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Problem: Find the cosine of the angle between the vectors \( \overrightarrow{A B} \) and \( \overrightarrow{A C} \). \[ A(0 ; 2 ;-4), \quad B(8 ; 2 ; 2), \quad C(6 ; 2 ; 4) \text {. } \]

1.1.15 Vector Algebra

0 $

Problem: Calculate the area of the parallelogram formed by the vectors \( \vec{a} \) and \( \vec{b} \). \[ \begin{array}{l} \vec{a}=3 \vec{p}+\vec{q}, \quad \vec{b}=\vec{p}-3 \vec{q}, \quad|\vec{p}|=7, \\ |\vec{q}|=2, \quad(\widehat{\vec{p} ; \vec{q}})=\frac{\pi}{4} . \end{array} \]

1.1.16 Vector Algebra

2 $

Problem: Write the decomposition of the vector \( \vec{v} \) into the vectors \( \vec{p}, \vec{q} \) and \( \vec{r} \). \[ \begin{array}{ll} \vec{v}=\{-1 ; 7 ; 0\}, & \vec{p}=\{0 ; 3 ; 1\}, \\ \vec{q}=\{1 ;-1 ; 2\}, & \vec{r}=\{2 ;-1 ; 0\} . \end{array} \]

1.1.17 Vector Algebra

1 $

Problem: Prove equalities using the properties of set operations and definitions of operations. Illustrate using Euler-Venn diagrams. a) \( A \cup(B \backslash C)=(A \cup B) \backslash(C \backslash A) \), b) \( A \times(B \cap C)=(A \times B) \backslash(A \times(B \backslash C)) \).

6.8.8 Set theory

1.5 $

Problem: Prove the identity: \( \overline{B \backslash A}=\bar{B} \cup A \).

6.8.9 Set theory

0 $

Problem: Prove equality using the properties of set operations and definitions of operations. Illustrate using Euler-Venn diagrams. \[ (A \cup B) \backslash(A \cap C)=(A \cap \bar{C}) \cup(\bar{A} \cap B) \text {. } \]

6.8.10 Set theory

1.5 $

Problem: Prove equalities using the properties of set operations and definitions of operations. \[ C \subseteq D \Rightarrow A \times C \subseteq B \times D \text {. } \]

6.8.11 Set theory

0 $

Problem: Prove equalities using the properties of set operations and definitions of operations. Illustrate using EulerVenn diagrams. a) \( A \backslash B=A \Delta(A \cap B) \), b) \( (A \cup C) \times B=(C \times B) \cup((A \cap C) \times B) \cup(A \times B) \).

6.8.12 Set theory

1.25 $

Problem: Find the Laplace transform of the signal by the given graph:

10.4.8 Laplace transform

2.5 $

Problem: Find the signal of the Laplace transform: \[ \frac{p-8}{(p+1)\left(p^{2}+2 p+10\right)} \text {. } \]

10.4.9 Laplace transform

2 $

Problem: Determine the circle of convergence of the given series. Find out if the series converges at the given point \( z_{1}, z_{2}, z_{3} \) (if yes, then how: absolutely or conditionally). Make the drawing. \begin{tabular}{|c|c|c|c|} \hline Series & \( z_{1} \) & \( z_{2} \) & \( z_{3} \) \\ \hline\( \sum_{n=1}^{\infty} \frac{(z+1-2 i)^{n}}{2^{n}(n+1) \ln ^{2}(n+1)} \) & 0 & \( 1+2 i \) & -1 \\ \hline \end{tabular}

10.8.1 Series with complex terms

4.99 $

Problem: Find all expansions of the given function \( f(z) \) in powers of \( z-a \) and indicate the domains of these expansions. Remark 1. For a multivalued function \( \sqrt[3]{z} \mathrm{p} \) we consider the branch, which that takes real values on the positive part of the real axis. Remark 2. For the multivalued function \( \arctan z \) we consider the branch that takes real values on the positive part of the real axis. In this case, there is a representation: \[ \begin{array}{l} \arctan z=\int_{0}^{z} \frac{d z}{1+z^{2}}=\frac{\pi}{2}+\int_{\infty}^{z} \frac{d z}{1+z^{2}} \\ f(z)=\frac{z-1}{\sqrt[3]{z^{3}-3 z^{2}+3 z}} \text { in powers of }(z-1) . \end{array} \]

10.8.2 Series with complex terms

3.75 $

Problem: Find all Laurent expansions in powers of \( z \). \[ \frac{3 z-4}{z^{3}-z^{2}-6 z} \text {. } \]

10.8.3 Series with complex terms

3.75 $

Problem: Let's find all Laurent expansions in powers of \( z-z_{0} \). \[ \frac{z+6}{z^{2}-4}, \quad z_{0}=1+2 i . \]

10.8.4 Series with complex terms

3.75 $

Problem: Expand \( f(z) \) into a Laurent series in a neighborhood of the point \( z_{0} \). \[ z^{2} \cos \frac{z}{z+2}, \quad z_{0}=-2 . \]

10.8.5 Series with complex terms

3.75 $

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