Explain the following terms with the help of a diagram, wherever needed: celestial sphere, zenith, circumpolar stars, diumal circle, resolving power of a telescope.

After about 1 billion year, the radius of a star is expected to increase by 100 times its present radius. If its temperature becomes half of what it is today, determine the change in its absolute magnitude.

The distance modulus of a star is-0.1. At what distance is it from the earth?

Using Rodrigue’s formula, obtain the expression for L₃ (x).

Using the following expression for a Bessel function of order m:

show that

i) 

ii) 

Obtain the first three terms in the expansion of the following function in a series of the form  :

Using the method of Laplace transform, solve the following initial value problem:

y"-10y'+9y=5t;    y(0)=-1, y'(0) = 2

Using the definition determine the Laplace transform of the function f(t) = e ^3t if it exists.

Determine the Fourier transform of the function f(t) = e ^-|t|. Hence deduce that:

Write Laurent series for the function

about the pole z =-2.

Using the Cauchy-Riemann equations, show that w = log z is analytic when z is not zero.

Evaluate the integral

Using the method of residues, evaluate the contour integral,

      where C is defined by |z| < 4.

Show that the set of all real numbers excluding zero is a group under multiplication. 

Define covariant and contravariant vectors. Show that velocity and acceleration are contravariant vectors.

For the quadratic equation 2x² + 4xy-y² = 24, write down the matrix of coefficients and diagonalize it. Recast it in new variables and identify the conic section it represents. 

Show that the eigenvectors belonging to distinct eigenvalues of a hermitian matrix are orthogonal to each other.

Determine the eigenvalues and eigenvectors of the following matrix A:

A uniform plane wave of 100 kHz travelling in free space strikes a large block of a material having  and Q = 0 normal to the surface. If the incident electric field vector is given by

Using Maxwell’s equations in free space, derive the wave equation for the xcomponent of the electric field vector.