# Contents

Geometry.
Geometry (simple).
Geometry (complex).
Vectors.
Algebra.
higher algebra.
Matrix.
Linear algebra.
10  Mathematical analysis.
11  Integrals.
12  Approximate calculation.
13  Informatics.
14  Discrete mathematics.
15  Coding.
16  Cryptography.
17  Differential equation. Here are the differential equations.
18  Theory of probability.
19  Graph theory.
20  Financial calculations.
21  Game theory.
22  Linear programming.
23  Economic and mathematical methods.
26  Sample.

Name:
Var.: 1. Group:           Day/Mo/Year:

Solve the equation:
 −3+(−4)·9−3·(−7)−6·x=0

 1:[x=−3]

/en/School tasks/Another simple equation, Internal name: ZsmpluriiZ Generate or Make Task ,

Name:
Var.: 2. Group:           Day/Mo/Year:

 36−(2 −x)2−(6−x)·(x+6)=0

 2:[x=1]

/en/School tasks/Another simple equation with fractions, Internal name: ZurSmpDrobZ Generate or Make Task ,

Name:
Var.: 3. Group:           Day/Mo/Year:

To find the roots of the equation
 x −7 x −9 = x −1 x−7

 3:

/en/School tasks/Another simple equation with a minus, Internal name: ZurSmpDrobiZ Generate or Make Task ,

Name:
Var.: 4. Group:           Day/Mo/Year:

To find the roots of the equation
 (−x − 4)2 − (x+2)2 = 0

 4:[−3]

Name:
Var.: 5. Group:           Day/Mo/Year:

2 diggers dig 9 meters of trench in 2 days. How many meters of trench will 8 diggers dig in 3 days?

 5:

Name:
Var.: 6. Group:           Day/Mo/Year:

Two turbo snails went to crawled for 85 km. The first speed at 24 km/h more than the second and she crawled to 6 an hour earlier. What were the speeds of turbo snails?

 6:[34, 10]

Name:
Var.: 7. Group:           Day/Mo/Year:

Two pipes fill the pool in 20 hours and the first pipe only fills it in 36 hours. How many hours will it take the second pipe only, to fill the pool?

 7:

Name:
Var.: 8. Group:           Day/Mo/Year:

In a rectangular triangle, the length of the hypotenuse is known: √{65} and the length of one leg: 4. Find the square of the triangle.

 8:

Name:
Var.: 9. Group:           Day/Mo/Year:

In triangle ABC the angle C is 90 degrees, sinA = [6/7], AC=5√{13}. Find AB.

 9:

/en/School tasks/The problem of the square equation, Internal name: ZsumproZ Generate or Make Task ,

Name:
Var.: 10. Group:           Day/Mo/Year:

The sum of the two numbers is 9.5 and their product is 22. Find these numbers.

 10:[4, 5.5]

Name:
Var.: 11. Group:           Day/Mo/Year:

 √ 8·x +12

1− x
=1

 11:[−1]

Name:
Var.: 12. Group:           Day/Mo/Year:

f(x) = x + 8 and g(x) = 7− x2. To find the value f( g (3) + 4 ).

 12:

/en/School tasks/Max-min on segment 1, Internal name: Zminmaxi1Z Generate or Make Task ,

Name:
Var.: 13. Group:           Day/Mo/Year:

Find the largest and the smallest value of the
y=−3·x+2 on the interval 3 ≤ x ≤ 15.

 13:[(3, −7), (15, −43)]

/en/School tasks/The equation with the module, Internal name: ZmodZ Generate or Make Task ,

Name:
Var.: 14. Group:           Day/Mo/Year:

Solve the equation:
 | x 2 +8| −15=0

 14:[x1=14; x2=−46]

/en/School tasks/The equation with the module is one more, Internal name: ZmodiZ Generate or Make Task ,

Name:
Var.: 15. Group:           Day/Mo/Year:

Solve the equation:
 |x−7|+4=6

 15:[x1=9, x2=5]

/en/School tasks/Max-min on segment 2, Internal name: Zminmaxi2Z Generate or Make Task ,

Name:
Var.: 16. Group:           Day/Mo/Year:

Find the largest and the smallest value of the
y=x2+6 on the interval −6 ≤ x ≤ 8.

 16:[(0, 6), (8, 70)]

Name:
Var.: 17. Group:           Day/Mo/Year:

Solve the equation
2.4+7.4·x

 √ 7.3·x + 0.4·x2
= 8.7

 17:[21.1054]

Name:
Var.: 18. Group:           Day/Mo/Year:

Write the answer in the form of an irreducible fraction.
 17 35 − 17 45

 5 63
=     (                     )

(                     )

 18:[34/25]

Name:
Var.: 19. Group:           Day/Mo/Year:

The price was first increased by 41%, and then reduced by 25%. What percentage is the final increase in price?

 19:[5.75%]

Name:
Var.: 20. Group:           Day/Mo/Year:

The solution with a weight of 2100 kg contains 16% salt. How much % salt will there be in solution after adding 1351 kg of water and 49 kg of salt?

 20:

/en/School tasks/Interest is the Addition of a solution (training), Internal name: ZpersentAtZ Generate or Make Task ,

Name:
Var.: 21. Group:           Day/Mo/Year:

Nothing

 21:

/en/School tasks/Percent Mixing of solutions, Internal name: ZpersentiZ Generate or Make Task ,

Name:
Var.: 22. Group:           Day/Mo/Year:

Mixed 5100 pounds 10 percent solution with 500 pounds 66 percent. What has the concentration of the solution become?

 22:[15%]

/en/School tasks/Percent Mixing of solutions (complex), Internal name: ZpersentiiZ Generate or Make Task ,

Name:
Var.: 23. Group:           Day/Mo/Year:

How many kilograms of 43 percent solution must be mixed with 500 kg 62 percent solution for getting 48 percent solution?

 23:[1400kg]

/en/School tasks/Equation with logarithms, Internal name: ZurlogZ Generate or Make Task ,

Name:
Var.: 24. Group:           Day/Mo/Year:

Solve the equation::
 52·log5 x +6·log5 ⎛⎝ 5x 25 ⎞⎠ −15=0

 24:[x=3]

/en/School tasks/Simple equation with logarithm, Internal name: ZurlogiZ Generate or Make Task ,

Name:
Var.: 25. Group:           Day/Mo/Year:

Solve the equation:
 log16 x +14 x+5 = 1 4

 25:[x=4]

Name:
Var.: 26. Group:           Day/Mo/Year:

Solve the inequality:
 1 x−3 ≤ 1 8

 26:[(−∞; 3)∪[11;∞)]

/en/School tasks/The inequality is simple, Internal name: ZneriiZ Generate or Make Task ,

Name:
Var.: 27. Group:           Day/Mo/Year:

Find the smallest integer solution of the inequality:
 1 x−7 < 1 −5

 27:

## 2Geometry.

/en/Geometry/The intersection of straight and plane, Internal name: ZplZ Generate or Make Task ,

Name:
Var.: 28. Group:           Day/Mo/Year:

Find the coordinates of the point of intersection of the plane passing through the point A=(−1;−4;1), B=(7;−8;3), C=(−4;−2;0) with the line passing through the point D=(10;−9;2), E=(19;−18;11).

 28:[(13;−12;5)]

/en/Geometry/The intersection of straight line and plane (training), Internal name: ZpltZ Generate or Make Task ,

Name:
Var.: 29. Group:           Day/Mo/Year:

Find the equation of the plane lying on the points A=(4;1;−1), B=(12;−5;3), C=(3;2;−2) and write it:
Find the parametric equation of the line passing through the point D=(−19;17;−13) and E=(−5;10;−6).
{
 x=
 +
 ·t
 y=
 +
 ·t
 z=
 +
 ·t

Find the coordinates of the point of intersection of this plane with this line.

 29:[(−13;14;−10)]

/en/Geometry/Intersection of straight line and plane (with check), Internal name: ZplttZ Generate or Make Task ,

Name:
Var.: 30. Group:           Day/Mo/Year:

empty

 30:[(−22;7;9)]

/en/Geometry/The image of a point on the line, Internal name: ZprlineZ Generate or Make Task ,

Name:
Var.: 31. Group:           Day/Mo/Year:

Find the coordinates of the projection of the point A=(−2,−2,−4) on the line passing through the points B=(−1,3,2) and C=(0,6,3).

 31:[(−3,−3,0)]

/en/Geometry/Symmetrical point with respect to a straight line, Internal name: ZsmlineZ Generate or Make Task ,

Name:
Var.: 32. Group:           Day/Mo/Year:

Find the coordinates of the point that is symmetric to the point A=(−1,2,−1) relative to the line passing through the points B=(1,−3,−1) and C=(4,−7,0).

 32:[(−3,0,−3)]

/en/Geometry/The projection of a point on the plane, Internal name: ZprplZ Generate or Make Task ,

Name:
Var.: 33. Group:           Day/Mo/Year:

Find the coordinates of the projection of the point A=(2, −8, −1) on the plane given by the equation −1·x+3·y−1·z−8=0.

 33:[(−1, 1, −4)]

/en/Geometry/Symmetrical point with respect to the plane, Internal name: ZsmplZ Generate or Make Task ,

Name:
Var.: 34. Group:           Day/Mo/Year:

Find the coordinates of the point, the symmetric point A=(−3, 5, 5) relative to the plane given by the equation 2·x−3·y−1·z+12=0.

 34:[(1, −1, 3)]

/en/Geometry/The intersection of lines on the plane, Internal name: ZprprZ Generate or Make Task ,

Name:
Var.: 35. Group:           Day/Mo/Year:

The first line passes through the points A=(0,−5) and B=(−1,−8). The second line passes through the points C=(5,8) and D=(6,10). Find the coordinates of the intersection point of these lines.

 35:[(3,4)]

/en/Geometry/The intersection of lines on the plane (complex), Internal name: ZprprxZ Generate or Make Task ,

Name:
Var.: 36. Group:           Day/Mo/Year:

The first line passes through the points A=(−10,6) and B=(−4,7). The second line passes through the points C=(14,4) and D=(19,5). Find the coordinates of the intersection point of these lines.

 36:[(194,40)]

/en/Geometry/The intersection of lines on the plane (with fractional numbers), Internal name: ZprprvZ Generate or Make Task ,

Name:
Var.: 37. Group:           Day/Mo/Year:

The first line passes through the points A=(−8;−4) and B=(−17;−1). The second line passes through the points C=(1;4) and D=(−3;3). Find the coordinates of the intersection point of these lines. (Tip: the answer will contain fractional numbers)

 37:[([(−125)/7];[(−5)/7]) ≈ (−17.857;−0.714)]

/en/Geometry/The intersection of straight lines on the plane (training), Internal name: ZuprprZ Generate or Make Task ,

Name:
Var.: 38. Group:           Day/Mo/Year:

Given four points: A=(−13,−5), B=(−5,−7), C=(15,−3) and D=(22,−5). Find:
(1) The coordinates of the vector AB=(          ;          ),
(2) parametric equation of the line passing through the points A and B: {
 x=
 +
 ·t1
 y=
 +
 ·t1

(3) The coordinates of the vector CD=(          ;          ),
(4) parametric equation of the line passing through the points C and D: {
 x=
 +
 ·t2
 y=
 +
 ·t2

(5) the coordinates of the intersection of these lines (           ;           ).

 38:[(267,−75)]

/en/Geometry/The intersection of lines in space, Internal name: ZprprprZ Generate or Make Task ,

Name:
Var.: 39. Group:           Day/Mo/Year:

The first line passes through the points A=(−3,−4,0) and B=(−3,−5,−1). The second line passes through the points C=(−7,−3,−7) and D=(−8,−3,−9). Find the coordinates of the intersection point of these lines.

 39:[(−3,−3,1)]

/en/Geometry/The intersection of straight lines in space (training), Internal name: ZprprprtZ Generate or Make Task ,

Name:
Var.: 40. Group:           Day/Mo/Year:

Given four points: A=(3,15,−9), B=(3,18,−11), C=(3,11,−5) and D=(3,13,−6). Find:
(1) The coordinates of the vector AB=(          ;          ;          ),
(2) parametric equation of the line passing through the points A and B: {
 x=
 +
 ·α
 y=
 +
 ·α
 z=
 +
 ·α

(3) The coordinates of the vector CD=(          ;          ;          ),
(4) parametric equation of the line passing through the points C and D: {
 x=
 +
 ·β
 y=
 +
 ·β
 z=
 +
 ·β

(5) the coordinates of the intersection of these lines (          ;           ;           ).

 40:[(3,3,−1)]

/en/Geometry/The intersection of straight lines in space (more training), Internal name: ZprprprtvZ Generate or Make Task ,

Name:
Var.: 41. Group:           Day/Mo/Year:

Given four points: A=(7,−1,−6), B=(10,0,−7), C=(−5,−4,3) and D=(−6,−4,5). Find:
(1) The coordinates of the vector AB=(          ;          ;          ),
(2) parametric equation of the line passing through the points A and B: {
 x=
 +
 ·α
 y=
 +
 ·α
 z=
 +
 ·α

(3) The coordinates of the vector CD=(          ;          ;          ),
(4) parametric equation of the line passing through the points C and D: {
 x=
 +
 ·β
 y=
 +
 ·β
 z=
 +
 ·β

(5) Equate x, y, z from the first equation of the line to x, y, z from the second equation of the line:
{
 +
 ·α =
 +
 ·β
 +
 ·α =
 +
 ·β
 +
 ·α =
 +
 ·β

(6) Solve this system of equations and find the values of α = (       ) and β = (       ).
(7) Substitute the value of α in the first equation of the line and find the values of x=(      ), y=(      ) and z=(      ).
(8) Substitute the value of β in the first equation of the line and find the values of x=(      ), y=(      ) and z=(      ).
(9) Find the coordinates of the intersection of these lines: (          ;           ;           ).

 41:[(−2,−4,−3)]

## 3Geometry (simple).

/en/Geometry (simple)/Straight line on plane, Internal name: ZoburiZ Generate or Make Task ,

Name:
Var.: 42. Group:           Day/Mo/Year:

Find: (1) the general equation of the line passing through the point A=(−4,−9) perpendicular to the vector α=(21,28)..
(2) Find the distance from this line to the point B=(−6,5).
(3) Write the equation of this line as y=k·x +b.

 42:[21·x +28·y +336=0, d=10, y=[(−3)/4]·x −12]

/en/Geometry (simple)/Straight line and two points of the plane, Internal name: ZlinedotsZ Generate or Make Task ,

Name:
Var.: 43. Group:           Day/Mo/Year:

Find relationship a/b if it is known that a straight line a·x + b·y + c = 0 passes through points with coordinates (−4;20) and (−1;8).

 43:[a/b=4, (-12 -3 12 )]

/en/Geometry (simple)/Three straight lines, Internal name: ZtriprZ Generate or Make Task ,

Name:
Var.: 44. Group:           Day/Mo/Year:

Draw lines defined by equations:
1) y=[2/3]·x−1
2) y=[(−3)/5]·x−1
3) y=[2/3]·x+2. 44:[ 2mm Picture Omitted]

/en/Geometry (simple)/Two straight lines, Internal name: ZdveprZ Generate or Make Task ,

Name:
Var.: 45. Group:           Day/Mo/Year:

Find the equation of the drawn line.
Draw a line given by the equation y=[1/5]·x +[(−17)/5]
Find the coordinates of the point where they intersect.
 2.7mm Picture Omitted
Answer: y=[((     ))/((     ))]·x + [((     ))/((     ))], intersect in: (    ,    ).

 45:[ 2mm Picture Omitted, y=[1/4]·x +[(−17)/4], (17,0)]

/en/Geometry (simple)/The vertices of a parallelogram on a plane, Internal name: Zparallelogram2Z Generate or Make Task ,

Name:
Var.: 46. Group:           Day/Mo/Year:

Find the coordinates of all the vertices of the parallelogram, if you know the coordinates of one vertex A=(−2,22) and the equation of its two sides: −8·x+5·y = 24 and −6·x−9·y+84=0.

 46:[(7,16), (−7,14), (2,8)]

/en/Geometry (simple)/Perpendicular line, Internal name: ZperppriamZ Generate or Make Task ,

Name:
Var.: 47. Group:           Day/Mo/Year:

The line is given by the y=[5/6]·x −[1/6] equation
(1) Write the general equation of this line,
(2) find the equation of a perpendicular line passing through the point (−11,11),
(3) find the point of intersection of these lines.

 47:[(−1,−1)]

/en/Geometry (simple)/The top of the square, Internal name: ZkvadratiZ Generate or Make Task ,

Name:
Var.: 48. Group:           Day/Mo/Year:

Find the coordinates of the vertices of a square, if you know the coordinates of one vertex (14,11) and the equation of one side y=[(−1)/7]·x +[41/7].

 48:[(13,4), (6,5), (7,12) or (20,3), (21,10)]

/en/Geometry (simple)/Distance from point to plane, Internal name: ZploskitochZ Generate or Make Task ,

Name:
Var.: 49. Group:           Day/Mo/Year:

Find the distance from the point A=(7,8,−9) to the plane passing through the point B=(−6,−8,−8) perpendicular to the vector α=(2,−4,4).

 49:

/en/Geometry (simple)/Is the point on the plane, Internal name: ZdotonplZ Generate or Make Task ,

Name:
Var.: 50. Group:           Day/Mo/Year:

At what value of z the point (4;5; z) lies on the plane 7·x+5·y−5·z−43=0?

 50:

/en/Geometry (simple)/Vector is parallel to plane, Internal name: ZvekparalplZ Generate or Make Task ,

Name:
Var.: 51. Group:           Day/Mo/Year:

At what value of z the vector (3; 1; z) is parallel to the plane 3·x−6·y−1·z+1=0?

 51:

/en/Geometry (simple)/The point of intersection of altitudes in a triangle, Internal name: ZvysintreugZ Generate or Make Task ,

Name:
Var.: 52. Group:           Day/Mo/Year:

The coordinates of two vertices of the triangle are given (−14; 8), (11; 33) and the points of intersection of heights (5; 15). Find the coordinates of the third vertex of the triangle.

 52:[(25; −5)]

## 4Geometry (complex).

/en/Geometry (complex)/Two vertices of a square, Internal name: ZkvadratZ Generate or Make Task ,

Name:
Var.: 53. Group:           Day/Mo/Year:

Two opposite vertices of square A=(5, 33, 22), C=(−13, −31, −26) and point E=(19, −22, −50) lying in the same plane as the square are given. Find the coordinates of the two remaining vertices.

 53:[(28, 10, −26), (−36, −8, 22)]

/en/Geometry (complex)/Three lines, Internal name: ZtrilineZ Generate or Make Task ,

Name:
Var.: 54. Group:           Day/Mo/Year:

The first line passes through points with coordinates (1,3,1) and (4,0,1). The second line passes through the points with coordinates (0,−3,−3) and (0,−12,−9). The third line passes through the point with coordinates (1,−2,−2) and crosses the first and second lines. Find the coordinates of the point of intersection of the first and third lines.

 54:[(2,2,1), (0,−6,−5)]

## 5Vectors.

/en/Vectors/Sum of vectors, Internal name: ZsumvektZ Generate or Make Task ,

Name:
Var.: 55. Group:           Day/Mo/Year:

Find the coordinates of the vector (2·a+b).
 2.7mm Picture Omitted

 55:[(14, 20)]

/en/Vectors/Vector of given length and direction 2, Internal name: ZvektIdir2Z Generate or Make Task ,

Name:
Var.: 56. Group:           Day/Mo/Year:

The vector CD is directed in the same direction as the vector AB and the vector length CD is √{1025}. Find the coordinates of D if A=(−3 ,9), B=(−7 ,4) and C=(8 ,−9).

 56:[(−12 ,−34)]

/en/Vectors/Vector of given length and direction 3, Internal name: ZvektIdir3Z Generate or Make Task ,

Name:
Var.: 57. Group:           Day/Mo/Year:

The vector CD is directed in the same direction as the vector AB and the vector length CD is √{272}. Find the coordinates of D if A=(1 ,7 ,4), B=(−1 ,10 ,2) and C=(−8 ,−5 ,9).

 57:[(−16 ,7 ,1)]

/en/Vectors/The fourth vertex of the parallelogram, Internal name: ZparalZ Generate or Make Task ,

Name:
Var.: 58. Group:           Day/Mo/Year:

The coordinates of three vertices of the parallelogram are given: A=(−4, −2), B=(−7, −7) and D=(−6, 1). Find the coordinates of the fourth vertex C.

