UPSC CSE Prelims 2025

Correct Option (4)

The problem requires determining the value of Q from the equation PP × PQ = RRSS, given that P, Q, R, S are distinct non-zero digits, with P ≤ 3 and Q ≤ 4.

The given equation can be translated into an algebraic form:

• PP represents 11P.
• PQ represents 10P + Q.
• RRSS represents 1100R + 11S.

Substituting these into the equation yields: 11P × (10P + Q) = 1100R + 11S.

Dividing both sides of the equation by 11 simplifies it to: P(10P + Q) = 100R + S.

Now, we apply the given constraints:

• P is a non-zero digit and P ≤ 3, so P can be 1, 2, or 3.
• R is a non-zero digit, meaning 100R + S must be at least 100 (specifically, at least 101, as S is also a non-zero digit).

Let's test the possible values for P:

• If P = 1: The left side becomes 1 × (10 × 1 + Q) = 10 + Q. Since Q ≤ 4, the maximum value is 10 + 4 = 14. This is less than 100, so P cannot be 1.
• If P = 2: The left side becomes 2 × (10 × 2 + Q) = 2(20 + Q). Since Q ≤ 4, the maximum value is 2(20 + 4) = 48. This is also less than 100, so P cannot be 2.
• Therefore, P must be 3.

Substitute P = 3 into the simplified equation: 3(10 × 3 + Q) = 100R + S, which simplifies to 3(30 + Q) = 100R + S.

Next, consider the constraints for Q:

• Q is a non-zero digit, Q ≤ 4, and Q must be distinct from P (which is 3).
• Thus, Q can be 1, 2, or 4.

Let's test these possible values for Q:

• If Q = 1: 3(30 + 1) = 3(31) = 93. This is less than 100, so Q cannot be 1.
• If Q = 2: 3(30 + 2) = 3(32) = 96. This is less than 100, so Q cannot be 2.
• If Q = 4: 3(30 + 4) = 3(34) = 102. This value fits the form 100R + S, where R = 1 and S = 2.

The unique set of distinct non-zero digits that satisfies all conditions is P=3, Q=4, R=1, S=2. These values also satisfy P ≤ 3 and Q ≤ 4.

Since Q = 4 is uniquely determined using only the information provided in the question, neither Statement I (R=1) nor Statement II (S=2) is necessary to answer the question.

Incorrect Options:

Options 1, 2, and 3 are incorrect because the value of Q can be definitively established through a systematic analysis of the given equation and its constraints alone. The information provided in Statement I (R=1) and Statement II (S=2) is consistent with the derived solution but not required for its determination.