problem stringlengths 1 13.6k | solution stringlengths 0 18.5k ⌀ | answer stringlengths 0 575 ⌀ | problem_type stringclasses 8
values | question_type stringclasses 4
values | problem_is_valid stringclasses 1
value | solution_is_valid stringclasses 1
value | source stringclasses 8
values | synthetic bool 1
class | __index_level_0__ int64 0 742k |
|---|---|---|---|---|---|---|---|---|---|
1613. Buffon tossed a coin 4040 times, and the head appeared 2048 times. Can the deviation of the number of heads from 2020 be considered random, or is it due to a systematic cause? | Solution. The discrepancy between Buffon's empirical frequency and the theoretical one can be considered random if the probability that the deviation of the number of heads from 2020 is equal to or greater than Buffon's in absolute value when a coin is tossed 4040 times is sufficiently large. Let $m$ be the number of h... | 0.3783 | Other | math-word-problem | Yes | Yes | olympiads | false | 30,653 |
1614. 600 corn seeds are planted with a probability of 0.9 for each seed to germinate. Find the boundary of the absolute value of the deviation of the frequency of germinated seeds from the probability $p=0.9$, if this boundary is to be guaranteed with a probability $P=0.995$. | Solution. We know that if $n$ is the number of independent trials and $p$ is the probability of the event occurring in a single trial, then for any $\varepsilon>0$ the approximate equality holds:
$$
P\left(\left|\frac{m}{n}-p\right|<\varepsilon\right) \approx \Phi\left(\varepsilon \sqrt{\frac{n}{p q}}\right)
$$
where... | 0.034 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 30,654 |
1615. On average, $85 \%$ of the items coming off the conveyor are of the first grade. How many items need to be taken so that with a probability of 0.997, the deviation of the frequency of first-grade items from 0.85 in absolute value does not exceed 0.01? | Solution. Here $p=0.85, q=1-0.85=0.15, \varepsilon=0.01$, $P=0.997, n$-? Since in the equality $P\left(\left|\frac{m}{n}-p\right|<\varepsilon\right) \approx$ $\approx \Phi\left(\varepsilon \sqrt{\frac{n}{p q}}\right)$ the probability $P$ on the left is known, we first solve the equation $\Phi(t)=P$. Let $t_{P}$ be the ... | 11475 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 30,655 |
1643. Given two independent random variables: $X$ - the number of heads appearing in 2 tosses of a five-kopeck coin, and $Y$ - the number of points that appear when rolling a die. Find the distribution law of the difference of these two random variables $X-Y$, the mathematical expectations and variances of the random v... | Solution. Since the random variable $X$ takes the value $m$ with probability $P_{m 2}=C_{2}^{m} p^{m} q^{2-m}$ (Bernoulli's formula),
where $m=0,1,2$ and $p=0.5$, the distribution table of this random variable is
| $x$ | 0 | 1 | 2 |
| :---: | :---: | :---: | :---: |
| $p(x)$ | $\frac{1}{4}$ | $\frac{1}{2}$ | $\frac{1}... | E(X-Y)=-2.5,\D(X-Y)=3.417 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 30,656 |
1644. A random variable $X$, which takes the value $m$ with probability
$$
P_{m n}=C_{n}^{m} p^{m} q^{n-m}(0 \leqslant p \leqslant 1 ; p+q=1 ; m=0,1, \ldots, n),
$$
is called binomial. Find the mathematical expectation and variance of such a random variable. | Solution. $X$ can be considered as the number of occurrences of some event $A$ in $n$ independent trials, each of which has the same probability of occurrence, equal to $p$. Let $X_{i}$ be the number of occurrences of event $A$ in the $i$-th trial. $X=\sum_{i=1}^{n} X_{i}$, where $X_{i}$ has the following distribution ... | E(X)=np,\;D(X)=npq | Algebra | math-word-problem | Yes | Yes | olympiads | false | 30,657 |
1645. A random variable that takes the value $m$ with probability
$$
P_{m}(\lambda)=\frac{\lambda^{m} e^{-\lambda}}{m!}(m=0,1, \ldots, m, \ldots ; \lambda>0)
$$
is said to be distributed according to the Poisson distribution. Find the mathematical expectation and variance of this random variable. | Solution. $\quad E X=\sum_{m=0}^{\infty} m P_{m}(\lambda)=\sum_{m=0}^{\infty} m \frac{\lambda^{m} e^{-\lambda}}{m!}=\sum_{m=1}^{\infty} \frac{\lambda^{m} e^{-\lambda}}{(m-1)!}$ (the first term is omitted, as it equals zero). In the obtained sum, we replace $m$ with $k+1$ and get
$E X=\sum_{m=1}^{\infty} \frac{\lambda^... | DX=\lambda | Algebra | math-word-problem | Yes | Yes | olympiads | false | 30,658 |
1648. Shooting is carried out at a certain target until the first hit without limiting the number of shots. Find the mathematical expectation and variance of the number of shots fired, if the probability of hitting with each shot is the same and equal to $p$. | Solution. First, let's find the probability that the shooting ends with the $(n+1)$-th shot (the first $n$ shots missed, and the $(n+1)$-th shot hit). Since the probability of the product of independent events is equal to the product of their probabilities, the desired probability is
$$
q_{-n}^{q q} \ldots q p=q^{n} p... | EX=\frac{1}{p},\quadDX=\frac{1-p} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 30,659 |
1663. Due to various reasons, the TVs that are delivered to the repair shop for repair usually have all kinds of malfunctions. Therefore, the time (service time) required to eliminate the malfunctions is a random variable $T$. Find the mathematical expectation and variance of the service time of TVs, if the probability... | Solution. Since the formulas for the expected value and variance involve the probability density function, we first find the density $p(t)$:
$$
p(t)=F^{\prime}(t)=\left\{\begin{array}{ll}
0 & \text { for } t \leqslant 0, \\
\left(1-e^{-v t}\right)^{\prime} & \text { for } t>0
\end{array}= \begin{cases}0 & \text { for ... | ET=\frac{1}{v},\quadDT=\frac{1}{v^2} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,660 |
1664. When studying the distribution of income or property, in some cases, each specific value of property can be considered as the result of the action of a large number of mutually independent factors, each of which causes some increment proportional to the already achieved level of income. In this case, the amount o... | Solution. By the known formula
$$
\begin{gathered}
E X=\int_{-\infty}^{+\infty} x p(x) d x=\int_{0}^{+\infty} x \cdot \frac{1}{x \sigma \sqrt{2 \pi}} \cdot e^{-\frac{(\ln x-a)^{2}}{2 \sigma^{2}}} d x= \\
=\frac{1}{\sigma \sqrt{2 \pi}} \int_{0}^{+\infty} e^{-\frac{(\ln x-a)^{2}}{2 \sigma^{2}}} d x
\end{gathered}
$$
In... | EX=e^{+\frac{\sigma^{2}}{2}},\quadDX=e^{2+\sigma^{2}}(e^{\sigma^{2}}-1) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 30,661 |
1665. It is known that the density of the distribution of a random variable $X$ has the form
$$
p(x)=\frac{1}{\sigma \sqrt{2 \pi}} e^{-\frac{x^{2}}{2 \sigma^{2}}}
$$
Find the density of the distribution of the random variable $Y=X^{2}$. | Solution. First, let's find the distribution function of the random variable $Y$. Since it does not take negative values, then $F(y)=P(Y0$. Then
$$
\begin{aligned}
F(y) & =P(Y0
\end{aligned}
$$
Therefore, the desired density function is given by the formula
$$
\begin{aligned}
\varphi(y)=F^{\prime}(y) & =\left\{\begi... | \frac{1}{\sigma\sqrt{2\piy}}\cdote^{-\frac{y}{2\sigma^{2}}} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,662 |
1666. Let the random variable $X$ have the probability density function
$$
p(x)=|x| e^{-x^{2}}
$$
What is the probability that this random variable will take a value in the interval $(0,1)$? | Solution. The desired probability $P(0<X<1)$ can be found using property 2 of the probability density function
$$
P\left(x_{1}<X<x_{2}\right)=\int_{x_{1}}^{x_{2}} p(x) d x
$$
In our case
$$
\begin{gathered}
P(0<X<1)=\int_{0}^{1}|x| e^{-x^{2}} d x=\int_{0}^{1} x e^{-x^{2}} d x=-\frac{1}{2} \int_{0}^{-1} e^{t} d t= \\... | 0.3160 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,663 |
1679. Let the random variable $X$ be distributed according to the normal law with the density
$$
p(x)=\frac{1}{\sigma \sqrt{2 \pi}} \cdot e^{-\frac{(x-a)^{2}}{2 \sigma^{2}}}
$$
It is required to find the probable deviation of this random variable, i.e., such a constant $E$ that
$$
P(|X-a|<E)=50 \%
$$ | Solution. $\quad P(|X-a|<E)=\Phi\left(\frac{E}{\sigma}\right)$.
