Assignment for M.Sc. Part I (Sem.I) Distance Mode .

Assignment for
M.Sc. Part I (Sem.I) Distance Mode.
Last Date for Assignment Submission: - 05th Nov. 2014
These assignments are to be submitted only by those students who have registered for the
Course M.Sc Part I (SEM-I) 2013 through Distance mode.
Sr. No.
Description
Page No.
01
Instructions for Submission of Assignments.
2
02
General instructions
3
03
Cover page format
4
05
M.Sc(Maths) Part I (Sem-I) Distance Mode Assignments
5-9
1
Instructions for Assignment Submission
Please note following instructions for submission of Assignments:
1. These assignments should be submitted only by those students who have
registered for M.Sc.(Maths) Part I(Sem.I) Distance mode, the examinations
to be held in Oct./Nov. 2014.
2. For each subject’s assignment the maximum marks obtainable are 30.
3. All Questions are compulsory.
4. For each course/paper/subject, assignment should be Hand Written on a
separate sheet. Only Blue Colored ink P en i s to be used for Assignment
Writing.
5. Only A4 / Journal/Assignment paper should be used for the Assignment
Writing.
6. A separate set should be made for each subject.
7. A cover page as per the format given below on page number 4 should be
attached on the top of the set for each subject.
8. Finally for a particular semester, one file should be made for all subjects.
9. Submit the Assignments to
Centre for Distance Education, Shivaji University, Kolhapur. Pin 416004.
Telephone: - (0231) 2693871, 2693771
E-mail:- cde_multi@unishivaji.ac.in
10. It is the student’s responsibility to ensure that the assignments reach the centre
on or before the due date. No excuses of any kind for late or non-submission of
assignments will be entertained.
If a student is unable to submit the assignment(s) in person, the student may at
his / her own risk submit the assignment(s) through an acquaintance, fellow
student or by courier.
If assignments are sent by Speed Post / courier, at the top of the envelope the
student should clearly write in BOLD letters
“ASSIGMET FOR M.Sc.(Maths) Part I (Sem.I) Oct./ov. 2014
DISTACE MODE”
2
General Instructions
a) Please note that the student has to obtain at least 12 marks out of 30 marks in
internal assignments and 36 marks out of 90 marks in university
examinations.
b) Students are advised that improvement in assignment marks is not permitted at
a later stage once the student gets the minimum passing marks or more (i.e.
12+marks). Hence the students are advised to try to score the maximum at the
first attempt.
c) Assignments should not be copied, should be clear, legible, well presented.
d) Illustrate your answer by giving suitable examples.
e) Draw graphs or diagrams wherever necessary.
f) Students are advised that in case two or more students’ assignments are too
similar in content, nature, the study center Co-Ordinator would at his / her
discretion decide on the quantum of marks to be awarded, irrespective of
how good the submitted assignments are. It is more than likely that the
minimum possible marks (if any) may be awarded to all such involved
assignments.
g) Students are also advised to quote sources (if any) of data, facts, sketches,
drawings etc in their assignments.
h) In case of any query contact
Coordinator
Centre for Distance Education,
Shivaji University, Kolhapur.
Telephone: - (0231) 2693871, 2693771
E-mail:- cde_multi@unishivaji.ac.in
i) Students should see their Namelist, PRN and Seat Nos./Hall Tickets on the
following website :
Website : online.shivajiuniversity.in (Download Hall Ticket for Distance
Education option - preferably through Google Chrome.)
Last Date for Assignment Submission:- 05th Nov. 2014
3
M.Sc. (Maths.) Part I – Sem. I
Oct./Nov. 2014
Distance Mode
Assignment for the Subject of
Subject Code:-
Paper Number: 1. Name of the Candidate :2. Name of the Study Centre
3. Address: -
_
Pin:-
Mobile No: -
4. Exam Seat Number: -
PRN Number :
5. Course: - M.Sc(Maths) Part I (Semester I) Distance Mode.
6. Date of Submission of Assignments: 7. Signature of Student: 8. Marks obtained out of 30:9. Signature of Evaluator of Assignment: -
4
Shivaji University, Kolhapur
M.Sc. (Part – I)(Semester – I) Examination, 2014 - 2015
MATHEMATICS ( Paper – MT 101)
Algebra
Home Assignment
Total Marks : - 30
. B. All Questions are compulsory.
1. a) State and prove Zassenhaus lemma.
[4]
b) Show that two subnormal series of a group have isomorphic refinements. [4]
2. a) State and prove Burnside theorem.
[4]
b) If G is a finite group then prove that G is a p – group if and only if
power of prime p.
G
is a
[4]
3. a) If F is a field then prove that F[x] is an Euclidean domain.
b) If R is UFD then prove that R[x] is UFD.
[4]
[3]
4. a) Prove that any homomorphic image of an R – module M is isomorphic with
its suitable quotient module.
[4]
b) Let M be an R – module. If K ⊂ N ⊂ M and if K is a direct summand of N
and N is direct summand of M then prove that K is direct summand of M.
[3]
5
M.Sc. (Part – I)(Semester – I) Examination, 2014
MATHEMATICS ( Paper – MT 102)
ADVACED CALCULUS
Home Assignment
Q.1 a) State and prove Dirichlet’s test for uniform convergence.
[4]
b) If {fn} is a sequence of functions defined by
fn(x) = n2x(1 – x)n, 0 < x < 1, x = 0, 1, 2, 3, . . .
1
f n ( x) dx ≠
then show that nlim
→∞∫
0
1
∫ f (x) dx .
[4]
0
Q.2 a) Define Taylor’s series about c generated by f. State and prove Bernstein’s
[4]
theorem.
∞
b) Show that if
∑a
n =0
∞
n
converges absolutely to the sum A and if
∑b
n
n=0
converges to the sum B, then prove that the Cauchy product of these
two series also converges.
[4]
Q.3 a) Define the directional derivative of function f. If f is differentiable at c,
[4]
then prove that f is continuous at c.
b) If both partial derivatives Drf and Dkf exists in an n – ball B(c ; δ) and if
both are differentiable at c, then prove that Dr , k f(c) = Dk, r f(c).
[3]
Q.4 a) Define curl and divergence of a vector field.
If f (x, y, z) = xy2z2i + z2cosy j + x2ey k is a vector field then compute
[4]
div f and curl f .
b) Find the surface area of the hemisphere.
6
[3]
M.Sc. (Part – I)(Semester – I) Examination, 2014 - 2015
MATHEMATICS ( Paper – MT 103)
Real Analysis
Home Assignment
Total Marks : - 30
. B. All Questions are compulsory.
Q.1 a) Let {Ek} be a countable collection of sets of real numbers then prove that
∗

