Higher mathematics [Topic 7-12]
Replenishment date: 01.04.2024
Contents: Higher mathematics 7-12.pdf (280.92 KB)
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Description
In ancient China, matrices were called...
Answer type: Single choice • with the choice of one correct answer from several proposed options
"smart rectangles"
"beautiful trapezoids"
"beautiful triangles"
"magic squares"
Gabriel Cramer published "Cramer's Rule" in...
Answer type: Single choice • with the choice of one correct answer from several proposed options
November 1781, XNUMX
November 1751, XNUMX
November 1741, XNUMX
November 1791, XNUMX
The graph of an odd function is symmetrical with respect to...
Answer type: Single choice • with the choice of one correct answer from several proposed options
y-axis
x-axis
origin
The graph of the solution to a differential equation is called... a curve
Response type: Text response
The graph of an even function is symmetrical with respect to...
Answer type: Single choice • with the choice of one correct answer from several proposed options
y-axis
x-axis
origin
Given a vector a = {2, 3, 2}. Find a vector x collinear to vector a and satisfying the condition (x, a) = 34. @6.1.png
Answer type: Single choice • with the choice of one correct answer from several proposed options
x = {4, 3, 4}
x = {7, 6, 7}
x = {4, 6, 4}
Given a matrix polynomial f(A) = 3A2– 5A + 2. We need to calculate its value. Give a solution method.
Answer type: Single choice • with the choice of one correct answer from several proposed options
Find the value of A², multiply by 3, multiply matrix A by -5, add the resulting matrices, add to it a matrix with elements of the main diagonal equal to 2.
Find the value of A², multiply by 3, multiply matrix A by -5, add the elements of the resulting matrices and add 2 to this value.
Find the inverse matrix, multiply it by 3, multiply matrix A by -5, add the elements of the resulting matrices and add 2 to this value.
Given an indefinite integral ∫ sinx cos5 xdx. Calculate its value.
Answer type: Single choice • with the choice of one correct answer from several proposed options
1/2 ⋅ tan(x²) + C.
−3¹⁻⁵ˣ / 5ln3 + C.
−cos⁶x / 6 + C.
Given a definite integral ∫ (√x /(1 + √x))dx, x=0..1. Calculate its value. @9.2.png
Answer type: Single choice • with the choice of one correct answer from several proposed options
√(3)π / 3 − ln2
1/3
2ln2 − 1
Given a matrix |A| =│(1, 0, 1), (2, 3, 5), (0, 4, 8)│. Is there an inverse of a given matrix and why? @2.1.png
Answer type: Single choice • with the choice of one correct answer from several proposed options
Exists because its determinant is nonzero.
Does not exist because the rank of the matrix is 3.
Exists because this matrix can be transposed.
Given a matrix A = ((1, 0, 1), (2, 3, 5), (0, 4, 8)) What is the determinant of this matrix? Will it be the same as the determinant of the transposed matrix? @2.2.png
Answer type: Single choice • with the choice of one correct answer from several proposed options
The determinant is 12, it will match.
The determinant is 12, it will not match.
The determinant is 24, it will match.
The determinant is 24, it will not match.
Given a system of equations {x₁ + 2 ⋅ x₂ − x₃ = 1, −3 ⋅ x₁ + x₂ + 2 ⋅ x₃ = 0, x₁ + 4 ⋅ x₂ + 3 ⋅ x₃ = 2 When solving the equation by the Gaussian method, what actions need to be performed? @07_0.jpg
Answer type: Single choice • with the choice of one correct answer from several proposed options
Write the extended system matrix; perform algebraic transformations; obtain an equivalent system of equations; calculate the value of free unknowns.
Write the extended system matrix; perform elementary transformations; obtain an equivalent system of equations; perform a reverse Gaussian move, calculating the values of the unknowns.
Write the extended system matrix; perform elementary transformations; obtain an equivalent system of equations; calculate the values of the unknowns by fitting.
Given a system of equations {x₁ + 2 ⋅ x₂ − x₃ = 1, −3 ⋅ x₁ + x₂ + 2 ⋅ x₃ = 0, x₁ + 4 ⋅ x₂ + 3 ⋅ x₃ = 2. How many solutions does this system of equations have and why? .
