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

Question: What are the values of $m$ and $n$, where $m$ and $n$ are natural numbers?

Statement-I : $m+n>mn$ and $m>n$.

Statement-II : The product of $m$ and $n$ is 24.

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

The question asks for the unique values of $m$ and $n$, given that $m$ and $n$ are natural numbers ($m,n\in \{1,2,3,...\}$).

Analysis of Statement-I:
Statement-I states: $m+n>mn$ and $m>n$.
Let us test some small natural numbers. If we set $n=1$, the inequality $m+n>mn$ becomes:
$m+1>m$
which is always true for any natural number $m$.
Since $m>n$, $m$ can be any natural number greater than 1 (i.e., $m\in \{2,3,4,...\}$).
For example, $(m,n)$ could be $(2,1)$, $(3,1)$, $(4,1)$, and so on. Since there are infinitely many such pairs, Statement-I alone is not sufficient to determine unique values for $m$ and $n$.

Analysis of Statement-II:
Statement-II states: The product of $m$ and $n$ is 24, which means:
$m·n=24$
Since $m$ and $n$ are natural numbers, the possible pairs for $(m,n)$ are:
$(1,24),(2,12),(3,8),(4,6),(6,4),(8,3),(12,2),(24,1)$
Since there are multiple possible solutions, Statement-II alone is not sufficient to determine unique values for $m$ and $n$.

Combining Statement-I and Statement-II:
Now we combine the conditions from both statements. We need a pair of natural numbers $(m,n)$ such that:
1) $m·n=24$ (From Statement-II)
2) $m+n>mn$ (From Statement-I)
3) $m>n$ (From Statement-I)
Substituting $mn=24$ into the inequality (2), we get:
$m+n>24$
Let us evaluate our list of candidate pairs from Statement-II where $m>n$:
- For $(m,n)=(6,4)$: $m+n=6+4=10$, which is not greater than 24.
- For $(m,n)=(8,3)$: $m+n=8+3=11$, which is not greater than 24.
- For $(m,n)=(12,2)$: $m+n=12+2=14$, which is not greater than 24.
- For $(m,n)=(24,1)$: $m+n=24+1=25$, which is greater than 24 ($25>24$).
Thus, the only pair that satisfies all conditions is $(m,n)=(24,1)$.
Since we obtain a unique solution when using both statements together, the question can be answered by using both statements together, but cannot be answered using either statement alone.