$\text{ Let }\mathrm{\alpha },\mathrm{\beta },\mathrm{\gamma }\text{ are real roots of the equation }{\mathrm{x}}^3+{\mathrm{ax}}^2+\mathrm{bx}+\mathrm{c}=0(\mathrm{a},\mathrm{b},\mathrm{c}\in \text{ Randa }\ne 0)\text{. If }$ the system of equation (in u,v and w) given by $\begin{array}{r}\mathrm{\alpha u}+\mathrm{\beta v}+\mathrm{\gamma w}=0\\ \mathrm{\beta u}+\mathrm{\gamma v}+\mathrm{\alpha w}=0\\ \mathrm{\gamma u}+\mathrm{\alpha v}+\mathrm{\beta w}=0\end{array}$ $\text{ Has non-trivial solutions, then the value of }{\mathrm{a}}^2/\mathrm{b}\text{ is \_\_\_\_\_\_\_\_}$

$\text{ Equation }{\mathrm{x}}^3+{\mathrm{ax}}^2+\mathrm{bx}+\mathrm{c}=0\text{ has roots }\mathrm{\alpha },\mathrm{\beta },\mathrm{\gamma }\text{. Therefore, }$
$\begin{array}{l}\mathrm{\alpha }+\mathrm{\beta }+\mathrm{\gamma }=-\mathrm{a}\\ \mathrm{\alpha \beta }+\mathrm{\beta \gamma }+\mathrm{\gamma \alpha }=\mathrm{b}\end{array}$
Since the given system of equations has non-trivial solutions, we have
$\left|\begin{array}{l}\mathrm{\alpha }\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{\beta }\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{\gamma }\\ \mathrm{\beta }\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{\gamma }\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{\alpha }\\ \mathrm{\gamma }\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{\alpha }\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{\beta }\end{array}\right|=0$
$\mathrm{Or} {\mathrm{\alpha }}^3+{\mathrm{\beta }}^3+{\mathrm{\gamma }}^3-3\mathrm{\alpha \beta \gamma }=0$
$\mathrm{Or}(\mathrm{\alpha }+\mathrm{\beta }+\mathrm{\gamma })\left[(\mathrm{\alpha }+\mathrm{\beta }+\mathrm{\gamma }{)}^2-3(\mathrm{\alpha \beta }+\mathrm{\beta \gamma }+\mathrm{\gamma \alpha })\right]=0$
$\Rightarrow -\mathrm{a}\left[{\mathrm{a}}^2-3\mathrm{b}\right]=0\text{ or }{\mathrm{a}}^2/\mathrm{b}=3$