Answer type: Single choice • with the choice of one correct answer from several proposed options
"smart rectangles"
"beautiful trapezoids"
"beautiful triangles"
"magic squares"
Gabriel Cramer published "Cramer's Rule" in...
Answer type: Single choice • with the choice of one correct answer from several proposed options
November 1781, XNUMX
November 1751, XNUMX
November 1741, XNUMX
November 1791, XNUMX
The graph of an odd function is symmetrical with respect to...
Answer type: Single choice • with the choice of one correct answer from several proposed options
y-axis
x-axis
origin
The graph of the solution to a differential equation is called... a curve
Response type: Text response
The graph of an even function is symmetrical with respect to...
Answer type: Single choice • with the choice of one correct answer from several proposed options
y-axis
x-axis
origin
Given a vector a = {2, 3, 2}. Find a vector x collinear to vector a and satisfying the condition (x, a) = 34. @6.1.png
Answer type: Single choice • with the choice of one correct answer from several proposed options
x = {4, 3, 4}
x = {7, 6, 7}
x = {4, 6, 4}
Given a matrix polynomial f(A) = 3A2– 5A + 2. We need to calculate its value. Give a solution method.
Answer type: Single choice • with the choice of one correct answer from several proposed options
Find the value of A², multiply by 3, multiply matrix A by -5, add the resulting matrices, add to it a matrix with elements of the main diagonal equal to 2.
Find the value of A², multiply by 3, multiply matrix A by -5, add the elements of the resulting matrices and add 2 to this value.
Find the inverse matrix, multiply it by 3, multiply matrix A by -5, add the elements of the resulting matrices and add 2 to this value.
Given an indefinite integral ∫ sinx cos5 xdx. Calculate its value.
Answer type: Single choice • with the choice of one correct answer from several proposed options
1/2 ⋅ tan(x²) + C.
−3¹⁻⁵ˣ / 5ln3 + C.
−cos⁶x / 6 + C.
Given a definite integral ∫ (√x /(1 + √x))dx, x=0..1. Calculate its value. @9.2.png
Answer type: Single choice • with the choice of one correct answer from several proposed options
√(3)π / 3 − ln2
1/3
2ln2 − 1
Given a matrix |A| =│(1, 0, 1), (2, 3, 5), (0, 4, 8)│. Is there an inverse of a given matrix and why? @2.1.png
Answer type: Single choice • with the choice of one correct answer from several proposed options
Exists because its determinant is nonzero.
Does not exist because the rank of the matrix is 3.
Exists because this matrix can be transposed.
Given a matrix A = ((1, 0, 1), (2, 3, 5), (0, 4, 8)) What is the determinant of this matrix? Will it be the same as the determinant of the transposed matrix? @2.2.png
Answer type: Single choice • with the choice of one correct answer from several proposed options
The determinant is 12, it will match.
The determinant is 12, it will not match.
The determinant is 24, it will match.
The determinant is 24, it will not match.
Given a system of equations {x₁ + 2 ⋅ x₂ − x₃ = 1, −3 ⋅ x₁ + x₂ + 2 ⋅ x₃ = 0, x₁ + 4 ⋅ x₂ + 3 ⋅ x₃ = 2 When solving the equation by the Gaussian method, what actions need to be performed? @07_0.jpg
Answer type: Single choice • with the choice of one correct answer from several proposed options
Write the extended system matrix; perform algebraic transformations; obtain an equivalent system of equations; calculate the value of free unknowns.
Write the extended system matrix; perform elementary transformations; obtain an equivalent system of equations; perform a reverse Gaussian move, calculating the values of the unknowns.
Write the extended system matrix; perform elementary transformations; obtain an equivalent system of equations; calculate the values of the unknowns by fitting.
Given a system of equations {x₁ + 2 ⋅ x₂ − x₃ = 1, −3 ⋅ x₁ + x₂ + 2 ⋅ x₃ = 0, x₁ + 4 ⋅ x₂ + 3 ⋅ x₃ = 2. How many solutions does this system of equations have and why? .
