Value of Complex Ratio with Cube Roots of Unity

To evaluate the given expression, we analyze the given conditions step-by-step.

Given equations:
$x = a + b + c$
$y = a + b\omega + c\omega^2$
$z = a + b\omega^2 + c\omega$

We compute the squared moduli of $x$, $y$, and $z$:

1. For $|x|^2$:
   $|x|^2 = x \bar{x} = (a+b+c)(\bar{a}+\bar{b}+\bar{c})$
   $|x|^2 = |a|^2 + |b|^2 + |c|^2 + (a\bar{b} + b\bar{a}) + (b\bar{c} + c\bar{b}) + (c\bar{a} + a\bar{c})$

2. For $|y|^2$:
   $|y|^2 = y \bar{y} = (a+b\omega+c\omega^2)(\bar{a}+\bar{b}\bar{\omega}+\bar{c}\bar{\omega}^2)$
   Since $\bar{\omega} = \omega^{-1}$, we expand this to get:
   $|y|^2 = |a|^2 + |b|^2 + |c|^2 + a\bar{b}\bar{\omega} + b\bar{a}\omega + b\bar{c}\bar{\omega} + c\bar{b}\omega + c\bar{a}\omega^2 + a\bar{c}\bar{\omega}^2$

3. For $|z|^2$:
   $|z|^2 = z \bar{z} = (a+b\omega^2+c\omega)(\bar{a}+\bar{b}\bar{\omega}^2+\bar{c}\bar{\omega})$
   $|z|^2 = |a|^2 + |b|^2 + |c|^2 + a\bar{b}\bar{\omega}^2 + b\bar{a}\omega^2 + b\bar{c}\omega + c\bar{b}\bar{\omega} + c\bar{a}\omega + a\bar{c}\bar{\omega}$

Summing these three equations yields:
$|x|^2 + |y|^2 + |z|^2 = 3(|a|^2 + |b|^2 + |c|^2) + a\bar{b}(1 + \bar{\omega} + \bar{\omega}^2) + b\bar{a}(1 + \omega + \omega^2) + b\bar{c}(1 + \bar{\omega} + \omega) + c\bar{b}(1 + \omega + \bar{\omega}) + c\bar{a}(1 + \omega^2 + \omega) + a\bar{c}(1 + \bar{\omega}^2 + \bar{\omega})$

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Case 1: Intended Question ($\omega$ as a non-real cube root of unity, i.e., $\omega = e^{i 2\pi/3}$)

If $\omega = e^{i 2\pi/3}$, then $1 + \omega + \omega^2 = 0$ and $1 + \bar{\omega} + \bar{\omega}^2 = 0$.
In this case, all cross-terms vanish completely:
$|x|^2 + |y|^2 + |z|^2 = 3(|a|^2 + |b|^2 + |c|^2)$

Thus, the ratio would be:
$\frac{|x|^2+|y|^2+|z|^2}{|a|^2+|b|^2+|c|^2} = 3$

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Case 2: Given Question ($\omega = e^{i\pi/3}$ as explicitly stated)

With $\omega = e^{i\pi/3}$, we have:
$1 + \omega + \omega^2 = 1 + e^{i\pi/3} + e^{i 2\pi/3} = 1 + \left(\frac{1}{2} + i\frac{\sqrt{3}}{2}\right) + \left(-\frac{1}{2} + i\frac{\sqrt{3}}{2}\right) = 1 + i\sqrt{3} \neq 0$

Since $1 + \omega + \omega^2 \neq 0$, the cross-terms do not vanish, and the ratio $\frac{|x|^2+|y|^2+|z|^2}{|a|^2+|b|^2+|c|^2}$ depends explicitly on the values of the complex numbers $a$, $b$, and $c$, rather than evaluating to a fixed constant.

Due to this typographical error in defining $\omega$ in the question paper, this question was awarded MARKS TO ALL.