UPSC CSE Prelims 2024

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

Question: What are the unique values of x and y, where x, y are distinct natural numbers?

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

EXPLANATION

The question requires determining unique values for x and y, given that x and y are distinct natural numbers.

Analysis of Statement-I: "x / y is odd."
If x and y are distinct natural numbers and their ratio x/y is an odd integer, multiple pairs of (x, y) are possible. For instance, (3, 1), (5, 1), (6, 2), (9, 3) all satisfy this condition. Therefore, Statement-I alone is insufficient to determine unique values for x and y.
Analysis of Statement-II: "xy = 12."
Given that x and y are distinct natural numbers and their product is 12, the possible pairs (x, y) are: (1, 12), (2, 6), (3, 4), (4, 3), (6, 2), and (12, 1). Since there are multiple possible pairs, Statement-II alone is insufficient to determine unique values for x and y.
Analysis of Both Statements Together:
Combining the conditions from both statements:
1. From Statement-II, the possible distinct natural number pairs (x, y) where xy = 12 are: (1, 12), (2, 6), (3, 4), (4, 3), (6, 2), and (12, 1).
2. Applying Statement-I (x/y is odd) to these pairs:
  - For (1, 12), x/y = 1/12 (not an odd integer).
  - For (2, 6), x/y = 2/6 = 1/3 (not an odd integer).
  - For (3, 4), x/y = 3/4 (not an odd integer).
  - For (4, 3), x/y = 4/3 (not an odd integer).
  - For (6, 2), x/y = 6/2 = 3 (which is an odd integer). This pair satisfies both conditions.
  - For (12, 1), x/y = 12/1 = 12 (which is an even integer, not odd).
Only the pair (x=6, y=2) satisfies both conditions. Thus, using both statements together allows for the determination of unique values for x and y.

Therefore, the question can be answered by using both statements together, but not by using either statement alone.

Option A: "The Question can be answered by using one of the Statements alone, but cannot be answered using the other Statement alone."
This is incorrect because, as demonstrated, neither Statement-I nor Statement-II alone provides unique values for x and y.
Option B: "The Question can be answered by using either Statement alone."
This is incorrect because both statements individually yield multiple possible pairs for (x, y), failing to provide unique values.
Option D: "The Question cannot be answered even by using both the Statements together."
This is incorrect because combining both statements successfully narrows down the possibilities to a single unique pair (x=6, y=2), thus answering the question.