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.

Discuss lasing action of a He-Ne laser.

A current is flowing in an infinitely long straight wire. Using Biot-Savart law, show that the resultant magnetic field at a point along a line perpendicular to the wire is inversely proportional to the distance of the point from the wire.

Define numerical aperture and angle of acceptance. An optical fiber has a numerical aperture of 0.20 and cladding refractive index of 1.59. Calculate the refractive index of the core material and the acceptance angle of the fibre in water whose refractive index is 1.33.

Discuss applications of lasers in medicine and communication.

A vertical single and double slits and illuminated by a point source. Discuss the salient features of their Fraunhofer diffraction patterns. Also, obtain an expression for intensity distribution in case of double slit.

The inclined faces of a glass biprism (µ = 1.5) make an angle of 1° with its base. The biprism is illuminated by a sodium lamp (2. = 589 nm) and the eye piece is at a distance of 1 m from the slit. A convex lens inserted between the biprism and the eye piece gives clear images of coherent sources in the focal plane of the eye piece. If the images are 0.4 cm apart in one case and 0.16 cm apart in the second case, calculate the width of interference fringes observed on the screen.