Additional Information
Given the function: z=x2-2xy2+y3. Find the second order partial derivatives for this function.Answer type: Single choice • with the choice of one correct answer from several proposed options
-6x+7y.
-4x+8y.
-4x+6y.
Given a differential equation: (2x / y²) ⋅ dx + (y² − 2x²) / y⁴ ⋅ dy = 0. Solve this equation. @11.1.png
Answer type: Single choice • with the choice of one correct answer from several proposed options
3x² / 2y³ + (−2) / y = C₁.
2x² / 2y³ + (−4) / y = C₁.
2x² / 2y³ + (−1) / y = C₁.
Given a differential equation: y´+2y=4x. Solve this equation.
Answer type: Single choice • with the choice of one correct answer from several proposed options
5x-2+C⋅e⁻²ˣ.
4x-1+C⋅e⁻²ˣ.
2x-1+C⋅e⁻²ˣ.
A second order linear differential equation is given: y´´-4y´+5y=0. Solve this equation.
Answer type: Single choice • with the choice of one correct answer from several proposed options
y = c₁e²ˣcos5x + c₂e²ˣsin5x.
y = c₁e²ˣcos3x + c₂e²ˣsin3x.
y = c₁e²ˣcos2x + c₂e²ˣsin2x.
A second order linear differential equation is given: y´´-4y´+5y=0. Solve this equation
Answer type: Single choice • with the choice of one correct answer from several proposed options
y = 2c₁eˣ + c₂ ⋅ xeˣ.
y = 3c₁eˣ + 2c₂ ⋅ xeˣ.
y = c₁eˣ + c₂ ⋅ xeˣ.
Given a linear differential equation of the second order: y´´+y´-2y=0. Give a solution to this equation.
Answer type: Single choice • with the choice of one correct answer from several proposed options
y=c₁⋅eˣ+c₂⋅e⁻²ˣ.
y=c₁⋅eˣ+2c₂⋅e⁻²ˣ.
y=2c₁⋅eˣ+c₂2e⁻²ˣ.
An ordinary first order differential equation is given: y´ + y/x = x² ⋅ y⁴. Give a solution to this equation. @11.png
Answer type: Single choice • with the choice of one correct answer from several proposed options
z=(-3⋅ln|x|+C)⋅x³.
z=(-6⋅ln|x|+C)⋅x².
z=(-4⋅ln|x|+C)⋅x³.
Vectors p and a are given. Find the unit vector of the vector p (a vector of unit length and the same direction as the vector p) perpendicular to the vector a and the axis OX ⋅ pª ⊥ a = {3, 6, 8} and pª ⊥ OX. @4.2.png
Answer type: Single choice • with the choice of one correct answer from several proposed options
pª = ±(0; −0,8; 0,6}
pª = ±(0; −0,6; 0,6}
pª = ±(0; −0,8; 0,3}
The following matrices are given: A₂ = ((1, 2), (3, 6)), B₂ = ((2, 6), (−1, 3)). An algebraic operation was performed on these matrices, as a result of which the matrix C₂ = ((3, 8), (2, 9)) was obtained. What algebraic operation was performed? @01_0.jpg
Answer type: Single choice • with the choice of one correct answer from several proposed options
Matrix-matrix multiplication
Matrix-to-matrix addition
Matrix difference
The following matrices are given: A₂ = ((1, 2), (3, 6)), B₂ = ((2, 6), (−1, 3)). An algebraic operation was performed on these matrices, as a result of which the matrix C₂ = ((3, 8), (2, 9)) was obtained. What algebraic operation was performed? @1,1.png
Answer type: Single choice • with the choice of one correct answer from several proposed options
Matrix-matrix multiplication.
Matrix-to-matrix addition.
Subtracting a matrix from a matrix.
Two planes intersect if they have...
Answer type: Single choice • with the choice of one correct answer from several proposed options
one common point
two common points
infinitely many common points
Two straight lines y1=7x+5 and y2=7x-5 on the plane...
Answer type: Single choice • with the choice of one correct answer from several proposed options
parallel
intersect
may intersect or be parallel