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 |
|---|---|---|---|---|---|---|---|---|---|
Example 17. Evaluate the probability of the event
$$
\left|\frac{k}{300}-\frac{1}{6}\right| \leq 0.01
$$
i.e., the probability that the frequency of occurrence of event $A$ in 300 trials deviates from the probability of event $A$ by no more than 0.01. | notfound | Inequalities | math-word-problem | Yes | Yes | olympiads | false | 33,142 | |
Example. Find all values of $\sqrt[3]{-27 i}$.
---
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | Solution.
1. The cube root of the complex number $z=-27 i$ has three different values, which are determined by the formula
$$
w=\sqrt[3]{-27 i}=\sqrt[3]{r} e^{i \frac{\varphi+2 \pi k}{3}}=\sqrt[3]{r}\left(\cos \frac{\varphi+2 \pi k}{3}+i \sin \frac{\varphi+2 \pi k}{3}\right)
$$
where $r=|-27 i|, \varphi=\arg (-27 i)... | \frac{3\sqrt{3}}{2}-\frac{3}{2}i,\quad3i,\quad-\frac{3\sqrt{3}}{2}-\frac{3}{2}i | Algebra | math-word-problem | Yes | Yes | olympiads | false | 33,143 |
Example 1. Investigate the analytical properties of the function $w=e^{z}$ and find its derivative. | Solution.
1. Find the real and imaginary parts of the function $e^{z}$, using the definition of the exponential function
$$
e^{z}=e^{x}(\cos y+i \sin y)
$$
We obtain
$$
u(x, y)=e^{x} \cos y, \quad v(x, y)=e^{x} \sin y
$$
2. Find the partial derivatives
$$
\begin{aligned}
\frac{\partial u}{\partial x}=e^{x} \cos y... | f^{\}(z)=e^{z} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 33,144 |
Example 2. Investigate the analytical properties of the function $w=|z|^{2}$ and find its derivative. | SOLUTION.
1. Find the real and imaginary parts of the function
$$
w=|z|^{2}=|x+i y|^{2}=x^{2}+y^{2}
$$
We obtain
$$
u(x, y)=x^{2}+y^{2}, \quad v(x, y)=0
$$
2. Find the partial derivatives
$$
\begin{array}{ll}
\frac{\partial u}{\partial x}=2 x, & \frac{\partial v}{\partial y}=0 \\
\frac{\partial u}{\partial y}=2 y... | f^{\}(0)=0 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 33,145 |
Example 3. Investigate the analytical properties of the function $w=\bar{z}$. | Solution.
1. Find the real and imaginary parts of the function
$$
w=\bar{z}=x-i y
$$
We get $u(x, y)=x, v(x, y)=-y$.
2. Find the partial derivatives
$$
\frac{\partial u}{\partial x}=1, \quad \frac{\partial v}{\partial y}=-1, \quad \frac{\partial u}{\partial y}=0, \quad \frac{\partial v}{\partial x}=0
$$
The parti... | notfound | Algebra | math-word-problem | Yes | Yes | olympiads | false | 33,146 |
Example. Find the analytic function $f(z)$, if
$$
u(x, y)=\operatorname{Re} f(z)=x^{3}-3 x y^{2}+2 y \quad \text { and } \quad f(i)=2
$$ | SOLUTION.
1. We find the partial derivatives of the function $u(x, y)$
$$
\frac{\partial u}{\partial x}=3 x^{2}-3 y^{2}, \quad \frac{\partial u}{\partial y}=-6 x y+2
$$
2. From the 2nd Cauchy-Riemann condition (1)
$$
\frac{\partial v}{\partial x}=-\frac{\partial u}{\partial y}=6 x y-2
$$
we find
$$
v=\int \frac{\... | f(z)=z^{3}-2iz+i | Algebra | math-word-problem | Yes | Yes | olympiads | false | 33,147 |
Example 2. Calculate the integral
$$
\int_{L}|z| \bar{z} d z
$$
where $L-$ is the upper semicircle $|z|=1, \operatorname{Re} z>0$ traversed counterclockwise. | Solution. In this case, it is convenient to use the parametric form of the curve $L$ as $z=e^{i t}(0 \leqslant t \leqslant \pi)$ and apply formula (2)
$$
\int_{L} f(z) d z=\int_{\alpha}^{\beta} f(z(t)) z^{\prime}(t) d t
$$
1. Find $\bar{z}=e^{-i t},|z|=1, d z=i e^{i t} d t$.
2. Substitute into the integrand and compu... | i\pi | Calculus | math-word-problem | Yes | Yes | olympiads | false | 33,148 |
Example. Compute the integral
$$
\int_{L} \sin ^{2} z d z
$$
where $L$ is the line segment from the point $z_{1}=0$ to the point $z_{1}=i$. | Solution. The function $f(z)=\sin ^{2} z$ is analytic everywhere, and, therefore, the integral does not depend on the path of integration and can be computed using the Newton-Leibniz formula (1).
1. We find the antiderivative $F(z)$ using the power reduction formulas
$$
\int \sin ^{2} z d z=\frac{1}{2} \int(1-\cos 2 ... | \frac{i}{4}(2-\operatorname{sh}2) | Calculus | math-word-problem | Yes | Yes | olympiads | false | 33,149 |
Example 1. Find the first few terms of the Taylor series expansion of the function $f(z)=\tan z$ in the neighborhood of the point $z=0$ (in powers of $z$). Specify the region where this expansion is valid. | Solution.
1. Find the derivatives of the function $f(z)=\operatorname{tg} z$ directly or using formulas:
$$
\begin{array}{ll}
f^{\prime}(z)=\frac{1}{\cos ^{2} z}=1+f^{2}(z), & f^{\prime \prime}(z)=2 f(z) f^{\prime}(z) \\
f^{\prime \prime \prime}(z)=2\left[f^{\prime 2}(z)+f(z) f^{\prime \prime}(z)\right], & f^{\prime ... | \operatorname{tg}z+\frac{2}{3!}z^{3}+\frac{16}{5!}z^{5}+\ldots,\quad|z|<\pi/2 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 33,150 |
Example. Expand the function
$$
f(z)=\frac{z+1}{(z-1)^{2}(z+2)}
$$
in powers of \( z \). Specify the region in which this expansion is valid. | ## Solution.
1. $f(z)$ is a proper rational fraction.
2. Decompose $f(z)$ into partial fractions:
$$
\frac{z+1}{(z-1)^{2}(z+2)}=\frac{1}{9} \frac{1}{z-1}+\frac{2}{3} \frac{1}{(z-1)^{2}}+\frac{1}{9} \frac{1}{z+2}
$$
3. Expand each partial fraction into a series in powers of $z$:
$$
\begin{aligned}
& \frac{1}{z-1}=-\... | Algebra | math-word-problem | Yes | Yes | olympiads | false | 33,151 | |
Example. Expand the function $f(z)=\sin z$ into a Taylor series in the neighborhood of the point $z=3$ (in powers of $z-3$), using tabulated expansions. Specify the region where the obtained expansion is valid. | Solution. Introduce a new variable $t=z-3$ and find the expansion of the function $\sin (t+3)$ in powers of $t$.
1. Express the function $\sin (t+3)$ in terms of functions that have tabulated expansions:
$$
\sin (t+3)=\sin 3 \cos t+\cos 3 \sin t
$$
2. Find the Taylor series expansion of the function $\sin (t+3)$, us... | notfound | Calculus | math-word-problem | Yes | Yes | olympiads | false | 33,152 |
Example. Expand into Laurent series in powers of $z$ the function
$$
f(z)=\frac{z+2}{z^{2}-2 z-3}
$$ | ## SOLUTION.
1. The fraction is proper.
We find the roots of the equation $z^{2}-2 z-3=0$. We have two simple roots $z_{1}=-1$ and $z_{2}=3$.
2. The points $z_{1}=-1$ and $z_{2}=3$ are singular points of the function $f(z)$ (at these points $f(z)$ is not analytic).
The rings of analyticity of the function $f(z)$:
... | \begin{aligned}&\frac{z+2}{z^{2}-2z-3}=\frac{1}{4}\sum_{n=0}^{\infty}[(-1)^{n+1}-\frac{5}{3^{n+1}}]z^{n},\quad|z|<1\\&\frac{z+2}{z^{2}-2z-3}=\sum_{} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 33,153 |
Example. Expand the function in a Laurent series in the neighborhood of its singular point
$$
f(z)=z \sin \frac{\pi z}{z-1}
$$
Find the region where the obtained expansion is valid. | Solution. The function $f(z)$ has a unique singular point $z=1$, therefore, it must be expanded into a Laurent series in powers of $z-1$.
1. We introduce an auxiliary variable $t=z-1$. We obtain
$$
f(t+1)=(t+1) \sin \frac{\pi t+1}{t}
$$
2. We transform the function $f(t+1)$ into a form that allows the use of tabulat... | f(z)=-\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(2n+1)!}\frac{1}{(z-1)^{2n}}-\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(2n+1)!}\frac{1}{(z-1)^{2n+1}},0<|z-1 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 33,154 |
Example 1. Find the zeros of the function $f(z)=e^{z}-1-z$ and determine their order. | Solution.
1. Find the zeros of the function $f(z)$ by solving the equation $e^{z}-1-z=0$. We get $z=0$.
2. Determine the order of the obtained zero $z=0$. For this, we use the Taylor series expansion of the function $f(z)$ in powers of $z$:
$$
e^{z}-1-z=\left(1+z+\frac{z^{2}}{2!}+\frac{z^{3}}{3!}+\ldots\right)-1-z=\f... | 0 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 33,155 |
Example 3. Find the zeros of the function $f(z)=1+\operatorname{ch} z$ and determine their order. | Solution.
