UPSC CSE Prelims 2024

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?

EXPLANATION

The question asks to determine if P has scored more marks than Q (i.e., P > Q?).

Statement I: P + Q = R + S. This statement alone is insufficient.
- Consider Case 1: P=10, Q=5, R=8, S=7. Here, P + Q = 15 and R + S = 15. In this case, P > Q.
- Consider Case 2: P=5, Q=10, R=7, S=8. Here, P + Q = 15 and R + S = 15. In this case, P < Q.
- Since both P > Q and P < Q are possible while satisfying Statement I, it cannot definitively answer the question.
Statement II: P + S > Q + R. This statement alone is also insufficient.
- Consider Case 1: P=10, Q=5, S=3, R=2. Here, P + S = 13 and Q + R = 7. P + S > Q + R is satisfied. In this case, P > Q.
- Consider Case 2: P=5, Q=10, S=12, R=6. Here, P + S = 17 and Q + R = 16. P + S > Q + R is satisfied. In this case, P < Q.
- Since both P > Q and P < Q are possible while satisfying Statement II, it cannot definitively answer the question.
Combining Statement I and Statement II:
From Statement I: P + Q = R + S ...(i)

From Statement II: P + S > Q + R ...(ii)

Adding (i) and (ii):

(P + Q) + (P + S) > (R + S) + (Q + R)

$\Rightarrow$ 2P + Q + S > Q + S + 2R

$\Rightarrow$ 2P > 2R

This implies P > R.

Even with both statements combined, we can only conclude that P scored more marks than R. This information does not allow us to determine the relationship between P and Q. Therefore, the question cannot be answered even by using both statements together.

Options 1, 2, and 3 are incorrect because, as demonstrated, neither Statement I alone nor Statement II alone is sufficient to answer the question. Furthermore, even when both statements are used together, the relationship between P and Q cannot be definitively established.