If $D=\left|\begin{array}{ccc}1 & 1 & 1 \\ 1 & 1+x & 1 \\ 1 & 1 & 1+y\end{array}\right|$ for $x \neq 0, y…
If $D=\left|\begin{array}{ccc}1 & 1 & 1 \\ 1 & 1+x & 1 \\ 1 & 1 & 1+y\end{array}\right|$ for $x \neq 0, y \neq 0$ then $D$ is
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divisible by neither $x$ nor $y$
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divisible by both $x$ and $y$
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divisible by $x$ but not $y$
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divisible by $y$ but not $x$
Solution
$D=\left|\begin{array}{ccc}1 & 1 & 1 \\ 1 & 1+x & 1 \\ 1 & 1 & 1+y\end{array}\right|$
$\mathrm{C}_2 \rightarrow \mathrm{C}_2-\mathrm{C}_1$ \& $\mathrm{C}_3 \rightarrow \mathrm{C}_3-\mathrm{C}_1$
$\left|\begin{array}{lll}
1 & 0 & 0 \\
1 & x & 0 \\
1 & 0 & y
\end{array}\right|=x y$.
Hence $D$ is divisible by both $x$ and $y$.
Asked in: JEE Main 2007
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