1. Find the zeros of the analytic function $f(z)$ by solving the equation
$$
1+\operatorname{ch} z=0
$$
Since
$$
\operatorname{ch} z=\cos i z
$$
we have the equation for determining the zeros:
$$
\cos i z=-1
$$
From this, $i z=\pi(2 k+1)$ and
$$
z_{k}=\pi i(2 k+1), \quad k=0, \pm 1, \ldots
$$
2. Dete... | z_{k}=i\pi(2k+1)\quad(k=0,\1,\ldots) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 33,156 |
Example 5. Find the zeros of the function
$$
f(z)=\frac{\sin ^{4} z}{z^{2}}
$$
and determine their order. | Solution.
1. Function
$$
f(z)=\frac{\sin ^{4} z}{z^{2}}
$$
is analytic for all $z \neq 0$ (the function $f(z)$ is not defined at $z=0$). We find the zeros of the function by solving the equation $f(z)=0$ (for $z \neq 0$).
We obtain
$$
z_{k}=\pi k, \quad k= \pm 1, \pm 2, \pm 3, \ldots
$$
The points $z_{k}=\pi k$ a... | z_{k}=\pik,(k=\1,\2,\3,\ldots)zerosofthe4thorderofthefunctionf(z) | Calculus | math-word-problem | Yes | Yes | olympiads | false | 33,157 |
Example 1. Find the residues at the singular points of the function
$$
f(z)=\frac{\tan z}{z^{2}-\pi z / 4}
$$ | SOLUTION. Let's represent the function $f(z)$ as
$$
f(z)=\frac{\operatorname{tg} z}{z^{2}-\pi z / 4}=\frac{\sin z}{z \cos z(z-\pi / 4)}
$$
1. We find the isolated singular points of the function $f(z)$. The singular points of the function are the zeros of the denominator, i.e.
$$
z=0, \quad z=\pi / 4, \quad z_{k}=\f... | \operatorname{res}_{0}\frac{\operatorname{tg}z}{z^{2}-\piz/4}=0,\quad\operatorname{res}_{\pi/4}\frac{\operatorname{tg}z}{z^{2}-\piz/4}=\frac{4}{\pi},\quad\operatorname{res}_{\pi/2+\pik}\frac | Calculus | math-word-problem | Yes | Yes | olympiads | false | 33,158 |
Example. Compute the integral
$$
\oint_{|z-1|=1} \frac{d z}{z^{4}+1}
$$ | Solution.
1. We find the singular points of the function $f(z)$. The singular points of the function
$$
f(z)=\frac{1}{z^{4}+1}
$$
are the zeros of the denominator. We find all values of $z_{k}=\sqrt[4]{-1}$
$$
z_{k}=e^{\frac{\pi+2 \pi k}{4}}=\cos \frac{\pi+2 \pi k}{4}+i \sin \frac{\pi+2 \pi k}{4}, \quad k=0,1,2,3
$... | -\frac{\sqrt{2}}{2}\pii | Calculus | math-word-problem | Yes | Yes | olympiads | false | 33,159 |
Example. Calculate the integral
$$
\int_{0}^{2 \pi} \frac{d x}{(5+4 \cos x)^{2}}
$$ | Solution.
1. Introduce the complex variable $z=e^{i x}$. In this case, the integration interval $[0,2 \pi]$ will map to the circle $|z|=1,0 \leqslant \arg z \leqslant 2 \pi$.
2. Calculate
$$
d z=i e^{i x} d x \Longrightarrow d x=\frac{d z}{i z}
$$
By Euler's formula
$$
\cos x=\frac{e^{i x}+e^{-i x}}{2}=\frac{1}{2}\... | notfound | Calculus | math-word-problem | Yes | Yes | olympiads | false | 33,160 |
Example. Compute the integral
$$
\int_{-\infty}^{+\infty} \frac{x+1}{x^{2}-2 x+2} \cos x d x
$$ | Solution. To solve the problem, it is sufficient to compute the improper integral
$$
\int_{-\infty}^{+\infty} \frac{x+1}{x^{2}-2 x+2} e^{i x} d x
$$
and use the formula
$$
\int_{-\infty}^{+\infty} \frac{x+1}{x^{2}-2 x+2} \cos x d x=\operatorname{Re} \int_{-\infty}^{+\infty} \frac{x+1}{x^{2}-2 x+2} e^{i x} d x
$$
1.... | \pie^{-1}(2\cos1-\sin1) | Calculus | math-word-problem | Yes | Yes | olympiads | false | 33,161 |
Example 1. Prove that the Heaviside function
$$
\eta(t)= \begin{cases}1, & t>0 \\ 0, & t<0\end{cases}
$$
is an original, and find its image (Laplace transform). | Solution.
1. Conditions a)-b) are obviously satisfied. Since the function $\eta(t)$ is bounded, its growth exponent $s=0$.
2. We find the image $F(p)$ using formula (1):
$$
F(p)=\int_{0}^{\infty} \eta(t) e^{-p t} d t=-\left.\frac{1}{p} e^{-p t}\right|_{0} ^{\infty}=\frac{1}{p}
$$
since from $\left|e^{-p t}\right|=e^... | \eta()\longleftrightarrow\frac{1}{p} | Calculus | proof | Yes | Yes | olympiads | false | 33,162 |
Example 2. Prove that the function
$$
f(t)=\eta(t) e^{a t}, \quad a \in \mathbb{C}
$$
is an original, and find its image (Laplace transform). | ## Solution.
1. Conditions a)-b) are obviously satisfied. The growth exponent $s$ of the function $f(t)=\eta(t) e^{a t}$ is $\operatorname{Re} a$.
2. We find the image $F(p)$ using formula (1):
$$
F(p)=\int_{0}^{\infty} e^{a t} e^{-p t} d t=-\left.\frac{e^{(a-p) t}}{a-p}\right|_{0} ^{\infty}=-\frac{1}{a-p}=\frac{1}{p... | notfound | Calculus | proof | Yes | Yes | olympiads | false | 33,163 |
Example 2. Find the image of the function
$$
f(t)=\cos t
$$ | Solution.
1. Represent $f(t)=\cos t$ as a linear combination of functions
$$
\cos t=\frac{e^{i t}+e^{-i t}}{2}=\frac{1}{2} e^{i t}+\frac{1}{2} e^{-i t}
$$
the images of which are known:
$$
e^{i t} \longleftrightarrow \frac{1}{p-i}, \quad e^{-i t} \longleftrightarrow \frac{1}{p+i}
$$
2. According to the linearity p... | notfound | Calculus | math-word-problem | Yes | Yes | olympiads | false | 33,164 |
Example. Find the image of the function
$$
f(t)=\cos \omega t \quad(\omega>0)
$$ | Solution. We have
$$
\cos t \longleftrightarrow \frac{p}{p^{2}+1}
$$
Then, by the similarity theorem $\forall \omega>0$
$$
\cos \omega t \longleftrightarrow \frac{1}{a} \frac{p / \omega}{(p / \omega)^{2}+1}=\frac{p}{p^{2}+\omega^{2}}
$$
Answer. $\quad \cos \omega t \longleftrightarrow \frac{p}{p^{2}+\omega^{2}}$.
... | \cos\omega\longleftrightarrow\frac{p}{p^{2}+\omega^{2}} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 33,165 |
Example. Find the image of the function
$$
f(t)=e^{-t} \cos 2 t
$$ | Solution. We have
$$
\cos 2 t \longleftrightarrow \frac{p}{p^{2}+4}
$$
Then, by the shift theorem for $a=1$
$$
e^{-t} \cos 2 t \longleftrightarrow \frac{p+1}{(p+1)^{2}+4}
$$
Answer. $\quad e^{-t} \cos 2 t \longleftrightarrow \frac{p+1}{p^{2}+2 p+5}$.
Conditions of the Problems. Find the image of the function $f(t)... | notfound | Calculus | math-word-problem | Yes | Yes | olympiads | false | 33,166 |
Example. Find the image of the function
$$
f(t)=\left[\begin{array}{ll}
1, & 0 \leqslant t \leqslant 1 \\
0, & t > 1
\end{array}\right.
$$ | Solution. Let's represent $f(t)$ as
$$
f(t)=\eta(t)-\eta(t-1)
$$
1. We have
$$
\eta(t) \longleftrightarrow \frac{1}{p}
$$
By the delay theorem for $\tau=1$
$$
\eta(t-1) \longleftrightarrow \frac{e^{-p}}{p}
$$
2. Using the linearity property, we find the desired image
$$
F(p)=\frac{1}{p}-\frac{e^{-p}}{p}
$$
Answ... | F(p)=\frac{1-e^{-p}}{p} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 33,167 |
Example 2. Find the image of the function
$$
f(t)=t \sin t
$$ | Solution.
1. We have
$$
\sin t \longleftrightarrow \frac{1}{p^{2}+1}=F(p)
$$
2. Compute the derivative
$$
F^{\prime}(p)=-\frac{2 p}{\left(p^{2}+1\right)^{2}}
$$
3. Find the image of the function $t \sin t$ using formula (1) for $n=1$:
$$
t \sin t \longleftrightarrow \frac{2 p}{\left(p^{2}+1\right)^{2}}
$$
Answer... | notfound | Calculus | math-word-problem | Yes | Yes | olympiads | false | 33,168 |
Example. Restore the original $f(t)$ from the image
$$
F(p)=\frac{1}{p^{3}-8}
$$ | ## Solution.
