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Oliy matematika fani

Total questions: 160

Worksheet time: 5hrs 50mins

Name
Class
Date
1.

For the sets M = {1, … , 20}, determine B \ A. Here A = {10, 13, 15}, B = {10, 11, 15}.

a)

{11}

b)

{10}

c)

{10,13}

d)

{15}

2.

For the sets M = {1, … , 20}, determine A ∩ B. Here A = {5, . . . , 15}, B = {12, . . . ,19}.

a)

{12,13,14,15}

b)

{11,12,13,14}

c)

{10,11,12,15}

d)

{10,11,12,13}

3.

For the sets M = {1, … , 20}, determine A \ B. Here A = {10,. . . , 15}, B = {11,. . . , 17}.

a)

{10}

b)

{10,11}

c)

{10,11,13,15}

d)

{15}

4.

The set formed by elements that belong to at least one of the sets A and B is called...

a)

Union

b)

Intersection

c)

Difference

d)

Cartesian product

5.

The set of all elements in A that are not in B is called...

a)

Difference

b)

Union

c)

Intersection

d)

Cartesian product

6.

Find the intersection of the sets.

a)

{3;4;5}

b)

{1;2;6}

c)

{3;8;4}

d)

{2;4;6}

7.

Find the union of the sets.

a)

{1;3;2;4;5;6;7;8;9;10}

b)

{1;3;2;4;7;8}

c)

{1;6;2;7;3;8}

d)

{11;12;13;1;2;3;21;22;23}

8.

If A=1,2,3,4,7,9 and B=2,3,4,8,10,11, find B\A.

a)

8,10,11

b)

1,2,3.4,6,8

c)

6,8

d)

2,4

9.

If A=1,2,3,4,7,9 and B=2,3,4,8,10,11, find A\B.

a)

1,7,9

b)

1,2,3.4,6,8

c)

1,3

d)

2,4

10.

If A=2,4,6,8 and B=1,2,3,4, find B\A.

a)

1,3

b)

1,2,3.4,6,8

c)

6,8

d)

2,4

11.

If A=2,4,6,8 and B=1,2,3,4, find AB.

a)

1,2,3.4,6,8

b)

1,3

c)

6,8

d)

2,4

12.

Find the solution of the following system.

a)

3,-1

b)

3,-2

c)

-1,2

d)

2,2

13.

Find the solution of the following system.

a)

No solution

b)

Infinitely many solutions

c)

3,-1,2

d)

-1,2,2

14.

In solving a system of linear equations using the Gauss method, the following rule applies: if the system is in 'triangular' form, it has...

a)

A unique solution

b)

No solution

c)

Infinitely many solutions

d)

Only the zero solution

15.

When solving a system of linear equations using the Gauss method, if it reaches a triangular form and does not maintain 0=0, then the system...

a)

Has a unique solution

b)

Has no solution

c)

Has a finite number of solutions

d)

Has infinitely many solutions

16.

When solving a system of linear equations using the Gauss method, if it reaches a trapezoidal form and does not maintain 0=1, then the system...

a)

Has infinitely many solutions

b)

Has no solution

c)

Has a finite number of solutions

d)

Has a unique solution

17.

Solve using the Kramer rule.

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18.

Solve the system using the Kramer rule.

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19.

Solve the system.

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20.

Solve the system using the Kramer rule.

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21.

Solve the system using the Gauss method.

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22.

Calculate the following determinant.

a)

1

b)

2

c)

3

d)

4

23.

Calculate the following determinant.

a)

20

b)

25

c)

15

d)

10

24.

If all elements of a row in a determinant are multiplied by a number...

a)

The determinant is multiplied by that number

b)

The determinant is multiplied by 3

c)

The determinant remains unchanged

d)

The determinant is multiplied by 2

25.

If one row of a determinant is a linear combination of the other rows...

a)

The determinant equals zero

b)

The determinant does not equal zero

c)

The determinant equals 1

d)

The determinant equals the product of the elements on the main diagonal.

26.

If all elements of one column of a determinant are multiplied by the algebraic complements of the corresponding elements of another column, the sum...

a)

Equals 0

b)

Is different from 0

c)

Equals 1

d)

Equals the determinant

27.

