# If  Option 1) divisible by $x$ but not $y$ Option 2) divisible by $y$ but not $x$ Option 3) divisible by neither $x$ nor  $y$ Option 4) divisible by both $x$ and   $y$

As we learnt in

Property of determinant -

If each element in a row ( or column ) of a determinant is written as the sum of two or more terms then the determinant can be written as the sum of two or more determinants

- wherein

$D=\begin{vmatrix} 1& 1&1\\ 1 & 1+x &1\\ 1&1 &1+y \end{vmatrix}$

$=1\left ( 1+x+y+xy-1 \right )-1\left ( 1+y-1 \right )+1\left ( 1-1-x \right )= xy$

= x + y + xy - y - x

= xy

which is divisible by both x and y

Option 1)

divisible by $x$ but not $y$

Incorrect Option

Option 2)

divisible by $y$ but not $x$

Incorrect Option

Option 3)

divisible by neither $x$ nor  $y$

Incorrect Option

Option 4)

divisible by both $x$ and   $y$

Correct Option

N

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