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Q.
Let $n$ be a positive integer and a complex number with unit modulus is a solution of the equation $Z^{n}+Z+1=0$ , then the value of $n$ can be
NTA AbhyasNTA Abhyas 2022
Solution:
Let, $Z_{1}$ satisfies $Z^{n}+1=-Z$
$\Rightarrow Z_{1}$ also satisfies $\left|Z^{n} + 1\right|=\left|- Z\right|$
$\Rightarrow Z_{1}$ also satisfies $\left|Z^{n} + 1\right|=1$
$\Rightarrow \left|Z_{1}^{n} + 1\right|=1$ … $\left(1\right)$
Now, $\left|Z_{1}\right|=1\Rightarrow \left|Z_{1}^{n}\right|=1$ … $\left(2\right)$
From $\left(1\right)$ & $\left(2\right)$ ,
we get, $Z_{1}^{n}$ must be the point of intersection of
$\left|Z\right|=1$ & $\left|Z^{n} + 1\right|=1$
$\Rightarrow Z_{1}^{n}=\omega $ or $\omega ^{2}$ {where, $\omega $ is non-real cube root of unity}
$\Rightarrow Z_{1}$ can be $\omega $ or $\omega ^{2}$
$\Rightarrow n$ is of the form of $3k+2$ .