C.S. (MAIN) EXAM, 2009

MATHEMATICS

Paper-II
Time Allowed: Three Hours Maximum Marks 300

INSTRUCTIONS

Each question is printed both in Hindi and in English.

Answers must be written in the medium specified in the Admission Certificate issued to you, which must be stated clearly on the cover of the answer-book in the space provided for the purpose. No marks will be given for the answers written in a medium other than that specified in the Admission Certificate.

Candidates should attempt Questions 1 and 5 which are compulsory, and any three of the remaining questions selecting at least one question from each Section.

Assume suitable data if considered necessary and indicate the same clearly. Symbols and notations carry usual meaning. unless otherwise indicated.

All questions carry equal marks.

SECTION-A

1. Attempt any FIVE o the following 2x5=60)

IR is the set of real numbers and IR is the set of positive real numbers, show that IR under addition and IR+ under multiplication ·) are isomorphic. Similarly if is the set of rational numbers and the set of positive rational numbers, are and isomorphic? Justify your answer. 12

Determine the number of homomorphisms from the additive group Z 15 to the additive group Z 10. is the cyclic group of order n). 12

State Rolle's theorem. Use it to prove that between two roots of ex cos x 1 there will be a root of e x sin x l. 2+10=12

<img src='./qimages/39-1d.jpg'> Wha t are the points of discontinuity of if any What are the points ,??.rhere f is not differentiable, if any Justify yours answers. 12

Let <img src='./qimages/39-1e.jpg'> Assume that the zeroes of the denominator are simple. Show that the sum of the residues of at its poles is equal t0 a n-1/bn 12

A paint factory produces both interior and exterior paint from two raw materials M1 and M2 • The basic data is as follows <img src='./qimages/39-1f.jpg'> A market survey indicates that the daily demand for interior paint cannot exceed that of exterior paint by more than 1 ton. The maximum daily demand of interior paint is 2 tons. The factory wants to determine the optimum product mix of interior and exterior paint that maximizes daily profits. Formulate the LP problem for this situation. 12

2. How many proper, non-zero ideals does the ring Z12 have Justify y our answer. How man y ideals does the ring Z 12 EB have Why
Show that the alternating group on four letters A4 has no subgroup of order 6. 15
Show that the series <img src='./qimages/39-2c.jpg'> converges. 15
Show that if f IR is a continuous function then for some real numbers C and C d. 15
3. Show that is a unique factorization domain that is not a principal ideal domain is the ring of integers). Is it possible to give an example of principal ideal domain that is not a unique fa ctorization domain is the ring of polynomials in the variable X with integer.) 15

How many elements does the quotient ring Z5[ have?Is 1t an 1ntegra oma1n . Just1 yours answers. 15
Show that <img src='./qimages/39-3c.jpg'> Justify all steps of your answer by quoting the theorems you are using. 15
Show that a bounded inf mite subset of IR must have a limit point. 15
4. If J3, y are real nu1nb ers such that a2 f3 2 y2 show that <img src='./qimages/39-4.jpg'>

SECTION-B
5. Answer any FIVE of the following (12x5=60)
Show that the differential equation of all cones which have their vertex at the origin is px qy z. Verify that this equation is satisfied by the surface yz zx xy 0. 12
Form the partial differential equation by eliminating the arbitrary function f given by f(x2 y2, z xy) 0. 6
Find the integral surface of x2p y2q z2 0 which passes through the curve xy X Z 1. 6

The equation x2 ax b 0 has two real roots a. and p. Show that the iterative method given by <img src='./qimages/39-5c1.jpg'>

Find the val ues of two valued bool ean variables D by solving the following simultaneous equations A+AB=O AB =AC AB AC CD CD where x denotes the complement of x. 6
Realize the following expression by using NAND gates only g b d a f where x denotes the complement of x. 6
Find the decimal equivalent of (357·32)8

The flat surface of a hemisphere of radius r is cemented to one flat surface of a cylinder of the same radius and of the same material. If the length of the cylinder be/ and the total mass be show that the moment of inertia of the combination about the axis of the cylinder is given by <img src='./qimages/39-5e.jpg'> 12

Two sources, each of strength m are placed at the points and a sink of strength 2 m is at the origin. Show that the stream l ines are the curves (x2 y2 a2(x2 Y2 Axy) where A is a variable parameter. Show also that the fluid speed at any point is (2ma2 r2 r3 where r l r2 and r3 are the distances of the points from the sources and the sink. 12

6(a)Find the characteristics of y2r x2 t 0 where r and t have their usual meanings. 1 5
(b)Solve (02 (2x2 xy y2) sin xy cos xy a a where D and represent and . 1 5 ax ay
A tightl y stretched string has i ts ends fixed at x 0 and x At time t the string is given a shape defined by where µ is a constant, and then released. Find the displacement of any point x of the string at time t 0. 30

7. Develop an algori thm for Regul a-Falsi method to find a root of 0 starting with two initial iterates x 0 and x I to the root such that sign sign 1 Take n as the maximum number of iterations all owed and eps be the prescribed error. 30
Using Lagrange interpolation formula, calculate the value of from the foll owing table of values of x and <img src='./qimages/39-7b.jpg'> 15
Find the value of l ·2) using Runge-Kutta fo urth order method with step size h O· 2 from the initial value problem xy 2. 15
8. A perfectly rough sphere of mass m and radius rests on the lowest point of a fixed spheri cal cavity of radius a. To the highest point of the movable sphere is attached a particle of mass and the •••• system is disturbed. Show that the oscillations are the same as those of a simple pendulum of length 7 <img src='./qimages/39-8a.jpg'> 30
An infinite mass of fluid is acted on by a force per unit mass directed to the origin. If initially r the fluid is at rest and there is a cavity in the form of the sphere r C in it, show that the cavity will be filled up after an interval of time<br><br> <img src='./qimages/39-8b.jpg'>

Note English version of the In structions is printed on the fr ont cover of this question pape r.