Obtain the Taylor series expansion of cos² z about z = a. 

Consider a triangle P in the z-plane with vertices at i, 1 – i, 1 + i. Determine the triangle P0 which mapped P under the transformation w = 3z + 4 – 2i. What is the relation between P and P0 and also calculate the area magnification.

Obtain the general solution of the two-dimensional Laplace’s equation in spherical polar coordinates given by

i) Show that the series  converges for |z|< 1 and find its sum.

ii) Obtain the analytic function whose real part u(x, y ) e^x cos y.

Two identical infinite non-conducting sheets having equal positive surface charge densities  are kept parallel to each other as shown in the Figure below. Determine the electric field at a point in (a) region A on the left of the sheet 1, (b) region B between the sheets and (iii) region C on the right of the sheets.

Consider three identical springs and two masses m₁ = m and m₂ = 2m , as shown in the figure. The motion of the two masses are constrained along the line joining them. Find the normal mode frequencies and the normal coordinates.

A particle is scattered by a rigid sphere of radius R , as given in the figure. Determine the differential scattering cross-section and the total scattering cross-section.

A particle moves under a central force,  . Is this an attractive or a repulsive force? Find the radius and energy of the circular orbits.

Using Cauchy’s residue theorem, evaluate the integral

The kinetic and potential energies for a mechanical system with two generalized coordinates q₁ and q₂ are 

i) Write down the Lagrangian for the system.

ii) Obtain the expressions for the generalized momenta.

iii) Derive the Euler Lagrange equations of motion for the system.

iv) Obtain the energy function for the system.

Show that the vectors  are linearly independent.

Using the D' Alembert’s Principle determine the equation of motion for an Atwood’s machine with masses M and 3M

Write down the equations of constraint/s for the following systems and classify them as holonomic/nonholonomic and scleronomic/rheonomic:

i) a particle of mass m sliding down the surface of a sphere of radius R without friction, under the action of gravity.

ii) a simple pendulum of length L for which the point of support is oscillating vertically with an angular speed   and amplitude A.

Show that g^ij is a contravariant tensor of rank 2.

A particle of mass m moving to the right with an initial velocity u collides elastically with a particle of unknown mass M coming from the opposite direction. After the collision m has a velocity u/2 at right angles to the incident direction, and M is deflected back making an angle of 45 degrees to its incident direction as shown below. Calculate the ratio M/m.

Diagonalized the matrix:

The eigenvalues of A are: λ₁ = 2, λ₂  =1 and λ₃ = -1.

Obtain the eigenvalues and the corresponding eigenvectors of the matrix:

Using the generating function, establish the relation between H_n (x) and H_n (-x). 

Using the recurrence relation, show that

भेद प्रवर्धक प्रकार के EVM की संरचना एवं कार्य समझाएं।