Question

If $\alpha ,\beta ,\gamma$ roots of the cubic $x^3-3x^2+2x+k=0$ $(k\in R)$ satisfying the relation $(\alpha +2)(\beta +2)(\gamma +2)=8$ the the value of ${}^{\prime }{k}^{\prime }$ equals ?

• A. $16$
• B. $8$
• C. $12$
• D. $4$

Answer 1

Answer: (A)

$16$

Given, $\alpha ,\beta ,\gamma$ are the roots of the equation $x^3-3x^2+2x+k=0$

By relation between roots and coefficient of cubic equation we get,

$\alpha +\beta +\gamma =\frac{-(-3)}{1}\Rightarrow \alpha +\beta +\gamma =3\alpha \beta +\beta \gamma +\gamma \alpha =\frac{2}{1}\Rightarrow \alpha \beta +\beta \gamma +\gamma \alpha =2\alpha \beta \gamma =\frac{-k}{1}\Rightarrow \alpha \beta \gamma =-kA/Q,(\alpha +2)(\beta +2)(\gamma +2)=8\Rightarrow \alpha \beta \gamma +2(\alpha \beta +\beta \gamma +\gamma \alpha )+4(\alpha +\beta +\gamma )=0\Rightarrow -k+2.2+4.3=0\Rightarrow k=16$