1. We write $F(p)$ as a sum of elementary fractions:
$$
F(p)=\frac{1}{p^{3}-8}=\frac{1}{(p-2)\left(p^{2}+2 p+4\right)}=\frac{1}{12} \frac{1}{p-2}-\frac{1}{12} \frac{p+4}{p^{2}+2 p+4}
$$
2. For each fraction, we find its original.
First fraction
$$
F_{1}(p)=\frac{1}{p-2}
$$
is the image of the functio... | f()=\frac{1}{12}e^{2}-\frac{1}{12}e^{-}\cos(\sqrt{3})-\frac{\sqrt{3}}{12}e^{-}\sin(\sqrt{3}) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 33,169 |
Example. Restore the original $f(t)$ from its image
$$
F(p)=\frac{p^{2}+p+1}{(p-1)(p+1)^{2}}
$$
using the decomposition theorem. | Solution.
1. Find the singular points of the function
$$
F_{1}(p)=e^{p t} F(p)=e^{p t} \frac{p^{2}+p+1}{(p-1)(p+1)^{2}}
$$
$F_{1}(p)$ has two poles: $p=1-$ a pole of the 1st order and $p=-1-$ a pole of the 2nd order.
2. Calculate the residues at these points
$$
\operatorname{res}_{p=1} \frac{\left(p^{2}+p+1\right)... | f()=\frac{3}{4}e^{}+\frac{1}{4}e^{-}-\frac{1}{2}e^{-} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 33,170 |
Example. Restore the original from its image
$$
\frac{1}{\left(p^{2}+1\right)^{2}}
$$ | Solution. In this case
$$
F(p)=G(p)=\frac{1}{p^{2}+1}
$$
1. We restore the original $f(t)$ from its image $F(p)$:
$$
\sin t \longleftrightarrow \frac{1}{p^{2}+1}
$$
By the convolution theorem
$$
\sin t * \sin t \longleftrightarrow \frac{1}{p^{2}+1} \cdot \frac{1}{p^{2}+1}
$$
## 2. We compute the convolution
$$
\... | f()=\frac{1}{2}\cos-\frac{1}{2}\sin | Algebra | math-word-problem | Yes | Yes | olympiads | false | 33,171 |
Example. Restore the original $f(t)$ from its image
$$
F(p)=\frac{e^{-2 p}}{\left(p^{2}+1\right)^{2}}
$$ | Solution. 1. We restore the original $r(t)$ from its image
$$
R(p)=\frac{1}{\left(p^{2}+1\right)^{2}}
$$
We have
$$
\sin t \longleftrightarrow \frac{1}{p^{2}+1}
$$
By the convolution theorem
$$
\sin t * \sin t \longleftrightarrow \frac{1}{p^{2}+1} \cdot \frac{1}{p^{2}+1}
$$
We compute the convolution
$$
\sin t *... | f()=\frac{1}{2}[(-2)\cos(-2)-\sin(-2)]\eta(-2) | Calculus | math-word-problem | Yes | Yes | olympiads | false | 33,172 |
Example. Solve the Cauchy problem
$$
x^{\prime \prime}+4 x=\cos 2 t, \quad x(0)=1, \quad x^{\prime}(0)=-1
$$ | Solution. Since $f(t)=\cos 2 t \eta(t)$ is the original, the solution $x(t)$ of the differential equation is also an original. Let's denote its image as $X(p)$.
## 1. Find the image of the left side of the equation.
By the theorem of differentiation of the original
$$
\begin{aligned}
x^{\prime}(t) & \longleftrightar... | x()=\frac{1}{4}\sin2+\cos2-\frac{1}{2}\sin2 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 33,173 |
Example. Solve the Cauchy problem
\[
\begin{cases}x^{\prime}=x+2 y-9 t, & x(0)=1 \\ y^{\prime}=2 x+y+4 e^{t}, & y(0)=2\end{cases}
\] | Solution. Assuming that the functions $x(t)$ and $y(t)$ are originals, denote their images as $X(p)$ and $Y(p)$.
1. By the theorem on differentiation of originals
$x^{\prime}(t) \longleftrightarrow p X(p)-x(0)=p X(p)-1, \quad y^{\prime}(t) \longleftrightarrow p Y(p)-y(0)=p Y(p)-2$.
By the linearity property, we find... | x()=2e^{3}-4e^{-}-2e^{}+5-3,\quady()=-4+6+4e^{-}+2e^{3} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 33,174 |
Example 2. Expand the function $y=x+1$ into a Fourier series on the interval $(0, \pi)$ with respect to the system $1, \cos \frac{1}{2} x, \cos \frac{3}{2} x, \ldots, \cos \frac{2 n+1}{2} x, \ldots$ | Solution. The required Fourier series expansion of the function $y=x+1$ on the interval $(0, \pi)$ with respect to the system $1, \cos \frac{1}{2} x, \cos \frac{3}{2} x, \ldots, \cos \frac{2 n+1}{2} x, \ldots$ is given by (6).
1. We check that
$$
\int_{0}^{\pi} f(x)^{2} d x<\infty
$$
The inequality holds because the... | x+1=\frac{4(\pi+1)}{\pi}\sum_{n=0}^{\infty}\frac{(-1)^{n}}{2n+1}\cos\frac{2n+1}{2}x,x\in(0,\pi) | Calculus | math-word-problem | Yes | Yes | olympiads | false | 33,177 |
Example 1. Expand the function $y=x+1$ into a Fourier series on the interval $(0,1)$ with respect to the system $1, \cos \pi x, \cos 2 \pi x, \ldots, \cos n \pi x, \ldots$ | Solution. The required Fourier series expansion of the function $y=x+1$ on the interval $(0,1)$ with respect to the system $1, \cos \pi x, \cos 2 \pi x, \ldots, \cos n \pi x, \ldots$ is given by (2).
1. We check that
$$
\int_{0}^{1} f(x)^{2} d x<\infty
$$
The inequality holds because the function $y=x+1$ is bounded ... | x+1=\frac{3}{2}+\frac{2}{\pi^{2}}\sum_{n=1}^{\infty}\frac{(-1)^{n}-1}{n^{2}}\cosn\pix | Calculus | math-word-problem | Yes | Yes | olympiads | false | 33,178 |
Example. Expand the function $y=x+1$ into a trigonometric Fourier series in complex form on the interval $(-2,2)$. | ## Solution.
1. We check that
$$
\int_{-2}^{2}|f(x)|^{2} d x<\infty
$$
The inequality holds since the function $y=x+1$ is bounded on $(-2,2)$.
2. We calculate the coefficients $c_{n}$ using the Euler-Fourier formulas (3). We have
$$
c_{0}=\frac{1}{4} \int_{-2}^{2}(x+1) d x=1, \quad c_{n}=\frac{1}{4} \int_{-2}^{2}(... | x+1=1+\frac{2i}{\pi}\sum_{\substack{n=-\infty\\(n\neq0)}}^{+\infty}\frac{(-1)^{n}}{n}e^{in\pix/2},x\in(-2,2) | Calculus | math-word-problem | Yes | Yes | olympiads | false | 33,180 |
Example 1. Expand the function $y=\arccos x+1$ into a Fourier series on the interval $(-1,1)$ using the system of Chebyshev polynomials. | Solution. Chebyshev polynomials are defined by the formula
$$
T_{n}(x)=\cos (n \arccos x), \quad n=0,1,2, \ldots
$$
They form an orthogonal basis in the space of functions defined on the interval $(-1,1)$, with the scalar product
$$
(u, v)=\int_{-1}^{1} u(x) v(x) \frac{1}{\sqrt{1-x^{2}}} d x
$$
The desired expansio... | \arccosx+1=\frac{\pi+2}{2}+\frac{2}{\pi}\sum_{n=1}^{\infty}\frac{(-1)^{n}-1}{n^{2}}T_{n}(x),x\in(-1,1) | Calculus | math-word-problem | Yes | Yes | olympiads | false | 33,181 |
Example 2. Expand the function $y=x^{3}$ into a Fourier series on the interval $(-1,1)$ using the system of Legendre polynomials. | Solution. The Legendre polynomials are defined by the formula
$$
P_{n}(x)=\frac{1}{2^{n} n!} \frac{d^{n}}{d x^{n}}\left(x^{2}-1\right)^{n}, \quad n=0,1,2, \ldots
$$
They form an orthogonal basis in the space of functions defined on the interval $(-1,1)$, with the scalar product
$$
(u, v)=\int_{-1}^{1} u(x) v(x) d x
... | x^{3}=\frac{3}{5}P_{1}(x)+\frac{2}{5}P_{3}(x) | Calculus | math-word-problem | Yes | Yes | olympiads | false | 33,182 |
Example 3. Expand the function $y=\alpha+\beta \cos \vartheta+\gamma \cos ^{2} \vartheta$ into a Fourier series on the interval $(0, \pi)$ with respect to the system $P_{0}(\cos \vartheta), P_{1}(\cos \vartheta), P_{2}(\cos \vartheta), \ldots$, where $P_{n}$ are Legendre polynomials. | Solution. The functions $P_{0}(\cos \vartheta), P_{1}(\cos \vartheta), P_{2}(\cos \vartheta), \ldots$ form an orthogonal basis in the space of functions defined in the interval $(0, \pi)$, with the scalar product
$$
(u, v)=\int_{0}^{\pi} u(\vartheta) v(\vartheta) \sin \vartheta d \vartheta
$$
The desired expansion of... | \alpha+\beta\cos\vartheta+\gamma\cos^{2}\vartheta=(\alpha+\frac{1}{3}\gamma)P_{0}(\cos\vartheta)+\betaP_{1}(\cos\vartheta)+\frac{2}{3}P_{2}(\cos\vartheta),0<\vartheta<\pi | Algebra | math-word-problem | Yes | Yes | olympiads | false | 33,183 |
Example 5. Expand the function $y=\left(1-x^{2}\right) x^{\nu}(\nu>-1)$ into a Fourier series on the interval $(0,1)$ with respect to the system of functions $J_{\nu}\left(\mu_{n} x\right), n=1,2, \ldots$, where $J_{\nu}$ is the Bessel function and $\mu_{n}$ is the zero of the function $J_{\nu}$. | Solution. Bessel functions $J_{\nu}(x)$ are defined by the equality
$$
J_{\nu}(x)=\sum_{m=0}^{\infty} \frac{(-1)^{m}}{m!\Gamma(m+\nu+1)}\left(\frac{x}{2}\right)^{2 m+\nu}
$$
The functions $J_{\nu}\left(z_{1} x\right), J_{\nu}\left(z_{2} x\right), \ldots$ form an orthogonal basis in the space of functions defined on t... | (1-x^{2})x^{\nu}=-4\sum_{n=1}^{\infty}\frac{J_{\nu+2}(\mu_{n})}{\mu_{n}J_{\nu-1}(\mu_{n})J_{\nu+1}(\mu_{n})}J_{\nu}(\mu_{n}x) | Calculus | math-word-problem | Yes | Yes | olympiads | false | 33,185 |
Example 6. Expand the function $y=x$ into a Fourier series on the interval $(0,1)$ with respect to the Haar function system. | Solution. Haar functions are defined by the equations
$$
\begin{aligned}
& h_{0}(x)=1, \\
& h_{1}(x)=\left[\begin{array}{rcl}
1 & \text { if } & 0<x<1 / 2, \\
-1 & \text { if } & 1 / 2<x<1,
\end{array}\right. \\
& h_{n}(x)=\left[\begin{array}{rl}
u_{n} & \text { if } \alpha_{n}<x<\beta_{n}, \\
-u_{n} & \text { if } \b... | \frac{1}{2}-\frac{1}{4}\sum_{n=1}^{\infty}\frac{1}{2^{3[\log_{2}n]/2}}h_{n}(x),x\in(0,1) | Calculus | math-word-problem | Yes | Yes | olympiads | false | 33,186 |
Example. Find the sine Fourier transform of the function $f(x)$ defined in the interval $(0, \infty)$:
$$
f(x)=\left[\begin{array}{ll}
0, & 0<x<1 \\
1, & 1<x<2 \\
0, & 2<x
\end{array}\right.
