MATHEMATICAL PHYSICS QUALIFYING
EXAM QUESTION AND CORRECT
ANSWERS (VERIFIED ANSWERS) PLUS
RATIONALES 2026 Q&A INSTANT
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1. The solution of the differential equation 𝑦 ′ = 𝑘𝑦with initial condition
𝑦(0) = 𝑦0 is
A. 𝑦 = 𝑦0 + 𝑘𝑡
B. 𝑦 = 𝑦0 𝑒 𝑘𝑡
C. 𝑦 = 𝑘𝑒 𝑦0𝑡
D. 𝑦 = 𝑦0 𝑡 𝑘
Answer: B
Rationale: This is a first-order linear ODE whose solution is exponential.
2. The Laplace transform of 1 is
A. 1/𝑠 2
B. 𝑠
C. 1/𝑠
D. 𝑒 −𝑠
Answer: C
∞
Rationale: ℒ{1} = ∫0 𝑒 −𝑠𝑡 𝑑𝑡 = 1/𝑠.
3. Which function is orthogonal to sin(𝑛𝑥)on [0, 𝜋]?
A. sin(𝑚𝑥)where 𝑚 ≠ 𝑛
B. cos(𝑛𝑥)
C. 1
, D. 𝑥
Answer: A
Rationale: Sine functions of different integer modes are orthogonal on
[0, 𝜋].
4. The determinant of a matrix equals zero implies
A. The matrix is orthogonal
B. The matrix is diagonal
C. The matrix is singular
D. The matrix is symmetric
Answer: C
Rationale: Zero determinant means the matrix is not invertible.
5. The Fourier transform of a Gaussian is
A. A delta function
B. A polynomial
C. Another Gaussian
D. Zero
Answer: C
Rationale: Gaussian functions are eigenfunctions of the Fourier transform.
6. The Sturm–Liouville form guarantees
A. Complex eigenvalues
B. Orthogonal eigenfunctions
C. Nonlinear solutions
D. Divergent series
Answer: B
Rationale: Sturm–Liouville theory ensures orthogonality under a weight
function.
7. The Green’s function satisfies
A. The homogeneous equation
B. Boundary conditions only
C. A delta-function source
D. No differential equation
, Answer: C
Rationale: Green’s functions solve inhomogeneous equations with delta
sources.
8. A Hermitian operator has
A. Complex eigenvalues
B. Negative norm
C. Real eigenvalues
D. No eigenfunctions
Answer: C
Rationale: Hermitian operators represent observables with real spectra.
9. The Cauchy–Riemann equations apply to
A. Real functions
B. Harmonic functions only
C. Analytic complex functions
D. Discontinuous functions
Answer: C
Rationale: They are the condition for complex differentiability.
10.The residue theorem is used to
A. Solve PDEs
B. Evaluate real integrals
C. Diagonalize matrices
D. Expand Fourier series
Answer: B
Rationale: Contour integration with residues simplifies real integrals.
11.The eigenvalues of a skew-symmetric real matrix are
A. Positive real
B. Negative real
C. Purely imaginary or zero
D. Always zero
EXAM QUESTION AND CORRECT
ANSWERS (VERIFIED ANSWERS) PLUS
RATIONALES 2026 Q&A INSTANT
DOWNLOAD PDF
1. The solution of the differential equation 𝑦 ′ = 𝑘𝑦with initial condition
𝑦(0) = 𝑦0 is
A. 𝑦 = 𝑦0 + 𝑘𝑡
B. 𝑦 = 𝑦0 𝑒 𝑘𝑡
C. 𝑦 = 𝑘𝑒 𝑦0𝑡
D. 𝑦 = 𝑦0 𝑡 𝑘
Answer: B
Rationale: This is a first-order linear ODE whose solution is exponential.
2. The Laplace transform of 1 is
A. 1/𝑠 2
B. 𝑠
C. 1/𝑠
D. 𝑒 −𝑠
Answer: C
∞
Rationale: ℒ{1} = ∫0 𝑒 −𝑠𝑡 𝑑𝑡 = 1/𝑠.
3. Which function is orthogonal to sin(𝑛𝑥)on [0, 𝜋]?
A. sin(𝑚𝑥)where 𝑚 ≠ 𝑛
B. cos(𝑛𝑥)
C. 1
, D. 𝑥
Answer: A
Rationale: Sine functions of different integer modes are orthogonal on
[0, 𝜋].
4. The determinant of a matrix equals zero implies
A. The matrix is orthogonal
B. The matrix is diagonal
C. The matrix is singular
D. The matrix is symmetric
Answer: C
Rationale: Zero determinant means the matrix is not invertible.
5. The Fourier transform of a Gaussian is
A. A delta function
B. A polynomial
C. Another Gaussian
D. Zero
Answer: C
Rationale: Gaussian functions are eigenfunctions of the Fourier transform.
6. The Sturm–Liouville form guarantees
A. Complex eigenvalues
B. Orthogonal eigenfunctions
C. Nonlinear solutions
D. Divergent series
Answer: B
Rationale: Sturm–Liouville theory ensures orthogonality under a weight
function.
7. The Green’s function satisfies
A. The homogeneous equation
B. Boundary conditions only
C. A delta-function source
D. No differential equation
, Answer: C
Rationale: Green’s functions solve inhomogeneous equations with delta
sources.
8. A Hermitian operator has
A. Complex eigenvalues
B. Negative norm
C. Real eigenvalues
D. No eigenfunctions
Answer: C
Rationale: Hermitian operators represent observables with real spectra.
9. The Cauchy–Riemann equations apply to
A. Real functions
B. Harmonic functions only
C. Analytic complex functions
D. Discontinuous functions
Answer: C
Rationale: They are the condition for complex differentiability.
10.The residue theorem is used to
A. Solve PDEs
B. Evaluate real integrals
C. Diagonalize matrices
D. Expand Fourier series
Answer: B
Rationale: Contour integration with residues simplifies real integrals.
11.The eigenvalues of a skew-symmetric real matrix are
A. Positive real
B. Negative real
C. Purely imaginary or zero
D. Always zero