Here $\Phi\left(\frac{E}{\sigma}\right)$ is the value of the Laplace function at the point $\frac{E}{\sigma}$. According to the problem, $\Phi\left(\frac{E}{\sigma}\right)=0.5$. From the table (appendix 2), we find the value of the argument $t$ for which ... | 0.675\sigma | Other | math-word-problem | Yes | Yes | olympiads | false | 30,665 |
1684. The average service life of the motor is 4 years. Estimate from below the probability that this motor will not serve for more than 20 years. | Solution. Let the random variable $X$ be the service life of the motor. From the problem statement, it follows that $E X=4$. We need to estimate from below the probability $P(X<20)$. This probability can be considered as the left side of Markov's inequality with $\xi=20$. Then
$$
P(X<20) \geqslant 1-\frac{E X}{\xi}=1-... | 0.8 | Other | math-word-problem | Yes | Yes | olympiads | false | 30,666 |
1685. The total amount of all deposits in a certain savings office is 20000 rubles, and the probability that a randomly selected deposit does not exceed 100 rubles is 0.8. What can be said about the number of depositors in this savings office? | Solution. Let $X$ be the amount of a randomly selected deposit, and $n-$ the number of all depositors. Then, from the condition of the problem, it follows that $E X=\frac{20000}{n}$. Since $P(X<100)=0.8$ and by Markov's inequality $P(X<100) \geqslant 1-\frac{E X}{100}$, then $0.8 \geqslant 1-\frac{20000}{n \cdot 100}$.... | n\leqslant1000 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 30,667 |
1692. A power station serves a network with 18000 lamps, the probability of each being turned on in the winter evening is 0.9. What is the probability that the number of lamps turned on in the network in the winter evening differs from its expected value in absolute value by no more than 200? | Solution. Since the number of turned-on lamps $X$ is a random variable distributed according to the binomial law, the mathematical expectation of this random variable is equal to the product of the number of all independent trials ($n=18000$) by the probability of the event occurring in a single trial $(p=0.9)$, i.e., ... | 0.955 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 30,668 |
1693. The distribution of the random variable $X$ is given by the following table:
| $x$ | 1 | 2 | 3 | 4 | 5 | 6 |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| $p(x)$ | 0.05 | 0.10 | 0.25 | 0.30 | 0.20 | 0.10 |
What is the probability that $|X-E X|<2$? Estimate this probability using Chebyshev's inequa... | Solution. First, we find the mathematical expectation and variance of the random variable $X$:
$E X=1 \cdot 0.05+2 \cdot 0.10+3 \cdot 0.25+4 \cdot 0.30+5 \cdot 0.20+6 \cdot 0.10=3.80;$
$E X^{2}=1^{2} \cdot 0.05+2^{2} \cdot 0.10+3^{2} \cdot 0.25+4^{2} \cdot 0.30+5^{2} \cdot 0.20+$ $+6^{2} \cdot 0.10=16.10$
$$
D X=E X... | 0.85 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 30,669 |
1694. The average length of a part is 50 cm, and the variance is 0.1. Using Chebyshev's inequality, estimate the probability that a randomly selected part will have a length of no less than 49.5 cm and no more than 50.5 cm. | Solution. Let $X$ be the length of a randomly taken part. From the condition of the problem, it follows that $E X=50$, and $D X=0.1$. It is required to estimate from below the value of $P(49.5<X<50.5)$. Since the inequalities $49.5<X<50.5$ are equivalent to the inequality $|X-50|<0.5$, it is necessary to estimate $P(|X... | 0.6 | Inequalities | math-word-problem | Yes | Yes | olympiads | false | 30,670 |
1701. To determine the average burning duration of light bulbs in a batch of 200 identical boxes, one bulb was taken from each box as a sample. Estimate from below the probability that the average burning duration of the 200 selected light bulbs differs from the average burning duration of the entire batch by less than... | Solution. Let $X_{i}$ be the burning duration of a light bulb taken from the $i$-th box. It is given in the problem that $D X_{i}<7^{2}=$
=49. Obviously, the average burning duration of a bulb in the sample is
$$
\frac{X_{1}+X_{2}+\ldots+X_{200}}{200}
$$
and the average burning duration of a bulb in the entire batch... | 0.9902 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 30,671 |
1702. How many times does one need to measure a given quantity, the true value of which is equal to $a$, to be able to state with a probability of at least 0.95 that the arithmetic mean of these measurements differs from $a$ in absolute value by less than 2, if the root mean square deviation of each measurement is less... | Solution. Let $X_{i}$ be the result of the $i$-th measurement. From the condition of the problem, it follows that $E X_{i}=a, D X_{i}<10^{2}=100$. Therefore,
$$
\frac{E X_{1}+E X_{2}+\ldots+E X_{n}}{n}=\frac{a+a+\ldots+a}{n}=a
$$
We need to find $n$ such that
$$
P\left(\left|\frac{X_{1}+X_{2}+\ldots+X_{n}}{n}-a\righ... | 500 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 30,672 |
1703. Determine whether Chebyshev's theorem is applicable to the sequence of independent random variables $X_{1}, X_{2}, \ldots$, $X_{n}, \ldots$, if $X_{n}$ has the following distribution:
| $x$ | $-5 n$ | 0 | $5 n$ |
| :---: | :---: | :---: | :---: |
| $p(x)$ | $\frac{1}{3 n^{2}}$ | $1-\frac{2}{3 n^{2}}$ | $\frac{1}... | Solution. Chebyshev's theorem is applicable to the sequence of random variables $X_{n}$ if there exists a constant $C$ such that $D X_{n}<C$ for any $n$. Let's check this:
$$
\begin{gathered}
E X_{n}=-5 n \cdot \frac{1}{3 n^{2}}+0 \cdot\left(1-\frac{2}{3 n^{2}}\right)+5 n \cdot \frac{1}{3 n^{2}}=0 ; E X_{n}^{2}= \\
=(... | \frac{50}{3} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 30,673 |
1713. 800 independent trials were conducted: in 200 of them, the probability of the expected outcome was 0.5; in 400 trials, this probability was 0.4; and, finally, in the remaining cases, the probability of a favorable outcome was 0.3. Estimate from below the probability that the deviation of the frequency of the expe... | Solution. First, we find the average probability of the event occurring in 800 trials. If $p_{i}$ denotes the probability of occurrence in the $i$-th trial, then
$$
\bar{p}=\frac{1}{n} \sum_{i=1}^{n} p_{i}=\frac{1}{800} \cdot(0.5 \cdot 200+0.4 \cdot 400+0.3 \cdot 200)=\frac{3.2}{8}=0.4
$$
The problem requires estimat... | 0.817 | Other | math-word-problem | Yes | Yes | olympiads | false | 30,674 |
1714. The wear of the machine in the production of some parts is such that the production of each part reduces the probability of producing a part of the highest grade by $1 \%$. What can be said, based on the Poisson theorem with a probability of no less than 0.8, about the number of parts of the highest grade in a ba... | Solution. If the probability that a part made by the $i$-th on a given machine will be of the highest grade is denoted by $p_{l}$, then
$$
\begin{gathered}
p_{1}=0.9 ; p_{2}=0.9-0.9 \cdot 0.01=0.9 \cdot 0.99 ; p_{3}=0.9 \cdot 0.99- \\
-0.9 \cdot 0.99 \cdot 0.01=0.9 \cdot(0.99)^{2} ; \ldots ; p_{i}=0.9 \cdot(0.99)^{i-1... | 46.7<<67.3 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 30,675 |
1726. Construct a discrete variational series and draw a polygon for the following distribution of 45 pairs of men's shoes sold by a store in one day:
| 39 | 41 | 40 | 42 | 41 | 40 | 42 | 44 | 40 | 43 | 42 | 41 | 43 |
| :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- |
| 39 | 4... | Solution. To construct a variational series, we arrange the different values of the feature in ascending order and record their frequencies under each of these values. For convenience in counting frequencies, the following notations are used:
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
| :---: | :---: | :---: | :-... | Other | math-word-problem | Yes | Yes | olympiads | false | 30,676 | |
1731. To determine the strength of the thread, 1000 tests were conducted, yielding the following results:
| Thread Strength, g | $180-190$ | $190-200$ | $200-210$ | $210-220$ | $220-230$ | $230-240$ | $240-250$ |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| Number of tests | 50 | 90 | 150 | 280 ... | Solution. First, we find the cumulative frequencies for each of the intervals of the given interval variation series:
$\gamma_{1}=\gamma(190)=m_{1}=50 ; \quad \gamma_{2}=\gamma(200)=m_{1}+m_{2}=50+90=140 ;$ $\gamma_{3}=290 ; \quad \gamma_{4}=570 ; \quad \gamma_{5}=790 ; \quad \gamma_{6}=910 ; \quad \gamma_{7}=1000$.
