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

$\begin{array}{l}11{\mathrm{z}}^{10}+10{\mathrm{iz}}^9+10\mathrm{iz}-11=0\\ \Rightarrow {\mathrm{z}}^9=\frac{11-10\mathrm{iz}}{11\mathrm{z}+10\mathrm{i}}Let z=x+iy\\ \Rightarrow {\mathrm{z}}^9=\frac{11-10i\left(x+iy\right)}{11\left(\mathrm{x}+\mathrm{iy}\right)+10\mathrm{i}}=\frac{11+10y-i\left(10x\right)}{11x+i(11y+10)}\\ \left|{\mathrm{z}}^9\right|=\sqrt{\frac{{\left(11+10y\right)}^2+100x^2}{121x^2+{(11y+10)}^2}}=\sqrt{\frac{121+100y^2+220y+100x^2}{121x^2+121y^2+100+220y}}\\ \left|{\mathrm{z}}^9\right|=\sqrt{\frac{121++220y+100\left(x^2+y^2\right)}{121\left(x^2+y^2\right)+100+220y}}\\ above is satisfied by x^2+y^2=1\iff \left|z\right|=1\end{array}$