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(200) is equal to

• A. 1
• B. 200
• C. -200

Answer 1

$f\left(X\right)=x(x+1)\left|\begin{array}{ccc}1& 1& 1\\ 2x& x-1& x\\ 3x(x-1)& (x-1)(x-2)& x(x-1)\end{array}\right|=x(x+1)(x-1)\left|\begin{array}{ccc}1& 1& 1\\ 2x& x-1& x\\ 3x& x-2& x\end{array}\right|$
Applying $C_2\to C_2-C_1$ and $C_3\to C_3-C_1$ then
$f\left(x\right)=x(x+1)(x-1)\left|\begin{array}{ccc}1& 0& 0\\ 2x& -x-1& -x\\ 3x& -2x-2& -2x\end{array}\right|=x^2(x+1{)}^2(x-1)\left|\begin{array}{ccc}1& 0& 0\\ 2x& -1& -1\\ 3x& -2& -2\end{array}\right|=0\therefore f(200)=0$