... | notfound | Other | math-word-problem | Yes | Yes | olympiads | false | 30,678 |
1734. The examination of the quality of yarn for strength gave the following results:
| Strength of thread, g | 120-140 | 140-160 | 160-180 | 180-200 |
| :--- | :---: | :---: | :---: | :---: |
| Number of cases | 1 | 6 | 19 | 58 |
| Strength of thread, $\boldsymbol{2}$ | 200-220 | 220-240 | 240-260 | 260-280 |
| :--... | Solution. Since the highest frequency $m_{k}=58$ corresponds to the interval $180-200$, then $X_{k}^{\prime}=180, m_{k-1}=19, m_{k+1}=53$ and
$$
M_{0} X=180+20 \cdot \frac{58-19}{(58-19)+(58-53)}=180+17.73=197.73
$$
To find the median, we construct the cumulative series:
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| :---: | :... | 197.73,202.26 | Other | math-word-problem | Yes | Yes | olympiads | false | 30,679 |
1737. Let the dairy farm of a collective farm have data on the milk yield of cows for the lactation period:
| Milk yield, kg | Number of cows | Milk yield, kg | Number of cows |
| :---: | :---: | :---: | :---: |
| $400-600$ | 1 | $1600-1800$ | 14 |
| $600-800$ | 3 | $1800-2000$ | 12 |
| $800-1000$ | 6 | $2000-2200$ | ... | Solution. The arithmetic mean will be found using the simplified formula
$$
\bar{X}=k \bar{U}+x_{0} \quad\left(U=\frac{X-x_{0}}{k}\right)
$$
by setting $x_{0}=1500$ and $k=200$. All necessary calculations are presented in the following table:
| Interval $X_{i}^{\prime}-X_{i}^{\prime \prime}$ | Frequency $m_{i}$ | M... | 1560 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 30,680 |
1751. Find the expression for the third-order central moment in terms of the initial moments. | Solution. $\quad \mu_{3}=\overline{(X-\bar{X})^{3}}=\overline{X^{3}-3 X^{2} \bar{X}+3 X(\bar{X})^{2}-(\bar{X})^{3}}=$
$$
=\bar{X}^{3}-3 \overline{X^{2}} \bar{X}+3 \bar{X}(\bar{X})^{2}-(\bar{X})^{3}=v_{3}-3 v_{1} v_{2}+2 v_{1}^{3}
$$ | v_{3}-3v_{1}v_{2}+2v_{1}^{3} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 30,682 |
1764. Out of 2000 parts, 400 were selected, the distribution of which by size is given by the following table:
| Size of parts, mm | $7,95-8,00$ | $8,00-8,05$ | $8,05-8,10$ | $8,10-8,15$ | $8,15-8,20$ | $8,20-8,25$ |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| Number of parts | 12 | 28 | 132 | 150 | 62... | Solution. First, we find the sample variance $\sigma^{2}$ using the method indicated in $\S 5$ of Chapter XVI: $\sigma^{2}=0.00272$. For repeated sampling, $\mu=\frac{\sigma_{0}}{\sqrt{n}}=\frac{\sigma}{\sqrt{n-1}}=\sqrt{\frac{0.00272}{400-1}}=0.00261$. Since $\sqrt{\frac{0.00272}{400}}=0.002608$, for large $n$, $\mu$ ... | 0.00234 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 30,684 |
1765. To determine the percentage of non-standard items in a batch with a volume of 10000, 500 items were selected, among which 10 were found to be non-standard. Find the sampling error in determining the proportion in cases of with-replacement and without-replacement sampling. | Solution. In the case of repeated sampling $\mu_{\text {d }}=\sqrt{\frac{p(1-p)}{n}} \approx$ $\approx \sqrt{\frac{w(1-w)}{n}}$. According to the condition $w=\frac{10}{500}=0.02$ and $n=500$. Therefore, $\mu_{\text {d }}=\sqrt{\frac{0.02 \cdot 0.98}{500}}=0.00626$.
For sampling without replacement
$$
\begin{gathered... | 0.00626 | Other | math-word-problem | Yes | Yes | olympiads | false | 30,685 |
1766. To determine the average percentage of raw protein in wheat grains, 626 grains were selected, and the examination showed that the sample mean is 16.8, and the sample variance is 4. What is the probability that the average percentage of raw protein differs from 16.8 by less than $0.2 \%$ in absolute value? | Solution. The volume of the general population is unknown to us. Therefore, we find the sampling error using the formula for repeated sampling $\mu \approx \frac{\sigma}{\sqrt{n-1}}$. In our case $n=626, \sigma^{2}=4$ and, consequently, $\mu \approx \frac{2}{\sqrt{625}}=\frac{2}{25}=0.08$. Since $P(|\tilde{X}-\bar{X}|<... | 0.98758 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 30,686 |
1767. Among the standard products of one factory, on average $15 \%$ belong to the second grade. With what probability can it be stated that the percentage of second-grade products among 1000 standard products of this factory differs from $15 \%$ by less than $2 \%$ in absolute value? | The solution is as follows. Before finding the desired probability, we need to calculate the sampling error $\mu_{\text {d }}: \mu_{\pi}=\sqrt{\frac{p(1-p)}{n}}=\sqrt{\frac{0.15 \cdot 0.85}{1000}}=$ $=\sqrt{0.0001275}=0.0113$, since $n=1000, p=15 \%=0.15$. It is known that $P(|w-p|<\varepsilon) \approx \Phi\left(\frac{... | 0.92327 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 30,687 |
1768. From a batch of 5000 electric lamps, 300 were selected using a simple random sampling without replacement. The average burning duration of the lamps in the sample was 1450 hours, and the variance was 40000. What is the probability that the average burning duration of the lamps in the entire batch is within the ra... | Solution. In the problem, we are given $N=5000, n=300, \tilde{X}=1450$, $\sigma^{2}=40000$. We need to find $P(1410<\bar{X}<1490)=P(|\bar{X}-1450|<40)$.
First, let's calculate the sampling error
$$
\mu \approx \sqrt{\frac{\sigma^{2}}{n-1}\left(1-\frac{n}{N}\right)}=\sqrt{\frac{40000}{299}\left(1-\frac{300}{5000}\righ... | 0.99964 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 30,688 |
1774. To determine the percentage of second-grade items in a batch, a random repeated sample of 100 units is taken. Determine the confidence limits for the percentage of second-grade items in the entire batch, which can be guaranteed with a confidence probability of \( P = 0.997 \), if 25 second-grade items were found ... | Solution. The frequency of the feature in the sample (sample proportion) $w=\frac{25}{100}=0.25$. Therefore, the sampling error
$$
\mu_{\text {d }} \approx \sqrt{\frac{w(1-w)}{n-1}}=\sqrt{\frac{0.25 \cdot 0.75}{100-1}}=0.0435
$$
We need to find such $\varepsilon$ that $P(|w-p|<\varepsilon)=0.997$. Since $P(|w-p|<\var... | 11.95to38.05 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 30,689 |
1775. To determine the average amount of deposits in a savings bank with 2200 depositors, a sample survey (sampling without replacement) of 111 deposits was conducted, which yielded the following results:
| Deposit amount, rubles | $10-30$ | $30-50$ | $50-70$ | $70-90$ | $90-110$ | $110-130$ |
| :---: | :---: | :---: ... | Solution. First, we find the sample mean and sample variance:
| $X_{i}^{\prime}-X_{i}^{\prime \prime}$ | $x_{i}$ | $m_{i}$ | $u_{i}=\frac{x_{i}-80}{20}$ | $m_{i} u_{i}$ | $m_{i} u_{i}^{2}$ |
| :---: | :---: | :---: | :---: | :---: | :---: |
| $10-30$ | 20 | 1 | -3 | -3 | 9 |
| $30-50$ | 40 | 3 | -2 | -6 | 12 |
| $50-7... | 86.4593.35 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 30,690 |
1776. What should be the sample size to ensure that when determining the percentage of non-standard items, the maximum error $\varepsilon=5\%$ is achieved with a confidence probability $P=0.99896$. | Solution. Given the confidence probability $P=$ $=0.99896$, we find the argument $t$ of the Laplace function $\Phi(t)$, for which $\Phi(t)=0.99896 ; t=3.28$. Therefore, the sample size, at which with a probability of 0.99896 it can be stated that the sample proportion differs from the population proportion in absolute ... | 1076 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 30,691 |
1777. It is planned to conduct a selective survey of 5000 light bulbs to determine the average duration of their burning. What should be the volume of the non-repeated sample to ensure with a probability of 0.9802 that the general average differs from the sample average by less than 15 hours in absolute value, if the g... | Solution. According to the table (Appendix 2), we find that the argument $t=2.33$ corresponds to the given confidence probability of 0.9802. For sampling without replacement, the sample size is determined by the formula
$$
n \approx \frac{N t^{2} \sigma_{0}^{2}}{N \varepsilon^{2}+t^{2} \sigma_{0}^{2}}
$$
In our case ... | 230 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 30,692 |
1786. The root mean square deviation of ten measurements of a certain quantity is 1 cm. With what probability can we assert that the representativeness error in absolute value is less than 3 mm? | Solution. The result of the measurement can be considered as a normal random variable, and the sample is small. Therefore, by formula (2) we have:
$$
P(|\hat{X}-\bar{X}|<3)=\Theta_{10-1}\left(\frac{3 \cdot \sqrt{10-1}}{10}\right)=\Theta_{9}(0.9)=0.608
$$
If we had used formula (1), we would have obtained
$$
P(|\tild... | 0.608 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 30,693 |
1792. On the exam for a given subject, the examiner asks the student only one question from one of the four parts of the course. Out of 100 students, 26 received a question from the first part, 32 from the second, 17 from the third, and the rest from the fourth. Can we accept the hypothesis that a student coming to the... | Solution. In this case: $m_{1}=26, m_{2}=32, m_{3}=17$, $m_{4}=25, p_{i}=0.25, \quad n=100, \quad n p_{i}=25(i=1,2,3,4)$. We find $\chi_{0}^{2}: \chi_{0}^{2}=\frac{(26-25)^{2}}{25}+\frac{(32-25)^{2}}{25}+\frac{(17-25)^{2}}{25}+\frac{(25-25)^{2}}{25}=4.56$. Since none of the parameters of the assumed distribution were e... | 4.56<7.815 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 30,694 |
