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

$(a)\;0\qquad(b)\;1\qquad(c)\;100\qquad(d)\;-100$

1 Answer

$f(x)=\begin{vmatrix}1 &x&x+1\\2x&x(x-1)&(x+1)x\\3x(x-1)&x(x-1)(x-2)&(x+1)x(x-1)\end{vmatrix}$
Apply $C_1\rightarrow C_1+C_2$
$\begin{vmatrix}x+1 &x&x+1\\(x+1)x&x(x-1)&(x+1)x\\(x+1)x(x-1)&x(x-1)(x-2)&(x+1)x(x-1)\end{vmatrix}$
$\Rightarrow 0$
[$C_1$ and $C_2$ are identical]
Which is free of $x$,so the function is true for all values of $x$
$\therefore$ At $x=100$
$f(x)=0$
$f(100)=0$
Hence (a) is the correct answer.
answered Nov 20, 2013 by sreemathi.v
 

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