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Question

The value of $$\begin{vmatrix} 2& a + b + c + d & ab + cd\\ a + b + c + d & 2(a + b)(c + d) & ab(c + d) + cd(a + b)\\ ab + cd & ab(c + d) + cd(a + b) & 2abcd\end{vmatrix}$$


A
0
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B
1
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C
1
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D
None of these
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Solution

The correct option is A $$0$$
Given $$\left| \begin{matrix} 2 & a+b+c+d &  ab+cd \\ a+b+c+d\qquad  & 2(a+b)(c+d)\qquad  & ab(c+d)+cd(a+b) \\ ab+cd & ab(c+d)+cd(a+b) & 2abcd \end{matrix} \right| $$ 

To solve this type, consider $$a=b=c=d=1$$.

Now we have,

$$= \left| \begin{matrix} 2 & 4 & 2 \\ 4 & 8 & 4 \\ 2 & 4 & 2 \end{matrix} \right| $$ 

$$ =2(8\times 2-4\times 4) - 4(4\times 2-2\times 4) + 2(4\times 4-8\times 2)$$

 $$ =0$$

Mathematics

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