Question

If $z_1, z_2, z_3 \in C$ satisfy the system of equations given by $\left| z _1\right|=\left| z _2\right|=\left| z _3\right|=1, z _1+ z _2+ z _3=1$ and $z _1 z _2 z _3=1$ such that $\operatorname{Im}\left( z _1\right)<\operatorname{Im}\left( z _2\right)<\operatorname{Im}\left( z _3\right)$, then find the value of $\left[\left|z_1+z_3{ }^2+z_3{ }^3\right|\right]$ where [] denotes the greatest integer function.

Answer 1

We have $\left.z _1 z _2+ z _2 z _3+ z _3 z _1= z _1 z _2 z _3\left(\frac{1}{ z _1}+\frac{1}{ z _2}+\frac{1}{ z _3}\right)=\overline{ z }_1+\overline{ z }_2+\overline{ z }_3=\overline{\left( z _1+ z _2+ z _3\right.}\right)=1$
$\therefore $ The cubic equation with roots $z _1, z _2$ and $z _3$ in $z$ will be
$\left( z - z _1\right)\left( z - z _2\right)\left( z - z _3\right)= z ^3- z ^2+ z -1=0 $
$\Rightarrow ( z -1)\left( z ^2+1\right)=0$
$ z =1, \pm i$
$\because \operatorname{Im}\left( z _1\right)<\operatorname{Im}\left( z _2\right)<\operatorname{Im}\left( z _3\right) $
$\therefore z _1=- i , z _2=1, z _3= i$
Now $\left|z_1+z_2^2+z_3^3\right|=\left|-i+1^2+i^3\right|=|1-2 i|=\sqrt{5}$
Hence $\left.\left[\left|z_1+z_2{ }^2+z_3{ }^3\right|\right]=[\sqrt{5}]=2\right]$