If D=| array ccc 1 & 1 & 1 1 & 1+x & 1 1 & 1 & 1+y array | for x ≠ 0, y ≠ 0 then D is

Correct answer: B. divisible by both x and y D=| array ccc 1 & 1 & 1 1 & 1+x & 1 1 & 1 & 1+y array | C_2 arrow C_2-C_1 \& C_3 arrow C_3-C_1 | array lll 1 & 0 & 0 1 & x & 0 1 & 0 & y array |=x y . Hence D is divisible by both x and y .

JEE Main 2007

Exam: JEE Main
Year: 2007
Subject: Mathematics
Chapter: Determinants

JEE Main 2007 · Mathematics · Determinants

If D=| array ccc 1 & 1 & 1 1 & 1+x & 1 1 & 1 & 1+y array | for x ≠...

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

Options

Adivisible by neither $x$ nor $y$
Bdivisible by both $x$ and $y$
Cdivisible by $x$ but not $y$
Ddivisible by $y$ but not $x$

Correct answer

B. divisible by both $x$ and $y$

Step-by-step 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

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