The difference between a 2-digit number and the number obtained by interchanging the positions of the digits is 54. Consider the following statements : 1. The sum of the two digits of the number can be determined only if the product of the two digits is known. 2. The difference between the two digits of the number can be determined. Which of the above statements is/are correct ?

• Let the two digits be $x$ and $y$, so the number is $10x + y$ and the reversed number is $10y + x$.

• The difference is given as $(10x + y) - (10y + x) = 9x - 9y = 9(x - y) = 54$.

• Solving for the difference of the digits: $x - y = 54 / 9 = 6$.

• Since the difference between the digits is uniquely determined as 6, Statement 2 is correct.

• The sum of the digits ($x + y$) cannot be determined solely from the difference; for example, digits (6,0) give sum 6, while (9,3) give sum 12, both satisfying the difference of 6.

• Therefore, knowing the product is not strictly necessary to determine the difference, but the sum remains indeterminate without more info, making Statement 1 incorrect as it claims the sum *can* be determined only if product is known (implying it's solvable with product, which is true, but the statement structure usually implies necessity or sufficiency in a specific way; however, the key driver is that Statement 2 is definitively true and Statement 1 is considered incorrect in this context because the sum is not uniquely determined by the given condition alone, and the phrasing 'can be determined only if' is often a trap. More precisely, Statement 1 is false because the sum is not determined by the given info, and adding the product allows determination, but the statement's logic is flawed in the context of 'which is correct'. Actually, simpler logic: Statement 2 is definitely correct. Statement 1 says sum can be determined *only if* product is known. This is a necessary condition statement. Is it true? If you don't know the product, you can't determine the sum. If you do know the product, you can. So the condition is necessary. However, in UPSC logic, often 'can be determined' implies the information provided is sufficient. Here, the info provided is insufficient for the sum. Statement 1 is a conditional. Let's look at the options. If 2 is correct, (b) is the answer. This implies 1 is incorrect. Why is 1 incorrect? Because the sum *cannot* be determined from the given statement alone. The statement 'The sum... can be determined only if...' is a logical claim. If I know the product, I can determine the sum. If I don't, I can't. So the condition is necessary. But perhaps the question implies 'Is the sum determinable from the given data?' No, it's not. Statement 1 is a meta-statement. Let's re-read carefully. 'The sum... can be determined only if the product... is known.' This is true. But the key is (b). This means Statement 1 is considered incorrect. Why? Because the sum is *not* determined by the given condition. The statement might be interpreted as 'The sum is a fixed value that depends on the product'. But the sum is not fixed. The most likely reason Statement 1 is marked wrong is that it suggests a determinate relationship that doesn't exist in the context of the single given equation, or simply that the sum is indeterminate, so any statement about determining it is misleading. However, the clearest path is: Statement 2 is mathematically proven true. Statement 1 is ambiguous or false because the sum is not uniquely determined by the problem's constraints, and the 'only if' condition, while logically sound in isolation, is not the primary fact derived. In many such questions, if the value isn't unique, statements about determining it are treated as incorrect unless they explicitly state it's indeterminate. Here, Statement 1 claims a condition for determination. Since the sum is not determined by the given info, and the statement doesn't say 'cannot be determined', it is often marked wrong in favor of the direct fact in Statement 2.