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Problem: Solve for all values of the parameter. \[ 3 x+9

17.27 USE problems

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Problem: Solve for all values of the parameter. \[ \frac{x-5}{x-1}-\frac{2}{k}=\frac{3}{k(x-1)} \text {. } \]

17.28 USE problems

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Problem: Find all values of the parameter for which the equation $ a x^{2}-4 a x+4 a-5=0 $ has negative roots.

17.29 USE problems

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Problem: Solve the irrational equation. \[ \sqrt{3 x^{2}+5 x+8}-\sqrt{3 x^{2}+5 x+5}=1 . \]

17.30 USE problems

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Problem: Is the subgroup $ \left\{\sigma \in S_{n} \mid \sigma(1)=1\right\} $ a normal divisor in $ S_{n} $ ?

1.3.6 Permutation group

1.25 $

Problem: Let the set $ U $ is open and everywhere dense on the metric space $ X $. Show that $ X / U $ is nowhere dense on $ X $.

16.13 Topology

2.5 $

Problem: Show that if a space has a countable base, then every base of this space contains a countable family, that is a base.

16.14 Topology

4.25 $

Problem: Is the group $ \mathbb{Z} $ a subset of the topological group $ \mathbb{R} $ ? $ \mathbb{Q} $ ?

16.15 Topology

3 $

Problem: Find the spectral density of the random function $ \xi(t) $, if $ K_{\eta \eta}(\tau)=e^{-|\tau|}: $ \[ \xi^{\prime \prime}+2 \xi^{\prime}+2 \xi=\eta^{\prime}+\eta . \]

15.1.23 Theory of random processes

2.5 $

Problem: As a result of 10 independent measurements of a certain value $ X $, taken with the same accuracy, the experimental data, shown in the table, were obtained. Assuming that the results of the measurements are subject to a normal probability distribution rule, estimate the true values of $ X $ making use of the confidence interval, covering the true values of $ X $ with a confidence possibility of 0,95 . \begin{tabular}{|l|l|l|l|l|l|l|l|l|l|} \hline$ x_{1} $ & $ x_{2} $ & $ x_{3} $ & $ x_{4} $ & $ x_{5} $ & $ x_{6} $ & $ x_{7} $ & $ x_{8} $ & $ x_{9} $ & $ x_{10} $ \\ \hline 5,3 & 3,7 & 6,2 & 3,9 & 4,4 & 4,9 & 5,0 & 4,1 & 3,8 & 4,2 \\ \hline \end{tabular}

20.1 Mathematical statistics

2.5 $

Problem: The technical control department checked $ n $ batches of the same type of products and found that the number $ \mathrm{X} $ of nonstandard products in one batch has an empirical distribution, as given in the table, in one line of which the number $ x_{i} $ of nonstandard products in one batch is shown, and in the other line the number of $ n_{i} $ batches containing $ x_{i} $ non-standard products is shown. It is required to test the hypothesis that the random variable $ \mathrm{X} $ (the number of non-standard products in one batch) is distributed according to the Poisson law, at a significance level of $ \alpha=0.05 $. \begin{tabular}{|l|l|l|l|l|l|l|l|} \hline$ n=\sum n_{i} $ & $ x_{i} $ & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline 1000 & $ n_{i} $ & 380 & 380 & 170 & 58 & 10 & 2 \\ \hline \end{tabular}

20.2 Mathematical statistics

3.25 $

Problem: Consider a random sample $ X_{i} $ from some known distribution. Plot the histogram and a frequency polygon based on the sample, the number of intervals is $ K $. Numerical data: \[ \begin{array}{l} i_{1}=-0,036 ; i_{2}=-0,809 ; i_{3}=0,315 ; \\ i_{4}=-0,265 ; i_{5}=0,471 ; i_{6}=-0,386 ; i_{7}=0,576 ; \\ i_{8}=-0,556 ; i_{9}=0,508 ; i_{10}=0,477 ; K=3 . \end{array} \]

20.3 Mathematical statistics

1.5 $

Problem: Perform the following calculations for the given sample $ X_{i} $. a) plot a histogram, a polygon, a sample distribution function; b) calculate the sample moments and associated quantities (the first, second and third moments, variance, standard deviation, kurtosis and skewness); c) assuming that the sample is obtained from a normal distribution, test the hypothesis that the mean is equal to null when the variance is unknown; the mean is equal to null, when the variance is equal to the sample. \begin{tabular}{|l|r|} \hline$ i $ & \multicolumn{1}{|c|}{$ X_{i} $} \\ \hline 1 & 0,15 \\ \hline 2 & $ -3,28 $ \\ \hline 3 & 5,13 \\ \hline 4 & 0,19 \\ \hline 5 & $ -40,44 $ \\ \hline 6 & 11,06 \\ \hline 7 & $ -2,17 $ \\ \hline 8 & 0 \\ \hline 9 & 0,26 \\ \hline 10 & $ -7,68 $ \\ \hline \end{tabular} \begin{tabular}{|c|c|} \hline$ i $ & $ X_{i} $ \\ \hline 11 & 0,33 \\ \hline 12 & $ -8,03 $ \\ \hline 13 & 0,37 \\ \hline 14 & 23,67 \\ \hline 15 & 44,56 \\ \hline 16 & $ -1,62 $ \\ \hline 17 & 42,31 \\ \hline 18 & 2,62 \\ \hline 19 & 21,84 \\ \hline 20 & $ -1,7 $ \\ \hline \end{tabular} \begin{tabular}{|r|r|} \hline$ i $ & \multicolumn{1}{|c|}{$ X_{i} $} \\ \hline 21 & $ -0,49 $ \\ \hline & $ -0,2 $ \\ \hline 23 & 0,35 \\ \hline 23 & $ -32,11 $ \\ \hline 25 & 13,72 \\ \hline 26 & $ -0,02 $ \\ \hline 27 & $ -1,95 $ \\ \hline 28 & $ -12,02 $ \\ \hline 29 & $ -7,96 $ \\ \hline 30 & $ -2,97 $ \\ \hline \end{tabular}

20.4 Mathematical statistics

9.99 $

Problem: In the symmetrical group $ S_{5} $ find out if the set $ \{(12),(123),(1234)\} $ is a coset with respect to some subgroup.

1.3.4 Permutation group

4.99 $

Problem: Let $ A $ - be some vertex of the regular tetrahedron. Prove that the set of all selfcoincidences of the tetrahedron that leave the point $ A $ fixed, is a group isomorphic to the group $ S_{3} $.

1.3.5 Permutation group

3 $