A question is given followed by two Statements I and II. Consider the Question and the Statements and mark the correct option. 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?

The given equation is PP x PQ = RRSS. We can expand this as: (11 x P) x (10 x P + Q) = 1100 x R + 11 x S Dividing by 11: P x (10 x P + Q) = 100 x R + S We are given the following constraints: 1. P, Q, R, S are distinct non-zero digits (i.e., from {1, 2, ..., 9}). 2. P = 100 + 1 = 101 (if S=1, but S must be distinct from P, Q, R). In any case, 100 x R + S will always be 100 or greater. Since 10 + Q is at most 19, it cannot equal 100 x R + S. Therefore, P cannot be 1. Case 2: P = 2 Substitute P=2 into the simplified equation: 2 x (10 x 2 + Q) = 100 x R + S 2 x (20 + Q) = 100 x R + S 40 + 2Q = 100 x R + S Constraints for Q: Q = 100. If Q = 3: 40 + 2(3) = 46. So, 100 x R + S = 46. This is impossible. If Q = 4: 40 + 2(4) = 48. So, 100 x R + S = 48. This is impossible. Therefore, P cannot be 2. Case 3: P = 3 Substitute P=3 into the simplified equation: 3 x (10 x 3 + Q) = 100 x R + S 3 x (30 + Q) = 100 x R + S 90 + 3Q = 100 x R + S Constraints for Q: Q = 100. If Q = 2: 90 + 3(2) = 96. So, 100 x R + S = 96. This is impossible. If Q = 4: 90 + 3(4) = 90 + 12 = 102. So, 100 x R + S = 102. This result (100 x R + S = 102) implies R = 1 and S = 2. Let's check if P=3, Q=4, R=1, S=2 satisfy all conditions: - P, Q, R, S are distinct non-zero digits: 3, 4, 1, 2 are all distinct and non-zero. (Satisfied) - P <= 3: P=3. (Satisfied) - Q <= 4: Q=4. (Satisfied) All conditions are met uniquely for P=3, Q=4, R=1, S=2. The question asks "what is Q equal to?". We found Q = 4. Since we were able to determine the unique value of Q (Q=4) using only the information provided in the question and its initial constraints, the statements are not required to answer the question. Therefore, the question can be answered even without using any of the Statements. The final answer is D