Matrices and Determinants 2 Question 16
18. If $f(x)=\begin{vmatrix}1 & x & x+1 \\ 2 x & x(x-1) & (x+1) x \\ 3 x(x-1) & x(x-1)(x-2) & (x+1) x(x-1)\end{vmatrix}$,
then $f(100)$ is equal to
(1999, 2M)
(a) 0
(b) 1
(c) 100
(d) -100
Show Answer
Answer:
Correct Answer: 18. (a)
Solution:
- Given,
$ f(x)=\begin{vmatrix} 1 & x & x+1 \\ 2 x & x(x-1) & (x+1) x \\ 3 x(x-1) & x(x-1)(x-2) & (x+1) x(x-1) \end{vmatrix} $
Applying $C_{3} \rightarrow C_{3}-\left(C_{1}+C_{2}\right)$
$ \begin{vmatrix} 1 & x & 0 \\ 2 x & x(x-1) & 0 \\ 3 x(x-1) & x(x-1)(x-2) & 0 \end{vmatrix} $
$ \therefore \quad f(x)=0 \Rightarrow f(100)=0 $
