What will come in the place of question (?) mark in following the question:

$6\frac{1}{2}-3\frac{1}{4}\times {\left(2-1\frac{1}{2}\right)}^2=7\frac{1}{2}+?$

The correct answer is -9.

The expression to evaluate is:

$6\frac{1}{2}-3\frac{1}{4}\times 2-1\frac{1}{2}=7\frac{1}{2}+?$

We need to find the value of ? by simplifying the Left-Hand Side (LHS) first, then solving for ?.

Step 1: Convert all mixed numbers to improper fractions.

$6\frac{1}{2}=\frac{13}{2}$
$3\frac{1}{4}=\frac{13}{4}$
$1\frac{1}{2}=\frac{3}{2}$
$7\frac{1}{2}=\frac{15}{2}$

Step 2: Apply BODMAS — handle Multiplication first.

$\frac{13}{4}\times 2=\frac{13}{\times }=\frac{26}{4}=\frac{13}{2}$

Step 3: Now substitute back and handle subtraction from left to right.

$\frac{13}{2}-\frac{13}{2}-\frac{3}{2}$

$=\frac{0}{2}-\frac{3}{2}=\frac{-3}{2}$

So the LHS simplifies to $\frac{-3}{2}$.

Step 4: Set LHS equal to RHS and solve for ?

$\frac{-3}{2}=\frac{15}{2}+?$

$?=\frac{-3}{2}-\frac{15}{2}=\frac{\mathrm{-3\; -\; 15}}{2}=\frac{-18}{2}=-9$

Therefore, the value of ? is -9.