### IAS MAths Question Paper 2006

SECTION 'A' Q. 1. Answer any/ive of the following:

(a) Let S be the set of all real numbers except -1. Define * on S

by

a*b = a+ b + ab Is (S, *) a group ? Find the solution of the equation

2*x*3=7inS. 12

(b) If 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

dx

12

1/2/1 „xl/2

0

f(x) =

x^'^(l-x)

(d) Prove that the function f defined by

1, when X is rational -1, when X is irrational

is nowhere continuous. 12

(e) Determine all bilinear transformations which map the half plane Im (z) > 0 into the unit circle 1 w j < 1. 12

(f) Given the programme

Maximize u = 5x + 2y subject to X + 3y < 12

3x - 4y < 9

7x + 8y < 20

THE TEAM VISION IAS

x,y>0

Write its dual in the standard form. 12 Q. 2. (a) (i) Let O (G) - 108. Show that there exists a normal

subgroup or order 27 or 9.

(ii) Let G be the set of all those ordered pairs (a, b) of

real numbers for which a ?^ 0 and define in G, an operation ® as

follows :

(a, b) O (c, d) = (ac, be + d)

Examine whether G is a group w.r.t. the operation 0 . If it is a group, is G abelian ? 10 (b) Show that

Z[V2 ] = {a+ V2b|a,bG Z}

is a Euclidean domain. 30 Q. 3. (a) A twice differentiable function f is such that f (a) = f (b) = 0 and f (c) > 0 for a < c < b. Prove that there is at least one value ^ , a < ^ < b for which f" © < 0. 20 (b) Show that the function given by

f(x,y) =

3 3

""'^^^^ (x,y)^(0,0)

2 2

X -hy^

0 : (x,y)>(0..0)

(i) is continuous at (0, 0).

(ii) possesses partial derivatives

f^ (0, 0) and f (0,0). 20 (c) Find the volume of the ellipsoid

2 2 2

x^ y z

a^ b^ c^ Q. 4. (a) With the aid of residues, evaluate

cos 29

_____n . 2 d0 , - 1< a < 1

15

(b) Prove that all the roots of - 5z^ + 12 = 0 lie between the

circles | z | 1 and | z ] = 2.

(c) Use the simplex method to solve the problem Vfaximize n= 2x + 3y subject to -2x + 3y < 2 3 x + 2y<5 x,y>0 SECTION *B' Q. 5. Answer any five of the following:

(a) Solve:

px (z - 2y2) = (z~ qy) {z-f-- 2x^)

(b) Solve:

-4—-— + 4

^^i;r^ = 2sin(3x + 2y)

(c) Evaluate

15

30

12

12

I

=1

e^^^dx

0

by the Simpson's rule

b

Ax

J

a

f (x) dx - — [f (X,) + 4f (x^) + 2f (X,)]

vdth

+ 4f (X3) +.....+ 2 f (X2,_ 2) + 4f (x,^_,) + f (x^^)]

2n-10,Ax = 0.1,Xo = 0,x^ = 0.1,...,x^^j=LO 12

(d) (i) Given the number 59.625 in decimal system. Write its binar}^ equivalent. 6

(ii) Given the number 3898 in decimal system. Write its equivalent in system base 8. 6

(e) Given points A (0, 0) and B {x^, y^) 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 = y (x) is the required curve find the function f (x, y z) such that the equation of motion can be written as

dx

— =f(x,y(xXy'(x)). 12

(f) A steady inviscid incompressible flow has a velocity field u = fx, v = -fy, w = 0 where f is a constant. Derive an e?q)ression for the pressure field p {x, y, z} if the pressure

pRO,0}=Poandg = -gi,. 12

Q. 6. (a) The deflection of a vibrating string of length /, is governed by the partial differential equation u^ = u^. The ends of the string are fixed at x = 0 and /. The initial velocity is zero. The initial displacement is given by

u(x, 0)- -,0<

/ ' 2

1 /

= 7(^-xX-

Find the deflection of the string at any instant of time. 30

(b) Find the surface passing through the parabolas z = 0, y^ = 4ax and z = 1, y^ = - 4ax and satisfying the equation

X—+2—=0 15

dK^ dx

(c) Solve the equation

dz dz p^x + q^y = z,p=-,q=by Charpit's method. 15 Q. 7. (a) If Q is a polynomial with simple roots a2,.- cxn and if P is a polynomial of degree < n, show that

h Q'(ock)(x-ai,)-

Hence prove that there exists a unique polynomial of degree < n with given values c^^ at the point aj^, k = 1, 2, ... n. 30

(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 x^ x^ and X3:

C*X-D

with

Q. 8. (a) 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. 30

(b) 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

log r - log b log a - log b

30