If $f\left(x\right)=\left|\begin{array}{ccc}1& x& x+2\\ 3x& x(x-2)& x(x+2)\\ 5x(x-2)& x(x-2)(x-4)& x\left(x^2-4\right)\end{array}\right|$ then $f\left(300\right)$ equals

Consider

$f\left(x\right)=\left|\begin{array}{ccc}1& x& x+2\\ 3x& x(x-2)& x(x+2)\\ 5x(x-2)& x(x-2)(x-4)& x\left(x^2-4\right)\end{array}\right|$

Taking x out from $R_2,x(x-2)$ from $R_3$ and $x + 2$ from $C_3$we have

$=x^2\left(x^2-4\right)\left|\begin{array}{ccc}1& x& 1\\ 3& x-2& 1\\ 5& x-4& 1\end{array}\right|$

Using $R_2:R_2-R_1,R_3:R_3-R_2$ we have

$f\left(x\right)=x^2\left(x^2-4\right)\left|\begin{array}{ccc}1& x& 1\\ 2& -2& 0\\ 2& -2& 0\end{array}\right|=0$

$\left[\therefore {\mathrm{R}}_2\text{ and }{\mathrm{R}}_3\text{ are identical }\right]$

$\Rightarrow \mathrm{f}\left(300\right)=0$