Question

Решите плиз, срочно нужно!!!

Answer 1

$2.\;\frac{2+3i}{1-2i}=a(2+3i)+b(1-2i)\\2+3i=a(2+3i)(1-2i)+b(1-2i)^2\\2+3i=a(8-i)+b(-3-4i)\\2+3i=8a-ai-3b-4bi\\2+3i=(8a-3b)+(-a-4b)i\\\begin{cases}-a-4b=3\\8a-3b=2\end{cases}\Rightarrow\begin{cases}a=-3-4b\\-24-32b-3b=2\end{cases}\Rightarrow\begin{cases}a=-\frac1{35}\\b=-\frac{26}{35}\end{cases}$

$3.\;z=\frac{-41+63i}{50}-\frac{6i+1}{1-7i}=\frac{(-41+63i)(1-7i)-50(6i+1)}{50(1-7i)}=\\=\frac{-41+350i-441i^2-300i-50}{50(1-7i)}=\frac{-91+441+50i}{50(1-7i)}=\frac{350+50i}{50(1-7i)}=\frac{50(7+i)}{50(1-7i)}=\frac{7+i}{1-7i}\\\frac{7+i}{1-7i}=a+bi\\7+i=(a+bi)(1-7i)\\7+i=(a+7b)+(-7a+b)i\\\begin{cases}a+7b=7\\-7a+b=1\end{cases}\Rightarrow\begin{cases}a+7+49a=7\\b=1+7a\end{cases}\Rightarrow\begin{cases}a=0\\b=1\end{cases}\Rightarrow\\\\z=i$