Q59. 59. 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?

We are given a multiplication puzzle: PP × PQ = RRSS, where P, Q, R, S are distinct non-zero digits, P ≤ 3, and Q ≤ 4. PP can be written as 11P. PQ can be written as 10P + Q. RRSS can be written as 1100R + 11S = 11(100R + S).

So, 11P × (10P + Q) = 11(100R + S), which simplifies to P × (10P + Q) = 100R + S. Let's test possible values for P (1, 2, 3) and Q (1, 2, 3, 4), ensuring P and Q are distinct: * If P = 1: Q can be 2, 3, 4.

The product 11 × (10+Q) would be 11×12=132, 11×13=143, 11×14=154. These are 3-digit numbers, not 4-digit RRSS. So P cannot be 1. * If P = 2: Q can be 1, 3, 4. The product 22 × (20+Q) would be 22×21=462, 22×23=506, 22×24=528.

These are 3-digit numbers, not 4-digit RRSS. So P cannot be 2. * If P = 3: Q can be 1, 2, 4 (Q cannot be 3 as digits must be distinct). * If Q = 1: 33 × 31 = 1023. This is not of the form RRSS (e.g., 1122, 2233).

Also, P=3, Q=1, R=1, S=2. R=Q, not distinct. * If Q = 2: 33 × 32 = 1056. Not of the form RRSS. * If Q = 4: 33 × 34 = 1122. This is of the form RRSS, where R=1 and S=2. Let's check distinct non-zero digits: P=3, Q=4, R=1, S=2.

All are distinct and non-zero. This is a valid solution. From this, we uniquely determine Q = 4. We found the value of Q without using Statement I or Statement II. Therefore, the question can be answered even without using any of the statements. This is a challenging number puzzle combined with data sufficiency.