 58:[For ABCD: (−9, −4), For ABDC: (−3, 6)]

/en/Vectors/The fourth vertex of the parallelogram in space, Internal name: Zparal3Z Generate or Make Task ,

Name:
Var.: 59. Group:           Day/Mo/Year:

The coordinates of three vertices of the parallelogram are given: A=(2, 2, −4), B=(7, 1, −7) and D=(3, −3, −9). Find the coordinates of the fourth vertex C.

 59:[For ABCD: (8, −4, −12), For ABDC: (−2, −2, −6)]

/en/Vectors/Two vertices of a square, Internal name: ZverkvadrZ Generate or Make Task ,

Name:
Var.: 60. Group:           Day/Mo/Year:

Two adjacent vertices of a square are given: (−4;5) and (41;−42). Find the coordinates of the remaining vertices.

 60:[(88;3), (43;50) and (−6;−87), (−51;−40)]

/en/Vectors/Division of the segment in relation to the 2-dimensional case, Internal name: Zdelotr2Z Generate or Make Task ,

Name:
Var.: 61. Group:           Day/Mo/Year:

The coordinates of two points A=(49, 42) and B=(174, 112) are given. Find the coordinates of the point C, which lies on the segment AB and divides it so that |AC|:|CB|=3:2.

 61:[(124, 84)]

/en/Vectors/Division of the segment in relation to the 3-dimensional case, Internal name: Zdelotr3Z Generate or Make Task ,

Name:
Var.: 62. Group:           Day/Mo/Year:

The coordinates of two points A=(−35, 41, −38) and B=(53, −43, 58) are given. Find the coordinates of the point C, which lies on the segment AB and divides it so that |AC|:|CB|=3:1.

 62:[(31, −22, 34)]

/en/Vectors/Division of a segment in relation to (training), Internal name: Zdelotr4Z Generate or Make Task ,

Name:
Var.: 63. Group:           Day/Mo/Year:

Nothing

 63:[(86, −82)]

/en/Vectors/Simple scalar product, Internal name: ZskalpriZ Generate or Make Task ,

Name:
Var.: 64. Group:           Day/Mo/Year:

Find the scalar product of the vector (−2; 3) with the vector (3; 4).

 64:

/en/Vectors/Vector orthogonal to the given, Internal name: ZortiZ Generate or Make Task ,

Name:
Var.: 65. Group:           Day/Mo/Year:

Find a number z that the vector (−7, 4, 1) is perpendicular to the vector (−2, 5, z).

 65:[−34]

/en/Vectors/Vector orthogonal to two data (with length), Internal name: ZortiiZ Generate or Make Task ,

Name:
Var.: 66. Group:           Day/Mo/Year:

Find the coordinates of the vector a, which is orthogonal to the vectors b = (−9, −3, −7) and c = (−3, −6, 1) and has a length of √{88}.

 66:[±(6, −4, −6)]

/en/Vectors/Vector orthogonal to two given, Internal name: ZortvZ Generate or Make Task ,

Name:
Var.: 67. Group:           Day/Mo/Year:

Find a non-zero vector perpendicular to the vector (5, −2, 1) and perpendicular to the vector (−6, 1, −1).

 67:[λ·(1, −1,−7)]

/en/Vectors/Vector orthogonal to two data (training), Internal name: ZortvtZ Generate or Make Task ,

Name:
Var.: 68. Group:           Day/Mo/Year:

1) Find two different solutions to a system of linear equations: {
 16·x
 −2·y
 −1·z
 =
 0
 9·x
 −1·y
 −1·z
 =
 0

2) Find a non-zero vector perpendicular to the vector (16, −2, −1) and perpendicular to the vector (9, −1, −1).

 68:[λ·(1, 7,2)]

/en/Vectors/Scalar product, Internal name: ZproizZ Generate or Make Task ,

Name:
Var.: 69. Group:           Day/Mo/Year:

The coordinates of vectors a, b in an orthonormal basis are given: a=(−1, −1), b=(1, 0). The coordinates of c, d in the basis a,b are given: c=(−1, −1), d=(3, 3).
Find the scalar product of the vector c with vector d.

 69:[−3]

/en/Vectors/Coordinates in another basis, Internal name: ZdotZ Generate or Make Task ,

Name:
Var.: 70. Group:           Day/Mo/Year:

Given the coordinates of the points A, B, C, D, E in Cartesian coordinate system: A=(2,3), B=(6,4), C=(9,4), D=(3,3), E=(17,5). Find the coordinates of E in the new coordinate system with the origin at D and the base vectors AB and BC.

 70:[(2,2)]

/en/Vectors/Area of a triangle on a plane, Internal name: ZploshZ Generate or Make Task ,

Name:
Var.: 71. Group:           Day/Mo/Year:

Find the area of the triangle, the coordinates of the vertices of which is (1,7), (−5,11) and (6,12).

 71:

## 6Algebra.

/en/Algebra/SLEQ is a very simple 2x2, Internal name: Zslu22Z Generate or Make Task ,

Name:
Var.: 72. Group:           Day/Mo/Year:

 8·x
 −1·y
 =
 26
 −4·x
 +1·y
 =
 −10

 72:[x=4, y=6]

/en/Algebra/SLEQ is a very simple 3x3, Internal name: Zslu33Z Generate or Make Task ,

Name:
Var.: 73. Group:           Day/Mo/Year:

 +1·y
 =
 −2
 −1·x
 +1·y
 +1·z
 =
 2
 +1·z
 =
 3

 73:[x=−1, y=−2, z=3]

/en/Algebra/SLEQ 3x3 with one solution, Internal name: Zslu33mZ Generate or Make Task ,

Name:
Var.: 74. Group:           Day/Mo/Year:

 5·x
 −5·y
 −7·z
 =
 24
 −3·x
 +4·y
 +5·z
 =
 −15
 3·x
 −3·y
 −4·z
 =
 14

 74:[x=3, y=1, z=−2]

/en/Algebra/SLEQ 3-unknowns 4-equations one solution, Internal name: ZsluZ Generate or Make Task ,

Name:
Var.: 75. Group:           Day/Mo/Year:

 6 ·x1
 −3 ·x2
 +5 ·x3
 =35
 −3 ·x1
 +2 ·x2
 −2 ·x3
 =−14
 −3 ·x1
 +1 ·x2
 −2 ·x3
 =−17
 −2 ·x1
 +2 ·x2
 −2 ·x3
 =−10

 75:[x1=4, x2=3, x3=4]

/en/Algebra/SLEQ 4-unknown, 5-equation, one solution, Internal name: ZsluuZ Generate or Make Task ,

Name:
Var.: 76. Group:           Day/Mo/Year:

 +1 ·x3
 −1 ·x4
 =−4
 −1 ·x1
 −4 ·x2
 −3 ·x3
 +1 ·x4
 =3
 −1 ·x1
 −5 ·x2
 −4 ·x3
 =2
 1 ·x1
 +2 ·x2
 +1 ·x3
 =1
 1 ·x1
 +5 ·x2
 +5 ·x3
 =−4

 76:[x1=1, x2=1, x3=−2, x4=2]

/en/Algebra/SLEQ with multiple solutions, Internal name: ZsluiZ Generate or Make Task ,

Name:
Var.: 77. Group:           Day/Mo/Year:

Find five different solutions of the equation system:

 1 ·x1
 +1 ·x2
 −3 ·x3
 =0
 1 ·x1
 +2 ·x2
 −5 ·x3
 =0

 77:[(1 x3,2 x3, x3)]

/en/Algebra/SLEQ 3x3 with multiple solutions, Internal name: Zsluii3Z Generate or Make Task ,

Name:
Var.: 78. Group:           Day/Mo/Year:

Find five different solutions to the system of equations:

 3 ·x1
 −1 ·x2
 +7 ·x3
 =3
 −2 ·x1
 +2 ·x2
 −2 ·x3
 =−2
 2 ·x1
 −1 ·x2
 +4 ·x3
 =2

 78:[Formula to verify the solution: (1−3 x3,−2 x3, x3)]

/en/Algebra/SLEQ 4x4 with multiple solutions, Internal name: Zsluii4Z Generate or Make Task ,

Name:
Var.: 79. Group:           Day/Mo/Year:

Find five different solutions to the system of equations:

 4 ·x1
 −3 ·x2
 +1 ·x3
 +7 ·x4
 =4
 −3 ·x1
 +1 ·x2
 −1 ·x3
 −8 ·x4
 =−3
 7 ·x1
 −6 ·x2
 +3 ·x3
 +12 ·x4
 =7
 4 ·x1
 −3 ·x2
 +2 ·x3
 +8 ·x4
 =4

 79:[Formula to verify the solution: (1−3 x4,−2 x4,−1 x4 ,x4)]

/en/Algebra/SLEQ 3-unknowns 4-equation of one-dimensional space solutions, Internal name: ZsluuuZ Generate or Make Task ,

Name:
Var.: 80. Group:           Day/Mo/Year:

Find the general solution of the equation system:

 2 ·x1
 −1 ·x2
 +1 ·x3
 −2 ·x4
 =0
 −1 ·x1
 +2 ·x2
 −1 ·x3
 +4 ·x4
 =0
 2 ·x1
 −3 ·x2
 +2 ·x3
 −5 ·x4
 =0
 −1 ·x1
 +2 ·x2
 −1 ·x3
 +4 ·x4
 =0

 80:[(1,−3,−3,1)·λ]

/en/Algebra/SLEQ with two-dimensional solution space, Internal name: ZsluiiiZ Generate or Make Task ,

Name:
Var.: 81. Group:           Day/Mo/Year:

Find two linearly independent solutions.

 5 ·x1
 +1 ·x2
 −11 ·x3
 +11 ·x4
 =0
 −4 ·x1
 −1 ·x2
 +9 ·x3
 −9 ·x4
 =0
 2 ·x1
 +1 ·x2
 −5 ·x3
 +5 ·x4
 =0

 81:[(2 x3−2 x4, 1 x3−1x4, x3,x4)]

/en/Algebra/New 2x2 SLEQ, Internal name: Znslu22Z Generate or Make Task ,

Name:
Var.: 82. Group:           Day/Mo/Year:

 −20 ·x
 −9 ·y
 =
 −21
 −11 ·x
 −5 ·y
 =
 −12

 82:[x=−3; y=9]

/en/Algebra/New 2x3 SLEQ, Internal name: Znslu23Z Generate or Make Task ,

Name:
Var.: 83. Group:           Day/Mo/Year:

 20 ·x
 −27 ·y
 =
 5
 −2 ·x
 + 3 ·y
 =
 1
 −19 ·x
 + 26 ·y
 =
 −3

 83:[x=7; y=5]

/en/Algebra/New 3x3 SLEQ, Internal name: Znslu33Z Generate or Make Task ,

Name:
Var.: 84. Group:           Day/Mo/Year:

 x
 −3 ·y
 −2 ·z
 =
 −1
 −x
 + 5 ·y
 + 3 ·z
 =
 8
 x
 −2 ·y
 −z
 =
 1

 84:[x=8; y=5; z=−3]

/en/Algebra/New 3x4 SLEQ, Internal name: Znslu34Z Generate or Make Task ,

Name:
Var.: 85. Group:           Day/Mo/Year:

 x
 −y
 −z
 =
 1
 −x
 + 2 ·y
 + 2 ·z
 =
 −1
 −2 ·x
 + 3 ·y
 + 3 ·z
 =
 −2
 −2 ·x
 + 3 ·y
 + 4 ·z
 =
 −1

 85:[x=1; y=−1; z=1]

/en/Algebra/New 5x5 SLEQ, Internal name: Znslu55Z Generate or Make Task ,

Name:
Var.: 86. Group:           Day/Mo/Year:

 3 ·x1
 −x2
 −x3
 −x4
 −x5
 =
 1
 5 ·x1
 −x2
 −3 ·x3
 −2 ·x4
 −x5
 =
 −1
 10 ·x1
 −3 ·x2
 −6 ·x3
 −4 ·x4
 −3 ·x5
 =
 7
 −10 ·x1
 + 2 ·x2
 + 7 ·x3
 + 4 ·x4
 + 2 ·x5
 =
 −1
 −14 ·x1
 + 3 ·x2
 + 10 ·x3
 + 6 ·x4
 + 4 ·x5
 =
 −8

 86:[x1=−3; x2=−4; x3=−3; x4=2; x5=−5]

/en/Algebra/New 5x6 SLEQ, Internal name: Znslu56Z Generate or Make Task ,

Name:
Var.: 87. Group:           Day/Mo/Year:

 −3 ·x1
 + 3 ·x2
 −2 ·x3
 + 3 ·x4
 −4 ·x5
 =
 −20
 −6 ·x1
 + 7 ·x2
 −x3
 + 7 ·x4
 −4 ·x5
 =
 −44
 3 ·x1
 −4 ·x2
 −x3
 −3 ·x4
 −x5
 =
 15
 −3 ·x1
 + 4 ·x2
 + 2 ·x3
 + 5 ·x4
 + x5
 =
 −26
 2 ·x1
 −2 ·x2
 + 2 ·x3
 −x4
 + 3 ·x5
 =
 9
 −2 ·x1
 + 3 ·x2
 + x3
 + 3 ·x4
 + x5
 =
 −14

 87:[x1=−7; x2=−8; x3=−9; x4=−1; x5=8]

/en/Algebra/SLEQ with fractions, Internal name: ZsludZ Generate or Make Task ,

Name:
Var.: 88. Group:           Day/Mo/Year:

Find the solution of the system of equations and write the answer in the form of irreducible fractions.
{
 [5/7]·x
 +[1/2]·y
 =[(−17)/28]
 [1/4]·x
 −[2/7]·y
 =[(−31)/70]

Answer: x= [((    ))/((    ))], y= [((    ))/((    ))].

 88:[x=[(−6)/5], y=[1/2].]

## 7higher algebra.

/en/higher algebra/Actions with complex numbers, Internal name: ZcompliZ Generate or Make Task ,

Name:
Var.: 89. Group:           Day/Mo/Year:

(5+4·i)·(1−i)+(−2+4·i)=x+3·i
x=

 89:

/en/higher algebra/Division of complex numbers, Internal name: ZdelcomplZ Generate or Make Task ,

Name:
Var.: 90. Group:           Day/Mo/Year:

[(−6+22·i)/(1+5·i)]=

 90:[4+2·i]

/en/higher algebra/The square root of a complex number, Internal name: ZsqrtCZ Generate or Make Task ,

Name:
Var.: 91. Group:           Day/Mo/Year:

√{−48 + 14·i}=

 91:[−1 −7·i; 1 + 7·i]

/en/higher algebra/Ordinary square with negative discriminant, Internal name: ZkvursZ Generate or Make Task ,

Name:
Var.: 92. Group:           Day/Mo/Year:

To find the roots of the equation
x2 +2·x +10=0.

 92:[−1±3·i]

/en/higher algebra/Square equation with complex numbers, Internal name: ZkvurZ Generate or Make Task ,

Name:
Var.: 93. Group:           Day/Mo/Year:

To find the roots of the equation:
(1 −1·i)·x2 + (5 −3·i)·x + ( 10) = 0

 93:[−3 + 1·i, −1 −2·i;]

/en/higher algebra/Square equation with complex numbers (homogeneous), Internal name: ZkvurrrZ Generate or Make Task ,

Name:
Var.: 94. Group:           Day/Mo/Year:

To find the roots of the equation:
x2 + (4 + 3·i)·x + (1 + 5·i) = 0

 94:[−1 −1·i, −3 −2·i;]

/en/higher algebra/Square equation with complex numbers (with simple discriminant), Internal name: ZkvurrZ Generate or Make Task ,

Name:
Var.: 95. Group:           Day/Mo/Year:

To find the roots of the equation:
(1 −1·i)·x2 + (2 + 8·i)·x + (−11 −3·i) = 0

 95:[2 −3·i, 1 −2·i;]

/en/higher algebra/Rational roots of polynomials, Internal name: Zratroot3Z Generate or Make Task ,

Name:
Var.: 96. Group:           Day/Mo/Year:

Find all the roots of the polynomial 9·x3 +12·x2 −80·x +64 and determine their multiplicity.

 96:[[4/3];k=2, −4;k=1]

/en/higher algebra/Inverse polynomial, Internal name: ZobrPolZ Generate or Make Task ,

Name:
Var.: 97. Group:           Day/Mo/Year:

Find (3·x2 +9·x +7)−1 in the ring factor P[x]/(−9·x3 −33·x2 −36·x −11).

 97:[−3·x2 −8·x −3]

/en/higher algebra/Inverse polynomial (with hint), Internal name: ZobrPoltZ Generate or Make Task ,

Name:
Var.: 98. Group:           Day/Mo/Year:

Find (3·x2 −12·x +10)−1 in the ring factor P[x]/(−9·x3 +27·x2 +5·x −27).
Hint: (−9·x3 +27·x2 +5·x −27)·(3·x −3)+(3·x2 −12·x +10)·(9·x2 −8)=1.

 98:[9·x2 −8]

/en/higher algebra/Symmetric polynomials, Internal name: ZsympoliZ Generate or Make Task ,

Name:
Var.: 99. Group:           Day/Mo/Year:

Express the polynomial  −2·x22x33 −2·x12x23 −2·x13x32 −11·x12x22x3 −2·x1x2x33 −2·x1x23x3 −2·x12x33 −11·x12x2x32 −2·x13x22 −2·x23x32 −11·x1x22x32 −2·x13x2x3 via elementary symmetric polynomials.

 99:[−2·s1 s22+2·s12 s3−5·s2 s3]

/en/higher algebra/Factorization, Internal name: ZrazlpoliZ Generate or Make Task ,

Name:
Var.: 100. Group:           Day/Mo/Year:

Find the complex roots of the polynomial x4 −18·x3 +81·x2 +902 and decompose it into a product of irreducible over R polynomials.

 100:[12±6·i, −3±6·i, x2 −24·x +180, x2 +6·x +45]

## 8Matrix.

/en/Matrix/Matrix operations, Internal name: ZmatroperZ Generate or Make Task ,

Name:
Var.: 101. Group:           Day/Mo/Year:

( 3 ·(
 −2
 2
 −1
 2
) + (
 2
 −2
 −1
 −1
) ) ·(
 −1
 1
 2
 −1
)=

101:[(
 12
 −8
 14
 −9
)]

/en/Matrix/Matrix multiplication, Internal name: ZzprmtrZ Generate or Make Task ,

Name:
Var.: 102. Group:           Day/Mo/Year:

(
 −3
 −3
 1
 −3
 −3
 2
) ·(
 3
 −1
 3
 −1
 1
 −2
) =

102:[(
 −6
 0
 −3
 6
 −4
 9
 −11
 5
 −13
)]

/en/Matrix/Cofactor, Internal name: ZalgdopiZ Generate or Make Task ,

Name:
Var.: 103. Group:           Day/Mo/Year:

The matrix A=(
 3
 3
 2
 −2
 2
 2
 2
 −2
 2
) is given.
Find the cofactor of
A32=         , A11=         .

