by
SECTION'A!
Q.1.Answer any five of the following:
Let S be the set of all real numbers except -1. Define on S by Is a group Find the solution of the equation 3 7 in S.

G is a group of real numbers under addition and N is the subgroup of G consisting of integers, prove that G/N is isomorphic to the group H of all complex numbers of absolute value 1 under multiplication.
(c)Examine the convergence of
<img src='./qimages/1147-1c.jpg'>

Prove that the function f defined by
when x is rational
when x is irrational
is nowhere continuous.

Determine all bilinear transformations which map the half plane into the unit circle

(f)Given the programme
Maximize 5x 2y
subject to x 3y 12
3x- 4y 9
7x 8y

Write its dual in the standard form.

Let 108. Show that there exists a normal subgroup or order 27 or 9.
Let G be the set of all those ordered pairs of real numbers for which a not equal to 0 and define in an operation as follows:

Examine whether G is a group w.r. t. the operation . If it is a
group, is G abelian
Show that
<img src='./qimages/1147-2b.jpg'>
is a Euclidean domain.

Q. A twice differentiable function f is such that 0 and 0 for a<c<b. Prove that there is at least one value s a s b for which 0.

Show that the function given by <img src='./qimages/1147-3b.jpg'> continuous at possesses partial derivatives and
Find the volume of the ellipsoid
x2/a2+y2/b2

Q.4.(a) With the aid of residues, evaluate
<img src='./qimages/1147-4a.jpg'>

Prove that all the roots of z7 5z3 12 0 lie between the circles and 2.

Use the simplex method to solve the problem
Maximize 2x 3y
subject to -2x 3y 2

X,y 0
SECTION'B'
Q. 5. Answer any five of the following
Solve:
px qy)
Solve:
<img src='./qimages/1147-5b.jpg'>

Evaluate
<img src='./qimages/1147-5c.jpg'>
by the Simpson's rule
<img src='./qimages/1147-5c1.jpg'>

Given the number 59.625 in decimal system. Write its binary equivalent.
Given the number 3898 in decimal system. Write its equivalent in system base 8.

Given points A and B y0) not in the same vertical, it is required to find a curve in the x-y plane joining A to B so that a particle starting from rest will traverse from A to B along this curve without friction in the shortest possible time. if y is the required curve find the function such that the equation of motion can be written as dx/dt f

A steady inviscid incompressible flow has a velocity field u=fx w=0 where f is a constant. Derive an expression for the pressure field p if the pressure p =P0 and g bar iz

Q. The deflection of a vibrating string of length is governed by the partial differential equation utt C2uxx. The ends of the string are fixed at x 0 and l. The initial velocity is zero. The initial displacement is given by
<img src='./qimages/1147-6a.jpg'>
Find the deflection of the string at any instant of time.

Find the surface passing through the parabolas y2=4ax and y2 -4ax and satisfying the equation
<img src='./qimages/1147-6b.jpg'>

(c)Solve the equation
<img src='./qimages/1147-6c.jpg'>
by Charpit's method.

Q.7.(a) If Q is a polynomial with simple roots alpha1, alpha2....alpha n and if P is a polynomial of degree show that <img src='./qimages/1147-7a.jpg'> Hence prove that there exists a unique polynomial of degree
with given values ck at the point alpha k,k 1,2,..n.

(b)Draw a programme outline and a flow chart and also write a programme in BASIC to enable solving the following system of 3 linear equations in 3 unknowns x1,x2 and x3
with
<img src='./qimages/1147-7b.jpg'>
Q. A particle of mass m is constrained to move on the surface of a cylinder. The particle is subjected to a force directed towards the origin and proportional to the distance of the particle from the origin. Construct the Hamiltonian and Hamilton's equations of motion.
Liquid is contained between two parallel planes, the free surface is a circular cylinder of radius a whose axis is perpendicular to the planes. All the liquid within a concentric circular cylinder of radius b is suddenly annihilated; prove that if P be the pressure at the outer surface, the initial pressure at any point on the liquid, distant r from the centre is
<img src='./qimages/1147-8b.jpg'>