CS MAINS
MATHEMATICS (PAPER-II)
ITime Allowed Three Hours I IMaximum Marks 250 I
QUESTION PAPER SPECIFIC INSTRUCTIONS
(Please read each of the 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 FNE questions in all.
Question Nos. 1 and 5 are compulsory and out of the remaining, THREE are to be attempted
choosing at least ONE question from each Section.
The number of marks 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 medium other than the authorized
one.
Assume suitable data, if considered necessary, and indicate the same clearly.
Unless otherwise indicated, symbols and notations carry their usual standard meanings.
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 Question-cum-Answer Booklet must be clearly struck off.
1 IP.T.a.
l§l1S-A I


SECTION-A
1. Let G be the set of all real 2 x 2 matrices y 0 where xz equal to) O. Show that G is a group under matrix multiplication. Let N denote the subset a 0 a E R}.
Is N a normal subgroup of Justify your answer. 10
Test the convergence of the improper integral src='./qimages/281-1b.jpg'><br><br>.

Prove that the £unction u iv, where
src='./qimages/281-1c.jpg'><br><br>
satisfies Cauchy-Riemann equations at the origin, but the derivative of f at z 0 does not exist. 10

Expand in Laurent series the function 1/z2(z about z and z =1. 10

Solve graphically
Maximize Z 6x1 5x2
subject to
2x1 x2
Xl x2 11
Xl 2x2 6
5x1 6x2 90


2. Show that Z7 is a field. Then find and in Z7.
src='./qimages/281-2b.jpg'><br><br>

Find the initial basic feasible solution to the following transportation problem by Vogel's approximation method. Also, fInd its optimal solution and the minimum transportation cost
Destinations
Dt D2 D3 ,D4 Supply
O1 6 4 1 5 14
Origins O2 8 9 2 7 16
O3 4 3 6 2 5
Demand 6 10 15 4


20
3. Show that the set bw; w3 where a and b are real numbers. is a field with respect to usual addition and multiplication. 15
src='./qimages/281-3b.jpg'><br><br>

Evaluate the integral <img src='./qimages/281-3c.jpg'>using residues.

4. Prove that the set Q(root b root 5 b (belongs to) is a commutative ring with identity. 15
Find the minimum value of x2 y2 z2 subject to the condition xyz a3 by the method of Lagrange multipliers. 15
Find all optimal solutions of the following linear programming problem by the simplex method
Maximize Z =30x1 24x2
subject to
5xI +4x2 200
Xl 32
x2 40
Xl, X2 0 20

SECTION-B
5. Solve the partial differential equation (2D2 =24 x). 10
Apply Newton-Raphson method to determine a root of the equation
cosx xex 0 correct up to four decimal places. 10
Use five subintervals to integrate integral 0 to 1 using trapezoidal rule. 10
Use only AND and OR logic gates to construct a logic circuit for the Boolean expression z xy UV. 10
Find the equation of motion of a compound pendulum using Hamilton's equations. 10
6. Reduce the equation<img src='./qimages/281-6a.jpg'> to canonical form.

Solve the system of equations
2x1 x2 7
xl 2x2 x3
x2 2x3
using Gauss-Seidel iteration method (Perform three iterations).
15


Use Runge-Kutta formula of fourth order to find the value of y at X 0·8, where dy/dx root y(O.4) 0.41. Take the step length h =0.2.

7. Find the deflection of a vibrating string (length pie, ends fixed, at ax
corresponding to zero initial velocity and initial deflection
k(sin x sin 2x)
15

Draw a flowchart for Simpson's one-third rule. 15

Given the velocity potential 1/2 lOg[(X a)2 a)2 ,determine the streamlines. 20

7 src='./qimages/281-8a.jpg'><br><br>

For any Boolean variables x and show that x xy =x. 15
Find Navier-Stokes equation for a steady laminar flow of a viscous in compressible fluid between two infinite parallel plates. 20