(r, theta) denotes r(cos theta + i sin theta) | vedclass

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(A) Given that $(r, 	heta) = r(\cos 	heta + i \sin 	heta) = re^{i	heta}$.Thus,$x = e^{i\alpha}$,$y = e^{i\beta}$,and $z = e^{i\gamma}$.Given $x +

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$3$

Vedclass · Answer

(A) Given that $(r, 	heta) = r(\cos 	heta + i \sin 	heta) = re^{i	heta}$.Thus,$x = e^{i\alpha}$,$y = e^{i\beta}$,and $z = e^{i\gamma}$.Given $x + y + z = 0$,we know that for complex numbers,if $x + y + z = 0$,then $x^3 + y^3 + z^3 = 3xyz$.Dividing by $xyz$,we get $\frac{x^2}{yz} + \frac{y^2}{xz} + \frac{z^2}{xy} = 3$.Substituting the exponential forms:$\frac{e^{2i\alpha}}{e^{i\beta}e^{i\gamma}} + \frac{e^{2i\beta}}{e^{i\alpha}e^{i\gamma}} + \frac{e^{2i\gamma}}{e^{i\alpha}e^{i\beta}} = 3$.$e^{i(2\alpha - \beta - \gamma)} + e^{i(2\beta - \alpha - \gamma)} + e^{i(2\gamma - \alpha - \beta)} = 3$.Taking the real part of both sides:$\cos(2\alpha - \beta - \gamma) + \cos(2\beta - \alpha - \gamma) + \cos(2\gamma - \alpha - \beta) = 3$.

List-$I$	List-$II$
$(i)$ $a \bar{a}$	$(A)$ $-\frac{\pi}{3}$
$(ii)$ $\arg \left(\frac{1}{\bar{a}}\right)$	$(B)$ $-i \sqrt{3}$
$(iii)$ $a - \bar{a}$	$(C)$ $2i / \sqrt{3}$
$(iv)$ $\operatorname{Im}\left(\frac{4}{3a}\right)$	$(D)$ $1$
	$(E)$ $\pi / 3$
	$(F)$ $\frac{2}{\sqrt{3}}$

$(r, \theta)$ denotes $r(\cos \theta + i \sin \theta)$. If $x = (1, \alpha)$,$y = (1, \beta)$,$z = (1, \gamma)$ and $x + y + z = 0$,then $\sum \cos (2\alpha - \beta - \gamma) = $

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