IAS MAINS 2010 MATHEMATICS Paper II

Section—A
1. Attempt any five of the following
If lambda1, lambda2, lambda3 are the eigenvalues of the matrix <img src='./qimages/910-1a.jpg'>
What is the null space of the differentiation transfonnation where P" is the space of all polynomials of degree §n over the real numbers? What is the null space of the second derivative as a transformation of Pn? What is the null space of the kth derivative? 12
A twice-differentiable function f is such that f f and f 0 for a c b. Prove that there is at least one point <img src='./qimages/910-1c.jpg'>
Does the integral <img src='./qimages/910-1d.jpg'> exist? lf so,find its value. 12
(e)Construct tho dual of the primal problem: Maximize 2. 2x1 x2 x3. subject to the constraints xl x2 x3 3x1 2x2 3x3=3, 3x2-6x3= and 0. 12

Let be the multiplicative group of non- zero reals and be the multi-- plicative group of n x n non-singular real matrices. Show that the quotient group GL(n. IR) and are iso- morphic where SL IR) {AG GL IR)‘/det 2‘: What is the centre of GL IR) 15
Let C..= f I lRl f is continuous}. Show C is a commutative ring with 1 under pointwise addition and multiplication. Determine whether C is an integral domain. Explain. 15
Define the function
x2sin1/x, if

if x O
Find Is continuous at x Justify your answer. 15

Consider the series<br><br> <img src='./qimages/910-2d.jpg'> Find the values of x for which it is convergent and also the sum function. Is the convergence uniform Justify your answer. l5
Consider the series

Consider the polynomial ring Show x3 2 is irreducible over Q. Let I be the ideal in generated by Then show that Q I is a field and that each element of it is of the form ao alt aztz ‘with an, ai, a2 in Qand t=x+I. 15

Show that the quotient ring is isomorphic to the ring Z/1021 where denotes the ring of Gaussian integers. 15

Let on for Find the limit function. Is the convergence unifonn Justify your answer. 15

Find the maxima, minima and saddle points of
the surface 15

Evaluate the line integral I f dz where z2, c is the boundary of the triangle with vertices A B C in that order. "
Find the image of the finite vertical strip x=5 to ofz-plane under exponential function. 15

<img src='./qimages/910-4b.jpg'> contour in this region. 15

(c)Determine an optimal transportation pro- gramme so that the transportation cost of 340 tons of a certain type of material from three factories F1, F2, F3 to five warehouses W1, W2, W3, W4, W5 is minimized. The five warehouses must receive 40 tons, 50 tons. 70 tons, 90 tons and 90 tons respectively. The availability of the ‘material at Fl, F2, F3 is 100 tons, 120 tons, 120 tons respectively. The transportation costs per ton from factories to warehouses are given in the table below <img src='./qimages/910-4c.jpg'>

Section ‘B’

5. Attempt any ?ve of the following:
Solve the PDE
(D2 xy 12

Find the surface satisfying the PDE (D2 2DD' D’2)Z 0 and the conditions that b2 y2 when x 0 and aZ x2 when y 0. 12

Find the positive root of the equation lOxe-x2-1=0 correct up to 6 decimal places by using Newton-Raphson method. Carry out compu- tations only for three iterations. 12

Suppose a computer spends 60 per cent of its time handling a particular type of computation when running a given program and its manufacturers make a change that improves its performance on that type of computatibn by a factor of 10. If the program takes 100 sec to execute, what will its execution time be after the change 6
If A®B=AB’+A’B, ?nd the value of x GB y QB z. 6

A uniform lamina is bounded by a parabolic arc of latus rectum 4a and a double ordinate at a distance b from the vertex. If b show that two of the princi- pal faxes at the end of a latus rectum are the tangent and normal there. 12

ln un incompressible fluid the vorticity at every point is constant in magnitude and direc- lion; show that the components of velocity w are solutions of Laplace’s "equation. l2

Solve the following partial differential equa-
tion
ZP Yq x
2s
by the method of characteristics. 20

(b)Reduce the following 2nd order partial differential equation into canonical fonn and find its general solution
x uxx. 2x2uxy-ux 0 . 20

(c)Solve the following heat equation
ut-uxx=0, t>O
u(O. O
x(2—x). 2. 20

Given‘ the system of equations
2x 3y 1
2x 4y 2
2y 6z Aw 4
4z Bw C
State the solvability and uniqueness condi- tions for the system. Give the solution when it exists. 20

Find the value of the integral <img src='./qimages/910-7b.jpg'> by using Simpson‘s 1/3-rule correct up to 4 decimal places. Take 8 subintervals in your computation. 20

Find the hexadecimal equivalent of the decimal number (587632)l0
(ii)For the given set of data points

write an algorithm to find the value of by using Lagrange’s interpolation formula. ‘
(iii)Using Boolean algebra, simplify the
following expressiolns
a a'b’c’d+ .... ..
yz xz
where x’ represents the complement of x.. 5+10+5

8.(a)A sphere of radius a and mass m rolls down a rough plane inclined at an angle a to the horizontal. If x_be the distance of the point of contact of the sphere from a fixed point on the plane, find the acceleration by using Hamilton's equations. 30

(b)When a pair of equal and opposite rectilinear vortices are situated in a long circular cylinder at equal distances from its axis, show that the path of each vortex is given by the equation (r2sin2teta—b2) (r2 -a2 4a2b2r2,sin2teta, teta being measured from the line through the centre perpendicular to the joint of the vortices. 30