Solve the initial value problem using the method of Laplace transform

Obtain the Fourier cosine transform of the function

Obtain the images of the line x = 0 and y = 0 under the transformation w = z² and prove that they intersect at right angles.

Write the Laurent series expansion of ez/ (z-1)² singularity and the region of convergence. about z = 1. Determine the type of

Show that the series  converges for IzI < 2 and find its sum.

Using the method of Residues, prove that

Define contravariant and covariant tensors of rank 2. Prove that vⁱ = g^ijaj transform contravariantly.

Obtain the eigenvalues and the orthonormal eigenvectors for the following real symmetric matrix:

Show that the set of all 2x2 Hermitian matrices form a four dimensional real vector space. Obtain a suitable basis for this vector space.

What are orthonormal vectors? Show that two non-null, orthogonal vectors are linearly independent.

Write an expression for generating function for Hermite polynomials. Using this expression evaluate the integral

Using the Rodrigue's formula of Legendre polynomials, obtain the values of P₃(x) and P₄(x).

Using the generating function for Bessel functions of the first kind and integral order, obtain the recurrence relation

The Helmholtz equation in Cartesian coordinates can be written as

Reduce it to three ODEs.

The 1-D wave equation for e.m. wave propagation in free space in given by (for )

Solve this equation if E_y = 0 at x = 0 and x = L.

Describe the various applications of genetic engineering.

Genetic engineering can be used for improving nutritional value of food. Explain in detail.

Describe Griffith's experiment to show transformation in DNA.

Describe in brief the following:

i) Chemical mutagens

ii) ABO Blood group system

iii) Autoimmune diseases

iv) Genetic counselling

Describe any two modern methods for the improvement of crop plants.