$$ | Solution.
1. Condition (1) is satisfied, since
$$
\int_{0}^{\infty}|f(x)| d x=\int_{1}^{2} d x=11.
$$
2. $f(x)=x$ when $0<x<1$.
3. $f(x)=e^{-x}$.
4. $f(x)=x e^{-x}$.
5. $f(x)=e^{-x} \cos x$.
6. $f(x)=x e^{-x} \cos x$.
7. $f(x)=\frac{x}{1+x^{2}}$.
8. $f(x)=\frac{x \cos x}{1+x^{2}}$.
9. $f(x)=x e^{-x^{2}}$.
10. $f(x)=0... | F(p)=\sqrt{\frac{2}{\pi}}\frac{1-\cosp}{p} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 33,187 |
Example 1. Find the complex Fourier transform of the function
$$
f(x)=e^{-b^{2} x^{2}}
$$
using the formula
$$
e^{-x^{2}} \longmapsto \frac{1}{\sqrt{2}} e^{-p^{2} / 4}
$$ | Solution. Representing the function $f(x)$ as $f(x)=e^{-(b x)^{2}}$, we use property 3. We obtain
$$
e^{-b^{2} x^{2}} \longmapsto \frac{1}{b} \frac{1}{\sqrt{2}} e^{-p^{2} / 4 b^{2}}
$$
Answer. $\quad F(p)=\frac{1}{b \sqrt{2}} e^{-p^{2} /(2 b)^{2}}$. | F(p)=\frac{1}{b\sqrt{2}}e^{-p^{2}/(2b)^{2}} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 33,190 |
Example 2. Find the complex Fourier transform of the function
$$
f(x)=\frac{x}{x^{2}+1}
$$
using the formula
$$
\frac{1}{x^{2}+1} \longmapsto \sqrt{\frac{\pi}{2}} e^{-|p|}
$$ | Solution. Representing the function $f(x)$ as
$$
f(x)=x \cdot \frac{1}{x^{2}+1}
$$
we use property 6 with $n=1$. We obtain
$$
\frac{x}{x^{2}+1} \longmapsto \frac{1}{i} \frac{d}{d p} \sqrt{\frac{\pi}{2}} e^{-|p|}=-\frac{1}{i} \sqrt{\frac{\pi}{2}} \operatorname{sign}(p) e^{-|p|}
$$
Answer. $F(p)=i \sqrt{\frac{\pi}{2}... | F(p)=i\sqrt{\frac{\pi}{2}}\operatorname{sign}(p)e^{-|p|} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 33,191 |
Example 3. Find the complex Fourier transform of the function
$$
f(x)=\frac{1}{x^{2}+2 x+2},
$$
using the formula
$$
\frac{1}{x^{2}+1} \longmapsto \sqrt{\frac{\pi}{2}} e^{-|p|}
$$ | Solution. Representing the function $f(x)$ as
$$
f(x)=\frac{1}{(x+1)^{2}+1}
$$
we use property 4 with $c=1$. We obtain
$$
\frac{1}{x^{2}+2 x+2} \longmapsto e^{-i p} \sqrt{\frac{\pi}{2}} e^{-|p|}
$$
Answer. $F(p)=\sqrt{\frac{\pi}{2}} e^{-|p|-i p}$.
4.4. Complex transformation of the function $\sum_{k-1}^{n} a_{k} x... | F(p)=\sqrt{\frac{\pi}{2}}e^{-|p|-ip} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 33,192 |
Example 1. Find the function $f(x)$, the sine Fourier transform of which is
$$
F(p)=\left[\begin{array}{ll}
1, & 0<p<1 \\
0, & 1<p<\infty
\end{array}\right.
$$ | ## Solution.
1. Check that condition (1) is satisfied:
$$
\int_{0}^{\infty}|F(p)|^{2} d p=\int_{0}^{1} d p=1<\infty
$$
2. Evaluating the integral on the right-hand side of (2), we get:
$$
f(x)=\sqrt{\frac{2}{\pi}} \int_{0}^{\infty} F(p) \sin p x d p=\sqrt{\frac{2}{\pi}} \int_{0}^{1} \sin p x d p=\sqrt{\frac{2}{\pi}... | f(x)=\sqrt{\frac{2}{\pi}}\frac{1-\cosx}{x} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 33,193 |
Example 2. Find the function $f(x)$, the cosine Fourier transform of which is
$$
F(p)=\left[\begin{array}{ll}
1, & 0<p<1 \\
0, & 1<p<\infty
\end{array}\right.
$$ | Solution.
1. Check that condition (1) is satisfied:
$$
\int_{0}^{\infty}|F(p)|^{2} d p=\int_{0}^{1} d p=1<\infty
$$
2. Calculating the integral on the right-hand side of (3), we get:
$$
f(x)=\sqrt{\frac{2}{\pi}} \int_{0}^{\infty} F(p) \cos p x d p=\sqrt{\frac{2}{\pi}} \int_{0}^{1} \cos p x d p=\sqrt{\frac{2}{\pi}} ... | f(x)=\sqrt{\frac{2}{\pi}}\frac{\sinx}{x} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 33,194 |
Example 3. Find the function $f(x)$, the complex Fourier transform of which is
$$
F(p)=\left[\begin{array}{ll}
1, & 0<p<1 \\
0, & 1<p<\infty
\end{array}\right.
$$ | SOLUTION.
1. We check that condition (4) is satisfied:
$$
\int_{-\infty}^{+\infty}|F(p)|^{2} d p=\int_{0}^{1} d p=1<\infty
$$
2. By computing the integral on the right-hand side of (5), we obtain:
$$
f(x)=\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{+\infty} F(p) e^{-i p x} d p=\frac{1}{\sqrt{2 \pi}} \int_{0}^{1} e^{-i p... | f(x)=\frac{1}{\sqrt{2\pi}}\frac{1-e^{-ix}}{ix} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 33,195 |
Example 4. Find the function $f(x)$, the complex Fourier transform of which is
$$
F(p)=-i p \ln \frac{1+p^{2}}{p^{2}}
$$ | ## Solution.
1. We check that condition (4) is satisfied:
$$
\begin{aligned}
\int_{-\infty}^{+\infty}|F(p)|^{2} d p=\int_{-\infty}^{+\infty} p^{2} \ln ^{2} & \frac{p^{2}+1}{p^{2}} d p=2 \int_{0}^{\infty} p^{2} \ln ^{2} \frac{p^{2}+1}{p^{2}} d p= \\
& =2 \int_{0}^{1} p^{2} \ln ^{2} \frac{p^{2}+1}{p^{2}} d p+2 \int_{1}... | f(x)=\sqrt{2\pi}\frac{e^{-|x|}(|x|+1)-1}{x^{2}}\operatorname{sign}(x) | Calculus | math-word-problem | Yes | Yes | olympiads | false | 33,196 |
Example 1. Determine the type of the equation
$$
u_{x x}-4 u_{x y}-21 u_{y y}+2 u_{x}-3 u_{y}+5 u=x^{2}
$$
and reduce it to the canonical form. | Solution.
