Question

If $\alpha ,\beta$ and $\gamma$ are the roots of the equation $x^3+px+q=0$ then the value of the determinant $\left|\begin{array}{ccc}\alpha & \beta & \gamma \\ \beta & \gamma & \alpha \\ \gamma & \alpha & \beta \end{array}\right|$ is ?

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

Answer 1

The correct option is D $0$

Given $\alpha ,\beta ,\gamma$ are roots of equation

$x^3+px+q=\mathrm{0...}\left(1\right)$

then using properties of quadratic equation.

$\alpha +\beta +\gamma =0,\alpha \beta \gamma =-q$ & $\alpha \beta +\beta \gamma +\gamma \alpha =+p$

then

y = $\left|\begin{array}{ccc}\alpha & \beta & \gamma \\ \beta & \gamma & \alpha \\ \gamma & \alpha & \beta \end{array}\right|=\alpha (\beta \gamma -{\alpha }^2)-\beta ({\beta }^2-\gamma \alpha )+\gamma (\alpha \beta -{\gamma }^2)$

$\Rightarrow y=\alpha \beta \gamma -{\alpha }^3-{\beta }^3+\alpha \beta \gamma -{\gamma }^3+\alpha {\beta }^3$

$y=3\alpha \beta \gamma =({\alpha }^3+{\beta }^3+{\gamma }^3)...\left(2\right)$

now $(\alpha +\beta +\gamma {)}^3={\alpha }^3+{\beta }^3+{\gamma }^3+3\alpha \beta \gamma (\alpha \beta +\beta \gamma +\gamma \alpha ).$

or

$(\alpha +\beta +\gamma {)}^3=({\alpha }^3+{\beta }^3+{\gamma }^3)+3\left[\right(\alpha +\beta +\gamma \left)\right(\alpha \beta +\beta \gamma +\gamma \alpha )-\alpha \beta \gamma ]$

$0={\alpha }^3+{\beta }^3+{\gamma }^3+3[0+q]$

$[{\alpha }^3+{\beta }^3+{\gamma }^3=-3q]$

From eq (2)

$y=-3q-(-3q)=0$

$\because [y=0]$

$\therefore$ value of given expression is zero.