\documentclass[12pt, letterpaper]{exam}
\author{kevin j.}
\usepackage{amsmath}
\usepackage{amssymb}
\title{EE Mathematics}
\date{2026}
\begin{document}
\maketitle

In order to practice latex, I tried recreating my latest mathematics exam
- second semester, 1st year undergraduate. Most equations I do not remember exactly. 
Literal numbers have been replaced with numbers (e.g. $a_1, a_2$, a, b, ...)

\begin{questions}
\question Solve for X given $A,B,C \in M_{2\times2}(\mathbb{R})$

\begin{gather}
        \begin{bmatrix}
        a_1 & a_2\\
        a_3 & a_4
        \end{bmatrix}
        X + 
        \begin{bmatrix}
        b_1 & b_2\\
        b_3 & b_4
        \end{bmatrix}
        = 
        \begin{bmatrix}
        c_1 & c_2\\
        c_3 & c_4
        \end{bmatrix}
\end{gather} 
\question What does $\det A$ tell us about the matrix (singularity, volume, etc.) and what is their exact relationship?

\question Let $A \in M_{3\times3}(\mathbb{R})$ and $\vec{x},\vec{b} \in M_{3\times1}(\mathbb{R})$ denote a system of linear equations in $\mathbb{R}^3$: 

\begin{gather}
        A \vec{x} = \vec{b}
\end{gather}

How many solutions are there if: 

        (1) $\operatorname{rank}(A) = 3$

        (2) $\operatorname{rank}(A) = 2, \operatorname{rank}([A|b]) = 3$

        (3) $\operatorname{rank}(A) = 2, \operatorname{rank}([A|b]) = 2$

\question What is the relationship between $\vec{x},\vec{y} \in M_{3\times1}(\mathbb{R})$ if $|\vec{x}\times\vec{y}| = 0$

\question Without calculating, think of a valid eigenvalue - eigenvector pair for the given matrix.

\begin{gather}
        A = 
        \begin{bmatrix}
        a & 0\\
        0 & b
        \end{bmatrix}
\end{gather}

\question Define a plane on which lies the given point $A(1,1,-1)$

\question Find the general solution to the differential equation: 

\begin{gather}
        \frac{d^3}{dx^3}y(x) = -4x
\end{gather}

\question Find the general solution to the differential equation: 

\begin{gather}
        \frac{d^3}{dx^3}y(x) - 3 \frac{d^2}{dx^2}y(x) + 3 \frac{d}{dx}y(x) - y(x) = 0
\end{gather}

\question Find a third-degree polynomial that approximates the function $f(x)=sin(2x)$

\question Graph the Fourier series expansion of the function $g(x)=3cos(x)$ with period $[-\frac{2}{\pi}, \frac{2}{\pi}]$

\noindent\rule{\textwidth}{1pt}

* Bonus questions 

\question Find the line $\gamma$ that lies in the intersection of two planes $\Pi$, $\Sigma$.

\begin{gather}
        \Pi: x + y = 0\\
        \Sigma: x - 2z = 1
\end{gather}

What is the vector form equation of $\gamma$? 

\question Write out the Fourier series expansion of $f(x)$ with period $P=1$. No need to write the Fourier coefficients also. 

\end{questions}

\end{document}
