Let omega neq 1 be a cube root of unity. Then | vedclass

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(B) We know that $|a + b\omega + c\omega^2|^2 = (a + b\omega + c\omega^2)(\overline{a + b\omega + c\omega^2})$.Since $\overline{\omega} = \omega^2$ an

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$3$

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(B) We know that $|a + b\omega + c\omega^2|^2 = (a + b\omega + c\omega^2)(\overline{a + b\omega + c\omega^2})$.Since $\overline{\omega} = \omega^2$ and $\overline{\omega^2} = \omega$,this becomes $(a + b\omega + c\omega^2)(a + b\omega^2 + c\omega)$.Expanding this,we get $a^2 + ab\omega^2 + ac\omega + ab\omega + b^2\omega^3 + bc\omega^2 + ac\omega^2 + bc\omega^4 + c^2\omega^3$.Using $\omega^3 = 1$ and $1 + \omega + \omega^2 = 0$,this simplifies to $a^2 + b^2 + c^2 - ab - bc - ca$.This expression can be written as $\frac{1}{2}[(a - b)^2 + (b - c)^2 + (c - a)^2]$.For distinct non-zero integers $a, b, c$,the smallest possible values for the differences are $1$ and $2$ (e.g.,$a=1, b=2, c=3$).Substituting these,we get $\frac{1}{2}[(1 - 2)^2 + (2 - 3)^2 + (3 - 1)^2] = \frac{1}{2}[1 + 1 + 4] = \frac{6}{2} = 3$.

Let $\omega \neq 1$ be a cube root of unity. Then the minimum value of the set $\{|a + b\omega + c\omega^2|^2 : a, b, c \text{ are distinct non-zero integers}\}$ is equal to:

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