1800. In the "Fundamentals of Chemistry" by D. I. Mendeleev, data on the solubility of sodium nitrate $\mathrm{NaNO}_{3}$ depending on the temperature of water are provided. The following number of conditional parts of $\mathrm{NaNO}_{3}$ dissolve in 100 parts of water at the corresponding temperatures:
| Temperature ... | Solution. To find the parameters $a$ and $b$ using the least squares method, it is necessary to solve system (3), in which in this case $x_{i}$ is the temperature of the solution; $y_{i}=$ the amount of sodium nitrate that dissolves in $100 \Omega$ of water at the given temperature; $\alpha_{1}=a ; \alpha_{2}=b$. The c... | 0.87x+57.5 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 30,696 |
1801. The production of industrial products $(Y)$ by a certain enterprise over the years of a seven-year plan $(X)$ is characterized by the following data:
| $X$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| $Y$, million rubles | 0.5 | 0.5 | 1.5 | 3.5 | 6.5 | 10.5 | ... | Solution. To find the parameters $a, b, c$ using the method of least squares, it is necessary to solve system (4), in which in this case $x_{i}$ is the year of the seven-year period; $y_{i}$ is the corresponding production. The coefficients of system (4) are calculated according to the following scheme:
| $x_{i}$ | $y... | =0.5;b=-1.5;=1.5 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 30,697 |
1811. From the general population, where the distribution of characteristics $X$ and $Y$ is normal, a sample of 530 units has been taken. The results of measuring characteristics $X$ and $Y$ of the sample members are presented in the following correlation table:
| $x$ | $15-$ | $25-3$ | $35-4$ | $5-5$ | $35-6$ | 5-7 |... | Solution. First, we replace the intervals of the values of features $X$ and $Y$ with their midpoints. As a result, we obtain a correlation table that has the form of the table placed at the beginning of this paragraph. Since the linear correlation coefficient does not change from changes in the origins of measurement a... | y-40.43=0.085(x-477.9);x-477.9=9.16(y-40.43);\sigma_{r}\approx0.009 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 30,698 |
1817. The distribution of features $X$ and $Y$ is presented in the following correlation table:
| | | 0 | 10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 | $m_{x}$ |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| -2 | | | | 1 | 2 | 1 | | | | 4 | |
| -1 | | | 1 | 3 ... | Solution. The calculation of correlation relationships will be carried out according to the following scheme (table on p. 410):
$$
\begin{aligned}
& \bar{X}=\frac{\Sigma_{x} m_{x} x}{\Sigma_{x} m_{x}}=\frac{8}{42}=0.190 ; \bar{X}^{2}=\frac{\Sigma_{x} m_{x} x^{2}}{\sum_{x} m_{x}}= \\
& =\frac{60}{42}=1.424 ; \sigma_{X}... | \eta_{y/x}=0,\eta_{x/y}=0.935,\rho_{y/x}=\rho_{x/y}=0,r(X,Y)=0 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 30,699 |
1820. The distribution of features $X, Y$, and $Z$ among members of a certain statistical population is given in the following table:
| $X$ | 1 | 1 | 2 | 2 | 3 | 3 | 3 | 4 | 5 | 5 | 6 | 7 | 7 | 8 |
| :---: | ---: | ---: | ---: | ---: | ---: | ---: | ---: | ---: | ---: | ---: | ---: | ---: | ---: | ---: |
| $Y$ | 5 | 3... | Solution. Since $R_{Z, X Y}, r_{X}(Y, Z)$, and $r_{Y}(X, Z)$ are defined through the coefficients of linear correlation between the features $X$ and $Y$, $X$ and $Z$, and $Y$ and $Z$, we first find these. All necessary calculations are visible from the table (see table on page 414):
$$
\begin{gathered}
\overline{X^{\p... | R_{Z,XY}=0.935,r_{Y}(Z,X)=-0.163,r_{X}(Z,Y)=-0.570 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 30,700 |
1.1. Compute the double integrals. Change the order of integration.
$$
\text { a) } \int_{0}^{2} d x \int_{0}^{3}\left(x^{2}+2 x y\right) d y
$$
b) $\int_{-2}^{0} d y \int_{0}^{y^{2}}(x+2 y) d x$
c) $\int_{0}^{5} d x \int_{0}^{5-x} \sqrt{4+x+y} d y$ | Solution.
a) First, we compute the inner integral, treating $x$ as a constant:
$$
\int_{0}^{3}\left(x^{2}+2 x y\right) d y=\left.\left(x^{2} y+x y^{2}\right)\right|_{y=0} ^{y=3}=3 x^{2}+9 x .
$$
We substitute this function into the inner integral and compute it using the Newton-Leibniz formula:
$$
\int_{0}^{2}\left... | 26,-\frac{24}{5},\frac{506}{3} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,701 |
1.2. Compute the double integral
$$
\iint_{\Omega}(x+y) d x d y
$$
where $\Omega$ is the triangle formed by the lines:
a) $x=0 ; y=0 ; x+y=3$;
b) $x=3 ; y=0 ; y=x$.
 | ## Solution.
a) The region of integration is shown in Fig. 1.19.
We compute the double integral over $\Omega$ by repeated integration:
$$
\begin{aligned}
& \iint_{\Omega}(x+y) d x d y=\int_{0}^{3} d x \int_{0}^{3-x}(x+y) d y=\left.\int_{0}^{3}\left(x y+\frac{y^{2}}{2}\right)\right|_{0} ^{3-x} d x= \\
& =\int_{0}^{3}... | \frac{27}{2} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,702 |
1.5. Evaluate the double integral
$$
\iint_{\Omega}(2 x+3 y+1) d x d y
$$
over the region $\Omega$, bounded by the triangle with vertices $A(1,3), B(-1,-1), C(2,-4)$. | Solution. The specified triangle is depicted in Fig. 1.26. We draw the segment $B D$, parallel to the $O X$ axis. Thus, the original triangle $A B C$ will be divided into two triangles $A B D$ and $D B C$. Next, we write the equations of the lines containing the sides of the triangle $A B, B C$, and $C A$:
 $\int_{-1}^{0} d x \int_{x+1}^{\sqrt{1-x^{2}}} f(x, y) d y$
b) $\int_{0}^{1} d x \int_{x}^{2-x^{2}} f(x, y) d y$
c) $\int_{1}^{2} d y \int_{1 / y}^{y} f(x, y) d x$;
d) $\int_{-2}^{1} d y \int_{y^{2}}^{4} f(x, y) d x$
e) $\int_{0}^{2} d x \int_{\sqrt{2 ... | Solution.
a) First, it is necessary to correctly depict the region of integration $\Omega$. For this, we will plot the graphs of two functions: $y=x+1, y=\sqrt{1-x^{2}}$ for $x \in[-1,0]$. The results are shown in Fig. 1.27.
 $\iint_{\Omega} x y^{2} d x d y$, where the region $\Omega$ is bounded by the circles $x^{2}+(y-1)^{2}=1$ and $x^{2}+y^{2}=4 y$
b) $\iint_{\Omega} e^{-x^{2}-y^{2}} d x d y$, where $\Omega-$ is the circle $x^{2}+y^{2} \leqslant R^{2}$. | Solution.
a) First, let's depict the region $\Omega$ in the Cartesian coordinate system (Fig. 1.32).
The equations of the boundary circles in polar coordinates are obtained after the substitution $x=\rho \cos \varphi, y=\rho \sin \varphi$:
$$
\begin{aligned}
x^{2}+(y-1)^{2}=1 \Rightarrow x^{2}+y^{2} & =2 y \Rightarr... | 0 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,706 |
1.8. Calculate the areas of figures bounded by lines in the Cartesian coordinate system:
a) $3 x^{2}=25 y ; 5 y^{2}=9 x$
b) $x y=4 ; x+y=5$
c) $y=e^{x} ; y=e^{2 x} ; x=1$
d) $x+y=1 ; x+3 y=1 ; x=y ; x+2 y=2$ | Solution. To calculate the areas of the specified figures $\Omega$, we will use formula (1.76)
$$
\mu(\Omega)=S_{\Omega 2}=\iint_{\Omega} d x d y
$$
a) First, we solve the equations $3 x^{2}=25 y$, $5 y^{2}=9 x$ simultaneously. Multiplying them, we get
$$
(x y)^{2}=15 x y .
$$
Thus, either $x y=0$ or $x y=15$. In t... | 7,\frac{15}{2}-8\ln2,\frac{(e-1)^{2}}{2},\frac{11}{12} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,707 |
1.9. Find the areas of figures bounded by lines in polar coordinates:
a) $\rho=\cos 2 \varphi$
b) $\rho=4 \sin \varphi, \rho=2 \sin \varphi$
c) $\rho=\sin 3 \varphi$. | Solution. When calculating the areas of figures bounded by lines in polar coordinates, the formula used is
$$
S_{\Omega}=\iint_{D} \rho d \rho d \varphi
$$
where $D$ is the domain of variation of the variables $\rho$ and $\varphi$.
 $\left(x^{2}+y^{2}\right)^{2}=2 y^{2}$
b) $\left(x^{2}+y^{2}\right)^{3}=x^{4}+y^{4}$. | Solution.
a) Let's switch to polar coordinates $x=\rho \cos \varphi ; y=$ $=\rho \sin \varphi$, we get the equation of the line bounding $\Omega$ in polar coordinates:
$$
\begin{gathered}
\rho^{4}=2 \rho^{2} \sin ^{2} \varphi \\
\rho^{2}=2 \sin ^{2} \varphi \\
\rho=\sqrt{2 \sin ^{2} \varphi}=\sqrt{2}|\sin \varphi|
\e... | \frac{3\pi}{4} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 30,709 |
1.11. Calculate the volumes of bodies bounded by the surfaces:
a) the cylinder $x^{2}+y^{2}=1$ and the planes $z=0, x+y+z=2$;
b) the planes $x+y+z=2, 3x+y=2, 3x+2y=4, z=0$, $y=0$;
c) the elliptic paraboloid $z=\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}$ and the planes $x=-1, x=1, y=-1, y=1$;
d) the sphere $x^{2}+y^{2}... | Solution. If the body $\Omega \in R_{3}$ is bounded from above by the surface $z=f(x, y) \geqslant 0, (x, y) \in \Omega_{\text {xy }} \subset R_{2}$, from below by the plane $z=0$, and on the sides by the cylindrical surface $\varphi(x, y)=0$, where $\varphi(x, y)$ is the equation of the line that is the boundary of th... | notfound | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,710 |
1.13. Find the mass of a square plate, the density of which at each point $M$ is proportional to the sum of the distances from this point to the diagonals of the plate. | The problem is solved. The plate is depicted in Fig. 1.37 in the chosen Cartesian coordinate system.