 103:[A32=−10, A11=8]

/en/Matrix/Determinant 3x3, Internal name: ZoprediiiZ Generate or Make Task ,

Name:
Var.: 104. Group:           Day/Mo/Year:

det

 −1
 2
 1
 −2
 1
 0
 0
 −1
 −1

=

 104:[−1]

/en/Matrix/Determinant 4x4, Internal name: ZopredivZ Generate or Make Task ,

Name:
Var.: 105. Group:           Day/Mo/Year:

det

 7
 6
 −4
 10
 5
 5
 −3
 5
 −8
 −7
 5
 −20
 5
 4
 −3
 15

=

 105:

/en/Matrix/Determinant 5x5, Internal name: ZopredvZ Generate or Make Task ,

Name:
Var.: 106. Group:           Day/Mo/Year:

det

 1
 0
 0
 0
 0
 1
 0
 0
 −2
 4
 2
 1
 2
 −1
 −4
 −1
 0
 0
 1
 0
 0
 1
 1
 1
 −4

=

 106:[−4]

/en/Matrix/Inverse 2x2 matrix, Internal name: ZobrattZ Generate or Make Task ,

Name:
Var.: 107. Group:           Day/Mo/Year:

Find the inverse matrix to the matrix
(
 8
 −13
 −19
 31
)

107:[(
 31
 13
 19
 8
);]

/en/Matrix/Inverse 3x3 matrix, Internal name: ZobratZ Generate or Make Task ,

Name:
Var.: 108. Group:           Day/Mo/Year:

Find the inverse matrix to the matrix
(
 0
 1
 1
 −1
 1
 −1
 0
 0
 1
)

108:[(
 1
 −1
 −2
 1
 0
 −1
 0
 0
 1
);]

/en/Matrix/Inverse 4x4 matrix, Internal name: ZobratttZ Generate or Make Task ,

Name:
Var.: 109. Group:           Day/Mo/Year:

Find the inverse matrix to the matrix
(
 1
 0
 0
 0
 0
 1
 −1
 −1
 0
 0
 1
 1
 1
 −1
 1
 2
)

109:[(
 1
 0
 0
 0
 0
 1
 1
 0
 1
 −1
 1
 −1
 −1
 1
 0
 1
)
]

/en/Matrix/Inverse 5x5 matrix, Internal name: ZobratvZ Generate or Make Task ,

Name:
Var.: 110. Group:           Day/Mo/Year:

Find the inverse matrix to the matrix
(
 1
 −1
 0
 0
 1
 0
 1
 0
 0
 −1
 0
 0
 1
 0
 0
 1
 −1
 0
 1
 1
 0
 0
 −1
 −1
 1
)

110:[(
 1
 1
 0
 0
 0
 −1
 1
 1
 1
 1
 0
 0
 1
 0
 0
 −1
 0
 0
 1
 0
 −1
 0
 1
 1
 1
)
]

/en/Matrix/Matrix equation 2x2 (training), Internal name: ZMatrEqqZ Generate or Make Task ,

Name:
Var.: 111. Group:           Day/Mo/Year:

Find A−1 and solve matrix equations
A=(
 1
 −1
 −1
 2
);A·X=(
 2
 −2
 −2
 5
);   Y·A=(
 1
 0
 1
 1
)

111:[A−1=(
 2
 1
 1
 1
), X=(
 2
 1
 0
 3
), Y=(
 2
 1
 3
 2
)]

/en/Matrix/Matrix equation 2x2, Internal name: ZMatrEq2Z Generate or Make Task ,

Name:
Var.: 112. Group:           Day/Mo/Year:

Find A−1 and solve the matrix equation.
A=(
 −1
 −1
 −1
 0
);   A·X = (
 −4
 −2
 −2
 −1
)

112:[A−1=(
 0
 −1
 −1
 1
), X=(
 2
 1
 2
 1
)]

/en/Matrix/Matrix equation 3x3, Internal name: ZMatrEq3Z Generate or Make Task ,

Name:
Var.: 113. Group:           Day/Mo/Year:

Find A−1 and solve the matrix equation.
A=(
 −1
 0
 0
 1
 1
 1
 0
 −1
 0
);   X·A = (
 1
 0
 1
 0
 −1
 0
 −1
 0
 0
)

113:[A−1=(
 −1
 0
 0
 0
 0
 −1
 1
 1
 1
), X=(
 0
 1
 1
 0
 0
 1
 1
 0
 0
)]

/en/Matrix/Matrix equation 4x4 , Internal name: ZMatrEq4Z Generate or Make Task ,

Name:
Var.: 114. Group:           Day/Mo/Year:

Find A−1 and solve the matrix equation.
A=(
 −1
 0
 0
 0
 0
 1
 1
 −1
 0
 −1
 0
 1
 1
 0
 0
 1
);   X·A = (
 −1
 1
 1
 −1
 0
 1
 1
 0
 0
 1
 1
 −1
 0
 0
 1
 1
)

114:[A−1=(
 −1
 0
 0
 0
 1
 0
 −1
 1
 0
 1
 1
 0
 1
 0
 0
 1
), X=(
 1
 1
 0
 0
 1
 1
 0
 1
 0
 1
 0
 0
 1
 1
 1
 1
)]

/en/Matrix/Matrix equation 5x5, Internal name: ZMatrEq5Z Generate or Make Task ,

Name:
Var.: 115. Group:           Day/Mo/Year:

Find A−1 and solve the matrix equation.
A=(
 1
 0
 0
 0
 0
 0
 1
 0
 1
 −1
 1
 0
 1
 0
 0
 0
 0
 1
 1
 0
 0
 −1
 0
 −1
 2
);   A·X = (
 1
 1
 1
 1
 1
 1
 0
 1
 1
 1
 1
 1
 2
 1
 1
 1
 1
 1
 1
 0
 −1
 1
 −1
 −1
 −1
)

115:[A−1=(
 1
 0
 0
 0
 0
 −1
 2
 1
 −1
 1
 −1
 0
 1
 0
 0
 1
 0
 −1
 1
 0
 0
 1
 0
 0
 1
), X=(
 1
 1
 1
 1
 1
 0
 0
 1
 0
 1
 0
 0
 1
 0
 0
 1
 1
 0
 1
 0
 0
 1
 0
 0
 0
)]

/en/Matrix/Matrix equation 6x6, Internal name: ZMatrEq6Z Generate or Make Task ,

Name:
Var.: 116. Group:           Day/Mo/Year:

Find A−1 and solve the matrix equation.
A=(
 −1
 0
 0
 0
 0
 1
 0
 1
 0
 0
 0
 0
 1
 0
 1
 1
 0
 −1
 0
 −1
 −1
 0
 0
 0
 0
 0
 0
 0
 1
 0
 0
 0
 0
 0
 1
 1
);   A·X = (
 −1
 0
 −1
 1
 0
 1
 0
 1
 1
 1
 0
 0
 2
 1
 3
 −1
 0
 0
 −1
 −1
 −2
 −1
 0
 0
 1
 0
 1
 0
 0
 1
 1
 1
 1
 1
 1
 2
)

116:[A−1=(
 −1
 0
 0
 0
 −1
 1
 0
 1
 0
 0
 0
 0
 0
 −1
 0
 −1
 0
 0
 1
 1
 1
 1
 0
 0
 0
 0
 0
 0
 1
 0
 0
 0
 0
 0
 −1
 1
), X=(
 1
 1
 1
 0
 1
 0
 0
 1
 1
 1
 0
 0
 1
 0
 1
 0
 0
 0
 0
 1
 1
 0
 0
 1
 1
 0
 1
 0
 0
 1
 0
 1
 0
 1
 1
 1
)]

## 9Linear algebra.

/en/Linear algebra/Matrix core, Internal name: ZkeriZ Generate or Make Task ,

Name:
Var.: 117. Group:           Day/Mo/Year:

Find a basis of the kernel of a matrix:

 −2
 1
 6
 −5
 −1
 1
 4
 −3
 −2
 1
 6
 −5

 117:[(2 x3−2 x4, −2 x3+1x4, x3,x4)]

Name:
Var.: 118. Group:           Day/Mo/Year:

Find the basis of the orthogonal complement to the set of vectors {(4, −2, 12, −4), (7, −3, 20, −8), (−2, 1, −6, 2)}.

 118:[(−2 x3+2 x4, 2 x3+2x4, x3,x4)]

/en/Linear algebra/Mirror reflection on the plane, Internal name: ZzerkZ Generate or Make Task ,

Name:
Var.: 119. Group:           Day/Mo/Year:

Find the matrix of the linear operator reflecting the mirror plane with respect to the line running along the vector with the coordinates (−4, −5).

119:[[1/41] (
 −9
 40
 40
 9
)]

/en/Linear algebra/Basis selection, Internal name: ZsbazisZ Generate or Make Task ,

Name:
Var.: 120. Group:           Day/Mo/Year:

From the columns of the matrix, select the basis of the space generated by the columns and present the remaining columns as a linear combination of these base columns.

 −5
 −4
 2
 −4
 −5
 2
 −1
 −6
 2
 −1
 1
 2
 2
 1
 2
 −2
 −1
 2
 −2
 −1

120:[(
 1
 0
 −2
 0
 1
 0
 1
 2
 0
 1
 0
 0
 0
 1
 −1
 0
 0
 0
 0
 0
)
]

/en/Linear algebra/The root of the 2x2 matrix in detail, Internal name: ZsqrM2Z Generate or Make Task ,

Name:
Var.: 121. Group:           Day/Mo/Year:

Nothing

121:[(
 1
 −2
 4
 7
)]

/en/Linear algebra/Root of the 2x2 matrix, Internal name: ZsqrM2iZ Generate or Make Task ,

Name:
Var.: 122. Group:           Day/Mo/Year:

Find a matrix A with positive eigenvalues such that A·A=(
 −15
 8
 −48
 25
)

122:[(
 −3
 2
 −12
 7
)]

/en/Linear algebra/Root of the 3x3 matrix detail, Internal name: ZsqrM3Z Generate or Make Task ,

Name:
Var.: 123. Group:           Day/Mo/Year:

Nothing

123:[(
 −1
 −4
 −4
 2
 5
 2
 0
 0
 3
)]

/en/Linear algebra/The square root of a 3x3 matrix, Internal name: ZsqrM3iZ Generate or Make Task ,

Name:
Var.: 124. Group:           Day/Mo/Year:

Find a matrix A with positive eigenvalues such that A·A=(
 14
 5
 10
 0
 9
 0
 −5
 −5
 −1
)

124:[(
 4
 1
 2
 0
 3
 0
 −1
 −1
 1
)]

/en/Linear algebra/Quadratic form 2x2, Internal name: ZkvfiiZ Generate or Make Task ,

Name:
Var.: 125. Group:           Day/Mo/Year:

The quadratic form (25·x2 −36·x·y +40·y2 )/13 is given. Find the orthogonal change of variables, after which it takes the canonical form.

125:[1·x2 +4·y2 , (
 3
 −2
 2
 3
)/√{13}]

/en/Linear algebra/Stupid vector prototype, Internal name: ZobrazVecZ Generate or Make Task ,

Name:
Var.: 126. Group:           Day/Mo/Year:

The linear operator is given by the matrix (
 −3
 3
 −2
 1
).
Find the image of the vector (
 −3
 3
Find the prototype of the vector (
 6
 1

126:[(
 18
 9
), (
 1
 3
)]

/en/Linear algebra/Vector image, Internal name: ZmatrandvectZ Generate or Make Task ,

Name:
Var.: 127. Group:           Day/Mo/Year:

 3.1mm Picture Omitted
Find the matrix B of the linear operator of the mapping vector a1 in e1 and the vector a2 in e2.

127:[A=(
 4
 −5
 −3
 3
), B=(
 −1
 [(−5)/3]
 −1
 [(−4)/3]
)]

/en/Linear algebra/The image of the squiggles, Internal name: ZlinoperiZ Generate or Make Task ,

Name:
Var.: 128. Group:           Day/Mo/Year:

3.1mm
 Picture Omitted
The linear operator is given by the matrix (
 1
 −1
 −1
 2
). Draw the image of a squiggle.

 128:[ 2mm Picture Omitted]

/en/Linear algebra/Search matrix by squiggle, Internal name: ZlinoperiiZ Generate or Make Task ,

Name:
Var.: 129. Group:           Day/Mo/Year:

3.1mm
 Picture Omitted
Linear operator converts a solid to a dotted squiggle. Find the matrix of the operator.

129:[(
 3
 4
 −4
 −3
)]

/en/Linear algebra/Matrix of transition to another basis, Internal name: ZmatrcoordchiZ Generate or Make Task ,

Name:
Var.: 130. Group:           Day/Mo/Year:

 3.1mm Picture Omitted
Find the matrix of translation of coordinates from the coordinates in the basis a1, a2 to the coordinates in the basis e1, e2.

130:[(
 1
 1
 2
 1
)]

/en/Linear algebra/Matrices of transition to other bases, Internal name: ZmatrcoordchiiZ Generate or Make Task ,

Name:
Var.: 131. Group:           Day/Mo/Year:

3.1mm
 Picture Omitted
Find all kinds of matrix translation of coordinates: Tstd← a = ( ) , Ta← std = ( ) ,
Tstd← b = ( ) , Tb← std = ( ) ,
Tb← a = ( ) , Ta← b = ( )
Find the coordinates of the vector c in different bases:
c=( )std, c=( )a, c=( )b

131:[Tstd← a=(
 −1
 −2
 −2
 1
), Ta← std=(
 [(−1)/5]
 [(−2)/5]
 [(−2)/5]
 [1/5]
), Tstd← b=(
 −8
 −1
 −6
 8
), Tb← std=(
 [(−4)/35]
 [(−1)/70]
 [(−3)/35]
 [4/35]
) Tb← a=(
 [1/7]
 [3/14]
 [(−1)/7]
 [2/7]
), Ta← b=(
 4
 −3
 2
 2
), c=(
 4
 1
)std, c=(
 [(−6)/5]
 [(−7)/5]
)a, c=(
 [(−33)/70]
 [(−8)/35]
)b]

/en/Linear algebra/Matrix of rotation in space (simple), Internal name: ZmatrpoviiZ Generate or Make Task ,

Name:
Var.: 132. Group:           Day/Mo/Year:

1. Find the matrix defines the rotation clockwiseby 90 degrees around the vector (0, 1, 0). (basis orthonormal and negative oriented)
2. Find the matrix defines the rotation counterclockwiseby 90 degrees around the vector (0, 0, 1). (the basis is orthonormal and positively oriented)
3. Find the matrix defines the rotation clockwiseby 90 degrees around the vector (1, 0, 0). (the basis is orthonormal and positively oriented)

132:[1. (
 0
 0
 −1
 0
 1
 0
 1
 0
 0
) 2. (
 0
 1
 0
 −1
 0
 0
 0
 0
 1
) 3. (
 1
 0
 0
 0
 0
 −1
 0
 1
 0
)]

/en/Linear algebra/Matrix of rotation in space, Internal name: ZmatrpoviZ Generate or Make Task ,

Name:
Var.: 133. Group:           Day/Mo/Year:

Find the matrix defines the rotation of the counterclockwise 90 degrees around the vector (1; 4; −8).

133:[[1/81]·(
 1
 −68
 −44
 76
 16
 −23
 28
 −41
 64
)]

/en/Linear algebra/Eigenvector, Internal name: ZsobvektZ Generate or Make Task ,

Name:
Var.: 134. Group:           Day/Mo/Year:

Find the eigenvalues and eigenvectors of the matrix (
 20
 6
 −72
 −22
)

134:[2(
 1
 −3
), −4(
 −1
 4
)]

/en/Linear algebra/Eigenvalue vector, Internal name: ZsobvektmZ Generate or Make Task ,

Name:
Var.: 135. Group:           Day/Mo/Year:

A=(
 −24
 14
 −42
 25
). Which eigenvalue corresponds to the eigenvector (
 −1
 −2
)?

 135:

/en/Linear algebra/Gram Schmidt orthogonalization, Internal name: ZOGSHZ Generate or Make Task ,

Name:
Var.: 136. Group:           Day/Mo/Year:

Apply the method of orthogonalization Gram-Schmidt to the vectors A=(−4,−2,1,0), B=(−7,−3,8,1), C=(2,2,−9,128).

 136:[A=(−4,−2,1,0), B=(1,1,6,1), C=(−4,−2,−20,126); B:=B−2·A; C: = C + A−2·B]

/en/Linear algebra/Axis of rotation, Internal name: ZortMatrZ Generate or Make Task ,

Name:
Var.: 137. Group:           Day/Mo/Year:

The orthogonal matrix is given. Find the axis of rotation and the cosine of the angle of rotation.
[1/13]·(
 12
 −3
 4
 3
 −4
 −12
 4
 12
 −3
)

 137:[(4,0,1), cosα = [(−4)/13]=−0.308]

/en/Linear algebra/Jordan canonical form, Internal name: ZJordZ Generate or Make Task ,

Name:
Var.: 138. Group:           Day/Mo/Year:

 6
 −2
 −1
 −2
 1
 0
 4
 1
 0
 0
 0
 0
 4
 0
 0
 4
 −4
 −3
 0
 2
 0
 0
 2
 0
 2

(Hint: eigenvalues 4 and 2)

 138:

/en/Linear algebra/Polar decomposition, Internal name: ZpolrazZ Generate or Make Task ,

Name:
Var.: 139. Group:           Day/Mo/Year:

Present the matrix A=(
 −99
 −243
 27
 −261
) as a product of A=B·C, where B - symmetric matrix by positive eigenvalues and C - orthogonal matrix.

139:[B=(
 225
 135
 135
 225
), C=[1/5]·(
 −4
 −3
 3
 −4
)]

/en/Linear algebra/Polar decomposition (with check), Internal name: ZpolrazvZ Generate or Make Task ,

Name:
Var.: 140. Group:           Day/Mo/Year:

Nothing

140:[B=(
 65
 45
 45
 185
), C=[1/5]·(
 3
 4
 −4
 3
)]

Name:
Var.: 141. Group:           Day/Mo/Year:

The line on the plane is given by the equation
 41·x2 +37·y2 −410·x +148·y −344=0.
Bring it to the canonical form, draw "old" and the canonical coordinate system and the line. Calculate the coordinates of the new center and focuses in the coordinate system OXY.

 141:[[((x −5)2)/37]+[((y +2)2)/41]=1 , F1(5,−4), F2(5,0)]

Name:
Var.: 142. Group:           Day/Mo/Year:

The equation of the line is given:
52x2+73y2−72xy=2500.
Find the orthogonal change of variables

 x=
 (        ) (        ) ·x1 + (        ) (        ) ·y1
 y=
 (        ) (        ) ·x1 + (        ) (        ) ·y1
after which the equation becomes the canonical equation of the quadric:
 ⎛⎝ x1 ⎞⎠ 2 ⎛⎝ y1 ⎞⎠ 2 =1
Draw this line. Draw asymptotes (if they exist).
4mm
 Picture Omitted

 142:[ 3.1mm Picture Omitted In replacement there is −3 and 4. Equation: 4x2+1y2=100 or (x/5)2+(y/ 10)2=1 ]

Name:
Var.: 143. Group:           Day/Mo/Year:

Find the radius of the circle: x2+y2−6·x−27=0.

 143:

/en/Linear algebra/Linear combination, Internal name: ZlinCombZ Generate or Make Task ,

Name:
Var.: 144. Group:           Day/Mo/Year:

Represent vector c = (−2;22;−4) as a linear combination of vectors a = (1;28;−4) and b = (−1;−41;6).

 144:[→c = −8 ·→a−6 ·→b]

/en/Linear algebra/The basis of the intersection, Internal name: ZbazPerZ Generate or Make Task ,

Name:
Var.: 145. Group:           Day/Mo/Year:

Find a basis of the intersection:
〈(
 3
 −4
 −5
) , (
 2
 −3
 −4
) , (
 6
 −9
 −12
) 〉∩〈(
 −5
 9
 14
) , (
 7
 −11
 −18
) 〉

 145:[λ·(−1, 2, 3)]

/en/Linear algebra/Regression line, Internal name: ZrglineiZ Generate or Make Task ,

Name:
Var.: 146. Group:           Day/Mo/Year:

Find the equation of the regression line and draw it.
 3.1mm Picture Omitted

Answer: y=(         )·x + (         )

 146:[ 2mm Picture Omitted y=[19/86]·x +[99/86] ≈ 0.22·x +1.15 ]

## 10Mathematical analysis.

/en/Mathematical analysis/Schedule shift, Internal name: ZgipZ Generate or Make Task ,

Name:
Var.: 147. Group:           Day/Mo/Year:

Draw a curve defined by the equation:
 y= 1 x+5 +7 147:[ 2mm Picture Omitted]

/en/Mathematical analysis/The limit of a fraction, Internal name: ZlimPolZ Generate or Make Task ,

Name:
Var.: 148. Group:           Day/Mo/Year:

limx→ −9 [(x2 +8·x −9)/(x2 +7·x −18)]=

 148:[[10/11] ≈ 0.909]

/en/Mathematical analysis/Limit with fraction to infinity, Internal name: ZlimPoliZ Generate or Make Task ,

Name:
Var.: 149. Group:           Day/Mo/Year:

limx→ ∞ [(−9·x5 −8·x4 −3)/(3·x6 −9·x2 +6)]=

 149:

/en/Mathematical analysis/Limit with roots, Internal name: ZlimiZ Generate or Make Task ,

Name:
Var.: 150. Group:           Day/Mo/Year:

 lim x→ ∞ ( √ 16·x2+58·x+8 − √ 16·x2+2·x+5 )=

 150:

/en/Mathematical analysis/The second remarkable limit, Internal name: ZlimiiZ Generate or Make Task ,

Name:
Var.: 151. Group:           Day/Mo/Year:

 lim x→ 2 ( (x2 −19·x + 35)[1/(x2 −5·x+6)] ⎞⎠ =

 151:[e15]

/en/Mathematical analysis/Asymptotes, Internal name: ZasimptotZ Generate or Make Task ,

Name:
Var.: 152. Group:           Day/Mo/Year:

Find the asymptotes of the graph of a function. Represent its behavior near the asymptotes
 y= ln(x+2) +5x2 −2x −9 x +1

 152:[ y = 5 x −7 , x = −2, f(−2 + 0 ) = +∞,   x = −1,  f(−1 −0 ) = +∞,  f(−1 +0 ) = −∞]

/en/Mathematical analysis/Tangent, Internal name: ZkasZ Generate or Make Task ,

Name:
Var.: 153. Group:           Day/Mo/Year:

Find the coordinates of the point of intersection of two tangents to the graph of the function x2 −7·x +4. The first tangent is drawn at the point with x=3, and the second at the point with x=1.

