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\begin{document}
\setcounter{page}{5} \noindent\textit{Algebra liniowa -- dr
Micha\l \ G\'ora\hfill Zestaw 3. Liczby zespolone}\vspace{-11pt}%
\newline\noindent\rule{\textwidth}{0.5pt}

\begin{description}
\item[Zadanie 1.] Niech $z,z_{1},z_{2}\in\mathbb{C}$. Podaj interpretacj\k{e}
geometryczn\k{a} nast\k{e}puj\k{a}cych liczb:
\[%
\begin{array}
[c]{llllll}%
\text{a) }z,\quad & \text{b) }\overline{z},\quad & \text{c) }z_{1}+z_{2},\quad
& \text{d) }z_{1}-z_{2},\quad & \text{e) }\left\vert z\right\vert ,\quad &
\text{f)* }z_{1}z_{2}%
\end{array}
.
\]


\item[Zadanie 2.] Dla liczb zespolonych uzasadnij poni\.{z}sze zale\.{z}no\'{s}ci:

\begin{itemize}
\item[a)] $\overline{z_{1}z_{2}}=\overline{z_{1}}\cdot\overline{z_{2}}$ oraz
$\overline{\left(  \frac{z_{1}}{z_{2}}\right)  }=\frac{\overline{z_{1}}%
}{\overline{z_{2}}}$, dla $z_{2}\neq 0$;

\item[b)] $z\overline{z}=\left\vert z\right\vert ^{2}$;

\item[c)] $\left\vert z_{1}z_{2}\right\vert =\left\vert z_{1}\right\vert
\left\vert z_{2}\right\vert $ oraz $\left\vert \frac{z_{1}}{z_{2}}\right\vert
=\frac{\left\vert z_{1}\right\vert }{\left\vert z_{2}\right\vert },$ dla
$z_{2}\neq 0$,

\item[d)] $\mathfrak{Re}z\leqslant\left\vert z\right\vert $ oraz
$\mathfrak{Im}z\leqslant\left\vert z\right\vert $;

\item[e)] $\left\vert z_{1}+z_{2}\right\vert \leqslant\left\vert
z_{1}\right\vert +\left\vert z_{2}\right\vert $;

\item[f)] $\left\vert \left\vert z_{1}\right\vert -\left\vert z_{2}\right\vert
\right\vert \leqslant\left\vert z_{1}-z_{2}\right\vert $.
\end{itemize}

\item[Zadanie 3.] Sprowad\'{z} do postaci algebraicznej (dwumiennej)
nast\k{e}puj\k{a}ce wyra\.{z}enia:

\begin{itemize}
\item[a)] $\frac{1}{1+\cos\frac{\pi}{3}+i\sin\frac{\pi}{3}}$;

\item[b)] $\frac{1-2i}{3i+2}$;

\item[c)] $\frac{1}{i}+\frac{2+i}{2-i}$;

\item[d)] $\frac{\left(  1+i\right)  ^{n}}{\left(  1-i\right)  ^{n-2}},$ dla
$n\in\mathbb{N}$;

\item[e*)] $\left(  \cos\alpha+i\sin\alpha\right)  ^{n},$ dla $n\in\mathbb{N}$;

\item[f*)] $\left(  \sin\alpha+i\cos\alpha\right)  ^{n},$ dla $n\in\mathbb{N}$.
\end{itemize}

\item[Zadanie 4.] Rozwi\k{a}\.{z} r\'{o}wnania z niewiadomymi $z\in
\mathbb{C};$ $x,y\in\mathbb{R}$:

\begin{itemize}
\item[a)] $z^{2}+\left(  3-2i\right)  z+1-3i=0,$

\item[b)] $\left\vert z\right\vert +z=1+i,$

\item[c)] $\left\{
\begin{array}
[c]{r}%
\left(  1+2i\right)  x+\left(  2-2i\right)  y=5+4i\\
\left(  3-i\right)  x+\left(  4+2i\right)  y=2+6i
\end{array}
\right.  ,$

\item[d)] $\left\{
\begin{array}
[c]{r}%
\left(  1-2i\right)  x-\left(  1-4i\right)  y=2-2i\\
\left(  -2-i\right)  x+\left(  2+2i\right)  y=-4-i
\end{array}
\right.  ,$

\item[e)] $\left(  z-2\right)  ^{2}+2\mathfrak{Im}z+z=1,$

\item[f)] $\frac{1+i}{z}=\frac{2-3i}{\overline{z}},$

\item[g)] $z^{4}+4z^{2}-5=0,$

\item[h)] $z^{4}+4z^{2}+8=0,$

\item[i)] $\left(  3-i\right)  x^{2}-\left(  3+2i\right)  x-\left(
1-i\right)  y=13-10i,$

\item[j)] $\left(  2+3i\right)  x^{2}-\left(  2+i\right)  x+\left(
4-4i\right)  y=8-17i.$
\end{itemize}

