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Mathematics Question on Determinants

If f(x)=∣1x;x;+1 2xx;(x−1)(x−1)x 3x(x−1)x;(x−1)(x−2)(x−1)∣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)f (100) is equal to

A

0

B

1

C

100

D

10

Answer

0

Explanation

Solution

f(x)=x(x+1)∣111 x−1x; 3x(x−1)(x−1)(x−2)x;(x−1)∣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}
C1∼C1−C3;C2∼C2−C3C_{1}\sim C_{1}-C_{3} ; C_{2}\sim C_{2}-C_{3}
=x(x+1)∣001 x−1x; 2x(x−1)−2(x−1)x;(x−1)∣=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(x+1)∣x−1 2x(x−1)−2(x−1)∣=0=x\left(x+1\right)\begin{vmatrix}x&-1\\\ 2x\left(x-1\right)&-2\left(x-1\right)\end{vmatrix}=0
∴f(100)=0\therefore f\left(100\right)=0