If one row of a determinant has all elements except one being zeros, such a determinant...

a)

Equals the product of that non-zero element and its algebraic complement

b)

Equals the sum of that non-zero element and its algebraic complement

c)

Equals 0

d)

Equals that non-zero element

28.

For determinants with two rows that are not identical and two rows that are proportional, the determinants are equal to...

a)

The second equals zero

b)

The first equals zero

c)

The first is different from zero, the second equals zero

d)

Both are different from zero

29.

A determinant has the following property: if a column of the determinant has elements that share a common factor...

a)

This factor can be factored out of the determinant

b)

Such a determinant equals zero

c)

This factor can be reduced

d)

Such a determinant equals that common factor

30.

Calculate the determinant.

a)

-3

b)

1

c)

10

d)

5

31.

If two columns of a determinant are swapped...

a)

Only its sign changes

b)

Its value remains unchanged

c)

It equals zero

d)

Its value...

32.

If we swap any two columns in a determinant, what happens?

a)

Only its sign changes

b)

Its value does not change

c)

It becomes zero

d)

Its value changes

33.

Calculate the determinant.

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34.

Calculate the determinant.

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35.

Calculate the determinant.

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36.

Calculate the determinant.

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37.

Calculate the determinant.

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38.

Calculate the determinant.

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39.

Calculate the determinant.

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40.

Calculate the determinant.

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41.

Calculate the determinant.

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42.

Show the solution of the system:

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43.

Show the solution of the system:

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44.

Calculate the product of matrices.

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45.

Find the inverse of the following matrix.

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46.

Find the inverse of the matrix 1211.

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47.

Calculate.

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48.

Determine the value of the determinant:

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49.

Determine the value of the determinant:

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50.

Show the solution of the system:

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51.

Find the equation of any straight line.

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52.

If it exists, how is the straight line positioned in the coordinate system?

a)

Parallel to the y-axis

b)

Parallel to the x-axis

c)

Parallel to the z-axis

d)

Passes through the origin

53.

If it exists, how is the straight line positioned in the coordinate system?

a)

Passes through the origin

b)

Parallel to the x-axis

c)

Parallel to the y-axis

d)

Parallel to the z-axis

54.

Which answer contains the formula for finding the distance from a point to a line?

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55.

Which answer contains the formula for finding the angle between two lines?

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56.

What is the condition for the parallelism of two lines?

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57.

What is the condition for the perpendicularity of two lines?

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58.

Find the equation of the line passing through two points.

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59.

Which answer contains the formula for finding the distance between two points?

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60.

Find the formula for converting from Cartesian coordinates to polar coordinates.

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61.

Find the general equation of the plane.

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62.

Find the coordinates of the point in polar coordinates.

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63.

The coordinate axes are rotated. Find the new coordinates of the point.

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64.

Write the equation of the line that forms a 120-degree angle with the x-axis and intersects the y-axis at point A(0;3).

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65.

Find the angle between two lines.

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66.

Write the equation of the plane perpendicular to the vector.

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67.

Find the area of the square with opposite vertices A(4;-2) and C(-1;3).

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68.

Given the vertices A(-3;2) and B(1;6) of an equilateral triangle, find the coordinates of vertex C.

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69.

Find the point Q that is at a distance d from point P(-2;7) on the y-axis.

4 lines
70.

Determine the polar coordinates of the point.

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71.

Find the symmetric point to a point with respect to a line.

4 lines
72.

Determine the direction vector of the following line.

a)

(-4;-3)

b)

(4;-3)

c)

(3;-4)

d)

(-3;4)

73.

Find the vector that passes through point Mo(2; -1) and point (3; -2).

4 lines
74.

Determine the direction vector of the following straight line?

a)

(-4;-3)

b)

(4;-3)

c)

(3;-4)

d)

(-3;4)

75.

Write the equation of the straight line passing through the point Mo(2; -1) and parallel to the vector (3; -2)?

4 lines
76.

Find the equation of the straight line that passes through the origin and is parallel to the line y=4x-3?

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77.

Calculate the angle between these straight lines?

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78.

Write the equation of the straight line that is perpendicular to the given straight line passing through the origin?

4 lines
79.

Find the distance from the point Mo(3;1; -1) to the plane 22x+4y-20z-45=0?

a)

3/2

b)

3

c)

5

d)

2

80.

Determine the angle coefficient of the following straight line? 9x+3y-1=0

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81.

