Linear Algebra and Analytic Geometry

Vectors are called collinear if:
Answer type: Single choice • with the choice of one correct answer from several proposed options

they lie either on the same line or on parallel lines and are co-directed
they lie on the same line
they lie on parallel lines
they lie on the same line and on parallel lines
Vectors are called coplanar if
Answer type: Single choice • with the choice of one correct answer from several proposed options

they are parallel to the same plane or lie in the same plane and co-directed.
they are perpendicular to each other.
they are pairwise parallel.
they are parallel to the same plane or lie in the same plane.
Choose the correct matrix determinants:
Answer type: Multiple choice • with the choice of several correct answers from the proposed options

The determinant of the transposed matrix is ​​equal to the determinant of the original matrix
The determinant exists for any matrix
Multiplying all elements of a row or column of a determinant by some number λ is equivalent to multiplying the determinant by this number
If the matrix contains a zero row (column), then the determinant of this matrix is ​​zero:
Choose the correct properties of the vector product:
Answer type: Multiple choice • with the choice of several correct answers from the proposed options

a × b = b × a
The module of the cross product of two vectors a and b is equal to the area of ​​the parallelogram built on these vectors.
If the vectors are collinear, then the cross product is zero.
Calculate the area of ​​a triangle with vertices at points A(4;5), B(6;2) and C(2;3):
Response type: Text response

The double cross product is calculated by the formula:
Answer type: Single choice • with the choice of one correct answer from several proposed options

[a, [b, c]] = b[a ⋅ c] + c[a ⋅ b]
[a, [b, c]] = b(a ⋅ c) − c(a ⋅ b)
[a, [b, c]] = b(a ⋅ c) + c(a ⋅ b)
[a, [b, c]] = b[a ⋅ c] − c[a ⋅ b]
The division of complex numbers a + bi and c + di in algebraic form is performed according to the formula:
Answer type: Single choice • with the choice of one correct answer from several proposed options

(ac − bd + i(ad − bc)) / (c² + d²)
(ac + bd + i(ad − bc)) / (c² − d²)
(ac + bd + i(ad − bc)) / (c² + d²)
(ac + bd + i(ad + bc)) / (c² + d²)
The division of complex numbers r₁(cos cos α₁ + i sin sin α₁) and r₂(cos cos α₂ + i sin sin α₂) in trigonometric form is performed by the formula:
Answer type: Single choice • with the choice of one correct answer from several proposed options

r₁/r₂ × (cos cos (φ₁ − φ₂) − i sin sin (φ₁ − φ₂))
r₁/r₂ × (sin sin (φ₁ − φ₂) + i cos cos (φ₁ − φ₂))
r₁/r₂ × (cos cos (φ₁ + φ₂) + i sin sin (φ₁ + φ₂))
r₁/r₂ × (cos cos (φ₁ − φ₂) + i sin sin (φ₁ − φ₂))
What pair of numbers can have the sum 10 + 4i:
Answer type: Multiple choice • with the choice of several correct answers from the proposed options

6 + 2i and 4 + 2i
8 + i and 2 + 3i
12 − i and −2 + 3i
−8 + 6i and −2 − 10i
What is the complex number obtained by multiplying 3 ⋅ (cos cos 5° + i sin sin 5°) and 4(cos cos 25° + i sin sin 25°):
Answer type: Multiple choice • with the choice of several correct answers from the proposed options

12 × (cos cos 30° + i sin sin 30°)
7 × (cos cos 30° + i sin sin 30°)
6√3+6i
3√3+3i
The canonical equation of a two-sheeted hyperboloid has the form:
Answer type: Single choice • with the choice of one correct answer from several proposed options

x²/a² + y²/b² - z²/c² = -1
x²/a² + y²/b² − z²/c² = 1
x²/a² + y²/b² − 2z² = 0
x²/a² + y²/b² + z² = 1
x²/a² − y²/b² = 2z
The canonical equation of the cone is:
Answer type: Single choice • with the choice of one correct answer from several proposed options

x²/a² + y²/b² - z²/c² = -1
x²/a² + y²/b² = 2z
x²/a² + y²/b² − 2z² = 0
x²/a² + y²/b² − z²/c² = 1
x²/a² − y²/b² = 2z