If $\mathrm{f}\left(\mathrm{x}\right)=\left|\begin{array}{ccc}1& \mathrm{x}& \mathrm{x}+1\\ 2\mathrm{x}& \mathrm{x}(\mathrm{x}-1)& (\mathrm{x}+1)\mathrm{x}\\ 3\mathrm{x}(\mathrm{x}-1)& \mathrm{x}(\mathrm{x}-1)(\mathrm{x}-2)& (\mathrm{x}+1)\mathrm{x}(\mathrm{x}-1)\end{array}\right|$, then the value of f(500) ________ .

Taking x common from R₂ and x(x - 1) common from R₃, wo get

$\mathrm{f}\left(\mathrm{x}\right)={\mathrm{x}}^2(\mathrm{x}-1)\left|\begin{array}{ccc}1& \mathrm{x}& \mathrm{x}+1\\ 2& \mathrm{x}-1& \mathrm{x}+1\\ 3& \mathrm{x}-2& \mathrm{x}+1\end{array}\right|$

Applying ${\mathrm{C}}_3\to {\mathrm{C}}_3-{\mathrm{C}}_2$, we get

$\mathrm{f}\left(\mathrm{x}\right)={\mathrm{x}}^2(\mathrm{x}-1)\left|\begin{array}{ccc}1& \mathrm{x}& 1\\ 2& \mathrm{x}-1& 2\\ 3& \mathrm{x}-2& 3\end{array}\right|=0$

Thus, f(500) = 0.