3) P=3 → PP=33, PQ=30+Q:
33×31 = 1023 (not of form RRSS)
33×32 = 1056 (no)
33×34 = 1122 → this is “RRSS” with R=1, S=2, and all digits {3,4,1,2} are distinct.
No other choice works. Hence the only solution is P=3, Q=4, R=1, S=2.
Thus Q=4 and it is uniquely determined from the question itself, without using either statement.
Therefore the correct answer is (d):
“The Question can be answered even without using any of the Statements.”
