If $\overrightarrow{\mathrm{x}},\overrightarrow{\mathrm{y}}$  are two non-zero and non-collinear vectors satisfying  $\left[(\mathrm{a}-2){\mathrm{\alpha }}^2+(\mathrm{b}-3)\mathrm{\alpha }+\mathrm{c}\right]\overrightarrow{\mathrm{x}}+\left[(\mathrm{a}-2){\mathrm{\beta }}^2+(\mathrm{b}-3)\mathrm{\beta }+\mathrm{c}\right]\overrightarrow{\mathrm{y}}+\left[(\mathrm{a}-2){\mathrm{\gamma }}^2+(\mathrm{b}-3)\mathrm{\gamma }+\mathrm{c}\right](\overrightarrow{\mathrm{x}}\times \overrightarrow{\mathrm{y}})=0$ where $\alpha ,\beta ,\gamma$ are three distinct real numbers, then find the value of $\left({\mathrm{a}}^2+{\mathrm{b}}^2+{\mathrm{c}}^2-4\right)$

$\text{ Since }\overrightarrow{x}\text{ and }\overrightarrow{y}\text{ are non-collinear vectors, therefore }\overrightarrow{x},\overrightarrow{y}\text{ and }\overrightarrow{x}\times \overrightarrow{y}\text{ are non-coplanar vectors. }$

$\left[(\mathrm{a}-2){\mathrm{\alpha }}^2+(\mathrm{b}-3)\mathrm{\alpha }+\mathrm{c}\right]+\left[(\mathrm{a}-2){\mathrm{\beta }}^2+(\mathrm{b}-3)\right.\mathrm{\beta \beta }+\mathrm{c}]\overrightarrow{\mathrm{y}}+\left[(\mathrm{a}-2){\mathrm{\gamma }}^2+(\mathrm{b}-3)\mathrm{\gamma }+\mathrm{c}\right](\overrightarrow{\mathrm{x}}\times \overrightarrow{\mathrm{y}})=0$

$\text{ Coefficient of each vector }\overrightarrow{\mathrm{x}},\overrightarrow{\mathrm{y}}\text{ and }\overrightarrow{\mathrm{x}}\times \overrightarrow{\mathrm{y}}\text{ is zero. }$

$\begin{array}{r}(\mathrm{a}-2){\mathrm{\alpha }}^2+(\mathrm{b}-3)\mathrm{\alpha }+\mathrm{c}=0\\ (\mathrm{a}-2){\mathrm{\beta }}^2+(\mathrm{b}-3)\mathrm{\beta }+\mathrm{c}=0\\ (\mathrm{a}-2){\mathrm{\gamma }}^2+(\mathrm{b}-3)\mathrm{\beta }+\mathrm{c}=0\end{array}$

The above three equations will satisfy if the coefficients of $\alpha ,\beta$ and  $\gamma$ are zero because $\alpha ,\beta$ and $\gamma$ are three distinct real numbers .

$\begin{array}{l}\mathrm{a}-2=0\text{ or }\mathrm{a}=2\\ \mathrm{b}-3=0\text{ or }\mathrm{b}=3\text{ and }\mathrm{c}=0\\ \therefore {\mathrm{a}}^2+{\mathrm{b}}^2+{\mathrm{c}}^2=2^2+3^2+0^2=4+9=13\end{array}$