PAPER II
SECTION A
1. Attempt any 5 of the following:

Show that a group of order 35 is cyclic.

Show that polynomial 25x4+9x3+3x+3 is irreducible over the field of rational numbers
Prove that the integral <img src='./qimages/1111-1c.jpg'> is convergent if and only if m>0

find all the positive values of a for which the series
<img src='./qimages/1111-1d.jpg'> converges.

Suppose that f and g are two analytic functions on the set beta of all complex numbers with
for 1.2.3 ....then show that for each z in beta
Using Simplex method
Maximize 45x1+80x2
Subject to
5x1+20x2<=400,
10x1+15x2<=450,
x1,x2>=0.

Show that a group of p2 is abelian, where p is a prime number:
Prove that a group of order 42 has a normal subgroup of order 7.

Prove that in the ring of polynomial over a field the ideal is maximal if and
only if the polynomial is irreducible over F

Show that every finite integral domain is a field.

Let F be a field wuh q elements let E be a finite extension of degree n over F. Show
that E has qn elements.

3.(a)Test uniform convergence of the series
<img src='./qimages/1111-3a.jpg'>

(b)obtain the maxima and minima of
x2 +y2 +z2 yz xz xy
subject to the condition
x2+y2+z2-2x+2y+6z+9=0

A solid hemisphere H of radius has density row depending on the distance R from the centre
and is given by:
row=k(2a
where k is a constant.
find the mass of the hemisphere, by the method of multiple integrals.

4.(a)(i)Show that when then function has the Laurent
series expansion in power of as <img src='./qimages/1111-4a1.jpg'>

establish, by contour integration, <img src='./qimages/1111-4a2.jpg'>

using Simple method maiximize
5x1+3x2 Subject to
5x1+2X2<=10,
3x1+8x2<=12 x1,X2

company has 3 factories A,B and C whfcb supply units 10 warehouses Y and Z. Every month
the capacities of the factories per month are 60, 70 and 80 units at A,B and C respectively. The
requirements of X,Y and Z per month are 50, 80 and 80 respectively. Thc necessary data in terms of
unit transpot1ation costs in rupees. factory capacities and warehouse requirements are given below:
<img src='./qimages/.jpg'>

find the minimum distribution cost
SECTION-B

5. Attempt any five of the following:
Find two complete integrals of the partial differential equation x2p2+y2q2-4=0
Find the solution of the equation
Find a real root of the equation x3-2x-5=0 by the metod of false position
Convert (100.85)10 into its binary equivalent.

Multiply the binary numbers 1111.0l)2 and (1101.1l)2 and check with its decimal
equivalent.
Find the moment of inertia of o circular wire about
a diameter, and
a line through the centre and perpendicular to its plane.

Show that the velocity potential pie=1/2 a(x2=y2-2z2) satisfied the laplacae equation. and
determine the stream lines.


6.(a)frame the partial differential equation by eliminating the arbitrary constants a and b from
log x +ay
Find the characteristic strip Of the equation
xp+yq-pq=0
and then find the equation of the integral surface through the curve y=0.

(c)Solve <img src='./qimages/1111-6c.jpg'>


find the cube polynomial which takes the following values:
1 1 and
Hence,or otherwise obtain

Given dy/dx where using the Runge-Kutta fourth order method, find
and Compare the approximate solution with its exact solution.
(e0.1=l.10517,e0.2= 1.2214),

A teacher conducts there tests TEST 1. TEST 2,and FINAL for 50 marks
each.Out of the marks scored in the two tests. TEST 1 and TEST 2 he takes the
better one and adds to theb marks scored in FINAL. so that the total marks scored will
be for n maximum 100,

The letter grades will be assigned depending on the marks scored as per the
following norm:
0-39:E
40-49
50-59
60-74
75-lOO
for each student data consisting of name, scores in TEST 1. TEST and
FINAL are given. Write a program in BASIC which will print out the names, total
marks scored and grade obtained for all 20 students in a class.

Draw a flow chart to examine whether a given number is a prime.

8.(a)A thin circular disc of mass M and radius a can turn feelv about a thin axis OA, which is perpendicular to its plane and passes through a point() of its circumference. the axis OA is compelled to move in a horizontal plane with angular velocity w about its end A. Show that the inclination theta to the vertical of the radius of the disc through O is cos universe unless w2<g/a and then theta is zero.

Show that src='./qimages/1111-8b1.jpg'>
are the velocity components of a possible liquid motion. Is this motion irrigational?

Prove that: <img src='./qimages/1111-8b2.jpg'>
Where v is the kinematic viscosity of the fluid and the Stream function for a two
dimensional motion of a viscous fluid.