Given $p(x, y)=K(x+y)$, therefore
$$
\begin{gathered}
M(\Omega)=\iint_{\Omega} K(x+y) d x d y=4 K \int_{0}^{a} d x \int_{0}^{a-x}(x+y) d y= \\
=\left.4 K \int_{0}^{a}\left(x y+\frac{y^{2}}{2}\right)\right|_{0} ^{a-x} ... | \frac{4}{3}K^{3} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,711 |
1.14. The plate is bounded by the parabola $y^{2}=x$ and its chord passing through the focus perpendicular to the axis of the parabola. Find the mass of the plate if the surface density at each point is inversely proportional to the distance from the point to the directrix of the parabola. | Solution. The plate is shown in Fig. 1.38 with the focus and directrix indicated (for more details, see Part I, Chapter 3).
According to the problem, \( p(x, y) = \frac{K}{\left(x + \frac{1}{4}\right)} \), therefore
$$
M(\Omega) = \iint_{\Omega} p(x, y) d x d y = K \int_{0}^{1 / 4} d x \int_{-\sqrt{x}}^{\sqrt{x}} \fr... | 2K(1-\frac{\pi}{4}) | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,712 |
1.16. Find the center of gravity of a plate bounded by the lines $x=0, y=0, \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1, y \geqslant 0, x \geqslant 0$, if the surface density $P=x y$. | Solution. The coordinates of the center of gravity of a plane figure $x_{C}, y_{C}$ are determined by the formulas:
$$
x_{C}=\frac{m_{y}}{M}, y_{C}=\frac{m_{x}}{M}
$$
where $m_{y}=\iint_{\Omega} p x d x d y$ - the static moment of the plate relative to the $O Y$ axis; $m_{x}=\iint_{\Omega} p y d x d y$ - the static m... | x_{C}=\frac{8}{15}\quady_{C}=\frac{8}{15}b | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,714 |
1.17. Find the centers of gravity of homogeneous bodies:
a) hemisphere $x^{2}+y^{2}+z^{2} \leqslant R^{2}, z \geqslant 0$
b) tetrahedron bounded by the planes $x+2 y+z=1$, $x=y=z=0$. | Solution.
a) For a homogeneous body $\Omega=\left\{(x, y, z):(x, y) \in \Omega_{\text {hom }}\right.$; $0 \leqslant z \leqslant f(x, y)\}$, the coordinates of the center of gravity are calculated using the formulas derived from (1.85) when $P(x, y, z)=1:$
$$
\begin{aligned}
& \iint x f(x, y) d x d y \\
& x_{c}=\frac{... | z_{}=\frac{3}{4}R | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,715 |
1.18. Calculate the areas of the surfaces:
a) of the cylinder $x^{2}+y^{2}=R^{2}$, bounded by the planes $z=0(z \geqslant 0), z-y=0$
b) of the cylinder $y^{2}+z^{2}=R^{2}, z \geqslant 0$, bounded inside the cylinder $x^{2}+y^{2}=R^{2}$;
c) of the paraboloid $x^{2}+y^{2}=6 z$, bounded inside the cylinder $x^{2}+y^{2}... | The problem is solved. If $z=f(x, y) ;(x, y) \in \Omega_{\text {Hoch }} \subset R_{2}$ is a piecewise-smooth surface, then its area is calculated using the formula (1.83)
$$
\mu(S)=\iint_{\Omega} \sqrt{1+z_{x}^{2}+z_{y}^{2}} d x d y
$$
In the general case, when the surface is given by the implicit equation $F(x, y, z... | 2\sqrt{3}(\sqrt{3}-1)\pi^{2} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,716 |
1.19. Find the area of the part of the surface of the cylinder $x^{2}+$ $+y^{2}=2 a x$, cut out by the cylinder $z^{2}=2 a(2 a-x)$. | Solution. As is known (see formula (1.83)), the area of a part of the surface $z=f(x, y)$ (the projection onto the $XOY$ plane is $\Omega_{\text{XOY}}$) is calculated by the formula
$$
\mu(S)=\iint_{\Omega_{X O Y}} \sqrt{1+z_{x}^{2}+z_{y}^{2}} d x d y
$$
In our case, the equation of the surface is $y= \pm \sqrt{2 a x... | 16^{2} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,717 |
1.20. Find the area of the part of the surface of the paraboloid $z=x^{2}-y^{2}$, bounded by the paraboloids $z=3 x^{2}+$ $+y^{2}-2$ and $z=3 x^{2}+y^{2}-4$. | Solution. To calculate the area, we also use the formula (1.83)
$$
\mu(S)=\iint_{\Omega_{X O Y}} \sqrt{1+z_{x}^{2}+z_{y}^{2}} d x d y
$$
where $\Omega_{X O Y}$ is the projection of the part of the hyperbolic paraboloid onto the $X O Y$ plane, determined by the conditions:
$$
\begin{aligned}
& x^{2}-y^{2}=3 x^{2}+y^{... | \frac{\pi}{6}(27-5\sqrt{5}) | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,718 |
1.21. Compute the double integral $\iint_{\Omega} \sqrt{\left|x^{2}-y^{2}\right|} d x d y$, where
$\Omega=\{(x, z):|x|+|y| \leqslant a\}$.
Fig. 1.42 | Solution. To compute this integral, it is convenient to introduce new coordinates $u=x+y, v=x-y$. This transformation establishes a one-to-one correspondence between the square $\Omega:|x|+|y| \leqslant a$ in the $X O Y$ plane and the square $D:|u| \leqslant a,|v| \leqslant a$ in the UOV plane with the same origin $O$.... | \frac{8}{9}^{3} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,719 |
1.23. Compute the triple integral:
a) $\iiint_{\Omega}(2 x+3 y+z) d x d y d z$, where $\Omega=\{(x, y, z): x \geqslant 0 ; y \geqslant 0$; $0 \leqslant z \leqslant 3 ; x+y \leqslant 3\}$
b) $\iiint_{\Omega}(1-x)^{3} \sqrt{1-y^{2}} d x d y d z$, where $\Omega=\{(x, y, z):|x| \leqslant 1 ;|y| \leqslant 1$; $|z| \leqsla... | Solution.
a) We reduce the calculation of this integral to a triple repeated integration:
$$
\begin{gathered}
\iiint_{\Omega}(2 x+3 y+z) d x d y d z=\int_{0}^{4} d x \int_{0}^{4-x} d y \int_{0}^{3}(2 x+3 y+z) d z= \\
=\left.\int_{0}^{4} d x \int_{0}^{4-x}\left(2 x z+3 y z+\frac{z^{2}}{2}\right)\right|_{0} ^{3} d y=\i... | 412,12\pi,\frac{2}{9} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,720 |
1.25. Find the mass of the substance filling the common part of two spheres $x^{2}+y^{2}+z^{2}=R^{2}$ and $x^{2}+y^{2}+z^{2}=2 R z$, if the volumetric density of the substance at each point is proportional to the distance from the plane XOY. | Solution. The projection of the body onto the XOY plane is a circle, the radius of which is determined from the equation $r^{2}_{0}=$ $=x^{2}+y^{2}=R^{2}-z^{2}$, where $z=\frac{R}{2}$ (this is obtained by solving the equations of the two spheres simultaneously). Thus, $r_{0}=\frac{\sqrt{3}}{2} R$. In this problem, as i... | \frac{-5\piK}{24}R^{4} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,721 |
### 1.26. Find the center of gravity of a hemisphere
$$
0 \leqslant z \leqslant \sqrt{R^{2}-x^{2}-y^{2}},
$$
where the volume density \( p \) of the material at each point is proportional to the distance from the origin. | S o l u t i o n. The coordinates of the center of gravity of the body $\Omega$ are calculated by the formulas
$$
X_{c}=\frac{m_{y z}}{m} ; \quad Y_{c}=\frac{m_{z x}}{m} ; \quad Z_{c}=\frac{m_{x y}}{m}
$$
where $m_{x y}=\iiint_{\Omega} z \cdot p(M) d x d y d z$ (static moment of the body $\Omega$ relative to the plane... | \frac{2}{5}R | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,722 |
1.27. The body $\Omega$ is bounded by the planes $x=2 ; y=0 ; y=1 ; z=0$ and the cylinder $z^{2}=6 x$. The volume density of the material at each point is proportional to the distance from the plane $X O Y$. Find the moment of inertia of the body $\Omega$ about the $z$-axis. | Solution. By definition, the moment of inertia of a non-uniform body
Fig. 1.43 of $\Omega$ relative to the $z$-axis is calculated by the formula
$$
I_{O z}=\iint_{\varsigma} \int_{2}\left(... | 14k | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,723 |
1.30. Investigate the absolute convergence of improper multiple integrals:
a) $\iint_{\Omega} \frac{\varphi(x, y)}{\left(x^{2}+y^{2}\right)^{p}} d x d y$, where $\Omega: x^{2}+y^{2} \leqslant 1$ and $\Omega: x^{2}+y^{2} \geqslant 1$; $\varphi(x, y)$ - a bounded function on $\Omega$;
b) $\iiint_{\Omega} \frac{\varphi(... | Solution.