1. Determine the coefficients of the equation $a_{11}, a_{12}$ and $a_{22}$. We have
$$
a_{11}=1, \quad a_{12}=-2, \quad a_{22}=-21
$$
2. Calculate the expression
$$
a_{12}^{2}-a_{11} a_{22}=4+21=25>0
$$
3. Since $a_{12}^{2}-a_{11} a_{22}>0$ for all $x, y$, equation (4) is of hyperbolic type in the entir... | u_{\xi\eta}=-\frac{10}{112}u_{\xi}+\frac{11}{112}u_{\eta}+\frac{5}{112}u-\frac{1}{112}(\frac{\eta-\xi}{10})^{2} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 33,197 |
Example 2. Determine the type of the equation
$$
u_{x x}-2 u_{x y}+u_{y y}+u_{x}-u_{y}+u=x y
$$
and reduce it to the canonical form. | Solution.
1. Determine the coefficients of the equation $a_{11}, a_{12}$, and $a_{22}$. We have
$$
a_{11}=1, \quad a_{12}=-1, \quad a_{22}=1
$$
2. Calculate the expression
$$
a_{12}^{2}-a_{11} a_{22}=1-1=0
$$
3. Since $a_{12}^{2}-a_{11} a_{22}=0$ for all $x, y$, equation (6) is of parabolic type in the entire $X O... | u_{\eta\eta}=u_{\eta}-u+(\xi-\eta)\eta | Algebra | math-word-problem | Yes | Yes | olympiads | false | 33,198 |
Example 3. Determine the type of the equation
$$
u_{x x}+2 u_{x y}+2 u_{y y}+6 u_{x}+6 u_{y}-3 u=x+y^{2}
$$
and reduce it to the canonical form. | Solution.
1. We determine the coefficients of the equation $a_{11}, a_{12}$, and $a_{22}$. We have
$$
a_{11}=1, \quad a_{12}=1, \quad a_{22}=2
$$
2. We compute the expression
$$
a_{12}^{2}-a_{11} a_{12}=1-2=-1<0
$$
3. Since $a_{12}^{2}-a_{11} a_{12}<0$ for all $x, y$, equation (8) is of elliptic type in the entire... | u(x,y)=C_{1}(y-(b+)x)+C_{2}(y-(b-)x) | Calculus | math-word-problem | Yes | Yes | olympiads | false | 33,199 |
Example. Find the general solution of the hyperbolic equation
$$
u_{x x}+4 u_{x y}-5 u_{y y}=0
$$ | ## Solution.
1. We reduce equation (5) to its canonical form. We obtain
$$
u_{\xi \eta}=0
$$
where $\xi=y-5 x, \eta=y+x$.
2. Integrating equation (6) with respect to $\eta$, we get
$$
u_{\xi}=C_{0}(\xi)
$$
Integrating equation (7) with respect to $\xi$, we get
$$
u(\xi, \eta)=\int C_{0}(\xi) d \xi+C_{2}(\eta) \e... | u(x,y)=C_{1}(y-)+C_{2}(y-)e^{-/b} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 33,200 |
Example. Find the general solution of the parabolic equation
$$
u_{x x}-2 u_{x y}+u_{y y}-u_{x}+u_{y}=0
$$ | Solution.
1. We reduce equation (4) to its canonical form. We obtain
$$
u_{\eta \eta}+u_{\eta}=0
$$
where $\xi=y-x, \eta=y$.
2. To find the general solution of equation (5), we form the corresponding characteristic equation $k^{2}+k=0$ and find its roots $k_{1}=0$ and $k_{2}=-1$.
Therefore, the general solution of... | notfound | Calculus | math-word-problem | Yes | Yes | olympiads | false | 33,201 |
Example. Find the general solution of the elliptic equation
$$
u_{x x}+4 u_{x y}+5 u_{y y}=0
$$
11 V.I. Afanasyev et al. | ## Solution.
1. We transform equation (3) into its canonical form. We obtain
$$
u_{\xi \xi}+u_{\eta \eta}=0
$$
where $\xi=y-2 x, \eta=x$.
2. Equation (4) means that $u(\xi, \eta)$ is the real (or imaginary) part of an analytic function $f(\xi+\eta i)$. Therefore,
$$
u(x, y)=\operatorname{Re} f(y-2 x+x i)
$$
Answe... | u(x,y)=\operatorname{Re}f(y-2x+xi) | Calculus | math-word-problem | Yes | Yes | olympiads | false | 33,202 |
Example. Solve the Dirichlet boundary value problem for Laplace's equation in a cylinder
$$
\begin{array}{ll}
\Delta u=0, \quad 0 \leqslant r<1, & 0<z<1 \\
\left.u\right|_{z=0}=1-r^{2}, & 0 \leqslant r<1 \\
\left.u\right|_{z=1}=0, & 0 \leqslant r<1 \\
\left.u\right|_{r=1}=0, & 0<z<1 .
\end{array}
$$ | notfound | Calculus | math-word-problem | Yes | Yes | olympiads | false | 33,203 | |
Example. Solve the Dirichlet boundary value problem for Laplace's equation in a sphere:
$$
\begin{gathered}
\Delta u=0, \quad 0 \leqslant r<1 \\
\left.u\right|_{r=1}=3 \cos ^{2} \vartheta
\end{gathered}
$$ | notfound | Calculus | math-word-problem | Yes | Yes | olympiads | false | 33,204 | |
Example. Solve the Dirichlet boundary value problem for the Helmholtz equation in a circle:
$$
\begin{gathered}
\Delta u+4 u=0, \quad 0 \leqslant r<1, \\
\left.u\right|_{r=1}=\sin ^{3} \varphi .
\end{gathered}
$$ | notfound | Calculus | math-word-problem | Yes | Yes | olympiads | false | 33,205 | |
Example. Find all $\lambda$ and $u$ such that
$$
\begin{aligned}
& \Delta u=\lambda u \quad \text { in the domain } \Omega=\left\{r_{1}^{2}<x^{2}+y^{2}<r_{2}^{2}\right\}, \\
& \alpha u+\beta \frac{\partial u}{\partial n}=0 \quad \text { on the boundary of the domain } \Omega,
\end{aligned}
$$
where $\alpha=1, \beta=0... | notfound | Calculus | math-word-problem | Yes | Yes | olympiads | false | 33,207 | |
Example. Solve the Dirichlet boundary value problem for the Poisson equation in an annulus:
$$
\begin{gathered}
\Delta u=r^{3} \cos \varphi, \quad 1<r<2, \\
\left.u\right|_{r=1}=\cos 2 \varphi \\
\left.\frac{\partial u}{\partial r}\right|_{r=2}=\sin 3 \varphi .
\end{gathered}
$$ | notfound | Calculus | math-word-problem | Yes | Yes | olympiads | false | 33,208 | |
Example. Solve the Dirichlet boundary value problem for Poisson's equation in a sphere
$$
\begin{gathered}
\Delta u=14 x y, \quad 0 \leqslant r<2 \\
\left.u\right|_{r=2}=14
\end{gathered}
$$ | Solution.
1. The particular solution of equation (6) is sought in the form
$$
u_{\text {p }}=E\left(x^{2}+y^{2}+z^{2}\right) x y
$$
Substituting it into equation $(6)$, we get
$$
6 E x y+6 E x y+2 E x y=14 x y
$$
Therefore, $E=1$ and
$$
u_{\text {p }}=\left(x^{2}+y^{2}+z^{2}\right) x y
$$
## 2. The function
$$
... | u(x,y,z)=(x^{2}+y^{2}+z^{2}-4)xy+14 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 33,209 |
Example. Solve the first mixed problem for the homogeneous wave equation on an interval
$$
\begin{gathered}
u_{t t}-4 u_{x x}=0, \quad x \in(0,1), \quad t \in(0, \infty) \\
u(x, 0)=0, \quad u_{t}(x, 0)=x(1-x) \\
u(0, t)=u(1, t)=0
\end{gathered}
$$ | Solution.
1. We find auxiliary solutions $v(x, t)$ of equation (5) in the form
$$
v(x, t)=X(x) T(t)
$$
with $v(0, t)=v(1, t)=0$, i.e., $X(0)=X(1)=0$. For this, we substitute the function $v(x, t)=X(x) T(t)$ into equation (5) and separate the variables. We obtain
$$
\frac{X^{\prime \prime}}{X}=\frac{T^{\prime \prime... | notfound | Calculus | math-word-problem | Yes | Yes | olympiads | false | 33,210 |
Example. Solve the first mixed problem for the inhomogeneous wave equation on an interval
$$
\begin{aligned}
& u_{t t}-u_{x x}=4 \sin ^{3} x, \quad x \in(0, \pi), t \in(0, \infty) \\
& u(x, 0)=0, \quad u_{t}(x, 0)=0, \quad x \in(0, \pi) \\
& u(0, t)=u(\pi, t)=0, \quad t \in(0, \infty)
\end{aligned}
$$ | notfound | Calculus | math-word-problem | Yes | Yes | olympiads | false | 33,211 | |
Example. Solve the first mixed problem for the homogeneous wave equation in a rectangle
$$
\begin{aligned}
& u_{t t}=4 \Delta u, \quad x \in(0,2), \quad y \in(0,1), \quad t \in(0, \infty) \\
& u(x, y, 0)=0, \quad u_{t}(x, y, 0)=(2-x)(1-y) x y \\
& u(0, y, t)=u(x, 0, t)=u(2, y, t)=u(x, 1, t)=0
\end{aligned}
$$ | notfound | Calculus | math-word-problem | Yes | Yes | olympiads | false | 33,212 | |
Example. Solve the first mixed problem for the heat equation on the interval:
$$
\begin{aligned}
& u_{t}=u_{x x}, \quad x \in(0,1), \quad t \in(0, \infty) \\
& u(x, 0)=\sin ^{3} 2 \pi x, \quad x \in(0,1) \\
& u(0, t)=u(1, t)=0, \quad t \in(0, \infty)
\end{aligned}
$$ | ## Solution.
