Question

If $f\left(x\right)=\left|\begin{array}{ccc}1& x& x+1\\ 2x& x(x-1)& x(x+1)\\ 3x(x-1)& x(x-1)(x-2)& x(x^2-1)\end{array}\right|$ then $f\left(200\right)$ is equal to ..............

Answer 1

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

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

$R_3\to R_3-R_2,R_2\to R_2-R_1$

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

$\Rightarrow f\left(x\right)=0$

$\Rightarrow f\left(200\right)=0$