IIT JAM & CSIR NET - Mathematical Sciences

Group, Ring & Field Theory question bank (Abstract Algebra & Complex Analysis)

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Q1.
[Abstract Algebra & Complex Analysis Problem #1] Regarding Cauchy-Riemann equations for an analytic function f(z) = u(x,y) + i v(x,y): Which of the following analytical formulations is correct?

(A)
∂u/∂x = -∂v/∂y and ∂u/∂y = ∂v/∂x
(B)
∂u/∂x = ∂v/∂y and ∂u/∂y = -∂v/∂x
(C)
∂u/∂x + ∂v/∂y = 0 and ∂u/∂y + ∂v/∂x = 0
(D)
∂^2u/∂x^2 + ∂^2v/∂y^2 = 0 strictly
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Q2.
[Abstract Algebra & Complex Analysis Problem #2] Regarding eigenvalues of a real symmetric matrix A: Which of the following analytical formulations is correct?

(A)
All eigenvalues are strictly real, and eigenvectors from distinct eigenvalues are orthogonal
(B)
Eigenvalues must always be positive integers
(C)
The matrix cannot be diagonalized under any orthogonal transformation
(D)
All eigenvalues are pure imaginary
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