1. We find auxiliary solutions of equation (5) in the form
$$
v(x, t)=X(x) T(t),
$$
where $v(0, t)=v(1, t)=0$, i.e., $X(0)=X(1)=0$. For this, we substitute the function $v(x, t)=X(x) T(t)$ into equation (5) and separate the variables:
$$
\frac{X^{\prime \prime}}{X}=\frac{T^{\prime}}{T}=\lambda
$$
Ther... | u(x,)=\frac{3}{4}e^{-4\pi^{2}}\sin2\pix-\frac{1}{4}e^{-36\pi^{2}}\sin6\pix | Calculus | math-word-problem | Yes | Yes | olympiads | false | 33,213 |
Example. Solve the Cauchy problem for the heat equation
\[
\begin{gathered}
u_{t}=u_{x x}, \quad x \in(-\infty,+\infty), \quad t \in(0,+\infty) \\
u(x, 0)=e^{-x} \quad x \in(-\infty,+\infty)
\end{gathered}
\] | Solution.
The solution to the Cauchy problem for the heat equation on a line is determined by the Poisson formula (3). For \(a=1, \alpha=0\) and \(\beta=-1\), we have
\[
u(x, t)=\frac{1}{2 \sqrt{\pi t}} \int_{-\infty}^{+\infty} e^{-\frac{(x-\xi)^{2}}{4 t}} e^{-\xi} d \xi
\]
Thus, the problem reduces to computing the... | u(x,)=e^{-x} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 33,215 |
Example. Two dice are thrown. Find the probability that the number of points on one die is twice the number on the other. | Solution. When the number of equally possible elementary outcomes is finite, the probability of an event is equal to the ratio of the number of elementary outcomes favorable to this event to the total number of elementary outcomes.
1. Define the space of elementary outcomes.
Number the dice. Then an elementary outcom... | \frac{6}{36} | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 33,216 |
Example. Find the probability that among six cards randomly taken from a deck of 36 cards, there will be exactly three black figure cards. | Solution. When the number of equally possible elementary outcomes is finite, the probability of an event is equal to the ratio of the number of elementary outcomes favorable to this event to the total number of elementary outcomes.
1. Define the space of elementary outcomes.
An elementary outcome is an arbitrary set ... | P(3)=\frac{546}{5797}\approx0.0942 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 33,217 |
Example. There are $n$ boxes and $r$ balls. The balls are randomly placed into the boxes. Find the probability that the $i$-th box will contain $k$ balls, and the boxes with numbers $1,2, \ldots, i-1$ will contain in total $l$ balls $(k+l \leqslant r)$. | Solution. When the number of equally possible elementary outcomes is finite, the probability of an event is equal to the ratio of the number of elementary outcomes favorable to this event to the total number of elementary outcomes.
1. Define the space of elementary outcomes.
Number all the boxes and all the balls. Th... | \mathrm{P}(A)=\frac{C_{r}^{k}C_{r-k}^{}(i-1)^{}(n-i)^{r-k-}}{n^{r}} | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 33,218 |
Example. The plane is divided into square cells with side $a$ by straight lines. The cells are numbered using pairs of integers $(m, n)$ (thus, the cell to the right will have the number $(m+1, n)$, and the cell above will have the number $(m, n+1)$). Find the probability that a coin of radius $r<a / 2$ thrown randomly... | Solution. Randomly tossing a coin is equivalent to randomly selecting a point where the center of the coin will be located. The desired probability is the ratio of the area of the favorable outcomes region in one cell to the area of the entire cell.
1. Define the sample space.
In this case, it consists of equal recta... | \frac{(-2r)^{2}}{3^{2}} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 33,220 |
Example. The probability of failure of the $i$-th conducting element of the circuit within time $T$, as shown in Fig. 6.3, is $p_{i}$. All elements of the circuit function independently of each other. Find the probability that the entire circuit will not fail within time $T$, i.e., that current can flow through the cir... | Solution. Let us introduce random events
$$
\begin{aligned}
& A=\left\{\begin{array}{c}
\text { the entire chain will not fail } \\
\text { during time } T
\end{array}\right\}, \\
& A_{i}=\left\{\begin{array}{c}
i \text {-th element of the chain will not fail } \\
\text { during time } T
\end{array}\right\}, i=1,2,3,4... | (1-p_{1})(1-p_{2}p_{3})(1-p_{4}) | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 33,221 |
Example 1. 10 Bernoulli trials are conducted with a success probability of $p$. Find the probability that there will be only two failures, and three successes between them. | Solution. The given event is caused by chains
| Nuuunuuuu | Uuunuuunu |
| :--- | :--- |
| unuuunuun | uuuunuunu |
| uunuunuun | uuuuunuun |
The probability of each such chain is $p^{8} q^{2}$, and the number of such chains is 6. Therefore, the required probability is $6 p^{8} q^{2}$.
Answer. $6 p^{8} q^{2}$. | 6p^{8}q^{2} | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 33,222 |
Example. An unlimited number of Bernoulli trials are conducted with the probability of success $p$. Find the probability that the first success will occur on the $n$-th trial, and the second success - after the $m$-th trial $(m>n)$. | Solution. The given event is caused by "chains" N...NUN...NU, where the first series of symbols N consists of the $(n-1)$-th element, and the second series consists of $k$ elements, where $k \geqslant m-n$. The probability of such a chain is $q^{n-1} p q^{k} p$, so the desired probability is
$$
\sum_{k=m-n}^{+\infty} ... | pq^{-1} | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 33,223 |
Example. A youth radio station, to attract listeners, holds prize and super-prize draws among them. Prizes are drawn over sixteen hours (one prize per hour), and super-prizes are drawn during four evening hours (also one super-prize per hour). The probability that listeners will win a prize is 0.3, and a super-prize is... | Solution. Super prize draws represent independent trials with two outcomes: if the listeners win a super prize - this is a success, if the super prize does not go to anyone - this is a failure. The probability of success in all trials is the same. A similar situation applies to prize draws. Therefore, in both cases (a)... | )0.2;b)0.9 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 33,224 |
Example. On average, 6 letters are sent to the department per week, of which 2 are from abroad. Find the probability that 13 letters will be sent in 2 weeks, of which 5 are from abroad. | SOLUTION. There are two independent simple streams of letters with intensities $\lambda_{1}=4$ (internal) and $\lambda_{2}=2$ (foreign). Let $\mu_{t}$ and $\tilde{\mu}_{t}$ be the number of events that occurred over time $t$ in the first and second streams, respectively. We are interested in the probability of the rand... | 0.0218 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 33,225 |
Example. A die is thrown. Let $m$ be the number of points that come up. Then $m$ shots are fired at a target with the probability of hitting the target in a single shot being $p$.
1. Find the probability that the target will be hit.
2. Given that the target is hit, find the probability that $m=6$. | Solution. Problem 1.
The first random experiment is the rolling of a die. Let us introduce random events $H_{m}=$ \{the number $m$ of points appears $\}, m=1, \ldots, 6$. These events do not intersect and uniquely describe all possible outcomes of the first experiment. The second random experiment is $m$ Bernoulli tri... | \sum_{=1}^{6}\frac{1}{6}(1-(1-p)^{}) | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 33,226 |
Example. Four coins are tossed. Let $\xi_{i}=1$ if the $i$-th coin lands heads up, and $\xi_{i}=0$ otherwise, $i=1,2,3,4$. Construct the distribution series of the random variable $\eta=\xi_{1}+\xi_{2}-\xi_{3}-\xi_{4}$ and find, using this series, the probability $\mathrm{P}(\eta<1)$. | SOLUTION.
1. Define the space of elementary outcomes.
The elementary outcomes of the considered random experiment are ordered sets of numbers $\left(n_{1}, n_{2}, n_{3}, n_{4}\right)$, where $n_{i}$ is either zero or one, $i=1,2,3,4$.
2. Define the set of possible values of $\eta$.
The possible values of the random... | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 33,227 | |
Example. A die is rolled. Let $\xi$ be the number of points that come up. Find the mathematical expectation and variance of the random variable $\eta=\min (\xi, 3)$.
---
Translation:
Example. A die is rolled. Let $\xi$ be the number of points that come up. Find the mathematical expectation and variance of the random ... | ## Solution.
1. We find the distribution series of the random variable $\eta$.
The elementary outcome of the considered random experiment is the number of points fallen $i$: $i=1, \ldots, 6$. The random variable $\eta$ on the $i$-th elementary outcome takes the value $\min (i, 3)$. We construct a table of elementary ... | \mathrm{M}\eta=5/2,\mathrm{D}\eta=7/12 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 33,229 |
Example. Two dice are thrown. Find the distribution of the random vector $(\xi, \eta)$ and the covariance of $\xi$ and $\eta$, where $\xi$ is the maximum of the numbers that appear, and $\eta$ is the number of times one appears. | Solution.
