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(100) is equal to

• A. 0
• B. 1
• C. 100
• D. -100

Answer 1

The correct option is A 0

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

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

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

operating : $R_1\to R_2-R_1$ & $R_2\to R_3-R_2$

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

As two rows of the determinant is zero

$\therefore$ Determinant $=0$

Hence, $f\left(x\right)=0$

$\therefore$ for any value of $x;f\left(x\right)=0$

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