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

(B) divisible by both $x$ and $y$

(A) divisible by neither x nor y

(C) divisible by $x$ but not $y$

JEE Main · 2007 · 29 Apr 2007 · Offline · Determinants

D = 111 1 1 + x 1 11 1 + y

for

x \neq = 0, y \neq = 0

then

D

Answer: (B)

Subtracting the first column from the others,

C_{2} \to C_{2} - C_{1}

and

C_{3} \to C_{3} - C_{1}

, turns the determinant into

111 0 x 0 00 y

. Expanding along the top row gives

D = x y

. Being the product

x y

, the determinant is divisible by both

x

and

y

JEE Main · 2007 · 29 Apr 2007 · Offline · Single correct option