m  U E k  ≤ ∑ m∗ ( E k ) .
k
 k
[4]
b) State and prove Borel – Cantelli Lemma.
[4]
Q.2 a) Prove that f is measurable if and only if inverse image of an open set is
measurable.
[4]
b) State and prove Fatou’s Lemma.
[4]
Q.3 a) Let f and g be integrable function over E then prove that
i) αf is integrable over E and
∫ αf = α ∫ f
E
E
∫f +g= ∫f + ∫g
ii) f + g is integrable over E and
E
E
E
[4]
b) Prove that if the function f is monotone on the open interval (a, b) then it is
differentiable almost everywhere on (a, b).
[3]
Q.4 a) Let f be the absolutely continuous on a closed bounded interval [ a, b ] , then
prove that f is the difference of two increasing absolutely continuous and
hence prove that f is a function of bounded variations on [ a, b ].
[4]
b) State and prove Holder’s Inequality.
7
[3]
M.Sc. (Part – I)(Semester – I) Examination, 2014 - 2015
MATHEMATICS ( Paper – MT 104)
Differential equations
Home Assignment
Total Marks : - 30
. B. All Questions are compulsory.
1. a) Find the solution of y′′′ – y′ = x.
[4]
b) Find two linearly independent power series solution of the following equation
y′′ – xy′ + y = 0 .
[3]
c) If φ1, φ2, φ3, . . . , φn are n solutions of L(y) = 0 then prove that they are
linearly independent if and only if W(φ1, φ2, φ3, . . . , φn)(x) ≠ 0, for all x in I.
[3]
2. a) Classify the singular points in the finite plane of the equation
[4]
x(x – 1)2(x + 2)y′′ + x2y′ – (x3 2x – 1)y = 0.
b) Show that
J0′(x) = – J1(x).
(xα Jα)′(x) = xαJα – 1(x)
Jα – 1(x) = Jα + 1(x) = 2 Jα′ (x).
[3]
c) Compute the first four successive approximations φ0, φ1, φ2, φ3 for
y′ = x2 + y2, y(0) = 0.
3. a) Find all solutions of the equation x2y′′ + xy′ + 4y = 1, for
x
> 0.
[3]
[4]
b) Let b1, b2, b3, . . . , bn be non – negative constants such that for all x in I
a i ( x ) ≤ bi , i = 1, 2, 3, . . . , n and define kby,
where k = 1+ b1 + b2 + b3 + ... + bn .
If x0 is a point in I and φ is a solution of L(y) = 0 on I then
0
0
≤ φ(x) ≤ φ(x0 ) e
prove that φ(x0 ) e
[3]
c) If one solution of x2y′′ – 7xy′ + 15 y = 0, for x > 0 is φ1(x) = x3, then find
another solution .
[3]
−k x − x
k x−x
8
M.Sc. (Part – I)(Semester – I) Examination, 2014 - 2015
MATHEMATICS ( Paper – MT 105)
Classical Mechanics
Home Assignment
Total Marks : - 30
. B. All Questions are compulsory.
1. a) Define Conservative force. Prove that if the forces acting on a particle are
conservative then total energy of a particle is conserved.
[4]
b) Find the expression for the kinetic energy as the quadratic function of
generalized velocities. Further show that when the constraints are
sceleronomic the kinetic energy is a homogeneous function of generalized
velocities and
∑ q&
j
j
∂T
= 2T.
∂ q& j
[4]
2. a) If the cyclic generalized coordinate qj is such that dqj represents rotation of the
system of particles around some axis nˆ then prove that the total angular
momentum is conserved along nˆ .
[4]
b) If f satisfies Euler′s Lagrange′s equation
total derivative
dg
dx
∂f
d ∂ f 


−
∂y
d x  ∂ y ′ 
= 0, ten prove that f is
for some function of x and y and conversely.
[4]
3. a) Derive Lagrange′s equation of motion for conservative system from
Hamilton′s principle.
[4]
[4]
b) Obtain the relation between ∆ and δ variation.
[3]
4. a) Show that the general displacement of a rigid body with one point fixed is a
rotation about some axis passing through the fixed point.
[4]
b) Find the kinetic energy of a rigid body rotating about a fixed point of the body
when the moments and products of inertia of the body relative to the set of
axes through fixed point are known.
[3]
9