C S MAINS 2015 MATHEMATICS (PAPER-II)

Section—A
1. How many generators are there of the cyclic group G of order Explain.
Taking a group of order where e is the identity, construct
composition tables showing that one is cyclic while the other is not.
5+5=10

Give an example of a ring having identity but a subring of this having a
different identity. 10

Test the convergence and absolute convergence of the series
src='./qimages/ 129-1c.jpg'>

Show that the function <img src='./qimages/ 129-1d.jpg'> is harmonic. Find its conjugate harmonic function y). Also, find the corresponding analytic function u iv in terms of 10

Solve the following assignment problem to maximize the sales 10
src='./qimages/ 129-1e.jpg'>

2. If R is a ring with unit element 1 and Q(Pie) is a homomorphism of R onto prove that is the unit element of 15

Is the function
src='./qimages/ 129-2b-i.jpg'>

Riemann integrable? If yes, obtain the value of <img src='./qimages/ 129-2b-ii.jpg'>

Find all possible Taylor’s and Laurent’s series expansions of the function <img src='./qimages/ 129-2c.jpg'> about the point z 0.

3. State Cauchy’s residue theorem. Using it, evaluate the integral
src='./qimages/ 129-3a.jpg'>

Test the series of functions <img src='./qimages/ 129-3b.jpg'>for uniform convergence. 15

Consider the following linear programming problem
src='./qimages/ 129-3c.jpg'>

Using the definition, find its all basic solutions. Which of these are
degenerate basic feasible solutions and which are non-degenerate basic
feasible solutions?

Without solving the problem, show that it has an optimal solution. Which of the basic feasible solution(s) is/are optimal? 20

4. Do the following sets form integral domains with respect to ordinary addition and multiplication? If so, state if they are fields
The set of numbers of the form b/2 with b rational squre root)
The set of even integers
The set of positive integers

Find the absolute maximum and minimum values of the function
<img src='./qimages/ 129-4b-i.jpg'> over the region <img src='./qimages/ 129-4b-ii.jpg'> 15

Solve the following linear programming problem by the simplex method. Write its dual. Also, write the optimal solution of the dual from the optimal table of the given problem 20
<img src='./qimages/ 129-4ci.jpg'>


5. Solve the partial differential equation
src='./qimages/ 129-5a.jpg'>

src='./qimages/ 129-5b.jpg'> 10

Find the principal (or canonical) disjunctive normal form in three variables r for the Boolean expression v Is the given Boolean expression a contradiction or a tautology? 10

Consider a uniform flow U0 in the positive x-direction. A cylinder of radius a is located at the origin. Find the stream function and the velocity potential. Find also the stagnation points. 10

Calculate the moment of inertia of a solid uniform hemisphere
x 2 y2 z2 a2, 0 with mass m about the OZ-axis. 10


6. Solve for the general solution p cos z sin where p dz/dx and q=dz/dy

Solve the plane pendulum problem using the Hamiltonian approach and show that H is a constant of motion. 15

Find the Lagrange interpolating polynomial that fits the following data <img src='./qimages/ 129-6c.jpg'>

7 . Find the solution of the initial-boundary value problem
src='./qimages/ 129-7a.jpg'>

Solve the initial value problem<img src='./qimages/ 129-7b.jpg'> in the interval 2.4] using the Runge-Kutta fourth-order method with step size h 0.2. 15

A Hamiltonian of a system with one degree of freedom has the form
src='./qimages/ 129-7c.jpg'>

where k are constants, q is the generalized coordinate and p is the
corresponding generalized momentum.

Find a Lagrangian corresponding to this Hamiltonian.
Find an equivalent Lagrangian that is not explicitly dependent on time.
10+10=20

8. Reduce the second-order partial differential equation

src='./qimages/ 129-8a.jpg'>

into canonical form. Hence, find its general solution. 15

Find the solution of the system

src='./qimages/ 129-8b.jpg'>

using Gauss-Seidel method (make four iterations). 15

In an axisymmetric motion, show that stream function exists due to equation of continuity. Express the velocity components in terms of the stream function. Find the equation satisfied by the stream function if the flow is irrotational. 20