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. ✓ UPSC's answer
Why the answer is (d)
• The equation is $PP \times PQ = RRSS$, which translates to $11P \times (10P + Q) = 1100R + 11S$.
• Dividing both sides by 11 gives $P(10P + Q) = 100R + S$.
• Since $P$ and $Q$ are non-zero digits with $P \le 3$ and $Q \le 4$, we test possible values for $P$.
• If $P=1$, $10+Q = 100R+S$. Since $Q \le 4$, max LHS is 14, so $R=0$, but $R$ must be non-zero. Invalid.
• If $P=2$, $2(20+Q) = 40+2Q = 100R+S$. Max LHS is 48, so $R=0$. Invalid.
• If $P=3$, $3(30+Q) = 90+3Q = 100R+S$. Since $Q \le 4$, $90 \le 90+3Q \le 102$.
• For $R=1$, $100+S = 90+3Q \Rightarrow 10 = 3Q - S$. Testing $Q \in \{1,2,3,4\}$: If $Q=4$, $12-S=10 \Rightarrow S=2$. This yields distinct non-zero digits $P=3, Q=4, R=1, S=2$.
• Thus, $Q=4$ is uniquely determined by the constraints in the question alone, making the statements unnecessary.
Why the other options are wrong
- (a) The Question can be answered by using one of the Statements alone, but cannot be answered…
- The question can be answered without any statements, so it is not dependent on one statement alone.
- (b) The Question can be answered by using either Statement alone.
- The question can be answered without any statements, so it is not dependent on either statement alone.
- (c) The Question can be answered by using both the Statements together, but cannot be…
- The question can be answered without any statements, so it does not require both statements together.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2025, held on 25 May 2025. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.