a) Due to the boundedness of $\varphi(x, y)$ on $\Omega (|\varphi(x, y)| \leqslant C$, $(x, y) \in \Omega$), the absolute convergence of the studied integral is determined by the convergence of the integral
$$
\iint_{\Omega} \frac{d x d y}{\left(x^{2}+y^{2}\right)^{p}}
$$
To investigate this convergence, w... | p>1 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,724 |
2.1. Compute the integral
$$
\int_{A B} \frac{d l}{\sqrt{|x|+|y|}}
$$
along the segment connecting points $A(0,-2)$ and $B(4,0)$. | Solution. The segment connecting points $A$ and $B$ can be represented by the parametric equations $x=4 t, y=-2+2 t, t \in[0,1]$. Therefore,
$$
d l=\sqrt{d x^{2}+d y^{2}}=\sqrt{16+4} d t=2 \sqrt{5} d t
$$
and the desired line integral reduces to a definite integral:
$$
\begin{gathered}
\int_{A B} \frac{d l}{\sqrt{|x... | 2(\sqrt{20}-\sqrt{10}) | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,725 |
2.2. Find the length of the arc $A B$ of the curve $e^{2 y}\left(e^{2 x}-1\right)=e^{2 x}+1$, if $x_{A}=1, x_{B}=2$. | Solution. The length of a smooth curve is calculated using a line integral of the first kind with the integrand function $f(x, y)=1$. First, let's find the explicit equation of the given curve:
$$
y=\frac{1}{2} \ln \frac{e^{2 x}+1}{e^{2 x}-1}
$$
Using this formula, we reduce the line integral to a definite integral:
... | \frac{1}{2}\ln(1+e^{4})-1 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,726 |
2.3. Find the center of gravity of the arc $AB$:
a) $AB: x^{2 / 3} + y^{2 / 3} = a^{2 / 3}$ (astroid), $A(0, a), B(a, 0)$, the linear density at each point of the arc is equal to its abscissa;
b) $AB: x = \cos t, y = \sin t, z = \frac{H}{2 \pi} t, t_{A} = 0, t_{B} = 2 \pi$ (helical line with pitch $H$), $\rho(x, y) =... | S o l u t i o n.
a) First, we find the mass of the line $A B$ using a line integral of the first kind, considering that $P(x, y)=x$. According to the problem statement,
$$
\begin{gathered}
M_{A B}=\int_{A B} p(x, y) d l \\
M_{A B}=\int_{A B} x d l=\int_{0}^{a} x \cdot \sqrt{1+\left(y^{\prime}\right)^{2}} d x= \\
=\in... | x_c=\frac{5}{8},\quady_c=\frac{5\pi}{128},\quadz_c=H\pi\sqrt{1+\frac{H^{2}}{4\pi^{2}}} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,727 |
2.4. Compute the line integrals of the second kind $\int_{L} P d x+Q d y$, if:
a) $P=y(x-y) ; Q=x ; L: y=2 x, x \in[0,1]$
b) $P=\left(x^{2}-y\right) ; Q=0 ; L-$ perimeter of the rectangle formed by the lines $x=0, y=0, x=1, y=2$, traversed clockwise;
c) $P=2 x(y-1) ; Q=x^{2} ; L-$ contour of the figure formed by the ... | Solution.
a) By substituting $y=2 x, d y=2 d x$, the curvilinear integral over the given segment is reduced to a definite integral:
$$
\begin{aligned}
& \int_{L} P d x+Q d y=\int_{0}^{1}(2 x(x-2 x) d x+x 2 d x)= \\
& =\int_{0}^{1}\left(2 x-2 x^{2}\right) d x=\left.\left(x^{2}-\frac{2}{3} x^{3}\right)\right|_{0} ^{1}=... | \frac{1}{3},-2,0 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,728 |
2.5. Compute the specified line integrals of the second kind:
a) $\int_{L} 2 y \sin 2 x d x - \cos 2 x d y$, where $L$ is any piecewise-smooth curve connecting the points $A\left(\frac{\pi}{2}, 2\right)$ and $B\left(\frac{\pi}{6}, 1\right)$.
b) $\int_{L} y x e^{x} d x - (x-1) e^{x} d y$, where $L$ is any piecewise-sm... | Solution.
a) It is not difficult to understand that the integrand is a complete differential of the function $u=-y \cos 2 x$, i.e.
$$
\begin{gathered}
\int_{L} 2 y \sin 2 x d x-\cos 2 x d y=\int_{L} d u=u(B)-u(A)= \\
=-\cos \frac{\pi}{3}+2 \cos \pi=-2-\frac{1}{2}=-\frac{5}{2}
\end{gathered}
$$
b) In this problem, th... | 0 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,729 |
2.6. Find the area of the figure bounded by the given line:
a) $x=2 \cos t-\cos 2 t, y=2 \sin t-\sin 2 t$ (cardioid);
b) $x=a \cos ^{3} t, y=a \sin ^{3} t, a>0$ (astroid);
c) $y^{2}=x^{2}+x^{3}$. | Solution. The areas of figures $\Omega$ bounded by closed piecewise-smooth lines $L$ without self-intersections can be calculated using formula (2.65):
$$
\mu(\Omega)=\oint_{L}-y d x+x d y
$$
where the contour $L$ is traversed counterclockwise.
a) In the case of a cardioid $t \in [0, 2\pi]$, $d x = (-2 \sin t + 2 \s... | 12\pi,\frac{3\pi^2}{4},\frac{16}{15} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,730 |
2.7. The electric field intensity at each point in space is directed towards the origin, and its magnitude is proportional to the distance of the point from the origin. Determine the change in energy of a charged particle when it is moved from $A(0,0,0)$ to $B(1,1,1)$. The charge of the particle is $Q$. The proportiona... | Solution. According to the problem, in each point of space $R_{3}$, a vector $\dot{\mathcal{A}}=(P, Q, R)=(-x,-y,-z)$ is defined.
The change in the energy of the particle is equal to the work of the field during the movement from $A$ to $B$ with the opposite sign. We will calculate the work of the field using a line i... | \frac{3q}{2} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,731 |
2.8. Check that the differential expression is a complete differential of some function $u(x, y)$ and determine it:
a) $\left(3 x^{2} y+1\right) d x+\left(x^{3}-1\right) d y$
b) $y^{2} e^{x y} d x+e^{x y}(1+x y) d y$. | Solution.
a) In this case, $P=3 x^{2} y+1, Q=x^{3}-1$. As is known ( $\S 2.5$ ), the differential expression $P d x+Q d y$ is a complete differential of some function $u(x, y)$ if $Q_{x}=P_{y}$. Let's check this condition:
$$
\begin{aligned}
& Q_{x}=\frac{\partial}{\partial x}\left(x^{3}-1\right)=3 x^{2} \\
& P_{y}=\... | )u(x,y)=x+(x^3-1)y+C\\b)u(x,y)=ye^{xy}+C | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,732 |
2.11. Compute the line integral of the second kind
\[
\int_{L} x^{2} y d x - y^{2} x d y
\]
where \( L: \left(x^{2}+y^{2}\right)^{2}=a^{2}\left(x^{2}-y^{2}\right)^{2}, y \geqslant 0 \) from \( A(0,0) \) to \( B(a, 0) \). | Solution. To get an idea of the curve $L$, called a lemniscate, it is easiest to switch to polar coordinates $x=r \cos \varphi, y=$ $=r \sin \varphi$ (Fig. 2.12).
$$
L: r^{2}=a^{2} \cos 2 \varphi, \varphi \in\left[0, \frac{\pi}{4}\right]
$$
 $\iint_{\sigma}|x| d S, \quad \sigma: x^{2}+y^{2}+z^{2}=1, \quad z \geqslant 0$
b) $\iint_{\sigma}\left(x^{2}+y^{2}\right) d S, \quad \sigma: x^{2}+y^{2}=2 z, \quad z=1$;
c) $\iint_{\sigma}\left(x^{2}+y^{2}+z^{2}\right) d S$, where $\sigma$ is the part of the cone... | Solution.
a) Let's represent the hemisphere $x^{2}+y^{2}+z^{2}=1, z \geqslant 0$ with parametric equations:
$x=\sin \theta \cos \varphi, \quad y=\sin \theta \sin \varphi, \quad z=\cos 0, \quad \varphi \in[0,2 \pi], \quad \theta \in\left[0, \frac{\pi}{2}\right]$.
