IIT JAM & CSIR NET - Life Sciences & Biotechnology

Population Ecology & Ecosystem Dynamics question bank (Ecology, Evolution & Biostatistics)

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Q1.
[Ecology, Evolution & Biostatistics 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)
∂^2u/∂x^2 + ∂^2v/∂y^2 = 0 strictly
(D)
∂u/∂x + ∂v/∂y = 0 and ∂u/∂y + ∂v/∂x = 0
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Q2.
[Ecology, Evolution & Biostatistics Problem #3] 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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