Question

Let $\alpha ,\beta ,\gamma$ be the roots of equations, $x^3+a{x}^2+bx+c=0,(a,b,c\in R$ and $a,b\ne 0).$ If the system of the equations (in $u,v,w)$ given by $\alpha u+\beta v+\gamma w=0;$ $\beta u+\gamma v+\alpha w=0;$ $\gamma u+\alpha v+\beta w=0$ has non-trivial solutions, then the value of $\frac{a^2}{b}$ is

• A. $5$
• B. $1$
• C. $0$
• D. $3$

Answer 1

The correct option is D $3$

$\alpha ,\beta ,\gamma$ be the roots of equations, $x^3+a{x}^2+bx+c=0$
For non-trivial solutions,
$\left|\begin{array}{ccc}\alpha & \beta & \gamma \\ \beta & \gamma & \alpha \\ \gamma & \alpha & \beta \end{array}\right|=0$
$\Rightarrow {\alpha }^3+{\beta }^3+{\gamma }^3-3\alpha \beta \gamma =0$
$\Rightarrow \alpha +\beta +\gamma [\alpha +\beta +{\alpha }^2-3\sum \alpha \beta ]=0$
$\Rightarrow (-a)[a^2-3b]=0$
$\Rightarrow a^2=3b(\because a\ne 0)$
$\Rightarrow \frac{a^2}{b}=3$