Question

All the roots of the equation $11z^{10}+10i{z}^9+10iz-11=0$ lie

• A. Inside $|z|=1$
• B. On $|z|=1$
• C. Outside $|z|=1$
• D. None of these

Answer 1

The correct option is B On $|z|=1$

$11z^{10}+10i{z}^9+10iz-11=0$
or $z^9(11z+10i)=11-10iz$
or $z^9=\frac{11-10iz}{11z+10i}$
or $|z^9|=\frac{|11i-10z|}{|11z+10i|}$
Now $|11i+10z{|}^2-|11z+10i{|}^2=21(1-|z|{)}^2$
For $|z|<1$
$|11i-10z{|}^2-|11z+10i{|}^2>0$
$\Rightarrow |z^9|=\frac{|11i-10z|}{|11z+10i|}>1$
i.e. $|z^9|>1$ which contradicts with $|z|<1$
For $|z|>1$ we get $|z^9|<1$
$\Rightarrow |z|=1$