A Question is given followed by two Statements I and II. Consider the Question and the Statements.

P, Q, R and S appeared in a test.

Question:
Has P scored more marks than Q?

Statement-I:
The sum of the marks scored by P and Q is equal to the sum of the marks scored by R and S.

Statement-II:
The sum of the marks scored by P and S is more than the sum of the marks scored by Q and R.

Which one of the following is correct in respect of the above Question and the Statements?

Correct Option: (d) The Question cannot be answered even by using both the Statements together

We need to determine if P > Q based on the given statements. From Statement-I: P + Q = R + S. This equation alone does not allow us to compare P and Q. Thus, Statement-I alone is insufficient. From Statement-II: P + S > Q + R. This inequality alone does not allow us to compare P and Q. Thus, Statement-II alone is insufficient. Combining both statements together: We have (1) P + Q = R + S and (2) P + S > Q + R. Adding (1) and (2) gives: (P + Q) + (P + S) > (R + S) + (Q + R) ⇒ 2P + Q + S > 2R + Q + S ⇒ 2P > 2R ⇒ P > R. Substituting R = P + Q - S from (1) into (2) gives: P + S > Q + (P + Q - S) ⇒ 2S > 2Q ⇒ S > Q. We have established that P > R and S > Q, but we still cannot establish a direct relationship between P and Q. For example, the values P=10, Q=6, R=8, S=8 satisfy both conditions and give P > Q. However, the values P=6, Q=10, R=4, S=12 also satisfy both conditions and give P < Q. Therefore, the question cannot be answered even by using both statements. Hence, the correct option is (d).