- Home
- Standard 12
- Mathematics
3 and 4 .Determinants and Matrices
hard
Let $\alpha, \beta, \gamma$ be the real roots of the equation, $x ^{3}+ ax ^{2}+ bx + c =0,( a , b , c \in R$ and $a , b \neq 0)$ If the system of 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 solution, then the value of $\frac{a^{2}}{b}$ is
A
$5$
B
$3$
C
$1$
D
$0$
(JEE MAIN-2021)
Solution
$\left|\begin{array}{lll}\alpha & \beta & \gamma \\ \beta & \gamma & \alpha \\ \gamma & \alpha & \beta\end{array}\right|=0$
$\Rightarrow-(\alpha+\beta+\gamma)\left(\alpha^{2}+\beta^{2}+\gamma^{2}-\sum \alpha \beta\right)=0$
$\Rightarrow-(-a)\left(a^{2}-2 b-b\right)=0$
$\Rightarrow a\left(a^{2}-3 b\right)=0$
$\Rightarrow a^{2}=3 b \Rightarrow \frac{a^{2}}{b}=3$
Standard 12
Mathematics
