Question

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

• A. 0
• B. 1
• C. 100
• D. 10

Answer 1

Answer: (A)

$f \left(x\right)=x\left(x+1\right)\begin{vmatrix}1&1&1\\ x&-1&x\\ 3x\left(x-1\right)&\left(x-1\right)\left(x-2\right)&x\left(x-1\right)\end{vmatrix}$
$C_{1}\sim C_{1}-C_{3} ; C_{2}\sim C_{2}-C_{3}$
$=x\left(x+1\right)\begin{vmatrix}0&0&1\\ x&-1&x\\ 2x\left(x-1\right)&-2\left(x-1\right)&x\left(x-1\right)\end{vmatrix}$
$=x\left(x+1\right)\begin{vmatrix}x&-1\\ 2x\left(x-1\right)&-2\left(x-1\right)\end{vmatrix}=0$
$\therefore f\left(100\right)=0$