(sqrt{3}+i)^7+(sqrt{3}-i)^7 is equal to (in s | vedclass

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(C) Let $z = \sqrt{3}+i$. Converting to polar form,$r = \sqrt{(\sqrt{3})^2 + 1^2} = 2$ and $	heta = 	an^{-1}(\frac{1}{\sqrt{3}}) = \frac{\pi}{6}$. S

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$-128$

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(C) Let $z = \sqrt{3}+i$. Converting to polar form,$r = \sqrt{(\sqrt{3})^2 + 1^2} = 2$ and $	heta = 	an^{-1}(\frac{1}{\sqrt{3}}) = \frac{\pi}{6}$. So,$z = 2(\cos \frac{\pi}{6} + i \sin \frac{\pi}{6})$ and $\bar{z} = 2(\cos \frac{\pi}{6} - i \sin \frac{\pi}{6})$. Using De Moivre's Theorem,$z^7 + \bar{z}^7 = 2^7(\cos \frac{7\pi}{6} + i \sin \frac{7\pi}{6}) + 2^7(\cos \frac{7\pi}{6} - i \sin \frac{7\pi}{6})$. $= 2^7(2 \cos \frac{7\pi}{6}) = 2^8 \cos(\pi + \frac{\pi}{6})$. $= -2^8 \cos(\frac{\pi}{6}) = -256 	imes \frac{\sqrt{3}}{2} = -128 \sqrt{3}$.

List-$I$ (Expression)	List-$II$ (Value)
$A$. $\omega^{1010} + \omega^{2000}$	$I$. $0$
$B$. $(1 + \omega - \omega^2)(1 - \omega + \omega^2)$	$II$. $1$
$C$. $(2 + \omega^2 + \omega^4)^5$	$III$. $-1$
$D$. $(3 + 5\omega + 3\omega^2)^3$	$IV$. $4$
	$V$. $8$

$(\sqrt{3}+i)^7+(\sqrt{3}-i)^7$ is equal to (in $\sqrt{3}$)

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