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Problem: Prove that the groups \[ \left\langle R \backslash\{0\}, \cdot,{ }^{-1}\right\rangle \text { and }\left\langle R_{>0}, \cdot,{ }^{-1}\right\rangle \] are not isomorphic.

1.6.4 Fields, Groups, Rings

2.06 $

Problem: Find the order of each element in the group of 6th roots of 1 .

1.6.5 Fields, Groups, Rings

3.85 $

Problem: Find the sign and the order of permutation \( \alpha \), and also find the permutation \( \alpha^{1644} \), where \[ \alpha=(02) \cdot(0982) \cdot(45) \cdot(3579) \cdot(2684) \cdot(1825) \text {. } \]

1.3.2 Permutation group

2.57 $

Problem: Find the number of distinct subgroups of order 7 in the permutation group \( S_{15} \).

1.3.3 Permutation group

3.85 $

Problem: On the set \( A=\left\{\langle a, b\rangle \mid a, b \in R, a^{2}+b^{2} \neq 0\right\} \) the operation \( \langle a, b\rangle *\langle c, d\rangle=\langle a c-b d, a d+b c\rangle \) is defined. Define the transition operation \( { }^{-1} \) to the inverse element so that the set \( \left\langle A, *,{ }^{-1}\right\rangle \) is a group.

1.6.3 Fields, Groups, Rings

5.14 $

Problem: Is the mapping \( f \) of the group \( \langle\mathbb{Z} / 4 \mathbb{Z},+,-\rangle \) to the group \( \langle\mathbb{Z},+,-\rangle \), homomorphic such that a) \( f(\overline{2})=2 \); b) \( f(\overline{3})=3 \).

1.6.6 Fields, Groups, Rings

3.08 $

Problem: Prove that the factor group \( \mathbb{R}^{*} / \mathbb{Q}^{*} \) is not cyclic.

1.6.7 Fields, Groups, Rings

3.85 $

30 Problem: Calculate the definite integral: \[ \int_{0}^{1} \frac{4 x^{3}+11 x^{2}+12 x+4}{(x+1)^{2}\left(x^{2}+2 x+2\right)} d x . \]

9.3.3.7 Definite Integrals

1.28 $

Problem: Calculate the indefinite integral: \[ \int \frac{x^{4}-3 x^{3}+15 x-25}{(x+2)(x-1)(x-3)} d x \]

9.3.1.6 Indefinite integrals

1.28 $

Problem: Calculate the indefinite integral: \[ \int \frac{-x^{4}-8 x^{3}-18 x^{2}-13 x-1}{(x+1)^{4}(x+2)} d x \]

9.3.1.7 Indefinite integrals

1.8 $

Problem: Solve the system of differential equations by bringing it to a second order differential equation: \[ \left\{\begin{array}{l} \frac{d x}{d t}=x+2 y+e^{\prime} \\ \frac{d y}{d t}=2 x+y \end{array} .\right. \]

8.3.19 Systems of ordinary differential equations

2.57 $

Problem: Find the general solution of the first-order partial differential equation: \[ x^{2} z \frac{\partial z}{\partial x}-y^{2} z \frac{\partial z}{\partial y}=x+y . \]

11.4.4 First order partial differential equations

3.08 $

Problem: Solve the first-order partial differential equation under the given additional conditions: \[ x \frac{\partial z}{\partial x}-z \frac{\partial z}{\partial y}=z+2 x^{2}, \quad y=\frac{1}{4}-x^{2} \Rightarrow z=x . \]

11.4.5 First order partial differential equations

3.85 $

Problem: Find the general solution of the system of linear partial differential equations of the first order: \[ \left\{\begin{array}{l} \frac{\partial u}{\partial x}-\frac{\partial v}{\partial y}=0 \\ \frac{\partial u}{\partial y}-\frac{\partial v}{\partial x}=0 \end{array}\right. \]

11.6.4 Systems of equations in partial derivatives of the first order

5.14 $

Problem: Find the temperature distribution \( u(r ; t) \) inside the ball \( r \leq l \), if the initial temperature distribution inside the ball \( \varphi(r) \) is given. \begin{tabular}{|c|c|} \hline\( \left.u\right|_{l} \) & \( \left.u\right|_{t=0}=\varphi \) \\ \hline\( \left.u\right|_{r=l}=0 \) & \( \left.u\right|_{t=0}=\frac{A}{l}\left(\frac{r^{2}}{12 l^{2}}-\frac{r}{4 l^{2}}\right) \) \\ \hline \end{tabular}

11.5.2.3 Fourier method

7.71 $

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