Question

If $\alpha ,\beta$ and $\gamma$ are the roots of the equation $x^3-3x^2+4x+4=0$ , then the value of $\left|\begin{array}{ccc}{\alpha }^2+1& 1& 1\\ 1& {\beta }^2+1& 1\\ 1& 1& {\gamma }^2+1\end{array}\right|$ is equal to

• A. $32$
• B. $16$
• C. $56$
• D. $64$

Answer 1

Answer: (C)

$\alpha +\beta +\gamma =3$
$\alpha \beta +\beta \gamma +\gamma \alpha =4$
$\alpha \beta \gamma =-4$
$\left|\begin{array}{ccc}{\left(\alpha \right)}^2+1& 1& 1\\ 1& {\left(\beta \right)}^2+1& 1\\ 1& 1& {\left(\gamma \right)}^2+1\end{array}\right|=\left({\left(\alpha \right)}^2+1\right)\left[\left({\left(\beta \right)}^2+1\right)\left({\left(\gamma \right)}^2+1\right)-1\right]-1\left[{\left(\gamma \right)}^2\right]+1\left(-{\left(\beta \right)}^2\right)$
$=\left({\left(\alpha \right)}^2+1\right)\left[{\left(\beta \right)}^2+{\left(\gamma \right)}^2+{\left(\beta \right)}^2{\left(\gamma \right)}^2\right]-{\left(\gamma \right)}^2-{\left(\beta \right)}^2$
$={\alpha }^2{\beta }^2+{\alpha }^2{\gamma }^2+{\alpha }^2{\beta }^2{\gamma }^2+{\beta }^2+{\gamma }^2+{\beta }^2{\gamma }^2-{\gamma }^2-{\beta }^2$
$={\left(\alpha \right)}^2{\left(\beta \right)}^2+{\left(\beta \right)}^2{\left(\gamma \right)}^2+{\left(\gamma \right)}^2{\left(\alpha \right)}^2+{\left(\alpha \beta \gamma \right)}^2$
$={\left(\alpha \beta +\beta \gamma +\gamma \alpha \right)}^2-2\alpha \beta \gamma \left(\alpha +\beta +\gamma \right)+{\left(\alpha \beta \gamma \right)}^2$
$=16+8\left(3\right)+16$
$=56$