1. Attempt any five parts

Define hermitian matrix. Show that the

<img src='./qimages/10401-1a.jpg'> is hermitian.

Define symmetric and antisymmetric tensors.

Show that each element in an abelian group is a class by itself.

Calculate the residue of the function



Show that <img src='./qimages/10401-1e.jpg'> for a unit circle with centre at the origin.

Obtain the Laplace transform of 5 2 e^3t.

Determine Fourier sine transform of the function

<img src='./qimages/10401-1g.jpg'><br><br>

Using the Rodrigues' formula for Legendre polynomials

<img src='./qimages/10401-1h.jpg'>

obtain the value of P2(x).

2. Attempt any two parts:

Determine the eigenvalues and eigenvectors of the following matrix:

<img src='./qimages/10401-2a.jpg'>

Prove that the eigenvalues of a hermitian matrix are real.

Define a cyclic group. Give one example.

Show that is a subgroup of the multiplicative group

3. Attempt any two parts

Using the method of residues, evaluate the contour integral <img src='./qimages/10401-3a.jpg'> where C is defined by

Evaluate the following integral

<img src='./qimages/10401-3b.jpg'>

Obtain the Taylor series expansion of cos^2 z about 0.

4. Attempt any two parts:

Obtain the Fourier transform of the function

<img src='./qimages/10401-4a.jpg'>

Determine the inverse Laplace transform of

Solve the following differential equation using Laplace transform method:

4y 4 1

5. Attempt any one part

Using the generating relation for Legendre polynomials

<img src='./qimages/10401-5a.jpg'>

derive the recurrence relation

(b) Using the representation

<img src='./qimages/10401-5b1.jpg'>

show that

<img src='./qimages/10401-5b2.jpg'>