04-BS-5 Advanced Mathematics Exam D
Question 1. Which of the following best defines a vector space?
A) A set with a single binary operation
B) A set closed under vector addition and scalar multiplication
C) A set of points in Euclidean space only
D) A collection of matrices with addition and multiplication
Answer: B
Explanation: A vector space is a set where vector addition and scalar multiplication are defined and
satisfy specific axioms, including closure, associativity, distributivity, and existence of additive identity and
inverses.
Question 2. Which property must a subspace of a vector space satisfy?
A) It contains the zero vector
B) It contains only non-zero vectors
C) It is finite-dimensional
D) It is orthogonal to the original space
Answer: A
Explanation: A subspace must contain the zero vector, be closed under addition and scalar multiplication,
and contain all linear combinations of its vectors.
Question 3. Which matrix operation is necessary to find the inverse of a matrix?
A) Addition
B) Determinant calculation
C) Row reduction to identity
D) Transposing
Answer: C
Explanation: To find the inverse, one typically performs row operations to reduce the matrix to the identity,
applying the same to the identity matrix to obtain the inverse.
Question 4. The determinant of a matrix indicates:
A) The rank of the matrix
B) Whether the matrix is invertible
C) The eigenvalues of the matrix
D) The trace of the matrix
, 04-BS-5 Advanced Mathematics Exam D
Answer: B
Explanation: A non-zero determinant indicates the matrix is invertible; a zero determinant means it is
singular and not invertible.
Question 5. Eigenvalues of a matrix are defined as the scalars λ satisfying:
A) Av=λvAv = \lambda vAv=λv for some non-zero vector v
B) Av=0Av = 0Av=0
C) A+λI=0A + \lambda I = 0A+λI=0
D) det(A−λI)=0\det(A - \lambda I) = 0det(A−λI)=0
Answer: D
Explanation: Eigenvalues are solutions to the characteristic equation det(A−λI)=0\det(A - \lambda I) =
0det(A−λI)=0, which determines the scalar λ for which A has a non-trivial eigenvector.
Question 6. A matrix is diagonalizable if:
A) It has distinct eigenvalues
B) It can be expressed as PDP−1PDP^{-1}PDP
−1
where D is diagonal
C) Its Jordan form is diagonal
D) Both B and C
Answer: D
Explanation: Diagonalization involves expressing the matrix as PDP−1PDP^{-1}PDP
−1
, where D is diagonal; this is equivalent to the matrix being similar to a diagonal matrix, which also relates
to its Jordan form.
Question 7. Inner product spaces generalize the concept of:
A) Length and angle in Euclidean space
B) Distance in metric spaces
C) Volume in multi-dimensional space
D) Norms in Banach spaces
Answer: A
, 04-BS-5 Advanced Mathematics Exam D
Explanation: Inner product spaces define notions of length (norm) and angles between vectors,
generalizing the Euclidean dot product.
Question 8. The Gram-Schmidt process is used to:
A) Compute eigenvalues
B) Orthonormalize a set of vectors
C) Find the inverse of a matrix
D) Determine the determinant
Answer: B
Explanation: The Gram-Schmidt process orthonormalizes a set of linearly independent vectors, producing
an orthonormal basis.
Question 9. The gradient of a function f(x,y)f(x, y)f(x,y) points in the direction of:
A) The steepest descent
B) The steepest ascent
C) Zero change
D) The direction of maximum curvature
Answer: B
Explanation: The gradient vector points in the direction of the greatest increase of the function, indicating
the steepest ascent.
Question 10. The value of a double integral over a region R represents:
A) The volume under the surface z=f(x,y)z = f(x, y)z=f(x,y)
B) The area of the region R
C) The flux across the boundary of R
D) The line integral along the boundary of R
Answer: A
Explanation: Double integrals over a region in the xy-plane compute the volume under the surface
z=f(x,y)z = f(x, y)z=f(x,y).
Question 11. Green's Theorem relates a line integral around a closed curve C to:
A) A surface integral over the enclosed region
B) A double integral over the region D bounded by C
, 04-BS-5 Advanced Mathematics Exam D
C) A triple integral in three dimensions
D) A divergence integral over a surface
Answer: B
Explanation: Green's Theorem converts a circulation integral around a closed curve into a double integral
over the region it encloses.
Question 12. Stokes' Theorem connects a surface integral of curl of a vector field to:
A) The divergence over the surface
B) The line integral of the vector field around its boundary
C) The gradient of the scalar potential
D) The Laplacian of the field
Answer: B
Explanation: Stokes' Theorem states that the surface integral of the curl of a vector field over a surface
equals the line integral of the field over the boundary curve.
Question 13. The divergence theorem (Gauss's theorem) states that the flux of a vector field through a
closed surface equals:
A) The volume integral of divergence over the enclosed volume
B) The surface integral of the field
C) The line integral along the boundary
D) Zero, if the field is solenoidal
Answer: A
Explanation: The divergence theorem relates the flux through a closed surface to the volume integral of
divergence inside the volume.
Question 14. A first-order differential equation can be written in the form:
A) dy/dx=f(x,y)dy/dx = f(x, y)dy/dx=f(x,y)
B) d2y/dx2=f(x,y)d^2 y/dx^2 = f(x, y)d
2
y/dx
2
=f(x,y)
C) d3y/dx3=f(x,y)d^3 y/dx^3 = f(x, y)d
Question 1. Which of the following best defines a vector space?