 153:[(2;−7)]

/en/Mathematical analysis/Tangents (details), Internal name: ZkasiZ Generate or Make Task ,

Name:
Var.: 154. Group:           Day/Mo/Year:

Two tangents are drawn to the graph of the function x2 −6·x +4. The first tangent is drawn at the point with x=3, and the second at the point with x=1. Find: the equations of these tangents and the point of intersection of these tangents with each other.

 154:[(2;−5)]

/en/Mathematical analysis/Min max on segment, Internal name: ZminmaxZ Generate or Make Task ,

Name:
Var.: 155. Group:           Day/Mo/Year:

Find the largest and smallest value of the function
y=x3+(−6)·x2 + (9)·x +(1) on the segment 1 ≤ x ≤ 5.

 155:[(3, 1), (5, 21)]

/en/Mathematical analysis/Extreme and inflection points, Internal name: ZdotextZ Generate or Make Task ,

Name:
Var.: 156. Group:           Day/Mo/Year:

f(x)=x3 −12·x2 +21·x −10. Find the maximum point, minimum point and inflection point.

 156:[Max=1, Min=7, inflection point=4]

/en/Mathematical analysis/Minimum and maximum in the area, Internal name: ZminmaxiiZ Generate or Make Task ,

Name:
Var.: 157. Group:           Day/Mo/Year:

Find the largest and smallest value of the function of two variables z=x2+8·x+y2−4·y in a triangle with vertices A=(−7, 1), B=(−3, 1) and C=(−3, 9).

 157:[(−4, 2, −20), (−3, 9, 30)]

/en/Mathematical analysis/Derivative, Internal name: ZdiferZ Generate or Make Task ,

Name:
Var.: 158. Group:           Day/Mo/Year:

([(cos(√x))/((x3)−6·x)])′=

 158:[]

/en/Mathematical analysis/The value of the derivative with a fraction, Internal name: ZproizdrZ Generate or Make Task ,

Name:
Var.: 159. Group:           Day/Mo/Year:

 f(x)= 7·x2 +3·x −9 x −1 .
Find f′(2).

 159:

/en/Mathematical analysis/The value derived from the root, Internal name: ZproizsqrtZ Generate or Make Task ,

Name:
Var.: 160. Group:           Day/Mo/Year:

f(x)=√{−4·x2 −20·x −20} . Find f′(−3).

 160:

/en/Mathematical analysis/Value of the second order derivative, Internal name: ZproizdvaZ Generate or Make Task ,

Name:
Var.: 161. Group:           Day/Mo/Year:

f(x)=x3 +2·x2 +3·x +3. To find f"(5).

 161:

/en/Mathematical analysis/Partial derivative, Internal name: ZpatdefZ Generate or Make Task ,

Name:
Var.: 162. Group:           Day/Mo/Year:

Calculate z′x.
z=[(e(x−5+y)sin(−5·x+y))/(x−5−3·y3)]

 162:[]

/en/Mathematical analysis/Simple private derivative, Internal name: ZpatdefiZ Generate or Make Task ,

Name:
Var.: 163. Group:           Day/Mo/Year:

The function of two variables is given: 2·x ·y+9·x+4·y. Find the value of the partial derivative: f′y(−5,−4).

 163:[−6]

/en/Mathematical analysis/Plotting function, Internal name: ZpiciZ Generate or Make Task ,

Name:
Var.: 164. Group:           Day/Mo/Year:

Draw the graph of function y=(−1) ·(x −2 )2 ·(x+1), specify the points of extremum and inflection points.
3.1mm
 Picture Omitted

 164:[ 2.5mm Picture Omitted]

## 11Integrals.

/en/Integrals/Simple definite integral, Internal name: ZsIntZ Generate or Make Task ,

Name:
Var.: 165. Group:           Day/Mo/Year:

−21(−9·x2 −8·x +3)  dx=

 165:[−6]

/en/Integrals/Rational integral 1, Internal name: Zinti1Z Generate or Make Task ,

Name:
Var.: 166. Group:           Day/Mo/Year:

∫[(− x +7)/(x2 +6·x +5)]  dx =

 166:[2ln(x+1)−3ln(x+5)]

/en/Integrals/Rational integral 2, Internal name: Zinti2Z Generate or Make Task ,

Name:
Var.: 167. Group:           Day/Mo/Year:

∫[(4·x3 −7·x2 −26·x +28)/(x2 − x −6)]  dx =

 167:[2x2−3x−1ln(x−3)−4ln(x+2)]

/en/Integrals/Rational integral 3, Internal name: Zinti3Z Generate or Make Task ,

Name:
Var.: 168. Group:           Day/Mo/Year:

∫[(−7·x2 −14·x +8)/((x+2)·(x2−4))]  dx =

 168:[−4 ln(x+2)+2/(x+2)−3ln(x−2)]

/en/Integrals/Rational integral 4, Internal name: Zinti4Z Generate or Make Task ,

Name:
Var.: 169. Group:           Day/Mo/Year:

∫[(−4·x −26)/(x2 +14·x +50)]  dx =

 169:[−2ln(x2 +14·x +50)+2arctg(x +7)]

/en/Integrals/Rational integral 5, Internal name: Zinti5Z Generate or Make Task ,

Name:
Var.: 170. Group:           Day/Mo/Year:

∫[(4·x −4)/(x2 −8·x +32)]  dx =

 170:[2ln(x2 −8·x +32)+3arctg((x −4)/4)]

/en/Integrals/Rational integral 6, Internal name: Zinti6Z Generate or Make Task ,

Name:
Var.: 171. Group:           Day/Mo/Year:

∫[(−4·x3 −27·x2 −52·x −14)/(x2 +6·x +10)]  dx =

 171:[−2·x2 −3·x +3ln(x2 +6·x +10)−2arctg(x +3)]

/en/Integrals/Integral by parts, Internal name: ZintiiZ Generate or Make Task ,

Name:
Var.: 172. Group:           Day/Mo/Year:

∫(−4·x+1)·e(2·x+5)  dx =

 172:[(−2x+[1/2]) e(2x+5)-1 e(2x+5 )]

/en/Integrals/Definite integral (simple), Internal name: ZointiZ Generate or Make Task ,

Name:
Var.: 173. Group:           Day/Mo/Year:

0π/66·[sinx/(cos3x)]   dx =

 173:

/en/Integrals/Integral square, Internal name: ZplintZ Generate or Make Task ,

Name:
Var.: 174. Group:           Day/Mo/Year:

Find the area of the figure bounded by lines y=2·x2 +12·x +15 and y=−2·x2 −8·x −1.

 174:

/en/Integrals/a figure of rotation, Internal name: ZobTelVrashZ Generate or Make Task ,

Name:
Var.: 175. Group:           Day/Mo/Year:

Find the volume of the body obtained by rotating around the axis OX region, bounded by straight lines y = [5/4]·x +[49/4], y = [(−1)/8]·x +[43/8], x=−9, x=3.

 175:[891·π]

/en/Integrals/Volume of the figure of rotation, Internal name: ZobTelVrashiZ Generate or Make Task ,

Name:
Var.: 176. Group:           Day/Mo/Year:

Find the volume of the body obtained by rotating around the axis OX region, a limited line of y=√{77+2·x} and square x=0, x=4, y=2·x + 5.

 176:[136·π]

/en/Integrals/Double integral, Internal name: ZdvintZ Generate or Make Task ,

Name:
Var.: 177. Group:           Day/Mo/Year:

Find the weight of the triangular plate, the coordinates of the vertices of which (3;0), (3;6), (0;0) and the relative weight of the substance is given by the function ρ = 9·y −8.

 177:

/en/Integrals/Triple integral, Internal name: ZtrintgZ Generate or Make Task ,

Name:
Var.: 178. Group:           Day/Mo/Year:

Find the weight of the air enclosed inside the pyramid ABCD. Vertex coordinates: A=(0;0;0), B=(1; 1; 12), C=(1; 1;0) and D=(0; 1;0). The relative weight of air is given by the formula ρ = 3·z +4.

 178:

/en/Integrals/Triple integral (complex), Internal name: ZtrintZ Generate or Make Task ,

Name:
Var.: 179. Group:           Day/Mo/Year:

Find the weight of the air enclosed inside the pyramid ABCD. Vertex coordinates: A=(0;0;0), B=(3; 1; 16), C=(3; 1;0) and D=(3;0 ;0). The relative weight of air at the top B is 4, at the vertex C is 3 and the axis OZ is directed upwards.

 179:

/en/Integrals/Curvilinear, Internal name: ZkrintZ Generate or Make Task ,

Name:
Var.: 180. Group:           Day/Mo/Year:

Make sure that the curvilinear integral
 ⌠⌡ (−20·x4·y5+15·x4)  dx+(−20·x5·y4)  dy
is independent of the integration path and calculate it from A=(−1,6) to B=(1,−6).

 180:

/en/Integrals/Simple curvilinear integral, Internal name: ZkrvintiZ Generate or Make Task ,

Name:
Var.: 181. Group:           Day/Mo/Year:

Calculate the curvilinear integral of the second kind from the vector field (y,x) in a straight line from the point (−5,4) to a point (0,9).

 181:

/en/Integrals/In polar coordinates, Internal name: ZpolkoorZ Generate or Make Task ,

Name:
Var.: 182. Group:           Day/Mo/Year:

Calculate the double integral ∫∫D (x2+y2+1)−2 dx dy with area D bounded by a circle of radius r1=2 and r2=5 with center at the origin and rays emerging from the origin at angles φ1=1.6 and φ2=2.6.

 182:[0.0807692]

/en/Integrals/Area of integration, Internal name: ZpredintZ Generate or Make Task ,

Name:
Var.: 183. Group:           Day/Mo/Year:

The area of integration in the double integral is bounded by lines: y + x2 − 8·x +28 = 0;   4·y +12·x=0;   2·y+16·x=0. Set integration limits

 ⌠⌡ 0 dx ⌠⌡ f(x,y) dy + ⌠⌡ 4 dx ⌠⌡ f(x,y) dy

 183:[ ∫02 dx ∫−8·x[(−12)/4]·x f(x,y) dy + ∫24 dx ∫−x2 + 8·x −28[(−12)/4]·x f(x,y) dy ]

## 12Approximate calculation.

/en/Approximate calculation/Interpolation, Internal name: ZinterpZ Generate or Make Task ,

Name:
Var.: 184. Group:           Day/Mo/Year:

Find a polynomial whose graph passes through (1;2), (2;10) and (3;22).

 184:[2·x2 +2·x −2]

/en/Approximate calculation/Roots of numbers, Internal name: ZpriblNZ Generate or Make Task ,

Name:
Var.: 185. Group:           Day/Mo/Year:

Calculate √{5.7} approximately, by Newton, with an accuracy of three decimal places. (Start approximation with x=1);

 185:[x2=3.35, x3=2.52575, x4=2.39125, ]

/en/Approximate calculation/The root of the polynomial is a Simple option, Internal name: ZnrootiZ Generate or Make Task ,

Name:
Var.: 186. Group:           Day/Mo/Year:

Find the root of the polynomial x3 −7·x +1 by Newton method. Make three iterations starting at x1=3.
 x 3 y

186:[
 x 3 2.65 2.575 y 7 1.06 0.049
]

/en/Approximate calculation/The root of the polynomial is a Complex variant, Internal name: ZnrootiiZ Generate or Make Task ,

Name:
Var.: 187. Group:           Day/Mo/Year:

The function y=x3 −8·x +15 is given. Find the maximum and minimum points. Find the intervals of increase and decrease. Find the root of the polynomial x3 −8·x +15 by Newton method with accuracy of three decimal places.

 187:[−3.504 ]

/en/Approximate calculation/Euler method, Internal name: ZdifureilerZ Generate or Make Task ,

Name:
Var.: 188. Group:           Day/Mo/Year:

This is the direction (more precisely, the coefficients of the slope) of the river in 4 points: k(0,1)=−0.15, k(0,2)=−0.05, k(1,1)=0.34, k(1,2)=0.16. The pilot starts from the point (0,0.56). Find (modified by the Euler method with conversion to within three decimal places) the finish point (2,?).
 11mm Picture Omitted

 188:[(2,1.468)]

## 13Informatics.

/en/Informatics/Transfer from one number system to another, Internal name: ZsistScislZ Generate or Make Task ,

Name:
Var.: 189. Group:           Day/Mo/Year:

In 18-th number system the number is 7H8. What is it equal to in 7-th system? What is equal in decimal?

 189:[10346, 2582]

/en/Informatics/Opposite integers, Internal name: ZprotivIntZ Generate or Make Task ,

Name:
Var.: 190. Group:           Day/Mo/Year:

Eight-bit integers are used. The additional code is 21. Find the hexadecimal entry of the additional code of the opposite number.

 190:[DF]

/en/Informatics/Opposite integers (from decimal), Internal name: ZprotivIntiZ Generate or Make Task ,

Name:
Var.: 191. Group:           Day/Mo/Year:

Eight-bit integers are used. Find the hexadecimal entry of the additional code for the number −53.

 191:[CB]

/en/Informatics/Fractional binary to decimal, Internal name: ZsistScislDrZ Generate or Make Task ,

Name:
Var.: 192. Group:           Day/Mo/Year:

In the binary system the number is 1101.0111
What is it in decimal?

 192:[13.4375=1101.0111]

/en/Informatics/Fractional decimal to binary, Internal name: ZsistScislDriZ Generate or Make Task ,

Name:
Var.: 193. Group:           Day/Mo/Year:

In the decimal system the number is 11.3125
What is it in binary?

 193:[11.3125=1011.0101]

/en/Informatics/Machine representation of half-precision floating point numbers, Internal name: ZIEEEhpbinZ Generate or Make Task ,

Name:
Var.: 194. Group:           Day/Mo/Year:

Binary16 (IEEE 754 half-precision binary floating-point format) is used. Left bit - sign, then five bits - order with shift 15 bit and 10 mantissa bit without one).
What is 4300 in the usual decimal notation?

 194:[3.5 = 0/10000/1 1 0 0 0 0 0 0 0 0]

/en/Informatics/Machine representation of half-precision floating point numbers but Vice versa, Internal name: ZIEEEhpbinRZ Generate or Make Task ,

Name:
Var.: 195. Group:           Day/Mo/Year:

Binary16 (IEEE 754 half-precision binary floating-point format) is used. Left bit - sign, then five bits - order with shift 15 bit and 10 mantissa bit without one).
What number corresponds to the usual number 6.5? (Write the answer in hexadecimal).

 195:[4680 = 0/10001/1 0 1 0 0 0 0 0 0 0]

/en/Informatics/Machine representation of half-precision numbers with a floating point in both directions, Internal name: ZIEEEhpbinRRZ Generate or Make Task ,

Name:
Var.: 196. Group:           Day/Mo/Year:

Binary16 (IEEE 754 half-precision binary floating-point format) is used. What number corresponds to the usual number 2.375? (Write the answer in hexadecimal).
What is 3A80 in the usual decimal notation?

 196:[40C0, 0.8125]

/en/Informatics/Machine representation of single precision floating point numbers, Internal name: ZIEEEhpbiniZ Generate or Make Task ,

Name:
Var.: 197. Group:           Day/Mo/Year:

Binary32 (IEEE 754 single-precision binary floating-point format) is used. Left a bit - sign, then eight bits - right shift 127 and 23 significand bits, without units).
What is C0B00000 in the usual decimal notation?

 197:[−5.5 = 1/10000001/0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

/en/Informatics/Entropy of two balls, Internal name: ZentropiZ Generate or Make Task ,

Name:
Var.: 198. Group:           Day/Mo/Year:

In the box is 7 white and 3 black balls. Calculate the uncertainty of the experience of extracting one ball

 198:[0.881]

/en/Informatics/Conditional entropy of two balls, Internal name: ZuslentriZ Generate or Make Task ,

Name:
Var.: 199. Group:           Day/Mo/Year:

In the box is 4 white and 3 black balls. A complex experience consists in the sequential execution of two simple experiments, each of which consists in extracting the ball without returning. Calculate the entropy of the first experience, conditional entropy and entropy of complex experience.

 199:[0.985, 1, 0.918, 0.965, 1.95]

/en/Informatics/Prefix code, Internal name: ZprefCodZ Generate or Make Task ,

Name:
Var.: 200. Group:           Day/Mo/Year:

The code is given by table a-100 b-11 c-101 d-0. Decode
 111011101000000110011010100100100

/en/Informatics/File compression, Internal name: ZcodAndCmprsZ Generate or Make Task ,

Name:
Var.: 201. Group:           Day/Mo/Year:

Answer: Table: a[        ] b[        ] c[        ] d[        ]
Result after compression:
After compression the file contains             bit.

 201:[ Table: a-010 b-011 c-00 d-1, compression Result: 0100110011100010100011100111010100100, after compression 37 bits.]

/en/Informatics/LZW compression, Internal name: ZlzwiZ Generate or Make Task ,

Name:
Var.: 202. Group:           Day/Mo/Year:

Initial dictionary: (0-a), (1-b), (2-c), (3-d).
Compress aababbccbbaa according to the algorithm LZW.

 202:[Compressed file: 0 0 1 5 1 2 2 1 6 0. Dictionary: (0-a) (1-b) (2-c) (3-d) (4-aa) (5-ab) (6-ba) (7-abb) (8-bc) (9-cc) (10-cb) (11-bb) (12-baa). ]

/en/Informatics/LZW compression with the calculation of the coefficient, Internal name: ZlzwivZ Generate or Make Task ,

Name:
Var.: 203. Group:           Day/Mo/Year:

Compress aababaaaaabaaaaaaabaaaaacbaaa by LZW algorithm and calculate compression ratio.
Dictionary: (0-00-a) (1-01-b) (2-10-c)
(3-11-        ) (4-100-        ) (5-101-        ) (6-110-        )
(7-111-        ) (8-1000-        ) (9-1001-        )
(10-1010-        ) (11-1011-        ) (12-1100-        )
(13-1101-        ) (14-1110-        ) (15-1111-        )
After compression (in decimal):
(in binary):
compression ratio:

 203:[Compressed file: 0 0 1 4 3 7 5 7 8 10 0 2 9 0 or 00 00 001 100 011 111 0101 0111 1000 1010 0000 0010 1001 0000. K=58/48 ≈ 1.21. Dictionary: (0-00-a) (1-01-b) (2-10-c) (3-11-aa) (4-100-ab) (5-101-ba) (6-110-aba) (7-111-aaa) (8-1000-aaab) (9-1001-baa) (10-1010-aaaa) (11-1011-aaaba) (12-1100-aaaaa) (13-1101-ac) (14-1110-cb) (15-1111-baaa). ]

/en/Informatics/LZW unclench without conflict, Internal name: ZlzwiiZ Generate or Make Task ,

Name:
Var.: 204. Group:           Day/Mo/Year:

Initial dictionary: (0-a), (1-b), (2-c), (3-d).
Unzip 0 1 2 1 4 0 0 6 1 1 using LZW algorithm.

 204:[Source file: abcbabaacbbb. Dictionary: (0-a) (1-b) (2-c) (3-d) (4-ab) (5-bc) (6-cb) (7-ba) (8-aba) (9-aa) (10-ac) (11-cbb) (12-bb). ]

/en/Informatics/LZW unclenching with conflict, Internal name: ZlzwiiiZ Generate or Make Task ,

Name:
Var.: 205. Group:           Day/Mo/Year:

Initial dictionary: (0-a), (1-b), (2-c), (3-d).
Unzip 1 1 0 5 1 2 4 10 9 2 using LZW algorithm.
(The problem has a solution. Even if you think that it does not).

 205:[Source file: bbababcbbbbbcbc. Dictionary: (0-a) (1-b) (2-c) (3-d) (4-bb) (5-ba) (6-ab) (7-bab) (8-bc) (9-cb) (10-bbb) (11-bbbc) (12-cbc). ]

/en/Informatics/Huffman coding, Internal name: ZcodAndCmprsiZ Generate or Make Task ,

Name:
Var.: 206. Group:           Day/Mo/Year:

Huffman code is used. The frequency of characters a-47.6% b-9.5% c-14.3% d-28.6%. Find the average length of the code, the average amount of information per sign of the primary alphabet, the relative redundancy of the code and encode aacab
Answer: Table: a[        ] b[        ] c[        ] d[        ]
Average length:
Average amount of information per sign:
The relative redundancy:
The result of the Coding:

 206:[Table: a-1 b-011 c-010 d-00, average code length: 1.76, relative redundancy: 0.0068, I: 1.75013, encoding result: 110101011.]

