Question

The number of positive integral solutions of the equation $\left|\begin{array}{ccc}{x}^3+1& x^{2}y& x^{2}z\\ x{y}^2& y^3+1& y^{2}z\\ x{z}^2& y{z}^2& z^3+1\end{array}\right|=11$ is

• A. 0
• B. 3
• C. 6
• D. 12

Answer 1

The correct option is A 0

$\left|\begin{array}{ccc}{x}^3+1& x^{2}y& x^{2}z\\ x{y}^2& y^3+1& y^{2}z\\ x{z}^2& y{z}^2& z^3+1\end{array}\right|=11$

$\Rightarrow \left|\begin{array}{ccc}{x}^3& x^{2}y& x^{2}z\\ x{y}^2& y^3& y^{2}z\\ x{z}^2& y{z}^2& z^3\end{array}\right|+\left|\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right|=11$

$\Rightarrow xyz\left|\begin{array}{ccc}{x}^2& x^2& x^2\\ y^2& y^2& y^2\\ z^2& z^2& z^2\end{array}\right|+1=11\Rightarrow x^3{y}^3{z}^3\left|\begin{array}{ccc}1& 1& 1\\ 1& 1& 1\\ 1& 1& 1\end{array}\right|=0$
Hence, option 'A' is correct.