Question

If $\alpha ,\beta ,\gamma$ are roots of $7x^3-x-2=0$, then the value of $\sum (\frac{\alpha }{\beta }+\frac{\beta }{\alpha })$, is:

• A. $1$
• B. $2$
• C. $-3$
• D. $-4$

Answer 1

The correct option is C $-3$

Given, $\alpha ,\beta ,\gamma$ are roots of $7x^3-x-2=0$
$\sum \alpha =0,\sum \alpha \beta =-\frac{1}{7}$ and $\alpha \beta \gamma =\frac{2}{7}$
Now, $(\alpha +\beta +\gamma )(\frac{1}{\alpha }+\frac{1}{\beta }+\frac{1}{\gamma })=3+\sum (\frac{\alpha }{\beta }+\frac{\beta }{\alpha })$
$0=3+\sum (\frac{\alpha }{\beta }+\frac{\beta }{\alpha })(\because \alpha +\beta +\gamma =0)$
$\Rightarrow \sum (\frac{\alpha }{\beta }+\frac{\beta }{\alpha })=-3$
Hence, option 'C' is correct.