Find the parameters k and b for the straight line x-2y+6=0?

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82.

Write the equation of the line given its general equation x-y+1=0 with respect to the segments?

4 lines
83.

Write the equation of the straight line passing through a point and the origin?

4 lines
84.

Determine the angle formed by the straight lines?

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85.

The vector product of vectors a and b is defined as c:

a)

c=[ab]

b)

[ab]c=0

c)

[a(b+c)]=[ab]+[ac]

d)

c=absin(a^b)

86.

Show the distribution law for the vector product.

a)

[a(b+c)]=[ab]+[ac]

b)

|[ab]|=ab

c)

[ab]=0

d)

c=absin(a^b)

87.

If the vector product of two vectors [ab] is multiplied by the third vector c, it is called...

a)

Mixed product

b)

Scalar product

c)

Vector product

d)

Two vector product

88.

What does the geometric meaning of the mixed product express?

a)

Volume of the parallelepiped

b)

Area of the parallelogram

c)

Volume of the prism

d)

Area of the triangle

89.

If the mixed product of vectors a, b, and c is equal to zero, it is said to be...

a)

Coplanar

b)

Collinear

c)

Perpendicular

d)

Parallel

90.

How many parts does the Oxyz coordinate system divide the space?

a)

8

b)

4

c)

6

d)

12

91.

Which plane does the point B(0;5;2) lie on?

(a)  

92.

What will be the signs of the coordinates of points located in different octants?

a)

Different

b)

Same

c)

Positive

d)

Negative

93.

What will be the coordinates of points lying in the coordinate plane?

a)

One coordinate is zero

b)

All coordinates are zero

c)

Two coordinates are zero

d)

All coordinates are different from zero

94.

What will be the coordinates of points lying on the coordinate axes?

a)

Two coordinates are zero

b)

One coordinate is zero

c)

All coordinates are zero

d)

All coordinates are different from zero

95.

What will be the coordinates of the point at the origin?

a)

All coordinates are zero

b)

All coordinates are different from zero

c)

Positive

d)

Negative

96.

Which plane does the point M(3;0;5) lie on?

a)

xz

b)

xy

c)

yz

d)

xyz

97.

Find the point symmetric to the point M(a;b;c) with respect to the coordinate planes.

(a)  

98.

Find the distance from the point A(12;-3;4) to the origin.

a)

13

b)

14

c)

15

d)

12

99.

Find the distance from the point A(5;4;-3) to the OX axis.

a)

5

b)

7

c)

8

d)

4

100.

What is the relationship between the cosines of the angles formed by a straight line with all three coordinate planes?

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101.

What is the condition for the parallelism of two straight lines in space? (the direction vectors of the lines)

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102.

What is the condition for the perpendicularity of two straight lines in space?

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103.

Find the canonical equation of a straight line in space.

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104.

Find the parametric equation of a straight line in space.

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105.

Find the formula for the angle between two straight lines in space.

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106.

What is a vector?

a)

A segment with direction

b)

A quantity defined by a scalar value

c)

A concept that is defined without definition

d)

A concept that is defined

107.

When are vectors equal if they are given by their coordinates?

a)

If corresponding coordinates are equal

b)

If the ratio of corresponding coordinates is equal

c)

If their directions are the same

d)

If their absolute values are equal

108.

What is called a vector that lies on a given straight line or on a straight line parallel to it?

a)

Direction vector

b)

Collinear vector

c)

Normal vector

d)

Coplanar vector

109.

What is called the vector perpendicular to the given line?

a)

Normal vector

b)

Directing vector

c)

Collinear vector

d)

Coplanar vector

110.

What condition must be met for two vectors to be collinear?

a)

If the ratio of their coordinates is equal

b)

If their directions are the same

c)

If they are perpendicular to each other

d)

If their absolute values are equal

111.

What does the geometric meaning of the vector product of two vectors express?

a)

Area of a parallelogram

b)

Area of a triangle

c)

Volume of a parallelepiped

d)

Area of a rhombus

112.

What are the coordinates of a vector called?

a)

The projections of this vector on all three coordinate axes

b)

The position of the point in space

c)

An undefined concept

d)

The values of x and y

113.

Find the projection of the vector a=4m+3n-p on the x-axis. (m=3i+5j+8k, n=2i-4j-7k, p=5i+j-4k)

a)

13

b)

14

c)

15

d)

16

114.

