If | * 20 c a - b - c & 2a & 2a 2b & b - c - a & 2b 2c & 2c & c -… - Vidyadip

MCQ

If $\left| {\begin{array}{*{20}{c}}
{a - b - c}&{2a}&{2a}\\
{2b}&{b - c - a}&{2b}\\
{2c}&{2c}&{c - a - b}
\end{array}} \right|$ $ = \left( {a + b + c} \right)\,{\left( {x + a + b + c} \right)^2}$ , $x \ne 0$ and $a + b + c \ne 0$, then $x$ is equal to

A
$abc$
✓
$ - 2 \left( {a + b + c} \right)$
C
$ 2 \left( {a + b + c} \right)$
D
$ - \left( {a + b + c} \right)$

✓

Answer

Correct option: B.

$ - 2 \left( {a + b + c} \right)$

b
${R_1} \to {R_1} + {R_2} + {R_3}$

$\left( {a + b + c} \right)\left| {\begin{array}{*{20}{c}}
1&1&1\\
{2b}&{b - a - c}&{2b}\\
{2c}&{2c}&{c - a - b}
\end{array}} \right|$

${c_3} \to {c_3} - {c_1},{c_2} \to {c_2} - {c_1}$

$ = \left( {a + b + c} \right)\left| {\begin{array}{*{20}{c}}
1&0&0\\
{2b}&{ - \left( {a + b + c} \right)}&0\\
{2c}&0&{ - \left( {a + b + c} \right)}
\end{array}} \right|$

$ = {\left( {a + b + c} \right)^3}$

$ = \left( {a + b + c} \right){\left( {x + a + b + c} \right)^2}$

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