Question

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

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

Answer 1

Answer: (A)

$0$

Solution:- (A) $0$

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

Taking $x,x(x-1)$ and $(x+1)$ as common from $R_2,R_3$ and $C_3$ respectively.

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

Applying $R_1\to R_1-R_2$

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

Applying $R_2\to R_2-R_3$

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

As we know that if any two row or column is a determent are same then the value of determent is $0$.

$\therefore f\left(x\right)=0$

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

Hence the required answer is $\left(A\right)0$.