Question

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

• A. $87$
• B. $97$
• C. $104$
• D. $222$

Answer 1

Answer: (C)

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 $3k+2$ .