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Problem: Find the kernel, image and the inverse operator for the operator \[ A x=\left(2 x_{1},-3 x_{2}, x_{3}, x_{4}, \ldots\right), A: l_{2} \rightarrow l_{2} \text {. } \]

19.3.7 Linear operators

3.08 $

Problem: Find all integers \( m \), for which the fraction can be reduced: \[ \frac{m^{3}-5 m^{2}+5 m-2}{m^{2}-3 m+1} \text {. } \]

12.2.15 Number theory

2.57 $

Problem: Solve the equation in natural numbers: \[ 6 x^{2}-y^{2}=5 z^{2} \text {. } \]

12.2.16 Number theory

10.28 $

Problem: Find the image of domain \( D \) for the function mapping \( w=w(z) \), where \[ w=e^{i z}, \quad D:\left\{\begin{array}{c} 0

10.5.6 Conformal mappings

3.08 $

(3) Problem: Calculate the line integral: \[ \int_{L}\left(x^{5}+8 x y\right) d L, \text { where } L: 4 y=x^{4}, 0 \leq x \leq 1 . \]

9.4.4 Curvilinear integrals

1.28 $

3 Problem: Calculate the line integral: \[ \int_{L} \sqrt{\frac{a^{2}}{b^{2}} y^{2}+\frac{b^{2}}{a^{2}} x^{2}} d L \] where \( L:\left\{\begin{array}{l}x=a \cos t \\ y=b \sin t\end{array}, \quad 0 \leq t \leq \frac{\pi}{2}\right. \).

9.4.5 Curvilinear integrals

1.28 $

(2) Problem: Calculate the line integral: \[ \int_{L}(x+y) d L, \text { where } L \text { is the petal of the } \] lemniscate located in the first quadrant.

9.4.6 Curvilinear integrals

2.57 $

Problem: Find the area of convergence of the series: \[ \sum_{n=1}^{\infty} \frac{x^{n}}{3 n+8} \text {. } \]

2.8.23 Power series

1.03 $

Problem: Find the area of convergence of the series: \[ \sum_{n=1}^{\infty} \frac{7^{n}(x+3)^{n}}{n} \]

2.8.24 Power series

1.03 $

Problem: Find the domain of convergence of the series: \[ \sum_{n=1}^{\infty} \frac{n x^{n}}{2^{n}} \text {. } \]

2.8.25 Power series

0 $

Problem: Find the domain of convergence of the series: \[ \sum_{n=1}^{\infty} \frac{(x-1)^{n}}{n !} \]

2.8.26 Power series

0 $

Problem: Find the domain of convergence of the series: \[ \sum_{n=1}^{\infty} \frac{n^{3}(x-2)^{3 n+1}}{n^{n+1}} \]

2.8.27 Power series

0.51 $

Problem: Find the domain of convergence of the series: \[ \sum_{n=1}^{\infty} \frac{(x+2)^{n-1}}{\left(2 n^{3}+1\right) 3^{n+1}} \]

2.8.28 Power series

0.77 $

Problem: Find the domain of convergence of the series: \[ \sum_{n=1}^{\infty} \frac{(-1)^{n} x^{2 n}}{(n+1) 5^{n}} \]

2.8.29 Power series

0.77 $

Problem: The auditor checks three accounts. The probability of correct registration of the account is equal to 0,9 . Find the probability of the events: \( A=\{3 \) accounts are correctly registered \( \} \), \( B=\{2 \) accounts are correctly registered \( \} \), \( C=\{1 \) account is correctly registered \( \} \), \( D=\{ \) at least 1 account is correctly registered \( \} \).

15.6.9 Definition and properties of probability

0.77 $

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