If the number $97251\ast 6$ is completely divisible by $11$, then the smallest whole number in place of $\ast$ will be:

A number is divisible by $11$ if the difference between the sum of digits at odd positions and the sum of digits at even positions is either $0$ or a multiple of $11$.
Let the missing digit be $x$. The digits of the number $97251x6$ from left to right are at positions $1,2,3,4,5,6,7$.

* Sum of digits at odd positions ($1^{\text{st}},3^{\text{rd}},5^{\text{th}},7^{\text{th}}$):

$$S_{\text{odd}}=9+2+1+6=18$$

* Sum of digits at even positions ($2^{\text{nd}},4^{\text{th}},6^{\text{th}}$):

$$S_{\text{even}}=7+5+x=12+x$$

Now, find the difference:

$$\text{Difference}=S_{\text{odd}}-S_{\text{even}}=18-(12+x)=6-x$$

For the number to be divisible by $11$, this difference must be $0$ or a multiple of $11$. Since $x$ is a single-digit whole number:

$$6-x=0⟹x=6$$