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  4. If z+(1/z)+1=0, then z2003+(1/z2003) is equal to

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Q. If $z+\frac{1}{z}+1=0,$ then $z^{2003}+\frac{1}{z^{2003}}$ is equal to

NTA AbhyasNTA Abhyas 2020Complex Numbers and Quadratic Equations

Solution:

Given equation can be written as $z^{2} + z + 1 = 0$
$\Rightarrow $ $z=\omega , \, \left(\omega \right)^{2}\left( \, roots\right)$
Now, $z^{2003}=\omega ^{2003}=\omega ^{2}$
$z^{2003}+\frac{1}{z^{2003}}=\omega ^{2}+\frac{1}{\omega ^{2}}=\omega ^{2}+\omega =-1$