CIVILS MAINS 2013 MATHEMATICS Paper II

Time allowed: Three Hours

Maximum Marks: 250

Question Paper Specific Instructions

Please read each ofthe following instructions carefully before attempting questions:
There are EIGHT questions divided in two SECTIONS and printed both in HINDI and in ENGLISH.

Candidate has to attempt FIVE questions in all.

Questions no. 1 and 5 are compulsory and out of the remaining, THREE are to be attempted choosing at least ONE from each section.

The number ofmarks carried by a question /part is indicated against it.
Answers must be written in the medium authorized in the Admission Certificate which must be stated clearly on the cover of this Question-cum-Answer Booklet in the space provided. No marks will be given for answers written in a medium other than the authorized one. . Assume suitable data, ifconsidered necessary, and indicate the same clearly.
Unless and otherwise indicated, symbols and notations carry their usual standard meaning. Attempts of questions shall be counted in chronological order. Unless struck off, attempt of a question shall be counted even if attempted partly. Any page or portion of the page left blank in the answer book must be clearly struck off.

SECTION A

<img src='./qimages/187-1a.jpg'> 10

Give an example of an infinite group in which every element has finite order. 10

<img src='./qimages/187-2c.jpg'> Is f Riemann integrable in the interval 2J Why Does there exist a function g such that Justify your answer. 10

Prove that if b ea 1 1 where a and b are positive and real, then the function zn e-a b eZ has n zeroes in the unit circle. 10

Maximize subject to and z =2x1 3x2 5xa xl x2 Xa 7 2x1 5x2 xa 10, O. 10

Q2. What are the orders of the following permutations in SlO <img src='./qimages/187-2c.jpg'>
What is the maximal possible order of an element in SlO Why? Give an example of such an element. How many elements will there be in 810 of that order? 13

Show that the series <img src='./qimages/187-2c1.jpg'> is uniformly convergent but not 1 n+x absolutely for all real values ofx. 13

Show that every open subset of R is a countable union of disjoint open intervals. 14

Q3. Let J bi Ia, b E be the ring of Gaussian integers (subring of C). Which of the following is J Euclidean domain, principal ideal domain, unique factorization domaiIl JustifY your answer.

Let RC =ring of all real valued continuous functions on under the operations <img src='./qimages/187-3b.jpg'>

Let y2 4xy 3x2 x3 1. At what points will have a maximum or minimum?

Let denote the integer part of the real number i.e., if n S x n 1 where n is an integer, then n. Is the function ftx) 3 Riemann integrable in If not, explain why. If it is integrable, 2 compute f integral to dx. 10

Q4.(a) Solve the minimum time assignment problem:
<img src='./qimages/187-4a.jpg'>

Using Cauchy's residue theorem, evaluate the integral
<img src='./qimages/187-4b.jpg'>

Minimize z 5xI 4x2 6xs 8x4 subject to the constraints <img src='./qimages/187-4c.jpg'>

SECTIONB

Q5. Form a partial differential equation by eliminating the arbitrary functions f and g from z y x 10

Reduce the equation a2z a2z a2z <img src='./qimages/187-5b.jpg'> to its canonical form when y. 10

In an examination, the number of students who obtained marks between . certain limits were given in the following <img src='./qimages/187-5c.jpg'> Using Newton forward interpolation formula, find the number of students whose marks lie between 45 and 50. 10

Prove that the necessary and sufficient condition that the vortex lines may be at right angles to the stream lines are <img src='./qimages/187-5d.jpg'> where I-l and cp are functions of t. 10

Four solid spheres C and each of mass m and radius are placed with their centres on the four corners of a square of side b. Calculate the moment of inertia of the system about a diagonal of the square. 10

Q6. Solve <img src='./qimages/187-6a.jpg'>


Find the surface which intersects the surfaces ofthe system z(x being a constant) orthogonally and which passes through the circle x2 y2 z =1. 15

A tightly stretched string with fixed end points x 0 and x l IS initially at rest in equilibrium position. If it is set vibrating by giving each point a velocity A. x find the displacement of the string at any distance x from one end at any time t. 20

Q7. Develop an algorithm for Newton Raphson method to solve 0 starting with initial iterate n be the number of iterations allowed, eps be the prescribed relative error and delta be the prescribed lower bound for 20

Use Euler's method with step size h 0·15 to compute the approximate value of correct up to five decimal places from the initial value problem x 2 15

The velocity of a train which starts from rest is given in the following table. The time is in minutes and velocity is in kmlhour. <img src='./qimages/187-7c.jpg'> Estimate approximately the total distance run in 30 minutes by using composIte SImpson,s1- 1/3 ru1e.15


Q8. Two equal rods AB and Be, each of length smoothly jointed at are suspended from A and oscillate in a vertical plane through A. Show that the periods of normal oscillations are 2 n rr where <img src='./qimages/187-8a.jpg'>

Iffluid fills the region of space on the positive side of the x-axis, which is a rigid boundary and if there be a source m at the point and an equal sink at CO, and if the pressure on the negative side be the same as the pressure at infinity, show that the resultant pressure on the boundary is <img src='./qimages/187-8b.jpg'>where p is the density of the fluid. 15


If n rectilinear vortices of the same strength K are symmetrically arranged as generators of a circular cylinder of radius a in an infinite liquid, prove that the vortices will move round the cylinder uniformly in time <img src='./qimages/187-8c.jpg'>Find the velocity at any point of the liquid. 20