The element of area in this representation is
$$
d S=... | \pi,\frac{4\pi}{15}(1+15\sqrt{3}),3\sqrt{2}\pi | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,735 |
2.13. Find the mass of the cylindrical surface $x^{2}+$ $+y^{2}=R^{2}$, bounded by the planes $z=0, z=H$, if at each point the surface density is inversely proportional to the square of the distance from it to the origin. | Solution. According to the problem, the surface density $p=\frac{k}{r^{2}}(k=$ const $), r^{2}=x^{2}+y^{2}+z^{2}$. The surface of the cylinder projects onto a rectangle $y \in[-R, R], z \in[0, H]$ on the YOZ plane. The desired mass can be expressed as a surface integral of the first kind:
$$
M=\iint_{\sigma} \frac{k d... | 2k\pi\operatorname{arctg}\frac{H}{R} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,736 |
### 2.14. Find the centers of gravity:
a) of the hemisphere $z=\sqrt{R^{2}-x^{2}-y^{2}}$, with surface density $p=k\left(x^{2}+y^{2}\right), k=$ const;
b) of the cone $a z^{2}=b^{2}\left(x^{2}+y^{2}\right), 0 \leqslant z \leqslant b, p=p_{0}=$ const. | Solution.
a) From symmetry considerations, it is clear that $x_{c}=y_{c}=0$. We will determine $z_{c}$ using a surface integral of the first kind:
$$
z_{c}=\frac{\iint_{\sigma} z \cdot p d S}{\iint_{\sigma} p d S}
$$
The area element of the hemisphere is
$$
d S=\sqrt{1+z_{x}^{2}+z_{y}^{2}} d x d y=\frac{R d x d y}{... | z_{}=\frac{3}{8}R | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,737 |
2.16. Compute the surface integral of the second kind
$$
\iint_{S} x d y d z+y d z d x+z d x d y
$$
where $S--$ the inner side of the ellipsoid $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1 ;$
$z \in[-H, H];$ | Solution. The ellipsoid $S$ can be written in parametric form as:
$$
\begin{aligned}
& x=a \sin \theta \cos \varphi \\
& y=b \sin \theta \sin \varphi, \quad \theta \in[0, \pi], \varphi \in[0,2 \pi] \\
& z=c \cos \theta
\end{aligned}
$$
We calculate the tangent vectors
$$
\tau_{\theta}=\left(x_{\theta}, y_{\theta}, z... | -4\pi | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,738 |
### 2.17. Compute the surface integral of the second kind
$$
\iint_{S}\left(x y^{2}+z^{2}\right) d y d z+\left(y z^{2}+x^{2}\right) d z d x+\left(z x^{2}+y^{2}\right) d x d y
$$
where $S-$ is the outer side of the upper ( $z \geqslant 0$ ) hemisphere $x^{2}+$ $+y^{2}+z^{2}=a^{2}$. | Solution. The projection $S$ onto the $X O Y$ plane is a circle of radius $a$:
$$
x^{2}+y^{2} \leqslant a^{2}
$$
We will convert the given surface integral into a double integral over the variables $x, y$. For this, we will again use formula (2.54). In our case,
$$
\begin{aligned}
& P=x y^{2}+z^{2}, Q=y z^{2}+x^{2},... | \frac{\pi^{4}}{20}(8+5) | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,739 |
2.18. Find the divergence of the vector field:
a) $\vec{A}=x y^{2} \vec{i}+x^{2} y \vec{j}+z^{3} \vec{k}$ at the point $M(1,-1,3)$;
b) the gradient of the function $\varphi=x y^{2} z^{2}$;
c) $\dot{A}=y z(4 x \vec{i}-y \vec{j}-z \vec{k})$. | Solution.
a) By definition (see 2.20)
$$
\operatorname{div} \vec{A}=\nabla \vec{A}=\frac{\partial}{\partial x}\left(x y^{2}\right)+\frac{\partial}{\partial y}\left(x^{2} y\right)+\frac{\partial}{\partial z} z^{3}=y^{2}+x^{2}+3 z^{2}
$$
From this, at $x=1, y=-1, z=3$ we get
$$
\left.\operatorname{div} \vec{A}\right|... | 29,2x(y^{2}+z^{2}),0 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,740 |
2.19. Find the flux of the vector field $\vec{r}=x \vec{i}+y \vec{j}+z \vec{k}$:
a) through the lateral surface of the cylinder $x^{2}+y^{2} \leqslant R^{2}$, $|z| \leqslant H$ (normal outward);
b) through the lateral surface of the cone $x^{2}+y^{2} \leqslant 4 z^{2}$, $z \in[0,1]$ (normal inward);
c) through the c... | Solution.
a) We use the Gauss-Ostrogradsky formula:
$$
\iint_{S} \vec{r} \cdot \vec{n} d S=\iiint_{V} \operatorname{div} \vec{r} \cdot d V
$$
Obviously, $S=S_{6 \text{ok}} \cup S_{\text {osn }}(S_{\text {bok }}, S_{\text {osn }}$ - the lateral surface and the base surface of the cylinder) and by the additivity prope... | \begin{aligned}& | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,741 |
2.20. Calculate the rotor of the following vector fields:
a) $\vec{A}=x \dot{i}-z^{2} \vec{j}+y^{2} \dot{k}$;
b) $\dot{A}=y z \dot{i}+x z \dot{j}+x y \dot{k}$. | Solution.
a) By definition
$$
\begin{aligned}
& \operatorname{rot} \dot{A}=\nabla \times \dot{A}=\left|\begin{array}{ccc}
i & j & \dot{k} \\
\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\
x & -z^{2} & y^{2}
\end{array}\right|=\dot{i}(2 y+2 z) \\
& \text { b) } \operatorname... | )\vec{i}(2y+2z),\quadb)\vec{0},\quad\varphi=xyz+C | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,742 |
2.21. Calculate the circulation of the vector field:
a) $\vec{A}=x^{2} y^{2} \vec{i}+\vec{j}+z \vec{k}$ along the circle $x^{2}+y^{2}=a^{2}, z=0$;
b) $\dot{A}=(x-2 z) \dot{i}+(x+3 y+z) \dot{j}+(5 x+y) \vec{k}$ along the perimeter of the triangle $A B C$ with vertices $A(1,0,0), B(0,1,0), C(0,0,1)$. | Solution.
a) For $z=0$ we obtain the planar field $\vec{A}=x^{2} y^{2} \dot{i}+\ddot{j}$, where $P=x^{2} y^{2}, Q=1$. To compute the circulation of this field along the circle $x^{2}+y^{2}=a^{2}$, we can use Green's formula (2.62):
$$
\begin{gathered}
\oint_{x^{2}+y^{2}=a^{2}} \vec{A} \cdot d \vec{l}=\iint_{x^{2}+y^{... | 0 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,743 |
2.22. Find $\operatorname{grad} U$, if:
a) $U=\sqrt{x^{2}+y^{2}+z^{2}}$;
b) $U=\frac{1}{x^{2}+y^{2}+z^{2}}$. | Solution.
a) The problem is easiest to solve by transitioning to spherical coordinates. In these coordinates, $U=r$ and using formulas $(2.80)$ we get
$$
\operatorname{grad} U=\frac{\partial U}{\partial r} \ddot{e}_{r}+\frac{1}{r} \frac{\partial U}{\partial \theta} \ddot{e}_{0}+\frac{1}{r \sin \theta} \frac{\partial ... | \begin{aligned}&)\operatorname{grad}U=\frac{x}{\sqrt{x^{2}+y^{2}+z^{2}}}\ddot{i}+\frac{y}{\sqrt{x^{2}+y^{2}+z^{2}}}\ddot{j}+\frac{z}{\sqrt{x^{2}+y^{2}+z^{2}}}\ddot | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,744 |
2.23. Find the angle between $\operatorname{grad} \ln \left(x^{2}+y^{2}+z^{2}\right)$ and $\operatorname{grad}(x y + y z + z x)$ at the point $M(1,1,-1)$. | Solution. Let's perform the necessary calculations:
$$
\begin{aligned}
& \operatorname{grad} \ln \left(x^{2}+y^{2}+z^{2}\right)=\frac{2}{x^{2}+y^{2}+z^{2}}(x \vec{i}+y \vec{j}+z \vec{k}) \\
& \left.\operatorname{grad} \ln \left(x^{2}+y^{2}+z^{2}\right)\right|_{M(1,1,-1)}=\frac{2}{3}(\vec{i}+\vec{j}-\vec{k}) \\
& \oper... | \alpha=\pi-\arccos(\frac{\sqrt{3}}{3}) | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,745 |
2.25. Find $\operatorname{rot} \dot{A}$, if:
a) $\dot{A}=\frac{y}{z} \dot{i}+\frac{z}{x} \dot{j}+\frac{x}{y} \dot{k}$;
b) $\dot{A}=r i f(r)$, where $f(r)$ is an arbitrary differentiable function $\left(r=\sqrt{x^{2}+y^{2}+z^{2}}\right)$. | Solution.
a) We proceed according to formula (2.21):
$$
\begin{gathered}
\operatorname{rot} \dot{A}=\left|\begin{array}{ccc}
\vec{i} & \ddot{j} & \dot{k} \\
\frac{\hat{z}}{\partial x} & \frac{\hat{c}}{\partial y} & \frac{\hat{c}}{\partial z} \\
\frac{y}{z} & \frac{z}{x} & \frac{x}{y}
\end{array}\right|=i\left(\frac{\... | notfound | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,747 |
2.26. Let $f(r)$ be an arbitrary differentiable function ( $\left.f(r) \in C^{1}(R)\right)$. Determine when
a) $\operatorname{div}(\operatorname{grad} f(r))=0\left(r=\sqrt{x^{2}+y^{2}+z^{2}}\right)$;
b) $\operatorname{div}(f(r) \cdot \dot{r})=0$. | Solution.
a) For solving this problem, it is convenient to use spherical coordinates and formulas expressing the gradient and divergence in these coordinates (see $2.79, 2.83$):
$$
\operatorname{grad} f(r)=\frac{\partial f}{\partial r} \dot{e}_{r}
$$
(only one component in the direction of the radius vector),
$$
\o... | f(r)=-\frac{C}{r} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,748 |
2.27. Check whether the field $\dot{A}$ is solenoidal $(\operatorname{div} \dot{A}=0)$. If it is, find its vector potential:
a) $\dot{A}=(y+z) \dot{i}+(x+z) \vec{j}+(x+y) \vec{k}$;
b) $\dot{A}=y e^{x^{2}} \dot{i}+2 y z \dot{j}-\left(2 x z y e^{x^{2}}+z^{2}\right) \ddot{k}$. | Solution.
a) $\operatorname{div} \ddot{A}=\frac{\partial(y+z)}{\partial x}+\frac{\partial(x+z)}{\partial y}+\frac{\partial(x+y)}{\partial z}=0$.
