Section
1. How many elements of order 2 are there in the group of order 16 generated by a and b such that the order of a is the order of b is 2 and bab-1 a

Let
<img src='./qimages/214-1b.jpg'>
Show that converges to a continuous function but not uniformly.
Show that the function defined by
src='./qimages/214-1c.jpg'>
is not analytic at the origin though it satisfies Cauchy-Riemann equations at the origin.

For each hour per day that Ashok studies mathematics, it yields him 10 marks and for each hour that he studies physics, it yields him 5 marks. He can study at most 14 hours a day and he must get at least 40 marks in each. Determine graphically how many hours a day he should study mathematics and physics each, in order to maximize his marks
Show that the series
src='./qimages/214-1e.jpg'>
is convergent.

2. How many conjugacy classes does the permutation group S5 of permutations 5 numbers have Write down one element in each class (preferably in terms of cycles).

Let
src='./qimages/214-2b.jpg'>
is not continuous at 0).
Use Cauchy integral formula to evaluate
src='./qimages/214-2c.jpg'>

Find the minimum distance of the line given by the planes 3x 4y 5z 7 from the origin, by the method oafn dLa xg r-anzg multipliers.
3. ls the ideal generated by 2 and X in the polynomial ring z of polynomials in a single variable X with coefficients in the ring of integers z. a principal ideal Justify your answer.

Let be differentiable on such that
src='./qimages/214-3b.jpg'>
Expand the function
in Laurent series valid for

1 3



Evaluate by contour integration
src='./qimages/214-3d.jpg'>

4. Describe the maximal ideals in the ring of
Gaussian integers b e(belongs to) z}. 20
Give an example of a function that is not Riemann integrable but I I is Riemann integrable. Justify.
By the method of Vogel, determine an initial basic feasible solution for the following transportation problem Products P1, P2, P3 and P4 have to be sent to destinations D1, D2 and D3. The cost of sending product Pi to destinations Dj is Cij where the matrix
src='./qimages/214-4c.jpg'>

The total requirements of destinations D1, D2 and D3 are given by 45, 45, 95 respectively and the availability of the products P1, P2, P3 and P4 are respectively 25, 35, 55 and 70.

Section
5. Solve the partial differential equation
z x+Y .
Use Newton-Raphson method to find the real root of the equation 3x cos x 1 correct to four decimal places.

Provide a computer algorithm to solve an ordinary differential equation dy in dx the interval b for n number of discrete points, where the initial value is alfa, using Euler's method.
Obtain the equations governing the motion of a spherical pendulum.
A rigid sphere of radius a is placed in a stream of fluid whose velocity in the undisturbed state is v. Determine the velocity of the fluid at any point of the disturbed stream.

6. Solve the partial differential equation px+qy 3z. 20
A string of length l is fixed at its ends. The string from the mid-point is pulled up to a height k and then released from rest.Find the deflection of the vibrating siring. 20
Solve the following system of simultaneous equations, using Gauss-Seidel iterative
method:
3x+20y-z=-18
20x+ y-2z 17
2x 3y 20z 25.
7.

src='./qimages/214-7a.jpg'>

The edge r a of a circular plate is kept at temperature f(theta). The plate is insulated so that there is no loss of heat from either surface. Find the temperature distribution in steady state.

In a certain examination, a candidate has to appear for one major and two minor subjects. The rules for declaration of results are: marks for major are denoted by MI and for minors by M2 and M3. If the candidate obtains 75% and above marks in each of the three subjects, the candidate is declared to have passed the examination in first class with distinction. If the candidate obtains 60% and above marks in each of the three subjects, the candidate is declared to have passed the examination in first class. If the candidate obtains 50% or above in major, 40% or above in each of the two minors and an average of 50% or above in all the three subjects put together, the candidate is declared to have passed the examination in second class. All those candidates, who have obtained 50% and above in major and 40% or above in minor, are declared to have passed the examination. If the candidate obtains less than 50% in major or less than 40% in any one of the two minors, the candidate is declared to have failed in the examinations. Draw a flow chart to declare the results for the above.

8. A pendulum consists of a rod of length 2a and mass to one end of which a spherical bob of radius a/3 and mass 15 m is attached. Find the moment of inertia of the pendulum
about an axis through the other end of the rod and at right angles to the rod.
about a parallel axis through the centre of mass of the pendulum. (Given The centre of mass of the pendulum is a/12 above the centre of the sphere.]
Show that x is a possible form for the velocity potential for an incompressible fluid motion. If the fluid velocity q 0 as r co, find the surfaces of constant speed.