Question

If $f\left(x\right)=\left|\begin{array}{ccc}1& x& x+1\\ 2x& x(x-1)& (x+1)x\\ 3x(x-1)& x(x-1)(x-2)& (x+1)x(x-1)\end{array}\right|$ then $f\left(500\right)$ is equal to

• A. 0
• B. 1
• C. 500
• D. -500

Answer 1

The correct option is A 0

$f\left(x\right)=\left|\begin{array}{ccc}1& x& x+1\\ 2x& x(x-1)& (x+1)x\\ 3x(x-1)& x(x-1)(x-2)& (x+1)x(x-1)\end{array}\right|$
$=x^2(x-1)\left|\begin{array}{ccc}1& x& x+1\\ 2& (x-1)& (x+1)\\ 3& (x-2)& (x+1)\end{array}\right|$
$f\left(x\right)=x^2(x-1)(x+1)\left|\begin{array}{ccc}1& x& 1\\ 2& (x-1)& 1\\ 3& (x-2)& 1\end{array}\right|$
$R_1\to R_1-R_2,R_2\to R_2-R_3$
$f\left(x\right)=x^2(x-1)(x+1)\left|\begin{array}{ccc}-1& 1& 0\\ -1& 1& 0\\ 3& (x-2)& 1\end{array}\right|$
$f\left(x\right)=0$
$\Rightarrow f\left(500\right)=0$