Question

If $f\left(x\right)=\left|\begin{array}{ccc}1& x& x+1\\ 2x& x(x-1)& x(x+1)\\ 3x(x-1)& x(x-1)(x-2)& x(x+1)(x-1)\end{array}\right|$ then $f\left(100\right)=$

• A. 0
• B. 1
• C. 100
• D. -100

Answer 1

The correct option is A 0

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

Taking $x$ common from $R_2$ and $x(x-1)$ from $R_3$, we get
$f\left(x\right)=x^2(x-1)\left|\begin{array}{ccc}1& x& x+1\\ 2& (x-1)& (x+1)\\ 3& (x-2)& (x+1)\end{array}\right|$

Applying $C_3\to C_3-C_2$
$f\left(x\right)=x^2(x-1)\left|\begin{array}{ccc}1& x& 1\\ 2& (x-1)& 2\\ 3& (x-2)& 3\end{array}\right|=0$
Hence $f\left(100\right)=0$