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