A) A set with a single binary operation
B) A set closed under vector addition and scalar multiplication
C) A set of points in Euclidean space only
D) A collection of matrices with addition and multiplication
Answer: B
Explanation: A vector space is a set where vector addition and scalar multiplication are defined and
satisfy specific axioms, including closure, associativity, distributivity, and existence of additive identity and
inverses.
Question 2. Which property must a subspace of a vector space satisfy?
A) It contains the zero vector
B) It contains only non-zero vectors
C) It is finite-dimensional
D) It is orthogonal to the original space
Answer: A
Explanation: A subspace must contain the zero vector, be closed under addition and scalar multiplication,
and contain all linear combinations of its vectors.
Question 3. Which matrix operation is necessary to find the inverse of a matrix?
A) Addition
B) Determinant calculation
C) Row reduction to identity
D) Transposing
Answer: C
Explanation: To find the inverse, one typically performs row operations to reduce the matrix to the identity,
applying the same to the identity matrix to obtain the inverse.
Question 4. The determinant of a matrix indicates:
A) The rank of the matrix
B) Whether the matrix is invertible
C) The eigenvalues of the matrix
D) The trace of the matrix
, 04-BS-5 Advanced Mathematics Exam D
Answer: B
Explanation: A non-zero determinant indicates the matrix is invertible; a zero determinant means it is
singular and not invertible.
Question 5. Eigenvalues of a matrix are defined as the scalars λ satisfying:
A) Av=λvAv = \lambda vAv=λv for some non-zero vector v
B) Av=0Av = 0Av=0
C) A+λI=0A + \lambda I = 0A+λI=0
D) det(A−λI)=0\det(A - \lambda I) = 0det(A−λI)=0
Answer: D
Explanation: Eigenvalues are solutions to the characteristic equation det(A−λI)=0\det(A - \lambda I) =
0det(A−λI)=0, which determines the scalar λ for which A has a non-trivial eigenvector.
Question 6. A matrix is diagonalizable if:
A) It has distinct eigenvalues
B) It can be expressed as PDP−1PDP^{-1}PDP
−1
where D is diagonal
C) Its Jordan form is diagonal
D) Both B and C
Answer: D
Explanation: Diagonalization involves expressing the matrix as PDP−1PDP^{-1}PDP
−1
, where D is diagonal; this is equivalent to the matrix being similar to a diagonal matrix, which also relates
to its Jordan form.
Question 7. Inner product spaces generalize the concept of:
A) Length and angle in Euclidean space
B) Distance in metric spaces
C) Volume in multi-dimensional space
D) Norms in Banach spaces
Answer: A
, 04-BS-5 Advanced Mathematics Exam D
Explanation: Inner product spaces define notions of length (norm) and angles between vectors,
generalizing the Euclidean dot product.
Question 8. The Gram-Schmidt process is used to:
A) Compute eigenvalues
B) Orthonormalize a set of vectors
C) Find the inverse of a matrix
D) Determine the determinant
Answer: B
Explanation: The Gram-Schmidt process orthonormalizes a set of linearly independent vectors, producing
an orthonormal basis.
Question 9. The gradient of a function f(x,y)f(x, y)f(x,y) points in the direction of:
A) The steepest descent
B) The steepest ascent
C) Zero change
D) The direction of maximum curvature
Answer: B
Explanation: The gradient vector points in the direction of the greatest increase of the function, indicating
the steepest ascent.
Question 10. The value of a double integral over a region R represents:
A) The volume under the surface z=f(x,y)z = f(x, y)z=f(x,y)
B) The area of the region R
C) The flux across the boundary of R
D) The line integral along the boundary of R
Answer: A
Explanation: Double integrals over a region in the xy-plane compute the volume under the surface
z=f(x,y)z = f(x, y)z=f(x,y).
Question 11. Green's Theorem relates a line integral around a closed curve C to:
A) A surface integral over the enclosed region
B) A double integral over the region D bounded by C
, 04-BS-5 Advanced Mathematics Exam D
C) A triple integral in three dimensions
D) A divergence integral over a surface
Answer: B
Explanation: Green's Theorem converts a circulation integral around a closed curve into a double integral
over the region it encloses.
Question 12. Stokes' Theorem connects a surface integral of curl of a vector field to:
A) The divergence over the surface
B) The line integral of the vector field around its boundary
C) The gradient of the scalar potential
D) The Laplacian of the field
Answer: B
Explanation: Stokes' Theorem states that the surface integral of the curl of a vector field over a surface
equals the line integral of the field over the boundary curve.
Question 13. The divergence theorem (Gauss's theorem) states that the flux of a vector field through a
closed surface equals:
A) The volume integral of divergence over the enclosed volume
B) The surface integral of the field
C) The line integral along the boundary
D) Zero, if the field is solenoidal
Answer: A
Explanation: The divergence theorem relates the flux through a closed surface to the volume integral of
divergence inside the volume.
Question 14. A first-order differential equation can be written in the form:
A) dy/dx=f(x,y)dy/dx = f(x, y)dy/dx=f(x,y)
B) d2y/dx2=f(x,y)d^2 y/dx^2 = f(x, y)d
2
y/dx
2
=f(x,y)
C) d3y/dx3=f(x,y)d^3 y/dx^3 = f(x, y)d
