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. $\left|z+\frac{1}{2}\right| \leq \frac{1}{2}$ for all $z \in S$
• B. $|z| \leq 2$ for all $z \in S$
• C. $\left|z+\frac{1}{2}\right| \geq \frac{1}{2}$ 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 $
$\Rightarrow|(Z+\frac{1}{2})^2+\frac{3}{4}| \leq|Z+\frac{1}{2}|^2+\frac{3}{4}$
$\Rightarrow 1 \leq|z+\frac{1}{2}|^2+\frac{3}{4} \Rightarrow|(z+\frac{1}{2})|^2 \geq \frac{1}{4} $
$\Rightarrow|z+\frac{1}{2}| \geq \frac{1}{2} $
$\text { also }|(z^2+z)+1|=1 \geq|| z^2+z|-1| $
$\Rightarrow|z^2+z|-1 \leq 1 $
$\Rightarrow|z^2+z| \leq 2 $
$\Rightarrow|| z^2|-| z|| \leq|z^2+z| \leq 2$
$\Rightarrow|r^2-r| \leq 2 $
$\Rightarrow|r=| z \mid \leq 2 ; \forall z \in S$
Also we can always find root of the equation $z^2+z+1=e^\theta ; \forall \theta \in R$ Hence set ' $S$ ' is infinite