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 $f\left(100\right)$ is equal to

• A. $0$
• B. $1$
• C. $100$
• D. $-100$

Answer 1

The correct option is A $0$

$f\left(x\right)=\left|\begin{array}{ccc}1& x& x+1\\ 2x& x(x-1)& x(x+1)\\ 3x(x-1)& x(x-1)(x-2)& x(x^2-1)\end{array}\right|$
$f\left(x\right)=x(x+1)(x-1)\left|\begin{array}{ccc}1& 1& 1\\ 2x& (x-1)& x\\ 3x& (x-2)& x\end{array}\right|$
$C_1\to C_1-C_2,C_2\to C_2-C_3$
$f\left(x\right)=x(x+1)(x-1)\left|\begin{array}{ccc}0& 0& 1\\ x+1& -1& x\\ 2(x+1)& -2& x\end{array}\right|$
$f\left(x\right)=-x(x+1{)}^2(x-1)\left|\begin{array}{ccc}0& 0& 1\\ 1& 1& x\\ 2& 2& x\end{array}\right|$
$\Rightarrow f\left(x\right)=0$

$\Rightarrow f\left(100\right)=0$