1. Determine the set of possible values of $(\xi, \eta)$.
Number the dice. An elementary outcome of the considered random experiment is an ordered pair of numbers $\left(n_{1}, n_{2}\right)$, where $n_{1}$ is the number of points that fell on the first die and $n_{2}$ on the second. On the elementary outcom... | -\frac{95}{108} | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 33,231 |
Example. Find the characteristic function of a discrete random variable $\xi$ having a geometric distribution (this means that $\xi$ takes values $n=1,2, \ldots$ and
$$
\mathrm{P}(\xi=n)=p q^{n-1}
$$
where $p>0, q>0, p+q=1)$. Using the characteristic function, find the mathematical expectation $\mathrm{M} \xi$ and th... | Solution. Let $a_{n}=n$, then $p_{n}=\mathrm{P}\left(\xi=a_{n}\right)=p q^{n-1}$, $n=1,2, \ldots$
1. We find the characteristic function $\varphi(t)$ of the discrete random variable $\xi$ using formula (1). We get
$$
\begin{aligned}
& \varphi(t)=\sum_{n=1}^{\infty} e^{i t a_{n}} p_{n}=\sum_{n=1}^{\infty} e^{i t n} p ... | \varphi()=\frac{p}{e^{-i}-q},\mathrm{M}\xi=\frac{1}{p},\quad\mathrm{D}\xi=\frac{q}{p^{2}} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 33,233 |
Example. Let $\xi$ be a continuous random variable with probability density
$$
p(x)=\left\{\begin{array}{cl}
\lambda e^{-\lambda x}, & x \geqslant 0 \\
0, & x<0
\end{array}\right.
$$
where $\lambda$ is a positive constant (exponential distribution). Let $\eta=\cos \xi$. Find $\mathrm{P}(\eta<0)$. | Solution. In this case, $f(x) = \cos x$. Let's find the set
$$
G = \{x: f(x)8\}$.
7. Uniform distribution $U[0, 4\pi]; \eta = \sin \xi; A = \{\eta > 0\}$.
9. Geometric distribution; $\eta = \sin (\pi \xi / 2); A = \{\eta > 0.2\}$.
10. Geometric distribution; $\eta = \cos (\pi \xi); A = \{\eta < 0\}$.
Answers. 1. 0.8.... | e^{-\lambda} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 33,234 |
Example. Let the random variable $\xi$ have a normal distribution $N(0,1)$, i.e., $\xi$ is a continuous random variable with probability density
$$
p(x)=\frac{1}{\sqrt{2 \pi}} e^{-x^{2} / 2}, \quad x \in(-\infty,+\infty)
$$
Find the mathematical expectation and variance of the random variable $\eta=10^{\xi}$. | Solution. In this case, $f(x)=10^{x}$. First, we find $\mathrm{M} \eta$:
$$
\mathrm{M} \eta=\int_{-\infty}^{+\infty} 10^{x} p(x) d x=\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{+\infty} 10^{x} e^{-x^{2} / 2} d x
$$
Notice that
$$
10^{x} e^{-x^{2} / 2}=e^{x \ln 10} e^{-x^{2} / 2}=e^{x \ln 10-x^{2} / 2}=e^{-(x-\ln 10)^{2} ... | notfound | Calculus | math-word-problem | Yes | Yes | olympiads | false | 33,235 |
Example. Let $(\xi, \eta)$ be a random vector with independent components, both distributed according to the same exponential law. Find $\mathrm{M}|\xi-\eta|$. | Solution. The probability density $p(x, y)$ of the random vector $(\xi, \eta)$ is (see p. 321)
$$
p(x, y)=\left\{\begin{array}{cl}
\lambda^{2} e^{-\lambda(x+y)}, & (x, y) \in D \\
0, & (x, y) \notin D
\end{array}\right.
$$
where $D=\{(x, y): x \geqslant 0, y \geqslant 0\}$. In the considered case, $f(x, y) = |x-y|$, ... | \frac{1}{\lambda} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 33,237 |
Example. A random variable $\xi$ is distributed according to the normal law $N(0,4)$. Find the probability of the random event
$$
A=\left\{\frac{1}{\xi}>\frac{1}{1+\xi}\right\}
$$
*) Using unified notations and names of functions $\varphi(x)$ and $\Phi(x)$, different authors define them by different formulas. | ## Solution.
1. We represent the event $A$ in the form (1). For this, we solve the inequality
$$
\frac{1}{x}>\frac{1}{x+1}
$$
We obtain
$$
x \in(-\infty,-1) \cup(0,+\infty) \text {. }
$$
Therefore, the random event $A=\{1 / \xi>1 /(1+\xi)\}$ can be written as
$$
A=\{\xi \in(-\infty,-1) \cup(0,+\infty)\} \text {. ... | 0.8085 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 33,238 |
Example. Random variables $\xi_{1}, \xi_{2}, \ldots$ are independent and distributed according to the Poisson law with parameter $a=1$. Let $S_{n}=$ $=\xi_{1}+\ldots+\xi_{n}, n=1,2, \ldots$ Find the probability $\mathrm{P}\left(2 S_{300}>S_{600}-30\right)$. | Solution. The random variable $S_{300}$ is approximately normally distributed as $N(300,300)$, since $n=300, \quad m=\mathrm{M} \xi_{1}=1$, $\sigma^{2}=\mathrm{D} \xi_{1}=1$. The random variable $S_{600}-S_{300}$ has exactly the same distribution, and the random variables $S_{300}$ and $S_{600}-S_{300}$ are independent... | 0.8897 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 33,239 |
Example 1. The random variable $\xi$ is distributed according to the normal law with parameters $a$ and $\sigma^{2}$. From the sample $x_{1}, \ldots, x_{n}$ of values of $\xi$, the empirical moments $M_{1}^{*}=\bar{x}=2.3$ and $M_{2}^{*}=\overline{x^{2}}=8.7$ have been determined. Using these moments, find the paramete... | ## Solution.
1. We calculate the theoretical moments $M \xi^{k}$ for $k=1,2$, expressing them in terms of the unknown parameters $a$ and $\sigma$. Since $\xi$ is normally distributed with parameters $a$ and $\sigma^{2}$, we have
$$
\begin{aligned}
& M \xi=\frac{1}{\sqrt{2 \pi \sigma^{2}}} \int_{-\infty}^{+\infty} x e... | =2.3,\sigma^{2}=3.41 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 33,247 |
Example 2. The random variable $\xi$ has a negative binomial distribution $\mathrm{P}(\xi=k)=C_{m+k-1}^{k} p^{m} q^{k}$. Here $m>1$ and $p \in(0,1)$ are parameters, $q=1-p$. From the sample $x_{1}, \ldots, x_{n}$ of values of $\xi$, the empirical moments $M_{1}=\bar{x}=1.21$ and $M_{2}=\overline{x^{2}}=3.54$ are determ... | Solution.
1. We calculate the theoretical moments $M \xi^{k}$ for $k=1,2$, expressing them through the unknown parameters $m$ and $p$. We have
$$
\begin{gathered}
M \xi=\sum_{k=1}^{\infty} k \mathrm{P}(\xi=k)=\sum_{k=1}^{\infty} k C_{m+k-1}^{k} p^{m} q^{k} \\
M \xi^{2}=\sum_{k=1}^{\infty} k^{2} \mathrm{P}(\xi=k)=\sum... | =1.0023,\quadp=0.3418 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 33,248 |
Example 1. A random variable $\xi$ is distributed according to the normal law with unknown parameters $a$ and $\sigma^{2}$. Determine these parameters from the sample $x_{1}, \ldots, x_{n}$ of values of $\xi$ using the method of maximum likelihood. | ## Solution.
1. The probability density function of the random variable $\xi$ is given by
$$
p\left(x ; a, \sigma^{2}\right)=\frac{1}{\sqrt{2 \pi \sigma^{2}}} e^{-(x-a)^{2} /\left(2 \sigma^{2}\right)}
$$
2. We define the likelihood function as
$$
L\left(a, \sigma^{2}\right)=\frac{1}{\sqrt{2 \pi \sigma^{2}}} e^{-\le... | =\frac{x_{1}+\ldots+x_{n}}{n},\quad\sigma^{2}=\frac{(x_{1}-)^{2}+\ldots+(x_{n}-)^{2}}{n} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 33,249 |
Example 2. How to find out how many fish are in the pond?
We catch $n_{a}=20$ fish, mark them, and release them back into the pond. After some time, sufficient for the marked fish to disperse throughout the pond, we catch $m=50$ fish. Suppose that among them, $k_{1}=7$ are marked. Determine the number of fish in the p... | ## Solution.
