Question

Let $S$ be the set of all complex numbers z satisfying $\left|z^2+z+1\right|=1$. Then which of the following statements is/are TRUE?

• A. $\left|z+\frac{1}{2}\right|\le \frac{1}{2}$ for all $z\in S$
• B. $|z|\le 2$ for all $z\in S$
• C. $\left|z+\frac{1}{2}\right|\ge \frac{1}{2}$ for all $z\in S$
• D. The set $S$ has exactly four elements

Answer 1

Answer: (B), (C)

$|Z^2+Z+1|=1$
$\Rightarrow |(Z+\frac{1}{2}{)}^2+\frac{3}{4}|=1$
$\Rightarrow |(Z+\frac{1}{2}{)}^2+\frac{3}{4}|\le |Z+\frac{1}{2}{|}^2+\frac{3}{4}$
$\Rightarrow 1\le |z+\frac{1}{2}{|}^2+\frac{3}{4}\Rightarrow |(z+\frac{1}{2}){|}^2\ge \frac{1}{4}$
$\Rightarrow |z+\frac{1}{2}|\ge \frac{1}{2}$
$\text{ also }|(z^2+z)+1|=1\ge ||z^2+z|-1|$
$\Rightarrow |z^2+z|-1\le 1$
$\Rightarrow |z^2+z|\le 2$
$\Rightarrow ||z^2|-|z||\le |z^2+z|\le 2$
$\Rightarrow |r^2-r|\le 2$
$\Rightarrow |r=|z\mid \le 2;\forall z\in S$
Also we can always find root of the equation $z^2+z+1=e^{\theta };\forall \theta \in R$ Hence set ' $S$ ' is infinite