03.01.2024
Part I
Complex Numbers
1
Express the following complex number in an exponential form (z = reiφ):
z = 3 + 4i (1)
2
What is the natural logarithm of z = reiφ? Don’t forget all possible values of φ.
Part II
Linear Algebra
3
Write the following linear system in the canonical representation a
4 − 2x + 2z = 0
y + z − 4x = 4 (2)
x + y = z
1
4 2
Figure 1: A vector point to (x, y) = (1, 1).
4
To achieve a deeper understanding of matrices, we’ll see how it’s used as a transformation in space. Assuming we have a vector on the real plane , we can represent its coordinates as a linear algebra vector by defining the first number of the vector to be the x-axis coordinates, and the second - y-axis coordinates (figure 1). The vector in the figure points to (x, y) = (1, 1).
4.1
What is the mathematical operator that can transform this vector so that it points to (x, y) = (2, 3)? Hint: A comment: there is more than one operator, try to find the general solution.
4.2
What is the mathematical operator that can rotate this vector to (x, y) = (−1, 1)? Try to find the most general form that solves all questions of this type.
3
Part III
Statistics and Probability
5
In a football game, a specific player has a probability of 0.5 to not score any goals in a match. He also has a 0.25 probability to score 1 goal, 0.15 probability to score 2, and a probability of 0.1 to score 3 goals. What is the expected value of goals in the coming football season, assuming the player will play 30 games?
6
Bob goes to the gym each day of the week with a probability of 40%. Alice promised to go to a movie with him only if he visited the gym at least 5 times in the past week. What are the chances they’ll see each other?