1. The random variable $\xi$ - the number of marked fish among $m$ caught - is determined by the probabilities
$$
\mathrm{P}(\xi=k)=p(k ; N)=\frac{C_{n_{a}}^{k} C_{N-n_{a}}^{m-k}}{C_{N}^{m}}=\frac{C_{20}^{k} C_{N-20}^{50-k}}{C_{N}^{50}}(k=0,1, \ldots, 50)
$$
2. We define the likelihood function
$$
L(N)... | 142 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 33,250 |
Example 1. To determine the parameters $a_{1}, a_{2}$, and $a_{3}$ in the formula $y=a_{1} x^{2}+a_{2} x+a_{3}$, values of $y$ were measured at different values of $x$. The obtained sample is
$$
\begin{array}{rrrrrrrrrr}
x_{k} & -1 & -0.75 & -0.5 & -0.25 & 0 & 0.25 & 0.5 & 0.75 & 1 \\
y_{k} & 6.01 & 5.07 & 4.30 & 3.56... | Solution. The function (1) is given by
$$
S\left(a_{1}, a_{2}, a_{3}\right)=\sum_{k=1}^{9}\left(y_{k}-a_{1} x_{k}^{2}-a_{2} x_{k}-a_{3}\right)^{2}
$$
1. We find the partial derivatives
$$
\begin{aligned}
\frac{\partial S}{\partial a_{1}} & =-2 \sum_{k=1}^{9}\left(y_{k}-a_{1} x_{k}^{2}-a_{2} x_{k}-a_{3}\right) x_{k}^... | 0.96x^{2}-1.97x+3.07 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 33,251 |
Example 2. To determine the parameters $a_{1}$ and $a_{2}$ in the formula $y=a_{1} e^{a_{2} x}$, values of $y$ were measured at different values of $x$. The obtained sample is
$$
\begin{array}{rrrrrrrrrr}
x_{k} & -1 & -0.75 & -0.5 & -0.25 & 0 & 0.25 & 0.5 & 0.75 & 1 \\
y_{k} & 7.14 & 4.46 & 2.81 & 1.68 & 0.99 & 0.50 &... | Solution. Since the function $y=a_{1} e^{a_{2} x}$ is not a polynomial, the point at which function (1) takes the smallest value can only be found using a numerical method. However, in this case,
$$
\ln y=a_{2} x+\ln a_{1}=a_{2} x+b
$$
Considering the errors $\xi_{k}$ in the measurement of $y_{k}$ to be small, we hav... | 1.15e^{-1.8x} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 33,252 |
Example. Independent measurements of the angles of a triangle $x_{1}, x_{2}, x_{3}$ gave the results $x_{1}^{*}=31^{\circ}, x_{2}^{*}=62^{\circ}, x_{3}^{*}=89^{\circ}$. It is known that $x_{1}+x_{2}+x_{3}-180^{\circ}=0$. Use this equality to refine the values of $x_{1}^{*}, x_{2}^{*}, x_{3}^{*}$, assuming that the meas... | Solution. Let $\Delta x_{1}^{*}, \Delta x_{2}^{*}, \Delta x_{3}^{*}$ denote the errors contained in the numbers $x_{1}^{*}, x_{2}^{*}, x_{3}^{*}$.
1. We find $\Delta x_{1}^{*}, \Delta x_{2}^{*}, \Delta x_{3}^{*}$ from the condition
$$
\left\{\begin{array}{l}
\left(\frac{\Delta x_{1}^{*}}{0.1^{\circ}}\right)^{2}+\left... | x_{1}=3040^{\},x_{2}=6140^{\},x_{3}=8820^{\} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 33,253 |
Example 1. The random variable $\xi$ is distributed according to the normal law with an unknown expected value $a$ and variance 1. There is a sample $x_{1}, \ldots, x_{100}$ of values of $\xi, \bar{x}=1.3$. Find the confidence interval for the expected value $a$ at the confidence level $p=0.98$. | ## Solution.
1. Find the function $f\left(x_{1}, \ldots, x_{100}, a\right)$, which has a known distribution function $F(x)$.
In this case,
$$
f\left(x_{1}, \ldots, x_{100}, a\right)=\frac{x_{1}+\ldots+x_{100}}{100}-a
$$
is distributed according to the normal law with mean 0 and variance $1 / 100$. Therefore,
$$
F(... | \in(1.067,1.533)_{0.98} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 33,255 |
Example. A random variable $\xi$ is distributed according to the normal law with unknown mathematical expectation $a$ and variance $\sigma^{2}$. There is a sample $x_{1}, \ldots, x_{10}$ of values of $\xi$. The sample mean is 1.17, and the sample variance is 0.25. Find the confidence intervals for $a$ and $\sigma^{2}$ ... | Solution. Let's introduce the notation for the sample mean and sample variance:
$$
M_{x}^{*}=\frac{x_{1}+\ldots+x_{10}}{10} \quad \text { and } \quad D_{x}^{*}=\frac{\left(x_{1}-M_{x}^{*}\right)^{2}+\ldots+\left(x_{10}-M_{x}^{*}\right)^{2}}{9} .
$$
1. The random variable
$$
T=f_{1}\left(x_{1}, \ldots, x_{10}, a\righ... | \in(1.029,1.311)_{0.98},\quad\sigma^{2}\in(0.114,0.889)_{0.96} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 33,257 |
Example. Given the grouped frequency distribution of some random variable $\xi$, found in the example of section 7.1 (p. 332). Using the Pearson's chi-squared test, verify the hypothesis $H_{0}: \mathrm{P}(\xi \leqslant x)=F(x)=1-e^{-\lambda x}$ against the hypothesis $H_{1}: \mathrm{P}(\xi \leqslant x) \neq F(x)$ at t... | SOLUTION.
1. Prepare a table for placing the calculation results.
Since the absolute frequencies $n_{7}^{*}, \ldots, n_{21}^{*}$ are less than 10, we will combine the grouping intervals 7 and 8, 9 and 10, as well as the intervals 11-21.
| $k$ | $x_{k}^{*}$ | $n_{k}^{*}$ | $x_{k-1}$ | $x_{k}$ | $n_{k}$ | $\left(n_{k}... | notfound | Other | math-word-problem | Yes | Yes | olympiads | false | 33,259 |
1.1. Find the matrix $C=2 A+B$, if
$$
A=\left(\begin{array}{ll}
2 & 1 \\
3 & 4
\end{array}\right), B=\left(\begin{array}{cc}
0 & -5 \\
-1 & 6
\end{array}\right)
$$ | S o l u t i o n. According to the definition of linear operations on matrices and their properties, we have:
$$
C=2\left(\begin{array}{ll}
2 & 1 \\
3 & 4
\end{array}\right)+3\left(\begin{array}{cc}
0 & -5 \\
-1 & 6
\end{array}\right)=\left(\begin{array}{cc}
4 & 2 \\
6 & 8
\end{array}\right)+\left(\begin{array}{cc}
0 &... | (\begin{pmatrix}4&-13\\3&26\end{pmatrix}) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 33,260 |
1.2. Given matrices
1) $A=\left(\begin{array}{lll}1 & 1 & 2 \\ 3 & 2 & 3 \\ 4 & 3 & 2\end{array}\right), B=\left(\begin{array}{cc}1 & 1 \\ -1 & 2 \\ 1 & 4\end{array}\right)$;
2) $A=\left(\begin{array}{lll}1 & 1 & 3 \\ 1 & 0 & 2\end{array}\right), B=\left(\begin{array}{ccc}1 & 3 & 2 \\ 1 & 0 & 2 \\ 1 & 0 & -1\end{array... | Solution. According to the definition of matrix multiplication, we have:
$$
\text { 1) } A \cdot B=\left(\begin{array}{ll}
1 \cdot 1+1 \cdot(-1)+2 \cdot 1 & 1 \cdot 1+1 \cdot 2+2 \cdot 4 \\
3 \cdot 1+2 \cdot(-1)+3 \cdot 1 & 3 \cdot 1+2 \cdot 2+3 \cdot 4 \\
4 \cdot 1+3 \cdot(-1)+2 \cdot 1 & 4 \cdot 1+3 \cdot 2+2 \cdot ... | (\begin{pmatrix}2&11\\4&19\\3&18\end{pmatrix}),(\begin{pmatrix}5&3&1\\3&3&0\end{pmatrix}) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 33,261 |
1.3. Calculate $\left(\begin{array}{cc}1 & 2 \\ -3 & 4\end{array}\right)^{3}$. | Solution. Based on the definition of the power of a matrix with a natural exponent, we have:
$$
\left(\begin{array}{cc}
1 & 2 \\
-3 & 4
\end{array}\right)^{3}=\left(\begin{array}{cc}
1 & 2 \\
-3 & 4
\end{array}\right)^{2} \cdot\left(\begin{array}{cc}
1 & 2 \\
-3 & 4
\end{array}\right)=\left(\begin{array}{cc}
-5 & 10 \... | (\begin{pmatrix}-35&30\\-45&10\end{pmatrix}) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 33,262 |
1.4. Find the value of the polynomial $f(A)$ of the matrix $A$:
$$
f(x)=x^{2}-5 x, A=\left(\begin{array}{cc}
1 & -1 \\
2 & 3
\end{array}\right)
$$ | $$
\begin{aligned}
f(A) & =\left(\begin{array}{cc}
1 & -1 \\
2 & 3
\end{array}\right)^{2}-5\left(\begin{array}{cc}
1 & -1 \\
2 & 3
\end{array}\right)=\left(\begin{array}{cc}
1 & -1 \\
2 & 3
\end{array}\right) \cdot\left(\begin{array}{cc}
1 & -1 \\
2 & 3
\end{array}\right)-5\left(\begin{array}{cc}
1 & -1 \\
2 & 3
\end{a... | (\begin{pmatrix}-6&1\\-2&-8\end{pmatrix}) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 33,263 |
1.7. Calculate the determinant
$$
\Delta=\left|\begin{array}{cccc}
3 & 1 & -1 & 2 \\
-3 & 1 & 4 & -5 \\
2 & 0 & 1 & -1 \\
3 & -5 & 4 & -4
\end{array}\right|
$$ | S o l u t i o n. Notice that the second column of the determinant already contains one zero element. Add to the elements of the second row the elements of the first row multiplied by -1, and to the elements of the fourth row - the elements of the first row multiplied by 5. We get:
$$
\Delta=\left|\begin{array}{cccc}
3... | 40 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 33,264 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.