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Problem list Free problems

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

17.27 USE problems

0 $

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

0 $

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

0 $

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

17.30 USE problems

0 $

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

1.28 $

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

2.57 $

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

4.37 $

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

16.15 Topology

3.08 $

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

2.57 $

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

2.57 $

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

3.34 $

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

1.54 $

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

10.28 $

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

5.14 $

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

3.08 $

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