Problem source

Sketch the following subsets of the complex plane using Argand diagrams.
Give reasons for your answers. 
\begin{questionparts}
\item $\{z:\mathrm{Re}((1+\mathrm{i})z)\geqslant0\}.$ 
\item $\{z: |z^{2}| \leqslant2,\mathrm{Re}(z^{2})\geqslant0\}.$
\item $\{z=z_{1}+z_{2}:\left|z_{1}\right|=2,\left|z_{2}\right|=1\}.$
\end{questionparts}

Solution source

\begin{questionparts}
\item Multiplication by $1+i$ rotates by $45^{\circ}$ anticlockwise

\begin{center}
    \begin{tikzpicture}[scale=0.4]
        \draw[->] (-12,0) -- (12,0) node[right] {Re};
        \draw[->] (0,-12) -- (0,12) node[above] {Im};

        \draw[thick,red] (-12,-12) -- (12,12);

        \fill[fill opacity=0.2, red] (-12,-12) -- (12,-12) -- (12,12) -- cycle;
    \end{tikzpicture}
\end{center}

\item $|z| \leq \sqrt{2}$, $\textrm{Re}(z^2) \geq 0$ means $\textrm{Re}{z} \geq \textrm{Im}{z}$


\begin{center}
    \begin{tikzpicture}[scale=1.5]
        \draw[->] (-2,0) -- (2,0) node[right] {Re};
        \draw[->] (0,-2) -- (0,2) node[above] {Im};

        \draw[dashed] (0,0) circle ({sqrt(2)});
        \draw[dashed] (-2,-2) -- (2,2);

        \filldraw[domain = -135:45, samples=180, variable = \x, fill opacity=0.2, red]  
            plot ({sqrt(2)*cos(\x)}, {sqrt(2)*sin(\x)});

        \node at ({sqrt(2)},0) {$\sqrt{2}$};
    \end{tikzpicture}
\end{center}

\item These are all points within $1$ unit from a circle radius $2$ units.

\begin{center}
    \begin{tikzpicture}[scale=1]
        \draw[->] (-4,0) -- (4,0) node[right] {Re};
        \draw[->] (0,-4) -- (0,4) node[above] {Im};

        \draw[dashed] (0,0) circle (2);
        \draw[dashed] ({2*cos(45)}, {2*sin(45)}) circle (1);
        \draw[dashed] ({2*cos(90)}, {2*sin(90)}) circle (1);
        \draw[dashed] ({2*cos(30)}, {2*sin(30)}) circle (1);

        \draw (0,0) circle (1);
        \draw (0,0) circle (3);

        \fill[red, fill opacity=0.2,even odd rule] (0,0) circle (1) (0,0) circle (3);
    \end{tikzpicture}
\end{center}
\end{questionparts}