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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?

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