Mathematics II - Exam Review (TH Köln, July 2024)

Practice flashcards covering complex number arithmetic, constrained optimization, and statistical quantiles based on the Mathematics II exam from TH Köln.

Simplify the product of the complex numbers: $(5 - i) \cdot (2 - i)$ as found in Task 1.

$$(5 \times 2) + (5 \times -i) + (-i \times 2) + (-i \times -i) = 10 - 5i - 2i + i^2 = 10 - 7i - 1 = 9 - 7i$$

Simplify the division of the complex numbers: $\frac{9 - 7i}{2 - 3i}$ as found in Task 1.

$$\frac{(9 - 7i)(2 + 3i)}{(2 - 3i)(2 + 3i)} = \frac{18 + 27i - 14i + 21}{4 + 9} = \frac{39 + 13i}{13} = 3 + i$$

In Task 1, what is the value of the complex expression $6e^{2\pi i}$?

$$6 (\cos(2\pi) + i\sin(2\pi)) = 6(1 + 0i) = 6$$

In Task 1, identify the result of the complex expression $\cos(\frac{1}{2}\pi) + i\sin(\frac{1}{2}\pi)$.

$$0 + i(1) = i$$

What are the objective function $f(x, y)$ and constraint $g(x, y)$ defined in the Task 2 optimization problem?

Objective function: $f(x, y) = 5 - x^2 - \frac{1}{2}y^2$; Constraint: $g(x, y) = x + y = 2$

According to Task 4, which specific quantiles must be determined for the data set $Z$?

The $25\%$, $50\%$ (median), and $75\%$ quantiles.

What confidence level is specified for the calculation in Task 4?

$$99\%$$

What significance level for the statistical test is mentioned in Task 4?

$$5\%$$

Based on the transcript, what are the four main mathematical topics covered in this exam?

1. Komplexe Zahlen (Complex Numbers), 2. Extremwertaufgaben (Optimization Problems), 3. Integralrechnung (Integral Calculus), 4. Wahrscheinlichkeitsrechnung (Probability/Statistics).