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ISI2016-DCG-3

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The value of  $\begin{vmatrix} 1+a& 1& 1& 1\\ 1&1+b &1 &1 \\ 1&1 &1+c &1 \\ 1&1 &1 &1+d \end{vmatrix}$ is

  1. $abcd(1+\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d})$
  2. $abcd(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d})$
  3. $1+\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}$
  4. None of these
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$\begin{vmatrix} 1+a& 1& 1& 1\\ 1&1+b &1 &1 \\ 1&1 &1+c &1 \\ 1&1 &1 &1+d \end{vmatrix}$

$=abcd\begin{vmatrix} \frac{1}{a}+1& \frac{1}{a}& \frac{1}{a}&\frac{1}{a}\\ \frac{1}{b}&\frac{1}{b}+1 &\frac{1}{b} &\frac{1}{b} \\ \frac{1}{c}&1 &\frac{1}{c}+1 &\frac{1}{c} \\ \frac{1}{d}&\frac{1}{d} &\frac{1}{d}&\frac{1}{d}+1 \end{vmatrix}$

$=abcd\begin{vmatrix} \frac{1}{a}+ \frac{1}{b}+ \frac{1}{c}+ \frac{1}{d}+1& \frac{1}{a}+ \frac{1}{b}+ \frac{1}{c}+ \frac{1}{d}+1& \frac{1}{a}+ \frac{1}{b}+ \frac{1}{c}+ \frac{1}{d}+1& \frac{1}{a}+ \frac{1}{b}+ \frac{1}{c}+ \frac{1}{d}\\ \frac{1}{b}&\frac{1}{b}+1 &\frac{1}{b} &\frac{1}{b} \\ \frac{1}{c}&1 &\frac{1}{c}+1 &\frac{1}{c} \\ \frac{1}{d}&\frac{1}{d} &\frac{1}{d}&\frac{1}{d}+1 \end{vmatrix}$

$=abcd( \frac{1}{a}+ \frac{1}{b}+ \frac{1}{c}+ \frac{1}{d}+1)\begin{vmatrix} 1& 1& 1&1\\ \frac{1}{b}&\frac{1}{b}+1 &\frac{1}{b} &\frac{1}{b} \\ \frac{1}{c}&1 &\frac{1}{c}+1 &\frac{1}{c} \\ \frac{1}{d}&\frac{1}{d} &\frac{1}{d}&\frac{1}{d}+1 \end{vmatrix}$

$=abcd( \frac{1}{a}+ \frac{1}{b}+ \frac{1}{c}+ \frac{1}{d}+1)\begin{vmatrix} 1& 0& 0&0\\ \frac{1}{b}&1 &0 &0\\ \frac{1}{c}&0&1 &0\\ \frac{1}{d}&0 &0&1 \end{vmatrix}$

$=abcd( \frac{1}{a}+ \frac{1}{b}+ \frac{1}{c}+ \frac{1}{d}+1)$

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