### 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