/en/Informatics/Hamming coding, Internal name: ZkodHemmZ Generate or Make Task ,

Name:
Var.: 207. Group:           Day/Mo/Year:

Code F3 with Hamming code with 8 information and 4 verification bits.

 207:[2E3]

/en/Informatics/Decoding the Hamming, Internal name: ZkodHemmiZ Generate or Make Task ,

Name:
Var.: 208. Group:           Day/Mo/Year:

Received through communication channel BB4. Select the information message and write in the 16th system. (Used the Hamming code with 8 data bits and 4 verification bits)

 208:[F4]

/en/Informatics/Encoding and Decoding in the Hamming, Internal name: ZkodHemmiiZ Generate or Make Task ,

Name:
Var.: 209. Group:           Day/Mo/Year:

Use the Hamming code with 8 data bits and 4 verification bits.
Find the code corresponding to the message 91.
Find the message corresponding to the code BAB (there is an error in the code).

 209:[231, DF]

## 14Discrete mathematics.

/en/Discrete mathematics/Automata and words, Internal name: ZatmAcceptZ Generate or Make Task , Moodle computable

Name:
Var.: 210. Group:           Day/Mo/Year:

Given the state machine
 1
 2
 3
 4
 5
 a
 1
 2
 1
 4
 5
 b
 1
 5
 2
 1
 4

with initial state: 3 and acceptable sates: 2, 3, 4, 5. Select accepted word
1) bbbaaab
2) baaabb
3) abaaab
4) abbbbb
5) ababba

 210:

/en/Discrete mathematics/Regular expressions and words, Internal name: ZregExAcceptZ Generate or Make Task , Moodle computable

Name:
Var.: 211. Group:           Day/Mo/Year:

A regular expression is given
 (ab∪ba)*(a∪b(b∪λ))
What word corresponds to it?
1) bbaaba
2) abbaaba
3) babaaa
4) bababa
5) babbab

 211:

/en/Discrete mathematics/Regular expressions and automata, Internal name: ZautomatZ Generate or Make Task , Construction of the machine on a regular expression. Random number of vertices

Name:
Var.: 212. Group:           Day/Mo/Year:

Find the minimum finite deterministic automaton that recognizes the language (ba)*bb.

212:[ (
 1
 2
 3
 4
 a
 4
 4
 2
 4
 b
 4
 3
 1
 4
), In: 2, Out: 1. ]

/en/Discrete mathematics/Regular expressions and automata (3 vertices), Internal name: ZautomatiZ Generate or Make Task , Building an automaton for the regular expression

Name:
Var.: 213. Group:           Day/Mo/Year:

Find the minimum finite deterministic automaton that recognizes the language (ba)*b.

213:[ (
 1
 2
 3
 a
 2
 3
 3
 b
 3
 1
 3
), In: 2, Out: 1. ]

/en/Discrete mathematics/Regular expressions and automata (4 vertices), Internal name: ZautomatiiZ Generate or Make Task , Building an automaton for the regular expression

Name:
Var.: 214. Group:           Day/Mo/Year:

Find the minimum finite deterministic automaton that recognizes the language a(a∪λ)∪b∪λ.

214:[ (
 1
 2
 3
 4
 a
 1
 4
 1
 3
 b
 1
 3
 1
 1
), In: 2, Out: 2, 3, 4. ]

/en/Discrete mathematics/Regular expressions and automata (5 vertices), Internal name: ZautomatiiiZ Generate or Make Task , Building an automaton for the regular expression

Name:
Var.: 215. Group:           Day/Mo/Year:

Find the minimum finite deterministic automaton that recognizes the language a(bb∪a∪λ)∪λ.

215:[ (
 1
 2
 3
 4
 5
 a
 1
 4
 1
 3
 1
 b
 1
 1
 1
 5
 3
), In: 2, Out: 2, 3, 4. ]

/en/Discrete mathematics/Regular expressions and automata (6 vertices), Internal name: ZautomatiiiiZ Generate or Make Task , Building an automaton for the regular expression

Name:
Var.: 216. Group:           Day/Mo/Year:

Find the minimum finite deterministic automaton that recognizes the language ba*∪a(a∪b)(a∪b)∪λ.

216:[ (
 1
 2
 3
 4
 5
 6
 a
 2
 5
 1
 4
 5
 3
 b
 2
 5
 1
 5
 5
 4
), In: 6, Out: 2, 4, 6. ]

/en/Discrete mathematics/Regular expressions and automata (7 vertices), Internal name: ZautomatiiiiiZ Generate or Make Task , Building an automaton for the regular expression

Name:
Var.: 217. Group:           Day/Mo/Year:

Find the minimum finite deterministic automaton that recognizes the language ba*∪ab*(ab*ab*a∪λ).

217:[ (
 1
 2
 3
 4
 5
 6
 7
 a
 2
 4
 3
 6
 5
 5
 1
 b
 1
 2
 5
 4
 5
 5
 3
), In: 7, Out: 1, 3, 6. ]

/en/Discrete mathematics/Minimization of the state machine, Internal name: ZminautZ Generate or Make Task , Six peaks. After minimization - four

Name:
Var.: 218. Group:           Day/Mo/Year:

To minimize the state machine
 1
 2
 3
 4
 5
 6
 a
 4
 6
 3
 6
 2
 3
 b
 5
 6
 6
 6
 3
 3
,
In: 1, Out: 2, 4, 5.

218:[
 (1)
 (24)
 (36)
 (5)
 a
 (24)
 (36)
 (36)
 (24)
 b
 (5)
 (36)
 (36)
 (36)
, In: (1), Out: (24), (5). ]

/en/Discrete mathematics/Minimization of the state machine (large), Internal name: ZminautiZ Generate or Make Task , Nine peaks. After minimization - five

Name:
Var.: 219. Group:           Day/Mo/Year:

To minimize the state machine
 1
 2
 3
 4
 5
 6
 7
 8
 9
 a
 2
 8
 3
 5
 6
 3
 5
 3
 5
 b
 4
 7
 3
 9
 6
 6
 8
 6
 5
,
In: 1, Out: 2, 4, 5, 7, 8, 9.

219:[
 (1)
 (24)
 (36)
 (58)
 (79)
 a
 (24)
 (58)
 (36)
 (36)
 (58)
 b
 (24)
 (79)
 (36)
 (36)
 (58)
, In: (1), Out: (24), (58), (79). ]

/en/Discrete mathematics/Pparsing grammar LL(1), Internal name: ZgrammLLiZ Generate or Make Task , Parsing LL(1) grammar

Name:
Var.: 220. Group:           Day/Mo/Year:

Given LL (1) grammar:
 FIRST S AB S ac A dS A cb B SS B bb

Define FIRST and construct a syntax tree for the word ddacbbbb.

 220:[ FIRST: cd, a, d, c, acd, b, Tree: (S (A d (S (A d (S ac)) (B bb))) (B bb)). ]

## 15Coding.

/en/Coding/Error correction, Internal name: ZprmatrrZ Generate or Make Task ,

Name:
Var.: 221. Group:           Day/Mo/Year:

The coding matrix K is given. Over the communication channel it is: 1 1 1 0 1 0 0 0 0 0 0 1 0 1 1 1 0 1 1 1 1 0 0 0 . Find the test matrix, correct errors and decode.
K=(
 1
 0
 0
 1
 0
 0
 1
 0
 0
 0
 0
 1
 1
 0
 0
 0
 0
 1
 1
 1
 0
 1
 1
 0
 0
 1
 0
 0
 1
 1
 0
 1
)    P=

 221:[(0 1 1 1, 1 1 0 1, 1 0 0 1)]

/en/Coding/Error correction (complex), Internal name: ZprmatrrrZ Generate or Make Task ,

Name:
Var.: 222. Group:           Day/Mo/Year:

The coding matrix K is given. Over the communication channel it is: 1 0 1 1 1 1 1 0 0 1 1 1 0 0 1 1 1 1 1 1 0 1 1 1 0 1 1 . Find the test matrix, correct errors and decode.
K=(
 0
 0
 0
 1
 0
 1
 0
 0
 1
 1
 1
 0
 0
 0
 0
 0
 1
 0
 0
 0
 0
 0
 1
 0
 1
 1
 1
 0
 0
 0
 0
 0
 0
 0
 1
 0
 0
 1
 0
 0
 1
 1
 1
 1
 0
)    P=

 222:[(0 1 0 1 1, 1 0 1 1 1, 0 1 1 1 0)]

/en/Coding/Error correction only, Internal name: ZprmatrriZ Generate or Make Task ,

Name:
Var.: 223. Group:           Day/Mo/Year:

The coding matrix K is given. Over the communication channel it is: 1 1 0 1 0 1 1 0 1 0 1 1 1 1 1 1 0 1 0 0 0 0 1 0 . Find the test matrix, correct errors.
K=(
 0
 0
 0
 1
 0
 1
 1
 1
 1
 0
 1
 1
 1
 0
 0
 0
 0
 0
 1
 0
 0
 1
 0
 0
 1
 1
 0
 0
 0
 0
 1
 0
)

 223:[(1 1 0 1 0 0 1 0, 1 0 1 1 1 0 1 1, 0 1 0 0 0 1 1 0)]

/en/Coding/Decoding, Internal name: ZBdecodiZ Generate or Make Task ,

Name:
Var.: 224. Group:           Day/Mo/Year:

The coding matrix K is given. Find the decoding matrix D and decode the code words aT=(1 1 1 1 1 1 0), bT=(0 1 0 0 0 1 0) and cT=(1 0 0 0 0 0 0).

K=

 0
 1
 1
 0
 1
 1
 1
 0
 1
 0
 1
 0
 0
 0
 1
 1
 1
 0
 1
 0
 1
 1
 1
 0
 0
 0
 0
 0

D =

(Da)T=
(Db)T=
(Dc)T=

 224:[(0 0 1 0), (1 1 1 1), (1 0 1 1)]

/en/Coding/The core of the matrix (simple), Internal name: ZBkeriZ Generate or Make Task ,

Name:
Var.: 225. Group:           Day/Mo/Year:

Find a basis of the kernel of a matrix over a field F2:

 1
 0
 1
 0
 1
 1
 1
 1
 1
 0
 1
 0

 225:[(1 x3 , 0 x3+1x4, x3,x4)]

## 16Cryptography.

/en/Cryptography/A small fraction, Internal name: ZBigFraciZ Generate or Make Task ,

Name:
Var.: 226. Group:           Day/Mo/Year:

Write the answer in the form of an irreducible fraction.
 152 187 − 95 119 = (                     ) (                     )

 226:[19/1309]

/en/Cryptography/Fractions Are Large, Internal name: ZBigFraciiZ Generate or Make Task ,

Name:
Var.: 227. Group:           Day/Mo/Year:

Write the answer in the form of an irreducible fraction.
 70906 68021 − 62995 60433 = (                     ) (                     )

 227:[293/15168683]

/en/Cryptography/Inverse number, Internal name: ZobratzZ Generate or Make Task ,

Name:
Var.: 228. Group:           Day/Mo/Year:

Solve the equation in Z5483:
1352·x + 4571=0

 228:[x=390]

/en/Cryptography/Inverse number (Prime), Internal name: Zobratz1Z Generate or Make Task ,

Name:
Var.: 229. Group:           Day/Mo/Year:

Solve the equation in Z41:
31·x + 14=0

 229:[x=26]

/en/Cryptography/Inverse number (very simple), Internal name: Zobratz2Z Generate or Make Task ,

Name:
Var.: 230. Group:           Day/Mo/Year:

Solve the equation in Z100:
49·x + 64=0

 230:[x=64]

/en/Cryptography/Encryption backpack, Internal name: ZrukzakiZ Generate or Make Task ,

Name:
Var.: 231. Group:           Day/Mo/Year:

The secret key for cryptography based on the backpack packing is given: (1587, 10000 and 34 72 148 305 618 1245 ) (1) Decrypt encryption 16321, (2) to generate the corresponding public key to encrypt, (3) encrypt message   010011.

 231:[b=523; Key=7782 7656 7404 9515 3214 1135 ; Crypt=12005; Mess=101001; ]

/en/Cryptography/Encryption by backpack (with hint), Internal name: ZrukzakiiZ Generate or Make Task ,

Name:
Var.: 232. Group:           Day/Mo/Year:

The secret key for cryptography based on the backpack packing is given: (4977, 10000 and 37 74 152 308 620 1246 ) (1) Decrypt encryption 21220, (2) to generate the corresponding public key to encrypt, (3) encrypt message 101100. (Hint: 10000·(−4436)+4977·(8913)=1)

 232:[b=8913; Key=9781 9562 4776 5204 6060 5598 ; Crypt=19761; Mess=010011; ]

/en/Cryptography/Chinese theorem on residues (small), Internal name: ZChainTheoriZ Generate or Make Task , Find a match in The Chinese theorem but the numbers are small, you can search through.

Name:
Var.: 233. Group:           Day/Mo/Year:

Given the isomorphism: Z5×Z13 → Z65. Find a match: (           ;          ) → (48) and (4;8) → (          ) .

 233:[(3;9 ) → (48) , (4;8 ) → (34) ]

/en/Cryptography/The Chinese theorem on residues, Internal name: ZChainTheorZ Generate or Make Task , Find a match in The Chinese theorem. The numbers are average, you need a calculator.

Name:
Var.: 234. Group:           Day/Mo/Year:

Given the isomorphism: Z53×Z59 → Z3127. Find a match: (           ;          ) → (1342) and (19;15) → (          ) .

 234:[(17;44 ) → (1342) , (19;15 ) → (2139) ]

/en/Cryptography/RSA, Internal name: ZrsaZ Generate or Make Task ,

Name:
Var.: 235. Group:           Day/Mo/Year:

Solve the equation x3=5573 in Z57949.
(Hint: 57949=167·347, 167 and 347 - Prime number, 167·(−187)+347·(90)=1, 62111 %167 = 89, 76120 %347 = 244, 21451 %347 = 262, 2152 %347 = 42, 62180 %167 = 16, 89130 %167 = 115, 21231 %347 = 76, 62198 %167 = 133.)

 235:

/en/Cryptography/RSA (simple), Internal name: ZrsaiZ Generate or Make Task ,

Name:
Var.: 236. Group:           Day/Mo/Year:

Solve the equation x3=7 in Z85.
(Hint: 85=5·17, 5 and 17 - Prime number, 5·(−10)+17·(3)=1, 23 %5 = 3, 711 %17 = 14.)

 236:

/en/Cryptography/RSA (without hint), Internal name: ZrsatZ Generate or Make Task ,

Name:
Var.: 237. Group:           Day/Mo/Year:

Solve the equation x3=49 in Z187.
(Hint: 187=17·11, 17 and 11 - Prime number.)

 237:

/en/Cryptography/El Gammal Encryption, Internal name: ZElGamalcZ Generate or Make Task ,

Name:
Var.: 238. Group:           Day/Mo/Year:

The El Gammal cryptosystem withZ31 and g=3 is given. Here is the table of degrees:
 0 1 2 3 4 5 6 7 8 9 0* 3 9 27 19 26 16 17 20 29 1* 25 13 8 24 10 30 28 22 4 12 2* 5 15 14 11 2 6 18 23 7 21

Secret key: 28.
1. Decrypt: crypt: 11, hint: 6.
2. Generate the public key. Answer: (               )
3. Encrypt message 19 (as random numbers use 4).
Answer: Crypt: (               ), Hint: (               ).

 238:[ Message: 24, public key: 7 Crypt: (18, 19).]

/en/Cryptography/Signed By El Gammal, Internal name: ZElGamalsZ Generate or Make Task ,

Name:
Var.: 239. Group:           Day/Mo/Year:

The El Gammal cryptosystem withZ31 and g=3 is given. Here is the table of degrees:
 0 1 2 3 4 5 6 7 8 9 0* 3 9 27 19 26 16 17 20 29 1* 25 13 8 24 10 30 28 22 4 12 2* 5 15 14 11 2 6 18 23 7 21

Secret key: 17.
Sign message 6 (as random numbers use 29).
Answer: Sign: (             ), hint: (             ).

 239:[ (21, 21).]

## 17Differential equation. Here are the differential equations.

/en/Differential equation/Approximate solution, Internal name: ZpriblduZ Generate or Make Task ,

Name:
Var.: 240. Group:           Day/Mo/Year:

Find an approximate solution to the Cauchy problem:
y"′=4·y′·y+4·y·x,
y(2)=9, y′(2)=−9, y"(2)=6. Write the answer in the form of a series of Taylor to the fourth power of the summand inclusive.

 240:[y=9−9·(x−2)+3·(x−2)2−42·(x−2)3+21·(x−2)4]

/en/Differential equation/Simple Cauchy task, Internal name: ZkoshiZ Generate or Make Task ,

Name:
Var.: 241. Group:           Day/Mo/Year:

Solve the Cauchy problem: x·y′+3=y, y(−9)=−69.

 241:[y=8x+3]

/en/Differential equation/Integrating factor, Internal name: ZIntMnozhZ Generate or Make Task ,

Name:
Var.: 242. Group:           Day/Mo/Year:

Find the general solution of the differential equation
 (3·x2·y3 )  dx+(5·x3·y2+4·y1)  dy = 0

 242:[1x3y5+1y4 = c, y2]

/en/Differential equation/Linear first order, Internal name: ZlinduiZ Generate or Make Task ,

Name:
Var.: 243. Group:           Day/Mo/Year:

Find the general solution of the differential equation: y′=[(y+8·x −7)/(x+3)]

 243:[y=8(x+3)ln(x+3)+C(x+3)+31]

/en/Differential equation/Linear homogeneous with post 2nd order coefficient, Internal name: Zlopk2Z Generate or Make Task ,

Name:
Var.: 244. Group:           Day/Mo/Year:

Find the general solution of the differential equation:
y" −4·y=0

 244:[C1·e−2x + C3·e2x ]

/en/Differential equation/Linear homogeneous with post coefficient of the 3rd order, Internal name: Zlopk3Z Generate or Make Task ,

Name:
Var.: 245. Group:           Day/Mo/Year:

Find the general solution of the differential equation:
y"′+y" −21·y′−45·y=0

 245:[C1·e−3x+C2·x ·e−3x + C3·e5x ]

/en/Differential equation/Linear mixed post coeff 2nd order, Internal name: Zlnpk2Z Generate or Make Task ,

Name:
Var.: 246. Group:           Day/Mo/Year:

Find the general solution of the differential equation:
y" −6·y′+9·y=45·x2 −96·x +34

 246:[C1·e3x+C2·x ·e3x + (5·x2 −4·x )]

/en/Differential equation/Linear mixed post coeff complex, Internal name: ZlnpkiZ Generate or Make Task ,

Name:
Var.: 247. Group:           Day/Mo/Year:

Find the general solution of the differential equation:
y" +5·y′+6·y=6·e−1x−50·cos(−4x)

 247:[C1·e−3x+C2·e−2x + (3·e−1x+1·cos(−4x)+2·sin(−4x))]

/en/Differential equation/The linear non-uniform post-coefficient is very complex, Internal name: ZlnpkiiZ Generate or Make Task ,

Name:
Var.: 248. Group:           Day/Mo/Year:

Find the general solution of the differential equation:
y" −5·y′+6·y=−30·x +7−1·e2x

 248:[C1·e3x+C2·e2x + (−5·x −3+1·x·e2x)]

/en/Differential equation/Linear inhomogeneous with post factors is very complex 2, Internal name: ZlnpkiiiZ Generate or Make Task ,

Name:
Var.: 249. Group:           Day/Mo/Year:

Find the general solution of the differential equation:
y" −6·y′=36·x2 −48·x +30

 249:[C1·e6x+C2 + (−2·x3 +3·x2 −4·x )]

/en/Differential equation/Homogeneous first order, Internal name: ZodnordifuriZ Generate or Make Task ,

Name:
Var.: 250. Group:           Day/Mo/Year:

Find the particular solution of the differential equation x·y ·y′ = 7 ·y2 −18 ·x2, satisfying the condition y(1)=5

 250:[y = √{ 3 ·x2 +22·x14} ]

/en/Differential equation/Allowing the reduction, Internal name: ZdifurUmenStepZ Generate or Make Task ,

Name:
Var.: 251. Group:           Day/Mo/Year:

Find the general solution of the differential equation: y·y" = 3(y′)2 − (y′)3

 251:[y + [(C1)/(y2)] = 3x + C2]

/en/Differential equation/Bernulli, Internal name: ZbernulliZ Generate or Make Task ,

Name:
Var.: 252. Group:           Day/Mo/Year:

Find the general solution of
 y′=−5·y + e−1 ·x ·y−5

 252:[ y6 = C·e−30·x + [6/29] ·e−1 ·x ]

/en/Differential equation/Bernulli with answer, Internal name: ZbernullimZ Generate or Make Task , Moodle computable

Name:
Var.: 253. Group:           Day/Mo/Year:

Find the general solution of
 y′=−6·y + e−4 ·x ·y−1
1. y1 = C·e6·x + [(−1)/8] ·e−4 ·x
2. y2 = C·e−12·x + [(−1)/8] ·e−4 ·x
3. y2 = C·e−12·x + [1/4] ·e−4 ·x
4. y1 = C·e6·x + [(−1)/4] ·e−4 ·x

 253:

/en/Differential equation/Ricatti, Internal name: ZricattiZ Generate or Make Task ,

Name:
Var.: 254. Group:           Day/Mo/Year:

Find the general solution of
 y′ = 5·x2·y ·(y − 6·x−3) −18·x−4

 254:[y = [33/(C·x−30 −5·x3)] + 6·x−3]

/en/Differential equation/Ricatti with answer, Internal name: ZricattimZ Generate or Make Task , Moodle computable

Name:
Var.: 255. Group:           Day/Mo/Year:

Find the general solution of
 y′ = 10·x−4·y ·(y − 5·x3) +15·x2
1. y = [( −47 )/(C·x−50 +5·x3)] + 5·x 3
2. y = [(−47)/(C·x−3 −50·x −50 )] + 5·x 3
3. y = [47/(C·x−50 −10·x−3)] + 5·x3
4. y = [ 47 /(C·x 3 +3·x 50)] + 5·x3

 255:

/en/Differential equation/Lagranj, Internal name: ZlagranjZ Generate or Make Task ,

Name:
Var.: 256. Group:           Day/Mo/Year:

Find the general solution of
 −3·y = y′·( −6+10 ·y′) ·x −27·ln|y′|

256:[ {
 x = (C +[27/10] ·ln|p + [(−3)/10]|) ·p−2 ,
 y = (2 + [(−10)/3]·p) ·(C + [27/10] ·ln|p + [(−3)/10]|) ·p−1 +9·ln|p| .
]

/en/Differential equation/Lagranj with answer, Internal name: ZlagranjmZ Generate or Make Task , Moodle computable

Name:
Var.: 257. Group:           Day/Mo/Year:

Find the general solution of
 −3·y = y′·( −6−7 ·y′) ·x −15·ln|y′|
1. {
 x = (C +[(−15)/7] ·ln|p |) ·(2 + [7/3] ·p)−2 ,
 y = (C + [(−15)/7]·ln|p |) ·p ·(2 +[7/3] ·p)−1 +5·ln|p| .

