Q. If $\alpha,\beta,\gamma$ are the roots of $x^{3} - 2x^{2} + 3x - 4 =$0, then the value of $\alpha^{2}\beta^{2} + \beta^{2}\gamma^{2} + \gamma^{2}\alpha^{2}$ is
A) -7
B) -5
C) -3
D) 0
Explanation
(a)
Here, $\alpha + \beta + \gamma = \frac{2}{1} = 2,\ \alpha\beta + \beta\gamma + \gamma\alpha = 3$
And $\alpha\beta\gamma = 4$
We know that
$\alpha^{2}\beta^{2} + \beta^{2}\gamma^{2} + \gamma^{2}\alpha^{2}$
$= (\alpha\beta + \beta\gamma + \gamma\alpha)^{2} - 2(\alpha\beta\gamma)(\alpha + \beta + \gamma)$
$= (3)^{2} - 2(4)(2) = - 7$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.