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**Hint:**Use the property of imaginary root of unity, \[1 + \omega + {\omega ^2} = 0\]. This formula can be used to find an equation for \[{\omega ^2}\]which can be substituted in \[z = a + b\omega + c{\omega ^2}\]. Use one of the values of i.e. \[\left( { - \dfrac{1}{2} + \dfrac{{\sqrt 3 }}{2}i} \right)\] to make z in imaginary number notation. Next, use the formula \[|z| = \sqrt {{x^2} + {y^2}} \]and calculate it. Then find minimum value of \[|z|\] by using the equations \[a = b = k\] and \[c = k + 1\]. Finally, find the value of \[[2|z|]\].

**Complete step by step solution:**Find equation for \[{\omega ^2}\]

Let \[z = a + b\omega + c{\omega ^2}\]

We know that \[1 + \omega + {\omega ^2} = 0\].

We will use this to find the value of \[{\omega ^2}\]to form a linear equation that we can substitute in \[z = a + b\omega + c{\omega ^2}\]and solve it like we solve linear equations. This is known as substitution method.

Now,

\[\begin{array}{l}

1 + \omega + {\omega ^2} = 0\\

\Rightarrow {\omega ^2} = - 1 - \omega

\end{array}\]

Substitute the value of \[{\omega ^2}\]in \[z\].

\[\begin{array}{l}

z = a + b\omega + c{\omega ^2}\\

\Rightarrow z = a + b\omega + c( - 1 - \omega )\\

\Rightarrow z = (a - c) + (b - c)\omega

\end{array}\]

Put value of \[\omega \] in \[z\]

For the equation \[z = a + b\omega + c{\omega ^2}\], we know that one of the values of is \[\left( { - \dfrac{1}{2} + \dfrac{{\sqrt 3 }}{2}i} \right)\].

Putting the value of \[\omega \] in \[z\], we get

\[\begin{array}{l}

z = (a - c) + (b - c)\left( { - \dfrac{1}{2} + \dfrac{{\sqrt 3 }}{2}i} \right)\\

z = \left( {a - \dfrac{b}{2} - \dfrac{c}{2}} \right) + \dfrac{{\sqrt 3 }}{2}(b - c)i

\end{array}\]

Compare with formula for \[|z|\]

We know, \[|z| = \sqrt {{x^2} + {y^2}} \]

So, here, \[|z| = \sqrt {{{\left( {a - \dfrac{b}{2} - \dfrac{c}{2}} \right)}^2} + \dfrac{3}{4}{{(b - c)}^2}} \]

\[|z| = \sqrt {\dfrac{1}{2}[{{(a - b)}^2} + {{(b - c)}^2} + {{(c - a)}^2}} \]

Find minimum value of \[|z|\]

To find minimum value of \[|z|\], we will put, \[a = b = k\] and \[c = k + 1\]

Then, \[|z| = \sqrt {\dfrac{1}{2}(0 + {1^2} + {1^2}} = 1\]

So, minimum value of \[|z|\] will be 1

Find value of \[[2|z|]\]

\[\begin{array}{l}

2|z| = 2 \times 1 = 2\\

[2|z|] = 2

\end{array}\]

**Note:**Complex numbers can be a difficult topic for beginners. Learn about the imaginary roots of \[\omega \] and its properties. Also remember the values of \[\omega \] to solve questions easily. Make proper use of formulas of modulus and complex numbers.

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