# Quiz #2

Below I am giving you a handful of problems. You don't need to do all of them, but you must do the first problem. Please explain your answers and include your explanation as comments in your file.

Question #1: Let st be your student id number. Enter the formula on the blackboard. The answer to the question determines which two other questions you should answer.

Question #2: Pick a small value $0 \leq \epsilon \leq 0.01$ and a large value of $d>60$ (make it so you can easily change these values). Let $z := (1-\epsilon) e^{2 \pi i/d}$. Look at the powers of $z$ and, using the command pointplot, plot the points [x, y] where $x$ and $y$ are the real and imaginary parts of $z^r$ for $0 \leq r \leq 5d$. Describe in words what you observe, in particular, give a description of how the list of points change through this sequence. Explain what happens when you change the values of $\epsilon$ and $d$.

Question #3: Pick a small value $0 \leq \epsilon \leq 0.1$ (make it so you can easily change this value). Let $z := 1 + \epsilon i$. Look at the powers of $z$ and, using the command pointplot, plot the points [x, y] where $x$ and $y$ are the real and imaginary parts of $z^r$ for $0 \leq r \leq 100$. Describe in words what you observe, in particular, give a description of how the list of points change through this sequence. Explain what happens when you change the values of $\epsilon$ and why.

Question #4: Pick a small value $0 \leq \epsilon \leq 0.1$ (make it so you can easily change this value). Let $z := (1+\epsilon) i$. Look at the powers of $z$ and, using the command pointplot, plot the points [x, y] where $x$ and $y$ are the real and imaginary parts of $z^r$ for $0 \leq r \leq 100$. Describe in words what you observe, in particular, give a description of how the list of points change through this sequence. Explain what happens when you change the values of $\epsilon$ and why.

Question #5: Pick a small value $0 \leq \epsilon \leq 0.1$ (make it so you can easily change this value). Let $z := (1+\epsilon)/\sqrt{2} + i(1+\epsilon)/\sqrt{2}$. Look at the powers of $z$ and, using the command pointplot, plot the points [x, y] where $x$ and $y$ are the real and imaginary parts of $z^r$ for $0 \leq r \leq 100$. Describe in words what you observe, in particular, give a description of how the list of points change through this sequence. Explain what happens when you change the values of $\epsilon$ and why.

Question #6: Find the number of solutions to the equation $x_1 + x_2 + x_3 + x_4 + x_5 = n$ where the $x_i$ are non-negative integers and $x_1 \leq x_2 \leq x_3 \leq x_4 \leq x_5$. Find the answer for $0 \leq n \leq 10$. Use this data to look up your answer in the Online Encyclopedia of Integer Sequences. Explain how you know your answer is right.

Question #7: Find the number of solutions to the equation $x_1 + x_2 + x_3 + x_4 + x_5 = n$ where the $x_i$ are non-negative integers and of the variables $x_1$, $x_2$ and $x_3$, exactly two of them are odd. Find the answer for $0 \leq n \leq 10$. Use this data to look up your answer in the Online Encyclopedia of Integer Sequences. Explain how you know your answer is right.

Question #8: Plot the solutions to the equations $y^3~~(mod~n) = x^2 + x -2~~(mod~n)$ for $0 \leq x,y < n$ and for $n = 88,89,90,91$ (each $n$ should be on a separate graph). What do you observe that is different about the graphs where $n$ is prime and $n$ is not prime.

NOTE THAT THE PENALTY FOR HANDING THIS IS LATE IS QUITE STRICT. I WANT YOU TO DO THIS IN THE ALOTTED TIME. You should be able to complete this quiz within the class time. If you finish after the class time your overall grade for this assignment will be reduced by 20% per hour (or part thereof). Make sure that your file is uploaded by 12:30pm. Upload your worksheet to the course moodle.

You are expected to work alone on this assignment. Any indication of academic dishonesty will result in a $0$ for the assignment and possible higher penalties.