C. S. (Main) Exam: 2011
Serial No. C J
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 ofthe answer-book in the space provided/or the purpose. No marks will be givenfor,the answers written in a medium other than that specified in the Admission Certificate.
Candidates should attempt Question Nos. 1 and 5 whicharecompulsory, andanythree oftheremaining selecting at least olle questionfrom each Section.
Assume suitable data if considered necessary and
indicate the-same clearly.
Symbols, and notations carry usual meaning, unless
othenvise indicated.

All questions carry equal marks.
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I
Show that the set G f2, f5, f·}
36of six transformations on the set of Complex numbers defined by
1-z

1/Z

and
is a non-a belian group of order 6 w.r.t. composition of mappings. 12

Let S and. f be defined by 1/x where 0 x 1 (in IR). Is f uniformly continuous on S Justify your answer. 12
If fez) u iv is an analytic function of z=x+iy and u-v ey-cos x+sin x/cos hy-cos x,find subject to t e condition,
Solve by Simplex method, the following
LP Problem:
Maximize, Z 5x1 3x2
Constraints, 3xt 5x2<-15
5x1 2x2 10
x2 0 12
Prove that a group ofPrime order is a belian. 6
How many generators are there of the cyclic group of order 8 6
Give an example ofa group G in which every proper subgroup is cyclic but the group it self is not cyclic.
Let f nx(l XE Examine the uniform convergence of on 1]. 15
the function is analytic and one, valued in prove that for 0 r <img images/1076-2c.jpg'> where is the real part.of f(a reiO).

Find the shortest distance from the origin to the' hyperbola x2 8xy 7y2 225 15

Let F be the set of all real valued. continuous functions defined on the closed interval 1]. Prove that is a Cornmutative Ring with unity with respect to addition and multiplication of functions defined pointvise as below:

and (f. x belongs where gE F. 15
Show that the series for which the sum of first n terms cannot be differentiated term-by-term at x o. What happens ·at x not equal 0 15
Evaluate by Contour integration, <img images/1076-3c.jpg'>
Find the Laurent series for the function with centre z=1.
4. Let a and b be elements of a group, with a2= b6= e and ab b4a
. Find the order of ab, and express its inverse in each of the forms ambn and bman• 20 ...
(Contd.)
(b).Show that if <img images/1076-4b.jpg'> then its derivative
<img images/1076-4b.jpg'>,for all x. 20
(c)·Write down· the dual of the following LP problem and hence solve it by graphical method:
Minimize, Z =6xt 4x2
Constraints, 2x1+ 1
3x1 4x2 1·5

SECTION-B
5. Solve the PDE (D2 -x2y
Solve the PDE
<img src=/q images/1076-5b.jpg'>
Calculate 5 dx/1+x (upto 3 places of decimal) by dividing the range into 8 equal parts by Simpson's 1/3 rd Rule.
Compute (3205)10 to the base 8.
Let A be an arbitrary but fixed Boolean algebra with operations V and' and the zero and the unit element denoted by 0 and 1 respectively. Let be elements of A. If y belongs to A be such that x y 0 and x v y 1 then prove that y 12
Let a be the radius of the base of a right circular cone of height h and mass M. Find the moment of inertia of that right circular cone about a line through the vertex perpendicular to the axis. 12
6. Find the surface4 satisfying a2z ax 2 6x 2 and touching z x3 y3 along its section by the pIane x y 1 O. 20
Solve <img src=/q images/1076-6b1.jpg'>
satisfying the boundary conditions
<img src=/q images/1076-6b2.jpg'>

Obtain temperature distribution in a unifoRM bar of unit length whose one end is kept at 1aoc· and the other end is insulated. Also it is given that 1 0 x 1. 20
7. A solid of revolution is formed by rotating about
the x-axis, the area between the x-axis, the line
x 0 and x 1 and a curve through the points
with the following co-ordinates:
x ·00 ·25 ·50 ·75 1
y 1 -9896 ·9589 ·9089 ·8415
Find the volume of the solid. 20
Find the logic circuit represents the following Boolean function. Find also an equivalent simpler circuit

Y z
1 1 1 1
1 1 0 0
1 0 1 0
1 0 0 0
0 1 1 1
0 1 0 0
0 0 1 0
O 0 0 0

Draw a flow chart for Lagrange.'s interpolation 20
The ends of a heavy, rod of length 2a are rigidly
attached to two light rings which can respectively slide on the thin smooth fixed horizontal and vertical wires Ox and ·Oy. The rod starts at an angle a to the horizon with an angular velocity
<img src=/q images/1076-8A1.jpg'> and moves downwards.
Show that it will strike the horizontal Wire at the end of time
<img src=/q images/1076-8A2.jpg'>
An infinite row of equidistant rectilinear vortices are at a distance a apart. The vortices are of the same numerical strength K but they are alternately of opposite signs. Find the' Complex function that determines the velocity potential.and the stream
function. 30