Therefore, the field $\vec{A}$ is solenoidal. Let's find its vector potential $\vec{B}$. As is known ( $\$ 2.4$ ), the vector potential is defined up to an additive term $\o... | \begin{aligned}& | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,749 |
2.28. Under the action of gravity $\dot{g}$, directed opposite to the $O Z$ axis, a body of unit mass slides down a spiral:
$$
x=a \cos \varphi, y=a \sin \varphi, z=b(2 \pi-\varphi)
$$
from point $A(a, 0,2 \pi b)$ to point $B(a, 0,0)$.
Find the work of the field during such a displacement. | Solution. The vector field $\vec{F}$ of gravitational force has components:
$$
\vec{F}=(0,0,-g)
$$
where $g \approx 9.81 \frac{\mathrm{M}}{c^{2}}$.
Obviously, $\operatorname{rot} \vec{F}=\overrightarrow{0}$, hence this field is potential. Let's determine its potential:
$$
\varphi=\int_{0}^{x} 0 d t+\int_{0}^{y} 0 d... | 2\pi | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,750 |
3.2. Represent the following complex numbers in trigonometric and exponential forms, choosing the interval of variation of the principal value $\operatorname{Arg} z$, i.e., $\arg z$, first the interval $[0,2 \pi)$, and then $[-\pi, \pi): z_{1}=1+i$, $z_{2}=1-\sqrt{3} i, z_{3}=-1+i, z_{4}=-1-\sqrt{3} i$. | Solution. Let's find the modulus of the complex number $z_{1}$. We have: $\left|z_{1}\right|=\sqrt{1^{2}+1^{2}}=\sqrt{2}$. Let's find $\varphi=\operatorname{Arg} z$. For this, we first find $\arg z$. By definition, $\operatorname{tg} \arg z=\frac{y_{1}}{x_{1}}$. In our case, $x_{1}=y_{1}=1$. Then $\operatorname{tg} \ar... | notfound | Algebra | math-word-problem | Yes | Yes | olympiads | false | 30,751 |
3.3. Calculate: a) $(1+i)^{11}$, b) $(1-i \sqrt{3})^{73}$. | Solution. Consider part a). Let $1+i=z_{0}$, then, by the condition, it is necessary to raise the number $z_{0}$ to the 11th power, which is most conveniently done if the number $z_{0}$ is represented in trigonometric or exponential form. Choose the interval of variation of the principal value $\operatorname{Arg} z$, i... | 2^{72}-2^{72}\sqrt{3}i | Algebra | math-word-problem | Yes | Yes | olympiads | false | 30,752 |
3.4. Calculate: a) $\sqrt[3]{1+i}$, b) $\sqrt[4]{-1-i \sqrt{3}}$. | Solution. Consider part a). Let $1+i=z_{0}$, then, by the condition, it is necessary to find the cube root of the number $z_{0}$. The function of a complex variable $f(z)=\sqrt[n]{z}, n \in \mathbb{R}$ is multivalued and in any domain $D$ of the complex plane, not containing contours that encircle the point $z=0$, allo... | \begin{pmatrix}\sqrt[3]{1+i}={\begin{pmatrix}2^{\frac{1}{6}}(\cos(\frac{\pi}{12}+2\pi)+i\sin(\frac{\pi}{12}+2\pi)),k=0,\quad\in\mathbb{Z}\} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 30,753 |
3.6. Find the mapping of the circle $x^{2}+y^{2}=4$ onto the Ouv plane using the function $w=3z+2$. | Solution. The function $w=3z+2$ maps circles centered at $z=0$ to circles centered at $w=0$; lines passing through the point $z=0$ to lines passing through the point $w=0$, indeed:
$$
w=3z+2=3\rho e^{i\varphi_{\rho}}+2 \Rightarrow \left[\begin{array}{c}
\rho=\rho_{0}=\text { const, } \\
w=\rho_{0} e^{i \varphi}+2,|w|=... | (x-2)^{2}+y^{2}=36 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 30,755 |
3.7. Using the function $w=z^{2}$ to map the lines $x=3$ and $y=2$ onto the Ouv plane. | Solution. The function $w=z^{2}$ maps circles centered at $z=0$ to circles centered at $w=0$; lines passing through the point $z=0$ - to lines passing through the point $w=0$, indeed:
$$
w=z^{2}=\rho^{2} e^{2 i \varphi} \Rightarrow\left[\begin{array}{l}
\rho=\rho_{0}=\text { const }, w=\rho_{0}^{2} e^{2 i(\rho},|w|=\t... | \begin{aligned}&3\xrightarrow{w}u=9-\frac{v^{2}}{36}\\&2\xrightarrow{w}u=\frac{v^{2}}{16}-4\end{aligned} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 30,756 |
3.8. Prove, using the definition of the limit of a sequence of complex numbers, that:
a) $\lim _{n \rightarrow \infty} \frac{n^{a}+i n^{b}}{n^{b}+i n^{a}}=\left\{\begin{array}{cc}1, & a=b ; \\ i, & a \neq b,\end{array}\right.$
where $a, b \in \mathbb{R}, n \in \mathbb{N}$
b) $\lim _{n \rightarrow x} \frac{\left(\sqr... | Solution.
a) Let us take a number $\varepsilon>0$ and estimate the difference $\left|z_{n}-a\right|$ for the considered sequence in three cases ($a=b$, $a<b$, $a>b$):
1st case $-a=b:$
for... | proof | Algebra | proof | Yes | Yes | olympiads | false | 30,757 |
3.10. Find the limits of the functions:
a) $\lim _{z \rightarrow 1-i} \frac{z^{2}-2 z+2}{z-1+i}$;
b) $\lim _{z \rightarrow 0} \frac{1}{z}\left\{\sqrt{z+i}-\frac{1}{\sqrt{2}}(1+i)\right\}$;
c) $\lim _{z \rightarrow 0} \frac{\sin 3 z}{z}$;
d) $\lim _{z \rightarrow z_{0}} \frac{\ln z-\ln z_{0}}{z-z_{0}}, z_{0} \neq 0+0 i$... | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,759 | ||
3.12. Given the real part $u(x, y)=x^{2}-y^{2}+2 x$ of a differentiable function $f(z)$. Find $f(z)$. | Solution.
Since the function $f(z)$ is differentiable, the Cauchy-Riemann conditions are satisfied, i.e.,
$$
\frac{\partial u(x, y)}{\partial x}=\frac{\partial v(x, y)}{\partial y}, \frac{\partial u(x, y)}{\partial y}=-\frac{\partial v(x, y)}{\partial x}
$$
or, in the context of our problem:
$$
\frac{\partial v(x, ... | f(z)=z^2+2z+C | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,761 |
3.13. Given the imaginary part $v(x, y)=3^{x} \sin (y \ln 3)$ of the analytic function $f(z)$. Find $f(z)$. | Solution.
Since the function $f(z)$ is analytic, the Cauchy-Riemann conditions are satisfied, i.e.,
$$
\frac{\partial u(x, y)}{\partial x}=\frac{\partial v(x, y)}{\partial y}, \quad \frac{\partial u(x, y)}{\partial y}=-\frac{\partial v(x, y)}{\partial x}
$$
For our problem, these conditions can be rewritten as:
$$
... | 3^{z}+C | Algebra | math-word-problem | Yes | Yes | olympiads | false | 30,762 |
3.14. Find such functions $\xi(x), \eta(x), \omega(y), \tau(y)$, so that the function of a complex variable
$$
f(z)=\left(\xi(x)+\omega(y)-3 x y^{2}\right)+i\left(\eta(x)+\tau(y)+3 x^{2} y+4 x y+5 x\right)
$$
is analytic. Find its form. | Solution. Since the function $f(z)$ must be analytic, it must satisfy the Cauchy-Riemann conditions, i.e.,
$$
\left\{\begin{array}{l}
\frac{\partial}{\partial x}\left(\xi(x)+\omega(y)-3 x y^{2}\right)=\frac{\partial}{\partial y}\left(\eta(x)+\tau(y)+3 x^{2} y+4 x y+5 x\right) \\
\frac{\partial}{\partial y}\left(\xi(x)... | f(z)=z^{3}+2z^{2}+5iz+z_{0} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 30,763 |
3.16. Calculate
$$
\int_{L} \frac{d z}{(z-a)^{n}}
$$
where $L$ is a circle of radius $R$ centered at the point $a$, traversed counterclockwise, and $n$ is an integer. | Solution. It is clear that $L=\{z \| z-a \mid=R\}$, hence $L=\{z \mid z=$ $\left.=R e^{i \varphi}+a, \varphi \in[0,2 \pi)\right\}$. Therefore,
$$
\begin{aligned}
& \int_{L} \frac{d z}{(z-a)^{n}}=\left\{\begin{array}{c}
z=R e^{i \varphi}+a \\
d z=i R e^{i \varphi} d \varphi \\
\varphi \in[0,2 \pi)
\end{array}\right\}=\... | 2\pii,n=1;0,n\neq1 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 30,765 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.