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Question

If $$f(x)=\begin{vmatrix} 1 & 2(x-1) & 3(x-1)(x-2) \\ x-1 & (x-1)(x-2) & (x-1)(x-2)(x-3) \\ x & (x-1)x & (x-1)(x-2)x \end{vmatrix}$$ then $$f(41)$$is 


A
41
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B
41
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C
0
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D
Noneofthese
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Solution

The correct option is A $$41$$
$$f(x)=\left| \begin{matrix} 1 & 2(x-1) & 3(x-1)(x-2) \\ x-1 & (x-1)(x-2) & (x-1)(x-2)(x-3) \\ x & (x-1)x & x(x-1)(x-2) \end{matrix} \right| \\ f(x)=(x-1)\left| \begin{matrix} 1 & 2 & 3(x-1)(x-2) \\ (x-1) & (x-2) & (x-1)(x-2)(x-3) \\ x & x & x(x-1)(x-2) \end{matrix} \right| \\ =x{ (x-1) }^{ 2 }(x-2)\left| \begin{matrix} 1 & 2 & 3 \\ (x-1) & (x-2) & (x-3) \\ 1 & 1 & 1 \end{matrix} \right| \\ =x{ (x-1) }^{ 2 }(x-2)[(1)-(4)+(3)]\\ =x{ (x-1) }^{ 2 }(x-2)(0)\\ =0\\ \therefore f(x)=0\\ \Rightarrow f(41)=0$$

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