Question

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

• A. $0$
• B. $1$
• C. $500$
• D. $-500$

Answer 1

The correct option is A $0$

Taking $x$ common from $R_2$ and $x(x-1)$ common from $R_3$,
we get
$f\left(x\right)=x^2(x-1)\left|\begin{array}{ccc}1& x& x+1\\ 2& x-1& x+1\\ 3& x-2& x+1\end{array}\right|\text{Applying}{C}_3\to C_3-C_2,f\left(x\right)=x^2(x-1)\left|\begin{array}{ccc}1& x& 1\\ 2& x-1& 2\\ 3& x-2& 3\end{array}\right|=0f\left(x\right)=0\Rightarrow f\left(500\right)=0.$