If a=4i+7j+3k and b=3i-5j+k, find the scalar product of vectors a and b.

a)

(ab)=-20

b)

(ab)=20

c)

(ab)=-10

d)

(ab)=10

115.

If a=3p-2q and b=p+4q, find the scalar product ab.

a)

ab=-5

b)

ab=5

c)

ab=-4

d)

ab=4

116.

Find the distance from a point to a straight line.

a)

8

b)

12

117.

Find the distance from the origin to a straight line.

a)

2

b)

5

c)

4

d)

3

118.

The geometric locus of points where the difference of distances from any point to two given points is constant is called...

a)

Hyperbola

b)

Parabola

c)

Circle

d)

Ellipse

119.

The geometric locus of points where the sum of distances from any point to two given points is constant is called...

a)

Ellipse

b)

Hyperbola

c)

Parabola

d)

Circle

120.

The geometric locus of points equidistant from a given straight line and a given point is called...

a)

Parabola

b)

Circle

c)

Ellipse

d)

Hyperbola

121.

Write the equation of the ellipse with semi-axes a=3, b=4.

4 lines
122.

Calculate the distance between the foci.

a)

5

b)

3

123.

Calculate the distance between the foci.

a)

10

b)

8

c)

12

d)

9

124.

Find the point that is 4 units away from the directrix of the parabola y=8x.

a)

(2; 4), (2; -4)

b)

(4;4), (4; - 4)

c)

(4;2), (-4;2)

d)

(-2;4), (2;4)

125.

Find y for the function.

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126.

Find y for the function.

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127.

Find y if it is.

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128.

Find y for the function.

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129.

Find y for the function.

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130.

Find y for the function.

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131.

Find y for the function.

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132.

Find y for the function.

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133.

Find y for the function.

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134.

Find the specific derivatives of the function U=xyz.

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135.

Find the specific derivatives of the function.

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136.

Find the specific derivatives of the function.

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137.

Find the specific derivatives of the function.

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138.

Find the definite integral.

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139.

Calculate the definite integral.

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140.

Calculate the definite integral.

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141.

Calculate the definite integral.

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142.

Calculate the definite integral.

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143.

Calculate the definite integral.

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144.

Calculate the definite integral.

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145.

Calculate the definite integral.

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146.

Calculate the definite integral.

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147.

Calculate the definite integral.

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148.

Find the volume of the solid formed by rotating the shape around the Ox axis.

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149.

If Z=du(0,1,2)=?

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150.

Which of the following is a homogeneous function?

a)

Z=x^2 +2xy-y^2

b)

Z=x^2 +x^2y-y^2

c)

Z=x^3 -xy+y^3

d)

Z=x^4 -2xy+y^2

151.

Take the partial derivative of Z=x^2+y^2 with respect to y.

a)

2y

b)

2x

c)

2x+2y

d)

2xy

152.

Take the partial derivative of Z=x^2+y^2 with respect to x.

a)

2x

b)

2y

c)

2x+2y

d)

2xy

153.

At points where local maximum and local minimum are achieved, the partial derivatives of the function are ...

a)

Equal to zero

b)

Equal to one

c)

Not existing

d)

Opposite signs

154.

Points where the partial derivatives are equal to zero are called ... points.

a)

Stationary

b)

Critical

c)

Maximum

d)

Minimum

155.

Find the sum of the series.

a)

1.5

b)

0.5

c)

1

d)

0

156.

Which of the following sets does not converge?

a)

[0,1)

b)

[0,2)

c)

(-1,1)

d)

[0,3)

157.

If a positive series is given, then it will be ... if it converges.

a)

Convergent

b)

Divergent

c)

Absolutely convergent

d)

Absolutely divergent

158.

If a positive series is given, then it will be ... if it diverges.

a)

Divergent

b)

Convergent

c)

Absolutely convergent

d)

Absolutely divergent

159.

To determine whether a given series converges or diverges, the terms of this series are compared with the corresponding terms of another known positive series. This criterion is called ...

a)

Comparison criterion

b)

D'Alembert criterion

c)

Cauchy criterion

d)

Cauchy's integral criterion

160.

Find the general solution of the differential equation.

a)

y=x^3+C

b)

y=3x^3+C

c)

y=6x+C

d)

y=x^3

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