If ⊕ ÷ ⊙ = 2 ;

⊕ ÷ Δ = 3;

⊙ + Δ = 5;

Δ × ⊗ =10,

Then, the value of (⊕-⊕ )² , is:

The correct option is 1.

Let us analyze the given system of mathematical equations step-by-step to find the values of each symbol.
We are given the following four relations:
1) $\oplus ÷\odot =2$ (which means $\oplus =2\times \odot$)
2) $\oplus ÷\Delta =3$ (which means $\oplus =3\times \Delta$)
3) $\odot +\Delta =5$
4) $\Delta \times \otimes =10$

First, from equations (1) and (2), we can express $\odot$ and $\Delta$ in terms of $\oplus$:
$\odot =\frac{\oplus }{2}$
$\Delta =\frac{\oplus }{3}$

Substitute these expressions into equation (3):
$\frac{\oplus }{2}+\frac{\oplus }{3}=5$

To solve for $\oplus$, find a common denominator (which is 6) for the fractions:
$\frac{3\oplus +2\oplus }{6}=5$
$\frac{5\oplus }{6}=5$

Multiplying both sides by 6 and dividing by 5 gives:
$\oplus =6$

Now we can determine the remaining variables:
From $\odot =\frac{\oplus }{2}$, we get $\odot =\frac{6}{2}=3$.
From $\Delta =\frac{\oplus }{3}$, we get $\Delta =\frac{6}{3}=2$.
From equation (4), $\Delta \times \otimes =10$, so $2\times \otimes =10$, which gives $\otimes =5$.

We want to evaluate the value of the expression ${\left(\oplus -\otimes \right)}^2$.
Using the values $\oplus =6$ and $\otimes =5$ (note: the term in the question is a subtraction of the symbol $\oplus$ and the symbol $\otimes$, representing ${\left(\oplus -\otimes \right)}^2$):
${\left(6-5\right)}^2=1^2=1$

Thus, the value of the expression is 1, which corresponds to option 1.