Question

If $x\ne y\ne z$ and $\left|\begin{array}{ccc}x& y& z\\ x^2& y^2& z^2\\ 1+x^3& 1+y^3& 1+z^3\end{array}\right|=0$ then xyz equals

• A. -1
• B. -2
• C. -3
• D. 0

Answer 1

The correct option is A -1

$\left|\begin{array}{ccc}x& y& z\\ x^2& y^2& z^2\\ 1+x^3& 1+y^3& 1+z^3\end{array}\right|=0$
$\left|\begin{array}{ccc}x& y& z\\ x^2& y^2& z^2\\ 1& 1& 1\end{array}\right|+\left|\begin{array}{ccc}x& y& z\\ x^2& y^2& z^2\\ x^3& y^3& z^3\end{array}\right|=0$
$\left|\begin{array}{ccc}x& y& z\\ x^2& y^2& z^2\\ 1& 1& 1\end{array}\right|+xyz\left|\begin{array}{ccc}1& 1& 1\\ x& y& z\\ x^2& y^2& z^2\end{array}\right|=0$
$\Rightarrow \left|\begin{array}{ccc}x& y& z\\ x^2& y^2& z^2\\ 1& 1& 1\end{array}\right|(1+xyz)=0$
$R_1\leftrightarrow R_2,R_2\leftrightarrow R_3$
$(1+xyz)\left|\begin{array}{ccc}x& y& z\\ x^2& y^2& z^2\\ 1& 1& 1\end{array}\right|=0$
$\Rightarrow xyz=-1$