Question

If $\alpha ,\beta ,\gamma$ are the roots of $x^3+p{x}^2+q=0$ where $q\ne 0$ then
$\Delta =\left|\begin{array}{ccc}1/\alpha & 1/\beta & 1/\gamma \\ 1/\beta & 1/\gamma & 1/\alpha \\ 1/\gamma & 1/\alpha & 1/\beta \end{array}\right|$ equals

• A. $-p/q$
• B. $1/q$
• C. $p^2/q$
• D. $0$

Answer 1

Answer: (D)

$0$

We have $\beta \gamma +\gamma \alpha +\alpha \beta =0$
We can write $\Delta$
$\frac{1}{{\alpha }^3{\beta }^3{\gamma }^3}\left|\begin{array}{ccc}\beta \gamma & \gamma \alpha & \alpha \beta \\ \gamma \alpha & \alpha \beta & \beta \gamma \\ \alpha \beta & \beta \gamma & \gamma \alpha \end{array}\right|$
$\frac{1}{{\alpha }^3{\beta }^3{\gamma }^3}\left|\begin{array}{ccc}\beta \gamma +\gamma \alpha +\alpha \beta & \gamma \alpha & \alpha \beta \\ \gamma \alpha +\alpha \beta +\beta \gamma & \alpha \beta & \beta \gamma \\ \alpha \beta +\beta \gamma +\gamma \alpha & \beta \gamma & \gamma \alpha \end{array}\right|$
Using $C_2\to C_2-C_1$
$=\frac{1}{{\alpha }^3{\beta }^3{\gamma }^3}\left|\begin{array}{ccc}0& \gamma \alpha & \alpha \beta \\ 0& \alpha \beta & \beta \gamma \\ 0& \beta \gamma & \gamma \alpha \end{array}\right|=0$ [all zero property]