Contents

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

Solve the equation:

−3+(−4)·9−3·(−7)−6·x=0

        
Answer:

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

36−(2 −x)2−(6−x)·(x+6)=0

        
Answer:

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

To find the roots of the equation

x −7 x −9 = x −1 x−7

        
Answer:

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

To find the roots of the equation

(−x − 4)2 − (x+2)2 = 0

        
Answer:

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?         
Answer:

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?         
Answer:

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?         
Answer:

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.         
Answer:

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

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

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

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

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

√ 8·x +12 1− x =1

        
Answer:

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

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

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.         
Answer:

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

Solve the equation:

| x 2 +8| −15=0

        
Answer:

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

Solve the equation:

|x−7|+4=6

        
Answer:

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.         
Answer:

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

Solve the equation

2.4+7.4·x √ 7.3·x + 0.4·x2 = 8.7

        
Answer:

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

Write the answer in the form of an irreducible fraction.

17 35 − 17 45 5 63 =     (                     ) (                     )

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?         
Answer:

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?         
Answer:

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

Nothing         
Answer:

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?         
Answer:

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?         
Answer:

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

Solve the equation::

52·log5 x +6·log5 ⎛ ⎝ 5x 25 ⎞ ⎠ −15=0

        
Answer:

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

Solve the equation:

log16 x +14 x+5 = 1 4

        
Answer:

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

Solve the inequality:

1 x−3 ≤ 1 8

        
Answer:

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

Find the smallest integer solution of the inequality:

1 x−7 < 1 −5

        
Answer:

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).         
Answer:

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.         
Answer:

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

empty

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).         
Answer:

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).         
Answer:

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.         
Answer:

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.         
Answer:

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.         
Answer:

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.         
Answer:

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)         
Answer:

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 (           ;           ).

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.         
Answer:

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 (          ;           ;           ).

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: (          ;           ;           ).

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.         
Answer:

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).         
Answer:

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.

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: (    ,    ).

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.         
Answer:

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.
Answer:

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].         
Answer:

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).         
Answer:

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?         
Answer:

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?         
Answer:

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.         
Answer:

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.         
Answer:

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.         
Answer:

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

Find the coordinates of the vector (2·a+b).

2.7mm Picture Omitted

        
Answer:

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).         
Answer:

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).         
Answer:

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.         
Answer:

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.         
Answer:

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.         
Answer:

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.         
Answer:

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.         
Answer:

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

Nothing         
Answer:

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

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

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).         
Answer:

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}.         
Answer:

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).         
Answer:

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).         
Answer:

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.         
Answer:

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.         
Answer:

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).         
Answer:

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

⎧ ⎪ ⎨ ⎪ ⎩   8·x   −1·y    =   26   −4·x   +1·y    =   −10

        
Answer:

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

⎧ ⎪ ⎨ ⎪ ⎩   +1·y    =   −2   −1·x   +1·y   +1·z    =   2   +1·z    =   3

        
Answer:

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

        
Answer:

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

        
Answer:

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

        
Answer:

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

        
Answer:

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

Answer:

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

Answer:

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

        
Answer:

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

        
Answer:

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

⎧ ⎪ ⎨ ⎪ ⎩   −20 ·x   −9 ·y    =   −21   −11 ·x   −5 ·y    =   −12

        
Answer:

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

⎧ ⎪ ⎨ ⎪ ⎩   20 ·x   −27 ·y    =   5   −2 ·x   + 3 ·y    =   1   −19 ·x   + 26 ·y    =   −3

        
Answer:

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

⎧ ⎪ ⎨ ⎪ ⎩   x   −3 ·y   −2 ·z    =   −1   −x   + 5 ·y   + 3 ·z    =   8   x   −2 ·y   −z    =   1

        
Answer:

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

        
Answer:

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

        
Answer:

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

        
Answer:

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= [((    ))/((    ))].

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

(5+4·i)·(1−i)+(−2+4·i)=x+3·i
x=

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

[(−6+22·i)/(1+5·i)]=

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

√{−48 + 14·i}=      

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

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

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

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

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

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

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         
Answer:

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.         
Answer:

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).         
Answer:

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.         
Answer:

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.         
Answer:

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.         
Answer:

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

( 3 ·(

−2	2
−1	2

) + (

2	−2
−1	−1

) ) ·(

−1	1
2	−1

)=

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

(

−3	−3
1	−3
−3	2

) ·(

3	−1	3
−1	1	−2

) =

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=         .

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

det ⎛ ⎜ ⎜ ⎜ ⎝ −1 2 1 −2 1 0 0 −1 −1 ⎞ ⎟ ⎟ ⎟ ⎠ =

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

det ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 7 6 −4 10 5 5 −3 5 −8 −7 5 −20 5 4 −3 15 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ =

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 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ =

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

Find the inverse matrix to the matrix
(

8	−13
−19	31

)         
Answer:

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

Find the inverse matrix to the matrix
(

0	1	1
−1	1	−1
0	0	1

)         
Answer:

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

)         
Answer:

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

)         
Answer:

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

)