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?

Statement-I: $\frac{x}{y}$ is odd.

Statement-II: $xy=12$

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$\mathrm{Which\; one\; of\; the\; following\; is\; correct\; in\; respect\; of\; the\; above\; Question\; and\; the\; Statements?}$

The correct option is: "The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone".

Let's analyze the question and the statements step-by-step to understand why this option is correct.

The question asks for the unique values of $x$ and $y$, given that $x$ and $y$ are distinct natural numbers (i.e., $x,y\in \{1,2,3,...\}$ and $x\ne y$).

Analyzing Statement-I alone:
Statement-I tells us that
$\frac{x}{y}$
is odd. There are infinitely many pairs of distinct natural numbers $x$ and $y$ that satisfy this condition (for example, $x=3,y=1$ gives $3$; $x=6,y=2$ gives $3$). Thus, Statement-I alone is not sufficient to find the unique values of $x$ and $y$.

Analyzing Statement-II alone:
Statement-II tells us that
$xy=12$.
Since $x$ and $y$ are distinct natural numbers, the possible pairs $(x,y)$ whose product is $12$ are:
$(1,12)$, $(12,1)$, $(2,6)$, $(6,2)$, $(3,4)$, and $(4,3)$.
Since there are multiple possible solutions, Statement-II alone is not sufficient to determine the unique values of $x$ and $y$.

Combining Statement-I and Statement-II:
Now, we look for a pair from our list of options in Statement-II that also satisfies Statement-I (i.e.,
$\frac{x}{y}$
must be an odd integer). Let's test each pair:
1. For $(x,y)=(1,12)$: $\frac{x}{y}=\frac{1}{12}$ (not an integer).
2. For $(x,y)=(12,1)$: $\frac{x}{y}=\frac{12}{1}=12$ (even integer).
3. For $(x,y)=(2,6)$: $\frac{x}{y}=\frac{2}{6}=\frac{1}{3}$ (not an integer).
4. For $(x,y)=(6,2)$: $\frac{x}{y}=\frac{6}{2}=3$ (odd integer). This works!
5. For $(x,y)=(3,4)$: $\frac{x}{y}=\frac{3}{4}$ (not an integer).
6. For $(x,y)=(4,3)$: $\frac{x}{y}=\frac{4}{3}$ (not an integer).

Thus, when using both statements together, there is only one unique solution: $x=6$ and $y=2$.
Consequently, the Question can be answered by using both the Statements together, but cannot be answered using either Statement alone.