Question

Let $\alpha , \, \beta $ and $\gamma $ are the roots of the equation $2x^{3}+9x^{2}-27x-54=0.$ If $\alpha ,\beta ,\gamma $ are in geometric progression, then the value of $\left|\alpha \right|+\left|\beta \right|+\left|\gamma \right|=$

• A. $\frac{19}{2}$
• B. $\frac{21}{2}$
• C. $13$
• D. $11$

Answer 1

Answer: (B)

Let, $\alpha=\frac{\beta}{r}$ and $\gamma=\beta r$
$\alpha \cdot \beta \cdot \gamma=\frac{\beta}{y} \cdot \beta \cdot \beta r=\frac{54}{2}=27$
$\Rightarrow \beta^{3}=27 \Rightarrow \beta=3$
$\alpha+\beta+\gamma=\frac{\beta}{r}+\beta+\beta r=-\frac{9}{2}$
$\Rightarrow \frac{3}{r}+3+3 r=-\frac{9}{2}$
$\Rightarrow \frac{1}{r}+1+r=-\frac{3}{2}$
$\Rightarrow 2+2 r+2 r^{2}=-3 r$
$\Rightarrow 2 r^{2}+5 r+2=0$
$\Rightarrow (2 r+1)(r+2)=0 \Rightarrow r=-\frac{1}{2},-2$
$\alpha=\frac{\beta}{r}=\frac{3}{-2}=\frac{-3}{2}$
$\beta=3$
$\gamma=\beta r=3(-2)=-6$
$|\alpha|+|\beta|+|\gamma|=\frac{3}{2}+3+6=\frac{21}{2}$