Question

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

• A. 0
• B. 1
• C. 100
• D. 10

Answer 1

Answer: (A)

$f\left(x\right)=x\left(x+1\right)\left|\begin{array}{ccc}1& 1& 1\\ x& -1& x\\ 3x\left(x-1\right)& \left(x-1\right)\left(x-2\right)& x\left(x-1\right)\end{array}\right|$
$C_1\sim C_1-C_3;C_2\sim C_2-C_3$
$=x\left(x+1\right)\left|\begin{array}{ccc}0& 0& 1\\ x& -1& x\\ 2x\left(x-1\right)& -2\left(x-1\right)& x\left(x-1\right)\end{array}\right|$
$=x\left(x+1\right)\left|\begin{array}{cc}x& -1\\ 2x\left(x-1\right)& -2\left(x-1\right)\end{array}\right|=0$
$\therefore f\left(100\right)=0$