\(\left(\frac{z-1}{2z-i}\right)^4-1=0\Leftrightarrow\left[{}\begin{matrix}\left(\frac{z-1}{2z-i}\right)^2=1\left(1\right)\\\left(\frac{z-1}{2z-i}\right)^2=i^2\left(2\right)\end{matrix}\right.\)

\(\left(1\right)\Leftrightarrow\left[{}\begin{matrix}\frac{z-1}{2z-i}=1\\\frac{z-1}{2z-i}=-1\end{matrix}\right.\) \(\Leftrightarrow\left[{}\begin{matrix}z-1=2z-i\\z-1=-2z+i\end{matrix}\right.\) \(\Leftrightarrow\left[{}\begin{matrix}z=-1+i\\z=\frac{1}{3}+\frac{1}{3}i\end{matrix}\right.\)

\(\left(2\right)\Leftrightarrow\left[{}\begin{matrix}\frac{z-1}{2z-i}=i\\\frac{z-1}{2z-i}=-i\end{matrix}\right.\) \(\Leftrightarrow\left[{}\begin{matrix}z-1=2iz+1\\z-1=-2iz-1\end{matrix}\right.\) \(\Rightarrow\left[{}\begin{matrix}z=\frac{2}{5}+\frac{4}{5}i\\z=0\end{matrix}\right.\)

\(\Rightarrow P=\frac{17}{9}\) (ném vào casio bấm)

What ?? Lớp 13 ??

Đặt \(t=z^2\), ta có phương trình \(t^2+at+1=0 \qquad (1)\)

\(\Delta =a^2-4\)

PT đã cho có 4 nghiệm \(\Leftrightarrow\) (1) phải có hai nghiệm phân biệt

\(\Leftrightarrow \Delta\ne 0\Leftrightarrow a\ne \pm2\)

Khi đó (1) có nghiệm \(t=\dfrac{-a\pm \sqrt{a^2-4}}{2}\).

Không mất tính tổng quát, ta có thể giả sử: \(z_1=z_3;z_2=z_4\)

Khi đó ta có:

\([(z_1^2+4)(z_2^2+4)]^2=441\\ \Leftrightarrow \left(\dfrac{-a+\sqrt{a^2-4}}{2}+4\right)\left(\dfrac{-a-\sqrt{a^2-4}}{2}+4\right)=441\)

\(\Leftrightarrow (-a+8)^2-(a^2-4)=4.441\\ \Leftrightarrow -16a+68=1764\\ \Leftrightarrow a=-106\)

\(z^2-4z+5=0\Rightarrow\left\{{}\begin{matrix}z_1+z_2=4\\z_1z_2=5\end{matrix}\right.\) theo hệ thức Viet

\(w=\dfrac{z_1+z_2}{z_1z_2}+i.z_1z_2\left(z_1+z_2\right)=\dfrac{4}{5}+i.5.4=\dfrac{4}{5}+20i\)

\(\Delta'=4-5=-1\Rightarrow\left[{}\begin{matrix}z_1=2+i\\z_2=2-i\end{matrix}\right.\)

\(\Rightarrow\left(z_1-1\right)^{2019}+\left(z_2-1\right)^{2019}=\left(i+1\right)^{2019}+\left(i-1\right)^{2019}\)

\(=\left(i+1\right)\left[\left(i+1\right)^2\right]^{1009}+\left(i-1\right)\left[\left(i-1\right)^2\right]^{1009}\)

\(=\left(i+1\right)\left(2i\right)^{1009}+\left(1-i\right)\left(-2i\right)^{1009}=\left(2i\right)^{1009}\left(i+1+i-1\right)=\left(2i\right)^{1009}.2i=\left(2i\right)^{1010}=-2^{1010}\)

=>D

P/s: Sry nó bị trôi thông báo nên toi ko để ý