lecture 2 equations.tex

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\title{Equations for Mentimeter presentation - Lecture 2 - Review}
\date{\today}
\author{Arne Dankers}

\begin{document}

Question 1 

\begin{align*}
    \sqrt{3} - j
\end{align*}

\begin{align*}
    \sqrt{3} + j
\end{align*}
    
\begin{align*}
    \sqrt{\frac{3}{2}} + \frac{1}{2}j     
\end{align*}
        
\begin{align*}           
    \sqrt{3} + 2j           
\end{align*}

Question 2 

\begin{align*}           
    2\sqrt{2}e^{j 3 \pi/ 4}           
\end{align*}

\begin{align*}           
    2e^{-j 3 \pi/ 4}        
\end{align*}

\begin{align*}           
    2\sqrt{3}e^{j 3 \pi/ 2}          
\end{align*}

\begin{align*}           
    2\sqrt{3}e^{j 2 \pi/ 3}          
\end{align*}

Question 3

\vspace{2cm}

\begin{align*}
    \frac{\beta}{s^2 + 2\alpha s + \beta^2}
\end{align*}

\vspace{2cm}


\begin{align*}
    \frac{\beta}{s^2 + \alpha s + \alpha^2 + \beta^2}
\end{align*}


\vspace{2cm}


\begin{align*}
    \frac{\beta}{(s+\alpha+j\beta)(s+\alpha-j\beta)}
\end{align*}


\vspace{2cm}


\begin{align*}
    \frac{\beta}{s^2 + 2\alpha s + \alpha^2 + \beta^2}
\end{align*}

\vspace{1cm}



Question 4 

\vspace{1cm}

\begin{align*}
    \frac{F(s)}{1+G(s)H(s)}R(s)
\end{align*}

\vspace{1cm}

\begin{align*}
    \frac{F(s)+G(s)}{1+G(s)H(s)}R(s)
\end{align*}

\vspace{1cm}

\begin{align*}
    \frac{F(s)+G(s)}{1-G(s)H(s)}R(s)
\end{align*}

\vspace{1cm}

\begin{align*}
    \frac{F(s)+G(s)}{G(s)+H(s)}R(s)
\end{align*}

\vspace{1cm}

Question 5

\vspace{1cm}

\begin{align*}
    \frac{H(s)}{1 + H(s)G(s)}
\end{align*}

\vspace{1cm}

\begin{align*}
    \frac{H(s)}{1 - H(s)G(s)}
\end{align*}

\vspace{1cm}

\begin{align*}
    \frac{1}{1 + H(s)G(s)}
\end{align*}

\vspace{1cm}

\begin{align*}
    \frac{G(s)H(s)}{1 + H(s)G(s)}
\end{align*}

\vspace{1cm}

Question 6a

\vspace{2cm}

\begin{align*}
    e^{-t} + e^{-2t}
\end{align*}

\vspace{2cm}

\begin{align*}
    e^{-2t} + e^{-3t}
\end{align*}

\vspace{2cm}

\begin{align*}
    e^{t} + e^{2t}
\end{align*}

\vspace{2cm}

Question 6

\vspace{2cm}

\begin{align*}
    2\sqrt{2}e^{-\sqrt{3}t}\cos(t-\frac{\pi}{4})
\end{align*}

\vspace{2cm}

\begin{align*}
    \sqrt{2}e^{-\sqrt{3}t}\cos(t-\frac{\pi}{4})
\end{align*}

\vspace{2cm}

\begin{align*}
    2\sqrt{2}e^{-\sqrt{3}t}\cos(t-\frac{\pi}{6})
\end{align*}

\vspace{2cm}

Question 7 

\begin{align*}
\begin{bmatrix} -2 & 1 \\ \frac{3}{2} & \frac{-1}{2} \end{bmatrix}
\end{align*}

\begin{align*}
    \begin{bmatrix} 2 & 1 \\ \frac{3}{2} & \frac{-1}{2} \end{bmatrix}
\end{align*}


\begin{align*}
    \begin{bmatrix} -2 & 1 \\ \frac{3}{2} & \frac{1}{2} \end{bmatrix}
\end{align*}

Question 8

\begin{align*}
    A^{-1}B^{-1}
\end{align*}

\begin{align*}
    B^{-1}A^{-1}
\end{align*}

\begin{align*}
    (BA)^{-1}
\end{align*}

Question 9

\begin{align*}
    \begin{bmatrix} \frac{1}{d_1} & & \\ & \frac{1}{d_2} & \\ & & \frac{1}{d_3} \end{bmatrix}
\end{align*}

\begin{align*}
    \begin{bmatrix} \frac{-1}{d_1} & & \\ & \frac{-1}{d_2} & \\ & & \frac{-1}{d_3} \end{bmatrix}
\end{align*}


\begin{align*}
    \begin{bmatrix} \frac{1}{d_3} & & \\ & \frac{1}{d_2} & \\ & & \frac{1}{d_1} \end{bmatrix}
\end{align*}

Question 10




\end{document}