If omega is a non-real root of the equation x | vedclass

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(B) We know that $1 + \omega^r + \omega^{2r}$ is a geometric series sum or can be evaluated based on the properties of cube roots of unity.For any int

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$3$

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(B) We know that $1 + \omega^r + \omega^{2r}$ is a geometric series sum or can be evaluated based on the properties of cube roots of unity.For any integer $r$,the expression $1 + \omega^r + \omega^{2r}$ follows the property:$1 + \omega^r + \omega^{2r} = \begin{cases} 3, & 	ext{if } r 	ext{ is a multiple of } 3 \ 0, & 	ext{if } r 	ext{ is not a multiple of } 3 \end{cases}$In the given summation $\sum_{r=1}^5 (1 + \omega^r + \omega^{2r})$,we evaluate for $r = 1, 2, 3, 4, 5$:For $r=1$: $1 + \omega + \omega^2 = 0$For $r=2$: $1 + \omega^2 + \omega^4 = 1 + \omega^2 + \omega = 0$For $r=3$: $1 + \omega^3 + \omega^6 = 1 + 1 + 1 = 3$For $r=4$: $1 + \omega^4 + \omega^8 = 1 + \omega + \omega^2 = 0$For $r=5$: $1 + \omega^5 + \omega^{10} = 1 + \omega^2 + \omega = 0$Summing these values: $0 + 0 + 3 + 0 + 0 = 3$.

If $\omega$ is a non-real root of the equation $x^3 - 1 = 0$,then the value of $\sum_{r=1}^5 (1 + \omega^r + \omega^{2r})$ is

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