MCQ
If $\omega $ is cube root of unity, then root of the equation $\left| {\begin{array}{*{20}{c}}
  {x + 2}&\omega &{{\omega ^2}} \\ 
  \omega &{x + 1 + {\omega ^2}}&1 \\ 
  {{\omega ^2}}&1&{x + 1 + \omega } 
\end{array}} \right| = 0$ is 
  • A
    $0$
  • B
    $\omega $
  • C
    $\omega ^2$
  • $-1$

Answer

Correct option: D.
$-1$
d
$\mathrm{R}_{1} \rightarrow \mathrm{R}_{1}+\mathrm{R}_{2}+\mathrm{R}_{3}$

$\Rightarrow(x+1)\left|\begin{array}{ccc}{1} & {1} & {1} \\ {\omega} & {x+1+\omega^{2}} & {1} \\ {\omega^{2}} & {1} & {x+1+\omega}\end{array}\right|$

$ = 0 \Rightarrow {({\text{x}} + 1)^3} = 0 \Rightarrow \boxed{{\text{x}} =  - 1}$

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