\item[Zadanie 5.] Na p\l aszczy\'{z}nie zespolonej zaznacz liczby zespolone
$z$, dla kt\'{o}rych

\begin{itemize}
\item[a)] liczba $\frac{z+4}{z-2i}$ jest rzeczywista,

\item[b)] liczba $\frac{z}{iz+4}$ jest czysto urojona,

\item[c)] liczba $\frac{\left(  z-a\right)  ^{2}\overline{z-a}}{z-a}-2$ jest niedodatnia,

\item[d)] liczba $\frac{z+i}{z-i}$ nie jest ujemna.
\end{itemize}

\item[Zadanie 6.] Na p\l aszczy\'{z}nie zespolonej zaznacz wszystkie liczby
zespolone $z$, kt\'{o}rych modu\l \ jest liczb\k{a} ca\l kowit\k{a} i dla kt\'orych liczba
$z^{2}+\left(  1+i\right)  z$ jest czysto urojona.
\end{description}

\noindent\rule{0.5\textwidth}{0.5pt}\\
\hspace{-5pt}
\noindent {\large Odpowiedzi:}


\begin{description}
\item[Zadanie 2:] c) Wskaz\'{o}wka: wykorzysta\'{c} w\l asno\'{s}ci a) i b);
e) Dla $z_{1}+z_{2}\neq0$ mamy:
\begin{align*}
1 &  =\mathfrak{Re}\left(  \frac{z_{1}}{z_{1}+z_{2}}+\frac{z_{2}}{z_{1}+z_{2}%
}\right)  =\mathfrak{Re}\left(  \frac{z_{1}}{z_{1}+z_{2}}\right)
+\mathfrak{Re}\left(  \frac{z_{2}}{z_{1}+z_{2}}\right)  \overset{\text{d)+c)}%
}{\leqslant}\\
&  \leqslant\frac{\left\vert z_{1}\right\vert }{\left\vert z_{1}%
+z_{2}\right\vert }+\frac{\left\vert z_{1}\right\vert }{\left\vert z_{1}%
+z_{2}\right\vert }\text{;}%
\end{align*}


f) $\left\vert z_{1}\right\vert =\left\vert z_{1}-z_{2}+z_{2}\right\vert
\overset{\text{e)}}{\leqslant}\left\vert z_{1}-z_{2}\right\vert +\left\vert
z_{2}\right\vert $, nast\k{e}pnie zamieni\'{c} rol\k{a} $z_{1}$ i $z_{2}$.

\item[Zadanie 3:] a) $\frac{1}{2}-i\frac{\sqrt{3}}{6};$ b) $-\frac{4}%
{13}-\frac{7}{13}i;$ c) $\frac{3}{5}-\frac{i}{5};$ d) $2i^{n-1};$ e) $\cos
n\alpha+i\sin n\alpha;$

f) $i^{n}\left(  \cos n\alpha-i\sin n\alpha\right)  =$
\[
=\left\{
\begin{array}
[c]{ll}%
\left(  -1\right)  ^{n/2}\cos n\alpha+i\left(  -1\right)  ^{1+n/2}\sin
n\alpha\text{,} & n=2k\\
\left(  -1\right)  ^{\left(  n-1\right)  /2}\sin n\alpha+i\left(  -1\right)
^{\left(  n-1\right)  /2}\cos n\alpha\text{,} & n=2k+1
\end{array}
\right.  ,\text{ dla }k\in\mathbb{N}.
\]


\item[Zadanie 4:] a) $-2+i,-1+i;$

b) $i;$

c) $\varnothing;$

d) $x=3,y=1;$

e) $\frac{3}{2}+i\left(  1+\frac{\sqrt{7}}{2}\right)  ,\frac{3}{2}+i\left(
1-\frac{\sqrt{7}}{2}\right)  ;$

f) $\varnothing;$

g) $-1,1,-i\sqrt{5},i\sqrt{5};$

h) $\left(  -1-\sqrt{2},-1+\sqrt{2}\right)  ,$ $\left(  -1+\sqrt{2}%
,-1-\sqrt{2}\right)  ,$ $\left(  -1+\sqrt{2},1+\sqrt{2}\right)  ,$

$\left(  -1-\sqrt{2},1-\sqrt{2}\right)  ;$

i) $\left(  x,y\right)  \in\{\left(  3,5\right)  $, $\left(  -\frac{1}%
{2},-10\frac{3}{4}\right)  \};$

j) $\varnothing;$

\item[Zadanie 5:] a) $\left\{  \left(  x,y\right)  :y=\frac{1}{2}x+2,\text{
}x\neq0\right\}  ;$

b)$\ \left\{  \left(  x,y\right)  :x=0,\text{ }y\neq4\right\}  ;$

c) $\left\{  z:0<\left\vert z-a\right\vert \leqslant\sqrt{2}\right\}  ;$

d) $\mathbb{C}\backslash\left\{  z\in\mathbb{C}:\mathfrak{Re}z=0,\text{
}\mathfrak{Im}z\in(-1,1]\right\}  .$
\end{description}

\end{document}