2. {
 x = (C + [(−13)/7]·ln|p |) ·(2 + [7/3] ·p)−2,
 y = (C + [(−13)/7]·ln|p |) ·p ·(2 + [7/3] ·p)−1 +5·ln|p| .

3. {
 x = (C + [(−13)/7]·ln|p + [3/7]|) ·p−2 ,
 y = (2 + [7/3]·p) ·(C + [(−13)/7]·ln|p + [3/7]|)·p−1 +5·ln|p|.

4. {
 x = (C +[(−15)/7] ·ln|p + [3/7]|) ·p−2 ,
 y = (2 + [7/3]·p) ·(C + [(−15)/7] ·ln|p + [3/7]|) ·p−1 +5·ln|p| .

 257:

/en/Differential equation/Klero, Internal name: ZkleroZ Generate or Make Task , Moodle computable

Name:
Var.: 258. Group:           Day/Mo/Year:

Which of the figures shows some solutions of the following differential equation
 y = y′·(x +6 ·y′+11 ) −12
 1.7mmPicture Omitted

 258:

/en/Differential equation/Ortogonal fem, Internal name: ZortFmZ Generate or Make Task , Moodle computable

Name:
Var.: 259. Group:           Day/Mo/Year:

One of the curves orthogonal to the family
 y 4 = C x + 8
passes through the point (1, √{ 78 }) and intersects the axis OY at the point (0,t). Find t.

 259:[√{10}]

/en/Differential equation/Linear Simple, Internal name: ZlinSmplZ Generate or Make Task , Moodle computable

Name:
Var.: 260. Group:           Day/Mo/Year:

A particular solution y(x) of the differential equation
 y′+4 y tanx = 5 cos 2 x ·sinx
satisfies the condition y([( 2 π)/3] ) = [27/64]. Find y(0) for the solution.

 260:

/en/Differential equation/Integrating factor simple, Internal name: ZintMnTstZ Generate or Make Task , Moodle computable

Name:
Var.: 261. Group:           Day/Mo/Year:

The general solution of the differential equation
 (8·x1·y3−20·x2)  dx+(6·x2·y2)  dy = 0
is written in the form Axαyβ + B xγ yδ = Const. Find the sum α+ β+ γ+ δ.

 261:

/en/Differential equation/System of homogeneous equations 2x2, Internal name: ZsisOdnorDifurZ Generate or Make Task ,

Name:
Var.: 262. Group:           Day/Mo/Year:

Find the solution of the system

 y1′ = 5·y1 +2·y2
 y2′ = −6·y1 +12·y2

262:[ (
 y1
 y2
) = C1 ·e9·x ·(
 1
 2
) + C2 ·e8·x ·(
 −2
 −3
)]

/en/Differential equation/System of homogeneous equations 2x2 with choice of the answer, Internal name: ZsisOdnorDifurmZ Generate or Make Task , Moodle computable

Name:
Var.: 263. Group:           Day/Mo/Year:

Find the solution of the system

 y1′ = −4·y1 +6·y2
 y2′ = −9·y1 +11·y2
1. (
 y1
 y2
) = C1 ·e5·x ·(
 −1
 −1
) + C2 ·e2·x ·(
 2
 3
)
2. (
 y1
 y2
) = C1 ·e2·x ·(
 −1
 −1
) + C2 ·e5·x ·(
 2
 3
)
3. (
 y1
 y2
) = C1 ·e2·x ·(
 −1
 −1
) + C2 ·e5·x ·(
 3
 2
)
4. (
 y1
 y2
) = C1 ·e5·x ·(
 −1
 −1
) + C2 ·e2·x ·(
 3
 2
)

 263:

/en/Differential equation/System of equations 2x2, Internal name: ZsisDifurZ Generate or Make Task ,

Name:
Var.: 264. Group:           Day/Mo/Year:

Find the solution of the system

 y1′ = 6 ·y1 −2 ·y2 −6 ·x +10
 y2′ = 6 ·y1 −1 ·y2 −3 ·x +5

264:[ (
 y1
 y2
) = C1 ·e3·x ·(
 2
 3
) + C2 ·e2·x ·(
 −1
 −2
) + x ·(
 0
 −3
) + (
 −1
 2
) ]

/en/Differential equation/System of equations 2x2 with choice of the answer, Internal name: ZsisDifurmZ Generate or Make Task , Moodle computable

Name:
Var.: 265. Group:           Day/Mo/Year:

Find the solution of the system

 y1′ = 8 ·y1 +1 ·y2 +16 ·x −20
 y2′ = 3 ·y1 +6 ·y2 +6 ·x −18
1. (
 y1
 y2
) = C1 ·e5·x ·(
 1
 −3
) + C2 ·e9·x ·(
 1
 1
) + x ·(
 2
 2
) + (
 −2
 0
)
2. (
 y1
 y2
) = C1 ·e9·x ·(
 1
 −3
) + C2 ·e5·x ·(
 1
 1
) + x ·(
 2
 2
) + (
 −2
 0
)
3. (
 y1
 y2
) = C1 ·e9·x ·(
 1
 −3
) + C2 ·e5·x ·(
 1
 1
) + x ·(
 −2
 0
) + (
 2
 2
)
4. (
 y1
 y2
) = C1 ·e5·x ·(
 1
 −3
) + C2 ·e9·x ·(
 1
 1
) + x ·(
 −2
 0
) + (
 2
 2
)

 265:

/en/Differential equation/Euler, Internal name: ZEuliiZ Generate or Make Task , Moodle computable

Name:
Var.: 266. Group:           Day/Mo/Year:

The differential equation
 x2·y" +4·x·y′+2·y=3·x−1
has a particular solution y=f(x). Find limx→ 0 ( f(x) ·x2) assuming that f(1)=0, f′(1)=6.

 266:[ lim = −3; y=−3·x−2 +3·x−1 +3 ·x−1 ·lnx]

/en/Differential equation/Step-down with tip, Internal name: ZLinPonZ Generate or Make Task , Moodle computable

Name:
Var.: 267. Group:           Day/Mo/Year:

The function y=xn for some n is a particular solution of
 x·y" + (− x +4) ·y′+ (2·x−1 −2) ·y = 0 .
Another particular solution y=f(x) satisfies the following conditions
 lim x → 0 f(x) xn = −3        and lim x → 0 ⎛⎝ f(x) xn ⎞⎠ ′ = −4.
Calculate
 lim x→ −∞ f(x) xn .

 267:

/en/Differential equation/Second order linear with sin and answer choice, Internal name: ZlnpkimZ Generate or Make Task , Moodle computable

Name:
Var.: 268. Group:           Day/Mo/Year:

The Cauchy problem is given:
y" +10·y′+24·y=−75·sin(3x), y(0)=3, y′(0)=−17.
What pattern corresponds to its solution?
1) y = ∗·e−4x+2·e ∗·x + ∗·cos( ∗·x)−1·sin( ∗·x)
2) y = ∗·e−4x−6·e ∗·x + ∗·cos( ∗·x)−6·sin( ∗·x)
3) y = ∗·e−4x−6·e ∗·x + ∗·cos( ∗·x)−1·sin( ∗·x)
4) y = ∗·e−4x+5·e ∗·x + ∗·cos( ∗·x)−1·sin( ∗·x)
5) y = ∗·e−4x+5·e ∗·x + ∗·cos( ∗·x)−6·sin( ∗·x)
6) y = ∗·e−4x+2·e ∗·x + ∗·cos( ∗·x)−6·sin( ∗·x)
(The sign ∗ hides some numbers).

 268:[4, y=−4·e−4x+5·e−6x +2·cos(3x)−1·sin(3x)]

/en/Differential equation/The approximate solution in the form of Taylor series Only the coefficient, Internal name: ZpribldumZ Generate or Make Task , Moodle computable

Name:
Var.: 269. Group:           Day/Mo/Year:

The solution of the Cauchy problem
 y"+(x−1)·y′+ y = 7·x+2,
y(0)=−7, y′(0)=−2 has the form
 y= ∞∑ n=0 cn·xn .
Find the coefficient c3.

 269:

## 18Theory of probability.

/en/Theory of probability/Full probability and Bayes formula, Internal name: ZpolverZ Generate or Make Task ,

Name:
Var.: 270. Group:           Day/Mo/Year:

Probability to pass the exam by answering a simple ticket is 5/7 while answering a difficult is - 1/6. The student chooses a ticket from a pack in which there are 8 simple and 9 difficult tickets.
(1) Find the probability that the student will pass the exam.
(2) it is Known that the student passed the exam, what is the probability that he passed with a simple ticket?

 270:[1:[101/238]=0.42437, 2:[80/101]=0.792079]

/en/Theory of probability/Formula Bernoulli, Internal name: ZbernuliZ Generate or Make Task ,

Name:
Var.: 271. Group:           Day/Mo/Year:

It is known that for the final death, Count Dracula needs at least three silver bullets. Van Helsing is the holder of 9 rounds, and the probability of hitting with every shot is 5/7. Find the probability of the victory of good over evil.

 271:[0.996847]

/en/Theory of probability/Basket balls, Internal name: ZterverZ Generate or Make Task ,

Name:
Var.: 272. Group:           Day/Mo/Year:

In the basket there are 5 white and 1 black balls. From the basket, 2 balls were taken. What is the probability that they are the same color?

 272:[[10/15]=[2/3]=0.667]

/en/Theory of probability/methods of choice, Internal name: ZzcnmZ Generate or Make Task ,

Name:
Var.: 273. Group:           Day/Mo/Year:

How many ways to choose  8 items from  2 items? How many ways are there to choose  8 items from  2 items?

 273:

/en/Theory of probability/Shooting the hare, Internal name: ZsumveriZ Generate or Make Task ,

Name:
Var.: 274. Group:           Day/Mo/Year:

Two shooters shoot one hare. The probability of hitting with first arrow is 0.44 and whith second it is 0.5. What is the probability that the hare will get shot? What is the probability that there will be two holes in the hare?

 274:[0.72;0.22]

/en/Theory of probability/Expected value and Variance, Internal name: ZdiskcviZ Generate or Make Task ,

Name:
Var.: 275. Group:           Day/Mo/Year:

Random value is given by table
 x
 1
 3
 8
 p
 0.5
 0.1
 0.4
Find the expected value and variance.

 275:[M=4, D=11]

/en/Theory of probability/Malchish-Kibalchish, Internal name: ZpulemetZ Generate or Make Task ,

Name:
Var.: 276. Group:           Day/Mo/Year:

The Boy-Kibalchish has 3080   ammo. Firing accuracy of the revolutionary machine gun Maxim is 0.22. For the death of the Chief Burzhuin, 666 bullets are enough. What is the probability of the victory of the world Revolution?

 276:[0.693]

/en/Theory of probability/De Moivre Laplace, Internal name: ZterminatorZ Generate or Make Task ,

Name:
Var.: 277. Group:           Day/Mo/Year:

It is known that to terminate the Terminator T-1000 is requires 200 hits. The accuracy of the Terminator T-800 is - 0.3. How many shots is it enough to make the Terminator T-800, to terminate his opponent with a probability of 0.94?

 277:

/en/Theory of probability/Confidence interval, Internal name: ZdovintZ Generate or Make Task ,

Name:
Var.: 278. Group:           Day/Mo/Year:

Of the 578 conducted, successful experiments ware 208. Find the confidence interval for the probability of success in one experience. (significance level 0.01).

 278:[0.314914,0.40741]

/en/Theory of probability/Confidence interval 1, Internal name: ZstatiZ Generate or Make Task ,

Name:
Var.: 279. Group:           Day/Mo/Year:

Produced 5 analyses of a certain substance. Test results: 3.37; 3.73; 4.27; 4.18; 4.09. The standard deviation for this type of analysis is 0.9. Significance level is 92%. Find the confidence interval of the measured value.

 279:[(3.2234;4.6326)]

/en/Theory of probability/Confidence interval 2, Internal name: ZstatiiZ Generate or Make Task ,

Name:
Var.: 280. Group:           Day/Mo/Year:

5 experiments were made. Results: 2.73; 2.88; 3.18; 3.15; 3.03. Significance level: 90%. Find the sample average, corrected standard deviation and confidence interval of the measured value.

 280:[2.994;0.211335; (2.81036;3.17764)]

/en/Theory of probability/Sample mean, Internal name: ZvibsredZ Generate or Make Task ,

Name:
Var.: 281. Group:           Day/Mo/Year:

The variation series is given: 1; 4; 8; 12; 15. Find the sample average.

 281:

/en/Theory of probability/Parameter in the density function, Internal name: ZtvFplotniZ Generate or Make Task ,

Name:
Var.: 282. Group:           Day/Mo/Year:

The density function of some random variable is given:
f(x)=

 0,
 x ∉ [6 ; 11]
 c·x,
 x ∈ [6 ; 11]
Find the value of the c parameter. Find the distribution function.

 282:[[2/85] ≈ 0.024]

/en/Theory of probability/The expected value of a continuous random variable, Internal name: ZtvMatOnsviZ Generate or Make Task ,

Name:
Var.: 283. Group:           Day/Mo/Year:

The density function of some random variable is given:
F(x)=

 0,
 x ≤ 3
 x2−9 55 ,
 3 < x ≤ 8
 1,
 x > 8
Find the expectation.

 283:[[194/33] ≈ 5.879]

## 19Graph theory.

/en/Graph theory/The problem of the appointment of 3x3, Internal name: Zkanmun3Z Generate or Make Task ,

Name:
Var.: 284. Group:           Day/Mo/Year:

Solve the problem of optimal assignment. Specify the final layout of the vertices.

 d e f a 6 13 12 b 4 9 12 c 12 12 15

 a - b - c - ∑ =

 284:[∑ = 37, ae, bf, cd]

/en/Graph theory/The problem of the appointment of 4x4, Internal name: Zkanmun4Z Generate or Make Task ,

Name:
Var.: 285. Group:           Day/Mo/Year:

Solve the problem of optimal assignment. Specify the final layout of the vertices.

 e f g h a 14 16 14 21 b 13 15 7 14 c 8 12 5 12 d 4 9 3 14

 a - b - c - d - ∑ =

 285:[∑ = 53, ag, be, cf, dh]

/en/Graph theory/The problem of the appointment of 5x5, Internal name: Zkanmun5Z Generate or Make Task ,

Name:
Var.: 286. Group:           Day/Mo/Year:

Solve the problem of optimal assignment. Specify the final layout of the vertices.

 f g h i j a 19 22 20 25 12 b 11 22 18 15 10 c 6 18 8 11 3 d 18 27 19 20 18 e 10 18 11 18 5

 a - b - c - d - e - ∑ =

 286:[∑ = 91, af, bh, cg, dj, ei]

/en/Graph theory/The problem of the appointment of 6x6, Internal name: Zkanmun6Z Generate or Make Task ,

Name:
Var.: 287. Group:           Day/Mo/Year:

Solve the problem of optimal assignment. Specify the final layout of the vertices.

 g h i j k l a 25 19 22 17 27 24 b 12 14 9 7 22 21 c 20 16 11 7 16 24 d 27 23 18 21 24 31 e 17 21 14 14 22 22 f 10 11 8 5 14 22

 a - b - c - d - e - f - ∑ =

 287:[∑ = 128, ai, bk, cg, dj, eh, fl]

/en/Graph theory/The problem of the appointment of 7x7, Internal name: Zkanmun7Z Generate or Make Task ,

Name:
Var.: 288. Group:           Day/Mo/Year:

Solve the problem of optimal assignment. Specify the final layout of the vertices.

 h i j k l m n a 22 11 23 21 13 17 29 b 16 6 22 10 8 10 19 c 19 11 22 14 12 20 26 d 23 14 32 20 22 22 34 e 26 22 33 23 19 23 28 f 10 3 15 8 6 9 23 g 20 8 16 10 9 15 26

 a - b - c - d - e - f - g - ∑ =

 288:[∑ = 150, ak, bj, cm, dl, ei, fn, gh]

/en/Graph theory/Ford-Falkerson, Internal name: ZffZ Generate or Make Task ,

Name:
Var.: 289. Group:           Day/Mo/Year:

The "system of roads" with the specified capacity connecting the cities s and t is given. Using the Ford-Falkerson algorithm, find the maximum flow and prove that it is indeed the maximum.
s  →(7,    )a, s  →(6,    )b, s  →(7,    )c, a  →(4,    )e, a  →(4,    )i, b  →(3,    )g, b  →(2,    )e, c  →(2,    )f, c  →(2,    )i, e  →(2,    )h, e  →(4,    )i, f  →(3,    )i, f  →(2,    )g, g  →(6,    )t, h  →(8,    )t, i  →(9,    )t,
flow=

 289:[]

## 20Financial calculations.

/en/Financial calculations/Size of payments, Internal name: ZrazmviplZ Generate or Make Task ,

Name:
Var.: 290. Group:           Day/Mo/Year:

Nothing

 290:[13548.8]

/en/Financial calculations/Mission, Internal name: ZkomandirZ Generate or Make Task ,

Name:
Var.: 291. Group:           Day/Mo/Year:

Nothing

 291:[15196.6]

/en/Financial calculations/Purchase, Internal name: ZpokupkvZ Generate or Make Task ,

Name:
Var.: 292. Group:           Day/Mo/Year:

Someone plans to buy an apartment in 251000 in 10 years. What amount should he or she put into his or her Bank account monthly if the annual Bank rate is 10%?

 292:[1255.85]

/en/Financial calculations/Rent replacement 1, Internal name: ZrentaiZ Generate or Make Task ,

Name:
Var.: 293. Group:           Day/Mo/Year:

Replace the semi-annual rent with payment 1800 and duration 10 years with quarterly duration 7 years. Annual rate 10%. Interest is charged 3 once a year at regular intervals.

 293:[1118.19]

/en/Financial calculations/Rent replacement 2, Internal name: ZrentaiiZ Generate or Make Task ,

Name:
Var.: 294. Group:           Day/Mo/Year:

Replace the annual rent with payment 1700 and duration 8 years semi-annual with payment 1400. Annual rate 14%. Interest is charged 2 once a year at regular intervals.

