IIT JAM & CSIR NET - Mathematical Sciences

Probability & Applied Statistics question bank (Differential Equations & Probability)

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Q1.
[Differential Equations & Probability Problem #1] Regarding eigenvalues of a real symmetric matrix A: Which of the following analytical formulations is correct?

(A)
The matrix cannot be diagonalized under any orthogonal transformation
(B)
All eigenvalues are pure imaginary
(C)
Eigenvalues must always be positive integers
(D)
All eigenvalues are strictly real, and eigenvectors from distinct eigenvalues are orthogonal
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Q2.
[Differential Equations & Probability Problem #2] 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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