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If $z_k=\cos\big(\large\frac{k\pi}{10})$$+i\sin(\large\frac{k\pi}{10})$,then $z_1z_2z_3z_4$ equals

$\begin{array}{1 1}(A)\;-1&(B)\;0\\(C)\;1&(D)\;2\end{array} $

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We have $z_k=\omega^k$
where $\omega=\cos\large\frac{\pi}{10}$$+i\sin\large\frac{\pi}{10}$
Thus $z_1z_2z_3z_4=\omega^1.\omega^2.\omega^3.\omega^4=\omega^{10}$
$\Rightarrow \cos(\large\frac{10\pi}{10})+$$i\sin(\large\frac{10\pi}{10})$
(By De-Moivre's theorem)
$\Rightarrow \cos\pi$
$\Rightarrow -1$
Hence (A) is the correct answer.
answered Apr 10, 2014 by sreemathi.v
 

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