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?

The correct option is: The Question cannot be answered even by using both the Statements together.

Let us analyze the problem step-by-step to understand why the question cannot be answered even when combining both statements.

We are given that four individuals, P, Q, R, and S, appeared in a test. Let their respective marks be denoted by the variables P, Q, R, and S.

Our objective is to determine if we can definitively answer the question: "Has P scored more marks than Q?" (i.e., is P > Q?).

Let's represent the information given in the statements mathematically:

Analysis of 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.
Mathematically, this can be written as:
P + Q = R + S
From this equation alone, we cannot determine the relative order of P and Q. For example, if R = 5 and S = 5, we could have P = 6 and Q = 4 (where P > Q), or P = 4 and Q = 6 (where P < Q). Both cases satisfy the equation. Thus, Statement-I alone is insufficient.

Analysis of 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.
Mathematically, this can be written as:
P + S > Q + R
This inequality alone also does not give a direct relationship between P and Q, as the values of S and R can vary. Thus, Statement-II alone is insufficient.

Combining Statement-I and Statement-II:
Let us consider both mathematical relations together:
1) P + Q = R + S (which can be rewritten as R = P + Q - S)
2) P + S > Q + R

Substitute the expression for R from equation (1) into inequality (2):

$P+S>Q+(P+Q-S)$

Simplify the inequality by expanding the right side:

$P+S>P+2Q-S$

Subtract P from both sides of the inequality:

$S>2Q-S$

Add S to both sides of the inequality:

$2S>2Q$

Divide by 2:

$S>Q$

Combining both statements only tells us that S > Q. It does not establish any direct comparison between P and Q. P could still be greater than, equal to, or less than Q depending on the value of R. Since we cannot determine whether P scored more marks than Q, the question cannot be answered even by using both statements together.