 294:[7.25508]

/en/Financial calculations/Rates and inflation, Internal name: ZinflstavZ Generate or Make Task ,

Name:
Var.: 295. Group:           Day/Mo/Year:

The Bank announced the Deposit 17 % per annum. The real rate at the end of the year is 6.5 %. Find quarterly inflation.

 295:[2.37857]

/en/Financial calculations/Rates and inflation 1, Internal name: Zinflstav1Z Generate or Make Task ,

Name:
Var.: 296. Group:           Day/Mo/Year:

The Bank announced the Deposit 18.5 % per annum. The expected rate of inflation 3 %. Find the real annual rate for the Bank's client.

 296:[5.28572]

/en/Financial calculations/Rates and inflation 2, Internal name: Zinflstav2Z Generate or Make Task ,

Name:
Var.: 297. Group:           Day/Mo/Year:

The Bank announced the Deposit 15 % per annum. The real rate at the end of the year is 3.5 %. Find quarterly inflation.

 297:[2.66901]

/en/Financial calculations/Rates and inflation 3, Internal name: Zinflstav3Z Generate or Make Task ,

Name:
Var.: 298. Group:           Day/Mo/Year:

The real interest rate on the Deposit at the end of the year was 3.5 %. Quarterly inflation 2.75 %. Find the nominal rate appointed by the Bank.

 298:[11.4621]

/en/Financial calculations/What is more profitable, Internal name: ZchtovZ Generate or Make Task ,

Name:
Var.: 299. Group:           Day/Mo/Year:

Find modern amounts and find out what amount is more profitable: (1) 1188 within two years to date or (2) 1292 4 years after today. Annual compound interest rate 2%.

 299:[A1=1236, B1=1193.61]

/en/Financial calculations/Annual interest, Internal name: ZgodprZ Generate or Make Task ,

Name:
Var.: 300. Group:           Day/Mo/Year:

The Bank promises 2.5% in 30 days. How much will it be per annum? (In year considered 365 days.)

 300:[35.0435]

## 21Game theory.

/en/Game theory/Pure strategy, Internal name: ZClnGameZ Generate or Make Task , search is solved by the minimax and Maximin. Moodle computable

Name:
Var.: 301. Group:           Day/Mo/Year:

Given matrix wins a zero-sum game:
 1
 −4
 7
 −2
 6
 −7
 −6
 8
 −1
 9
 4
 7
 2
 −6
 −4
 −8
 6
 5
 8
 −2
 −6
 −3
 −7
 7
 4
 0
 6
 −9
Is there a solution in pure strategies (Yes/no)? If there is, find the price of the game.

 301:[−6, (2,7)]

/en/Game theory/The expectation of the mixed strategy, Internal name: ZgameMeZ Generate or Make Task , Moodle computable

Name:
Var.: 302. Group:           Day/Mo/Year:

Given a matrix game:

 2
 5
 3
 −4

.
What will be the worst expectation of winning from the left player when using a mixed strategy (0.8;0.2)?

 302:[2.2]

/en/Game theory/Mixed strategy, Internal name: ZgamethiZ Generate or Make Task , 2x3, solved by picture

Name:
Var.: 303. Group:           Day/Mo/Year:

Find the solution to the game in mixed strategies.
(
 3
 −3
 −2
 −3
 5
 0
)

 303:[[(−3)/4], ([3/8], [5/8]), ([1/4], 0, [3/4])]

/en/Game theory/Dominant strategies, Internal name: ZDomGameZ Generate or Make Task , First remove the dominant, then 2x2 picture

Name:
Var.: 304. Group:           Day/Mo/Year:

Given matrix wins a zero-sum game:
(
 3
 4
 1
 5
 1
 6
 4
 2
 3
)
To find the optimal strategy and the price of the game.

304:[ The strategy of the left of the player: (
 [5/8]
 [3/8]
 0
), Strategy the top player: (
 0
 [5/8]
 [3/8]
) price games: [23/8]]

## 22Linear programming.

/en/Linear programming/The simplex method is complex, Internal name: Zsimpl1Z Generate or Make Task ,

Name:
Var.: 305. Group:           Day/Mo/Year:

Solve the problem of linear programming:
L(x)=−2·x1−2·x2−11·x3 → max
{
 x1
 − 1·x2
 + x3
 =2
 −2·x1
 + 3·x2
 ≤ 18
 −1·x1
 + 2·x2
 ≥ 2

x1 ≥ 0, x2 ≥ 0, x3 ≥ 0.

 305:[(6, 4, 0,18, 0, 0), L(x)=−20]

/en/Linear programming/Transport problem, Internal name: ZtransiZ Generate or Make Task ,

Name:
Var.: 306. Group:           Day/Mo/Year:

Goods from warehouses B1, B2, B3 are delivered to consumers A1, A2, A3. Prices of transportation of goods are shown in the table:
 A1=90 A2=10 A3=70 B1=30 19 13 18 B2=50 16 11 20 B3=90 12 8 13

Make a transportation plan in which the transport costs are minimal and find these costs.

306:[xij=(
 0
 0
 30
 40
 10
 0
 50
 0
 40
) , L=2410]

/en/Linear programming/Dual problem, Internal name: ZsimpliZ Generate or Make Task ,

Name:
Var.: 307. Group:           Day/Mo/Year:

Write a dual problem to the problem of linear programming and solve it graphically. Restore the solution to the original problem.
L(x)=4·x1+4·x2+11·x3−5·x4 → min
{
 −2x1
 +2 x2
 −5x3
 +1x4
 =15
 −1x1
 +5 x2
 +1x3
 −3x4
 ≥ −17

x1 ≥ 0, x2 ≥ 0, x4 ≥ 0.

 307:[X=(0,0,−2,5), Y=(−2,1), L=−47]

/en/Linear programming/Graphical method, Internal name: ZoptplanZ Generate or Make Task ,

Name:
Var.: 308. Group:           Day/Mo/Year:

For the production of 2 types of goods a and B requires 3 types of resources. The consumption of each resource for the production of a unit of goods and the monthly supply of this resource is given in the table
 A B monthly supply I 1 10 25 II 6 8 52 III 9 5 55

The profit from selling unit a is 3. and with sale item B is 3. Find the (1) monthly plan for the release of goods, giving the maximum income and (2) this maximum income. (3) Is it possible to reduce the stock of one of the resources without changing the optimal plan, and by how much?
(2):
(3):

 308:[(5,2), 21, 2−6]

/en/Linear programming/Graphical method beautiful, Internal name: ZLPgrfiZ Generate or Make Task ,

Name:
Var.: 309. Group:           Day/Mo/Year:

For the production of two types of goods A and B requires three types of resources. The consumption of each resource for the production of a unit of goods and the monthly supply of this resource is shown in the table
 A B monthly supply I 7 8 164 II 1 5 89 III 2 1 34

Profit from the sale of unit A is 33 and from the sale of unit B is 57. Solve the problem by graphical method. 3.5mm
 Picture Omitted
To find the monthly production plan of the products giving the maximum revenue and the maximum revenue.

 309:[ 2.7mm Picture Omitted, A - 4, B - 17, Profit - 1101.]

/en/Linear programming/Graphic dual problem, Internal name: ZLPgrfiiZ Generate or Make Task ,

Name:
Var.: 310. Group:           Day/Mo/Year:

The problem of linear programming is given:
L(y)=223 ·y1 + 144 ·y2 + 132 ·y3 → min
{
 14 ·y1
 + 5 ·y2
 + 9 ·y3
 ≥
 32
 9 ·y1
 + 7 ·y2
 + 2 ·y3
 ≥
 13

yi ≥ 0.
Formulate the dual problem:
L(x)=                                               → max
{

xi ≥ 0.
Solve it by geometric method:
3.5mm
 Picture Omitted
Answer: x1=           , x2=            , L(x)=
By solving the dual problem to find a solution to the direct problem:
Answer: y1=           , y2=            , y3=

 310:[ 2.7mm Picture Omitted, xi=(14; 3), L(x)=L(y)=487, yi=(1; 0; 2)]

/en/Linear programming/Only dual 3x3, Internal name: ZLP33dZ Generate or Make Task ,

Name:
Var.: 311. Group:           Day/Mo/Year:

The problem of linear programming
L(x)=10 ·x1 + 21 ·x2 + 10 ·x3 → max
{
 5 ·x1
 + 11 ·x2
 + 2 ·x3
 ≤
 52
 x1
 + 2 ·x2
 + 3 ·x3
 ≤
 10
 x1
 + 5 ·x2
 + x3
 ≤
 18

xi ≥ 0
and its solution is given: x1=6; x2=2; x3=0.
Formulate the dual problem
L(t)=                                               → min
{

ti ≥ 0
and find its solution.

 311:[t1=1; t2=5; t3=0; max=min=102]

/en/Linear programming/Dual 4x4 only, Internal name: ZLP44dZ Generate or Make Task ,

Name:
Var.: 312. Group:           Day/Mo/Year:

The problem of linear programming
L(x)=9 ·x1 + 17 ·x2 + 9 ·x3 + 6 ·x4 → max
{
 2 ·x1
 + 2 ·x2
 + 2 ·x3
 + 2 ·x4
 ≤
 20
 3 ·x1
 + 3 ·x2
 + 3 ·x3
 + x4
 ≤
 19
 2 ·x1
 + 2 ·x2
 + 2 ·x3
 + x4
 ≤
 14
 x1
 + 5 ·x2
 + x3
 + 2 ·x4
 ≤
 35

xi ≥ 0
and its solution is given: x1=0; x2=5; x3=0; x4=4.
Formulate the dual problem
L(t)=                                               → min
{

ti ≥ 0
and find its solution.

 312:[t1=0; t2=5; t3=1; t4=0; max=min=109]

/en/Linear programming/Simplex method 2x2 simple, Internal name: ZLP22Z Generate or Make Task ,

Name:
Var.: 313. Group:           Day/Mo/Year:

Solve the direct and dual linear programming problem.
L(x)=3 ·x1 + 2 ·x2 → max
{
 x1
 + x2
 ≤
 6
 7 ·x1
 + x2
 ≤
 44

xi ≥ 0.

 313:[x1=6; x2=0; t1=3; t2=0; max=18]

/en/Linear programming/The simplex method for a simple 3x3, Internal name: ZLP33Z Generate or Make Task ,

Name:
Var.: 314. Group:           Day/Mo/Year:

Solve the direct and dual linear programming problem.
L(x)=2 ·x1 + 37 ·x2 + 17 ·x3 → max
{
 x1
 + 2 ·x2
 + x3
 ≤
 14
 3 ·x1
 + 9 ·x2
 + 4 ·x3
 ≤
 60
 x1
 + 4 ·x2
 + 6 ·x3
 ≤
 58

xi ≥ 0.

 314:[x1=0; x2=4; x3=6; t1=5; t2=3; t3=0; max=250]

/en/Linear programming/The simplex method is a simple 4x4, Internal name: ZLP44Z Generate or Make Task ,

Name:
Var.: 315. Group:           Day/Mo/Year:

Solve the direct and dual linear programming problem.
L(x)=19 ·x1 + 4 ·x2 + 7 ·x3 + 9 ·x4 → max
{
 2 ·x1
 + x2
 + 3 ·x3
 + x4
 ≤
 12
 2 ·x1
 + 3 ·x2
 + x3
 + 5 ·x4
 ≤
 40
 x1
 + 3 ·x2
 + 3 ·x3
 + x4
 ≤
 14
 11 ·x1
 + 3 ·x2
 + 2 ·x3
 + 5 ·x4
 ≤
 63

xi ≥ 0.

 315:[x1=3; x2=0; x3=0; x4=6; t1=4; t2=0; t3=0; t4=1; max=111]

/en/Linear programming/The simplex method is a simple 5x5, Internal name: ZLP55Z Generate or Make Task ,

Name:
Var.: 316. Group:           Day/Mo/Year:

Solve the direct and dual linear programming problem.
L(x)=27 ·x1 + 3 ·x2 + 9 ·x3 + 12 ·x4 + 32 ·x5 → max
{
 2 ·x1
 + x2
 + x3
 + x4
 + 2 ·x5
 ≤
 22
 2 ·x1
 + x2
 + 3 ·x3
 + 2 ·x4
 + x5
 ≤
 24
 3 ·x1
 + 3 ·x2
 + x3
 + x4
 + 3 ·x5
 ≤
 30
 2 ·x1
 + 3 ·x2
 + 3 ·x3
 + x4
 + 3 ·x5
 ≤
 28
 x1
 + 3 ·x2
 + x3
 + 3 ·x4
 + 3 ·x5
 ≤
 42

xi ≥ 0.

 316:[x1=2; x2=0; x3=0; x4=6; x5=6; t1=4; t2=0; t3=3; t4=5; t5=0; max=318]

/en/Linear programming/The simplex method is a simple 3x5, Internal name: ZLP35Z Generate or Make Task ,

Name:
Var.: 317. Group:           Day/Mo/Year:

Solve the direct and dual linear programming problem.
L(x)=23 ·x1 + 2 ·x2 + 10 ·x3 + 9 ·x4 + 2 ·x5 → max
{
 2 ·x1
 + 2 ·x2
 + 2 ·x3
 + x4
 + 3 ·x5
 ≤
 14
 x1
 + 2 ·x2
 + 3 ·x3
 + 3 ·x4
 + 3 ·x5
 ≤
 27
 3 ·x1
 + 3 ·x2
 + 2 ·x3
 + x4
 + 2 ·x5
 ≤
 18

xi ≥ 0.

 317:[x1=4; x2=0; x3=0; x4=6; x5=0; t1=4; t2=0; t3=5; max=146]

## 23Economic and mathematical methods.

/en/Economic and mathematical methods/Demand point, Internal name: ZeconomZ Generate or Make Task ,

Name:
Var.: 318. Group:           Day/Mo/Year:

nothing

 318:[(3, 2, 6)]

/en/Economic and mathematical methods/Suppliers and traders, Internal name: ZmekoniZ Generate or Make Task ,

Name:
Var.: 319. Group:           Day/Mo/Year:

Suppliers and traders decided to unite in one company. The company's income is set by the function Y=3575·x13 /11·x27/11, where x1 - number of sellers, x2 - number of suppliers. Salary seller 325, supplier - 2275. Find the optimal composition of the company, maximizing profits.

 319:[x1=81, x2=27, W=8775]

/en/Economic and mathematical methods/Street vendor, Internal name: ZmekoniiZ Generate or Make Task ,

Name:
Var.: 320. Group:           Day/Mo/Year:

A street vendor buys goods at a price of 3 per piece. Sales of y is associated with assign them a price of v by the formula y=1200−80·v. What is the optimal amount of goods should be purchased by the seller and what should be the optimal price of selling the goods?

 320:[y=480, v=9]

/en/Economic and mathematical methods/Price and costs, Internal name: ZmekoniiiZ Generate or Make Task ,

Name:
Var.: 321. Group:           Day/Mo/Year:

The price of v of the company's products is associated with the sales volume y dependence v(y)=66−1·y, the cost of production I(y)=1·y3+(−22)·y2 +(102)·y +(0). Find the optimal volume of sales, the price of goods, income and costs at maximum profit.

 321:[y=4, v=62, I=120, W=128]

/en/Economic and mathematical methods/Cross-sectoral balance (simple), Internal name: ZleonteviZ Generate or Make Task ,

Name:
Var.: 322. Group:           Day/Mo/Year:

On the island of Chunga-Chang in the production of 1 ton of coconuts natives eat 400 kg. coconuts and 600 kg. bananas. In the production of 1 ton. bananas they eat 100 kg. coconuts and 700 kg. bananas.
You need to create a matrix of direct costs and find the matrix of total material costs:

 (  )       (  )
Find out what the harvest of coconuts (           ) and bananas.(           ) need to plan for export from Chung-Chang 40 t. coconuts and 30 t. bananas.

322:[ (
 0.4
 0.1
 0.6
 0.7
) - direct cost matrix, (
 2.5
 0.83
 5
 5
) - full cost matrix, 125 T. coconuts and 350 T. bananas. ]

/en/Economic and mathematical methods/Interindustry balance, Internal name: ZobmenZ Generate or Make Task ,

Name:
Var.: 323. Group:           Day/Mo/Year:

The matrix of direct costs A and the vector of the final product Y are given. To find a vector of the gross domestic product of X. Create a scheme of inter-sectoral balance.
Y=(
 100
 200
 400
), A=(
 0.3
 0.1
 0.2
 0.3
 0.1
 0.2
 0.1
 0.3
 0.1
)

323:[X=(
 402.174
 502.174
 656.522
),
Z=(
 120.652
 50.2174
 131.304
 120.652
 50.2174
 131.304
 40.2174
 150.652
 65.6522
)]

/en/Economic and mathematical methods/Budget set, Internal name: ZbudgetZ Generate or Make Task ,

Name:
Var.: 324. Group:           Day/Mo/Year:

Nothing

 324:[(16, 8)]

/en/Economic and mathematical methods/Price supply and demand, Internal name: ZcsprZ Generate or Make Task ,

Name:
Var.: 325. Group:           Day/Mo/Year:

Given the dependence of demand on price: D(p)=496−8·p and the dependence of supply on price: S(p)=119+5·p. Find an equilibrium price (           ) and revenue at an equilibrium price (           ). Is the equilibrium state stable? (           ). Find the price at which the revenue from the sale of goods is maximum (           ) and find this maximum revenue (           ).

 325:[ Equilibrium price:29 Revenues at the Equilibrium price: 7656 is stable. Price at maximum revenue:31 Maximum revenue:7688 ]

/en/Economic and mathematical methods/Selling stuff, Internal name: ZshtuchkiZ Generate or Make Task ,

Name:
Var.: 326. Group:           Day/Mo/Year:

Someone decided to buy "things" in bulk at 50 and resell to students. He did two surveys to study demand. In the first 9 students from 17 has agreed to buy the thing for 80. At the second 7 from 17 agreed to buy for 130. The city is home to 7000 students. At what price is it best to resell things to students? (answer:                      ). How many "things" will they buy? (answer:                       ). How much can you earn? (answer:                       ).

 326:[ Demand function: y=−16.48x+5024.4, price: 177.4, quantity  2101, revenue: 267667]

/en/Economic and mathematical methods/Sly selling stuff, Internal name: ZshtuchkiiZ Generate or Make Task ,

Name:
Var.: 327. Group:           Day/Mo/Year:

Someone decided to buy "things" in bulk at 70 and resell to students in a cunning way: in the morning at the entrance at one price and in the evening at the exit at another, at a discount. He did two surveys to study demand: In the first survey 7 students from 13 agreed to buy a "thing" for 110, On the second survey 3 of 13 agreed to buy for 150. In University come 2000 students. Morning at what price (answer:                       ) and at what evening (answer:                       ) is best Someone to resell stuff to students? How many "things" will they buy? (answer:                       ). How much can Someone make from this? (answer:                       ).

