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

Equation ${\mathrm{x}}^3+{\mathrm{ax}}^2+\mathrm{bx}+\mathrm{c}=0$ has roots $\mathrm{\alpha },\mathrm{\beta },\mathrm{\gamma }$. Therefore,
$\begin{array}{r}\mathrm{\alpha }+\mathrm{\beta }+\mathrm{\gamma }=-\mathrm{a}\\ \mathrm{\alpha \beta }+\mathrm{\beta \gamma }+\mathrm{\gamma \alpha }=\mathrm{b}\end{array}$
Since the given system of equations has non-trivial solutions,
we have

$\begin{array}{l}\left|\begin{array}{lll}{\mathrm{\alpha }}_0& \mathrm{\beta }& \mathrm{\gamma }\\ \mathrm{\beta }& \mathrm{\gamma }& \mathrm{\alpha }\\ \mathrm{\gamma }& \mathrm{\alpha }& \mathrm{\beta }\end{array}\right|=0\\ \text{ or }{\mathrm{\alpha }}^3+{\mathrm{\beta }}^3+{\mathrm{\gamma }}^3-3\mathrm{\alpha \beta \gamma }=0\\ \text{ or }(\mathrm{\alpha }+\mathrm{\beta }+\mathrm{\gamma })\left[{\mathrm{\alpha }}^2+{\mathrm{\beta }}^2+{\mathrm{\gamma }}^2-\mathrm{\alpha \beta }-\mathrm{\beta \gamma }-\mathrm{\gamma \alpha }\right]=0\\ \text{ or }(\mathrm{\alpha }+\mathrm{\beta }+\mathrm{\gamma })\left[(\mathrm{\alpha }+\mathrm{\beta }+\mathrm{\gamma }{)}^2-3(\mathrm{\alpha \beta }+\mathrm{\beta \gamma }+\mathrm{\gamma \alpha })\right]=0\\ \Rightarrow -\mathrm{a}\left[{\mathrm{a}}^2-3\mathrm{b}\right]=0\text{ or }{\mathrm{a}}^2/\mathrm{b}=3\end{array}$