Question

Let $\alpha, \beta, \gamma$ be the roots of the cubic $x ^3+ ax ^2+ bx + c =0$, which (taken in given order) are in G.P. If $\alpha$ and $\beta$ are such that $\begin{vmatrix}2 & 1 & 2 \\ 1+\alpha & \alpha & \beta \\ 4-\beta & 3-\beta & \alpha+1\end{vmatrix}=0$, then
The value of $a+b+c$ equals

• A. 0
• B. 1
• C. -1
• D. 2

Answer 1

Answer: (C)

$c _1 \rightarrow c _1- c _2, \quad c _2 \rightarrow c _2- c _1, \quad c _3 \rightarrow c _3-2 c _1$
$\begin{vmatrix}2 & 1 & 2 \\ 1+\alpha & \alpha & \beta \\ 4-\beta & 3-\beta & \alpha+1\end{vmatrix}=\begin{vmatrix}1 & 1 & 2 \\ 1 & \alpha & \beta \\ 1 & 3-\beta & \alpha+1\end{vmatrix}=\begin{vmatrix}1 & 0 & 0 \\ 1 & \alpha-1 & \beta-2 \\ 1 & 2-\beta & \alpha-1\end{vmatrix}$
$=(\alpha-1)^2+(\beta-2)^2=0 \Rightarrow \alpha=1, \beta=2, \gamma=4$
$\therefore$ the cubic equation is $x ^3-7 x ^2+14 x -8=0$