 327:[Demand function: y=−15.375x+2768.25, the price in the morning: 143.4, price in the evening: 106.7, the quantity: 1128, income: 62067.5]

/en/(not checked) Test tasks/Convergence of series, Internal name: ZintshZ Generate or Make Task ,

Name:
Var.: 328. Group:           Day/Mo/Year:

Find the interval of convergence of the series:
 n=∞∑ n=1 (5·x−1)n·(3·n8−9) (6)n·(9·n8.9−7·n)

 328:[[−1;[7/5])]

Name:
Var.: 329. Group:           Day/Mo/Year:

 329:

Name:
Var.: 330. Group:           Day/Mo/Year:

 330:

## 26Sample.

Name:
Var.: 331. Group:           Day/Mo/Year:

 331:

29/11/2020 19:03:31
1:[x=−3] 2:[x=1] 3: 4:[−3] 5: 6:[34, 10] 7: 8: 9: 10:[4, 5.5] 11:[−1] 12: 13:[(3, −7), (15, −43)] 14:[x1=14; x2=−46] 15:[x1=9, x2=5] 16:[(0, 6), (8, 70)] 17:[21.1054] 18:[34/25] 19:[5.75%] 20: 21: 22:[15%] 23:[1400kg] 24:[x=3] 25:[x=4] 26:[(−∞; 3)∪[11;∞)] 27: 28:[(13;−12;5)] 29:[(−13;14;−10)] 30:[(−22;7;9)] 31:[(−3,−3,0)] 32:[(−3,0,−3)] 33:[(−1, 1, −4)] 34:[(1, −1, 3)] 35:[(3,4)] 36:[(194,40)] 37:[([(−125)/7];[(−5)/7]) ≈ (−17.857;−0.714)] 38:[(267,−75)] 39:[(−3,−3,1)] 40:[(3,3,−1)] 41:[(−2,−4,−3)] 42:[21·x +28·y +336=0, d=10, y=[(−3)/4]·x −12] 43:[a/b=4, (-12 -3 12 )] 44:[ 2mm
Picture Omitted
] 45:[ 2mm
Picture Omitted
, y=[1/4]·x +[(−17)/4], (17,0)] 46:[(7,16), (−7,14), (2,8)] 47:[(−1,−1)] 48:[(13,4), (6,5), (7,12) or (20,3), (21,10)] 49: 50: 51: 52:[(25; −5)] 53:[(28, 10, −26), (−36, −8, 22)] 54:[(2,2,1), (0,−6,−5)] 55:[(14, 20)] 56:[(−12 ,−34)] 57:[(−16 ,7 ,1)] 58:[For ABCD: (−9, −4), For ABDC: (−3, 6)] 59:[For ABCD: (8, −4, −12), For ABDC: (−2, −2, −6)] 60:[(88;3), (43;50) and (−6;−87), (−51;−40)] 61:[(124, 84)] 62:[(31, −22, 34)] 63:[(86, −82)] 64: 65:[−34] 66:[±(6, −4, −6)] 67:[λ·(1, −1,−7)] 68:[λ·(1, 7,2)] 69:[−3] 70:[(2,2)] 71: 72:[x=4, y=6] 73:[x=−1, y=−2, z=3] 74:[x=3, y=1, z=−2] 75:[x1=4, x2=3, x3=4] 76:[x1=1, x2=1, x3=−2, x4=2] 77:[(1 x3,2 x3, x3)] 78:[Formula to verify the solution: (1−3 x3,−2 x3, x3)] 79:[Formula to verify the solution: (1−3 x4,−2 x4,−1 x4 ,x4)] 80:[(1,−3,−3,1)·λ] 81:[(2 x3−2 x4, 1 x3−1x4, x3,x4)] 82:[x=−3; y=9] 83:[x=7; y=5] 84:[x=8; y=5; z=−3] 85:[x=1; y=−1; z=1] 86:[x1=−3; x2=−4; x3=−3; x4=2; x5=−5] 87:[x1=−7; x2=−8; x3=−9; x4=−1; x5=8] 88:[x=[(−6)/5], y=[1/2].] 89: 90:[4+2·i] 91:[−1 −7·i; 1 + 7·i] 92:[−1±3·i] 93:[−3 + 1·i, −1 −2·i;] 94:[−1 −1·i, −3 −2·i;] 95:[2 −3·i, 1 −2·i;] 96:[[4/3];k=2, −4;k=1] 97:[−3·x2 −8·x −3] 98:[9·x2 −8] 99:[−2·s1 s22+2·s12 s3−5·s2 s3] 100:[12±6·i, −3±6·i, x2 −24·x +180, x2 +6·x +45] 101:[(
 12
 −8
 14
 −9
)] 102:[(
 −6
 0
 −3
 6
 −4
 9
 −11
 5
 −13
)] 103:[A32=−10, A11=8] 104:[−1] 105: 106:[−4] 107:[(
 31
 13
 19
 8
);] 108:[(
 1
 −1
 −2
 1
 0
 −1
 0
 0
 1
);] 109:[(
 1
 0
 0
 0
 0
 1
 1
 0
 1
 −1
 1
 −1
 −1
 1
 0
 1
)
] 110:[(
 1
 1
 0
 0
 0
 −1
 1
 1
 1
 1
 0
 0
 1
 0
 0
 −1
 0
 0
 1
 0
 −1
 0
 1
 1
 1
)
] 111:[A−1=(
 2
 1
 1
 1
), X=(
 2
 1
 0
 3
), Y=(
 2
 1
 3
 2
)]
112:[A−1=(
 0
 −1
 −1
 1
), X=(
 2
 1
 2
 1
)]
113:[A−1=(
 −1
 0
 0
 0
 0
 −1
 1
 1
 1
), X=(
 0
 1
 1
 0
 0
 1
 1
 0
 0
)]
114:[A−1=(
 −1
 0
 0
 0
 1
 0
 −1
 1
 0
 1
 1
 0
 1
 0
 0
 1
), X=(
 1
 1
 0
 0
 1
 1
 0
 1
 0
 1
 0
 0
 1
 1
 1
 1
)]
115:[A−1=(
 1
 0
 0
 0
 0
 −1
 2
 1
 −1
 1
 −1
 0
 1
 0
 0
 1
 0
 −1
 1
 0
 0
 1
 0
 0
 1
), X=(
 1
 1
 1
 1
 1
 0
 0
 1
 0
 1
 0
 0
 1
 0
 0
 1
 1
 0
 1
 0
 0
 1
 0
 0
 0
)]
116:[A−1=(
 −1
 0
 0
 0
 −1
 1
 0
 1
 0
 0
 0
 0
 0
 −1
 0
 −1
 0
 0
 1
 1
 1
 1
 0
 0
 0
 0
 0
 0
 1
 0
 0
 0
 0
 0
 −1
 1
), X=(
 1
 1
 1
 0
 1
 0
 0
 1
 1
 1
 0
 0
 1
 0
 1
 0
 0
 0
 0
 1
 1
 0
 0
 1
 1
 0
 1
 0
 0
 1
 0
 1
 0
 1
 1
 1
)]
117:[(2 x3−2 x4, −2 x3+1x4, x3,x4)] 118:[(−2 x3+2 x4, 2 x3+2x4, x3,x4)] 119:[[1/41] (
 −9
 40
 40
 9
)] 120:[(
 1
 0
 −2
 0
 1
 0
 1
 2
 0
 1
 0
 0
 0
 1
 −1
 0
 0
 0
 0
 0
)
] 121:[(
 1
 −2
 4
 7
)] 122:[(
 −3
 2
 −12
 7
)] 123:[(
 −1
 −4
 −4
 2
 5
 2
 0
 0
 3
)] 124:[(
 4
 1
 2
 0
 3
 0
 −1
 −1
 1
)] 125:[1·x2 +4·y2 , (
 3
 −2
 2
 3
)/√{13}] 126:[(
 18
 9
), (
 1
 3
)] 127:[A=(
 4
 −5
 −3
 3
), B=(
 −1
 [(−5)/3]
 −1
 [(−4)/3]
)] 128:[ 2mm
Picture Omitted
] 129:[(
 3
 4
 −4
 −3
)] 130:[(
 1
 1
 2
 1
)] 131:[Tstd← a=(
 −1
 −2
 −2
 1
), Ta← std=(
 [(−1)/5]
 [(−2)/5]
 [(−2)/5]
 [1/5]
), Tstd← b=(
 −8
 −1
 −6
 8
), Tb← std=(
 [(−4)/35]
 [(−1)/70]
 [(−3)/35]
 [4/35]
) Tb← a=(
 [1/7]
 [3/14]
 [(−1)/7]
 [2/7]
), Ta← b=(
 4
 −3
 2
 2
), c=(
 4
 1
)std, c=(
 [(−6)/5]
 [(−7)/5]
)a, c=(
 [(−33)/70]
 [(−8)/35]
)b] 132:[1. (
 0
 0
 −1
 0
 1
 0
 1
 0
 0
) 2. (
 0
 1
 0
 −1
 0
 0
 0
 0
 1
) 3. (
 1
 0
 0
 0
 0
 −1
 0
 1
 0
)] 133:[[1/81]·(
 1
 −68
 −44
 76
 16
 −23
 28
 −41
 64
)] 134:[2(
 1
 −3
), −4(
 −1
 4
)] 135: 136:[A=(−4,−2,1,0), B=(1,1,6,1), C=(−4,−2,−20,126); B:=B−2·A; C: = C + A−2·B] 137:[(4,0,1), cosα = [(−4)/13]=−0.308] 138: 139:[B=(
 225
 135
 135
 225
), C=[1/5]·(
 −4
 −3
 3
 −4
)] 140:[B=(
 65
 45
 45
 185
), C=[1/5]·(
 3
 4
 −4
 3
)] 141:[[((x −5)2)/37]+[((y +2)2)/41]=1 , F1(5,−4), F2(5,0)] 142:[
3.1mm
Picture Omitted
In replacement there is −3 and 4. Equation: 4x2+1y2=100 or (x/5)2+(y/ 10)2=1 ] 143: 144:[c = −8 ·a−6 ·b] 145:[λ·(−1, 2, 3)] 146:[ 2mm
Picture Omitted
y=[19/86]·x +[99/86] ≈ 0.22·x +1.15 ] 147:[ 2mm
Picture Omitted
] 148:[[10/11] ≈ 0.909] 149: 150: 151:[e15] 152:[ y = 5 x −7 , x = −2, f(−2 + 0 ) = +∞,   x = −1,  f(−1 −0 ) = +∞,  f(−1 +0 ) = −∞] 153:[(2;−7)] 154:[(2;−5)] 155:[(3, 1), (5, 21)] 156:[Max=1, Min=7, inflection point=4] 157:[(−4, 2, −20), (−3, 9, 30)] 158:[] 159: 160: 161: 162:[] 163:[−6] 164:[ 2.5mm
Picture Omitted
] 165:[−6] 166:[2ln(x+1)−3ln(x+5)] 167:[2x2−3x−1ln(x−3)−4ln(x+2)] 168:[−4 ln(x+2)+2/(x+2)−3ln(x−2)] 169:[−2ln(x2 +14·x +50)+2arctg(x +7)] 170:[2ln(x2 −8·x +32)+3arctg((x −4)/4)] 171:[−2·x2 −3·x +3ln(x2 +6·x +10)−2arctg(x +3)] 172:[(−2x+[1/2]) e(2x+5)-1 e(2x+5 )] 173: 174: 175:[891·π] 176:[136·π] 177: 178: 179: 180: 181: 182:[0.0807692] 183:[ ∫02 dx ∫−8·x[(−12)/4]·x f(x,y) dy + ∫24 dx ∫−x2 + 8·x −28[(−12)/4]·x f(x,y) dy ] 184:[2·x2 +2·x −2] 185:[x2=3.35, x3=2.52575, x4=2.39125, ] 186:[
 x 3 2.65 2.575 y 7 1.06 0.049
] 187:[−3.504 ] 188:[(2,1.468)] 189:[10346, 2582] 190:[DF] 191:[CB] 192:[13.4375=1101.0111] 193:[11.3125=1011.0101] 194:[3.5 = 0/10000/1 1 0 0 0 0 0 0 0 0] 195:[4680 = 0/10001/1 0 1 0 0 0 0 0 0 0] 196:[40C0, 0.8125] 197:[−5.5 = 1/10000001/0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] 198:[0.881] 199:[0.985, 1, 0.918, 0.965, 1.95] 200:[bcbdaddddbddbdcddaa] 201:[ Table: a-010 b-011 c-00 d-1, compression Result: 0100110011100010100011100111010100100, after compression 37 bits.] 202:[Compressed file: 0 0 1 5 1 2 2 1 6 0. Dictionary: (0-a) (1-b) (2-c) (3-d) (4-aa) (5-ab) (6-ba) (7-abb) (8-bc) (9-cc) (10-cb) (11-bb) (12-baa). ] 203:[Compressed file: 0 0 1 4 3 7 5 7 8 10 0 2 9 0 or 00 00 001 100 011 111 0101 0111 1000 1010 0000 0010 1001 0000. K=58/48 ≈ 1.21. Dictionary: (0-00-a) (1-01-b) (2-10-c) (3-11-aa) (4-100-ab) (5-101-ba) (6-110-aba) (7-111-aaa) (8-1000-aaab) (9-1001-baa) (10-1010-aaaa) (11-1011-aaaba) (12-1100-aaaaa) (13-1101-ac) (14-1110-cb) (15-1111-baaa). ] 204:[Source file: abcbabaacbbb. Dictionary: (0-a) (1-b) (2-c) (3-d) (4-ab) (5-bc) (6-cb) (7-ba) (8-aba) (9-aa) (10-ac) (11-cbb) (12-bb). ] 205:[Source file: bbababcbbbbbcbc. Dictionary: (0-a) (1-b) (2-c) (3-d) (4-bb) (5-ba) (6-ab) (7-bab) (8-bc) (9-cb) (10-bbb) (11-bbbc) (12-cbc). ] 206:[Table: a-1 b-011 c-010 d-00, average code length: 1.76, relative redundancy: 0.0068, I: 1.75013, encoding result: 110101011.] 207:[2E3] 208:[F4] 209:[231, DF] 210: 211: 212:[ (
 1
 2
 3
 4
 a
 4
 4
 2
 4
 b
 4
 3
 1
 4
), In: 2, Out: 1. ] 213:[ (
 1
 2
 3
 a
 2
 3
 3
 b
 3
 1
 3
), In: 2, Out: 1. ] 214:[ (
 1
 2
 3
 4
 a
 1
 4
 1
 3
 b
 1
 3
 1
 1
), In: 2, Out: 2, 3, 4. ] 215:[ (
 1
 2
 3
 4
 5
 a
 1
 4
 1
 3
 1
 b
 1
 1
 1
 5
 3
), In: 2, Out: 2, 3, 4. ] 216:[ (
 1
 2
 3
 4
 5
 6
 a
 2
 5
 1
 4
 5
 3
 b
 2
 5
 1
 5
 5
 4
), In: 6, Out: 2, 4, 6. ] 217:[ (
 1
 2
 3
 4
 5
 6
 7
 a
 2
 4
 3
 6
 5
 5
 1
 b
 1
 2
 5
 4
 5
 5
 3
), In: 7, Out: 1, 3, 6. ] 218:[
 (1)
 (24)
 (36)
 (5)
 a
 (24)
 (36)
 (36)
 (24)
 b
 (5)
 (36)
 (36)
 (36)
, In: (1), Out: (24), (5). ] 219:[
 (1)
 (24)
 (36)
 (58)
 (79)
 a
 (24)
 (58)
 (36)
 (36)
 (58)
 b
 (24)
 (79)
 (36)
 (36)
 (58)
, In: (1), Out: (24), (58), (79). ] 220:[ FIRST: cd, a, d, c, acd, b, Tree: (S (A d (S (A d (S ac)) (B bb))) (B bb)). ] 221:[(0 1 1 1, 1 1 0 1, 1 0 0 1)] 222:[(0 1 0 1 1, 1 0 1 1 1, 0 1 1 1 0)] 223:[(1 1 0 1 0 0 1 0, 1 0 1 1 1 0 1 1, 0 1 0 0 0 1 1 0)] 224:[(0 0 1 0), (1 1 1 1), (1 0 1 1)] 225:[(1 x3 , 0 x3+1x4, x3,x4)] 226:[19/1309] 227:[293/15168683] 228:[x=390] 229:[x=26] 230:[x=64] 231:[b=523; Key=7782 7656 7404 9515 3214 1135 ; Crypt=12005; Mess=101001; ] 232:[b=8913; Key=9781 9562 4776 5204 6060 5598 ; Crypt=19761; Mess=010011; ] 233:[(3;9 ) → (48) , (4;8 ) → (34) ] 234:[(17;44 ) → (1342) , (19;15 ) → (2139) ] 235: 236: 237: 238:[ Message: 24, public key: 7 Crypt: (18, 19).] 239:[ (21, 21).] 240:[y=9−9·(x−2)+3·(x−2)2−42·(x−2)3+21·(x−2)4] 241:[y=8x+3] 242:[1x3y5+1y4 = c, y2] 243:[y=8(x+3)ln(x+3)+C(x+3)+31] 244:[C1·e−2x + C3·e2x ] 245:[C1·e−3x+C2·x ·e−3x + C3·e5x ] 246:[C1·e3x+C2·x ·e3x + (5·x2 −4·x )] 247:[C1·e−3x+C2·e−2x + (3·e−1x+1·cos(−4x)+2·sin(−4x))] 248:[C1·e3x+C2·e2x + (−5·x −3+1·x·e2x)] 249:[C1·e6x+C2 + (−2·x3 +3·x2 −4·x )] 250:[y = √{ 3 ·x2 +22·x14} ] 251:[y + [(C1)/(y2)] = 3x + C2] 252:[ y6 = C·e−30·x + [6/29] ·e−1 ·x ] 253: 254:[y = [33/(C·x−30 −5·x3)] + 6·x−3] 255: 256:[ {
 x = (C +[27/10] ·ln|p + [(−3)/10]|) ·p−2 ,
 y = (2 + [(−10)/3]·p) ·(C + [27/10] ·ln|p + [(−3)/10]|) ·p−1 +9·ln|p| .
] 257: 258: 259:[√{10}] 260: 261: 262:[ (
 y1
 y2
) = C1 ·e9·x ·(
 1
 2
) + C2 ·e8·x ·(
 −2
 −3
)] 263: 264:[ (
 y1
 y2
) = C1 ·e3·x ·(
 2
 3
) + C2 ·e2·x ·(
 −1
 −2
) + x ·(
 0
 −3
) + (
 −1
 2
) ] 265: 266:[ lim = −3; y=−3·x−2 +3·x−1 +3 ·x−1 ·lnx] 267: 268:[4, y=−4·e−4x+5·e−6x +2·cos(3x)−1·sin(3x)] 269: 270:[1:[101/238]=0.42437, 2:[80/101]=0.792079] 271:[0.996847] 272:[[10/15]=[2/3]=0.667] 273: 274:[0.72;0.22] 275:[M=4, D=11] 276:[0.693] 277: 278:[0.314914,0.40741] 279:[(3.2234;4.6326)] 280:[2.994;0.211335; (2.81036;3.17764)] 281: 282:[[2/85] ≈ 0.024] 283:[[194/33] ≈ 5.879] 284:[∑ = 37, ae, bf, cd] 285:[∑ = 53, ag, be, cf, dh] 286:[∑ = 91, af, bh, cg, dj, ei] 287:[∑ = 128, ai, bk, cg, dj, eh, fl] 288:[∑ = 150, ak, bj, cm, dl, ei, fn, gh] 289:[] 290:[13548.8] 291:[15196.6] 292:[1255.85] 293:[1118.19] 294:[7.25508] 295:[2.37857] 296:[5.28572] 297:[2.66901] 298:[11.4621] 299:[A1=1236, B1=1193.61] 300:[35.0435] 301:[−6, (2,7)] 302:[2.2] 303:[[(−3)/4], ([3/8], [5/8]), ([1/4], 0, [3/4])] 304:[ The strategy of the left of the player: (
 [5/8]
 [3/8]
 0
), Strategy the top player: (
 0
 [5/8]
 [3/8]
) price games: [23/8]] 305:[(6, 4, 0,18, 0, 0), L(x)=−20] 306:[xij=(
 0
 0
 30
 40
 10
 0
 50
 0
 40
) , L=2410] 307:[X=(0,0,−2,5), Y=(−2,1), L=−47] 308:[(5,2), 21, 2−6] 309:[ 2.7mm
Picture Omitted
,
A - 4, B - 17, Profit - 1101.] 310:[ 2.7mm
Picture Omitted
,
xi=(14; 3), L(x)=L(y)=487, yi=(1; 0; 2)] 311:[t1=1; t2=5; t3=0; max=min=102] 312:[t1=0; t2=5; t3=1; t4=0; max=min=109] 313:[x1=6; x2=0; t1=3; t2=0; max=18] 314:[x1=0; x2=4; x3=6; t1=5; t2=3; t3=0; max=250] 315:[x1=3; x2=0; x3=0; x4=6; t1=4; t2=0; t3=0; t4=1; max=111] 316:[x1=2; x2=0; x3=0; x4=6; x5=6; t1=4; t2=0; t3=3; t4=5; t5=0; max=318] 317:[x1=4; x2=0; x3=0; x4=6; x5=0; t1=4; t2=0; t3=5; max=146] 318:[(3, 2, 6)] 319:[x1=81, x2=27, W=8775] 320:[y=480, v=9] 321:[y=4, v=62, I=120, W=128] 322:[ (
 0.4
 0.1
 0.6
 0.7
) - direct cost matrix, (
 2.5
 0.83
 5
 5
) - full cost matrix, 125 T. coconuts and 350 T. bananas. ] 323:[X=(
 402.174
 502.174
 656.522
),
Z=(
 120.652
 50.2174
 131.304
 120.652
 50.2174
 131.304
 40.2174
 150.652
 65.6522
)] 324:[(16, 8)] 325:[ Equilibrium price:29
Revenues at the Equilibrium price: 7656
is stable.
Price at maximum revenue:31
Maximum revenue:7688 ] 326:[ Demand function: y=−16.48x+5024.4, price: 177.4, quantity  2101, revenue: 267667] 327:[Demand function: y=−15.375x+2768.25, the price in the morning: 143.4, price in the evening: 106.7, the quantity: 1128, income: 62067.5] 328:[[−1;[7/5])] 329: 330: 331:

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On 29 Nov 2020, 19:03.