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

Assignment for M.Sc. (Maths) Part I (Sem. - II) Distance Mode. Last Date for Assignment Submission: - 10 th October, 2016 16 th Mar. 2016 These assignments are to be submitted only by those students who have registered for the Course M.Sc Part I (SEM-II) 2016 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-II) Distance Mode Assignments 5-7

Assignment Questions for M.Sc. (Mathematics) Part I (Sem. II) Oct./Nov. 2016 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. II) Distance mode Repeaters, the examinations to be held in Oct./Nov. 2016. 2. For each subject s assignment the maximum marks obtainable are 30. 3. All Questions are compulsory for each course/paper/subject, assignment should be Hand Written on a separate sheet. 4. Only Blue Colored Ink Pen is 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 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 ASSIGNMENT FOR M.Sc.(Maths) Part I (Sem.II) Oct./Nov. 2016 DISTANCE MODE

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 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:- 10 th October, 2016 16 th Mar. 2016

M.Sc. (Maths.) Part I Sem. II Oct./Nov. 2016 Distance Mode Assignment for the Subject of Paper Number: - Subject Code:- 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 II) Distance Mode. 6. Date of Submission of Assignments: - 7. Signature of Student: - 8. Marks obtained out of 30:- 9. Signature of Evaluator of Assignment: -

M.Sc. (Maths) Part I (Sem. II) Assignment Questions Q.1 Sub. : Linear Algebra (61381) Show that H om (V, V) is an algebra over F. Q.2 Prove that two nilpotent transformations are similar if and only if they have same invariants. Q.3 Find orthonormal basis of subspace of R 5 spanned by υ 1 = (1, 1, 1, 0, 1), υ 2 = (1, 0, 0, -1, 1), υ 3 = (3, 1, 1, -2, 3), υ 4 = (0, 2, 1, 1, -1) Q.4 Find all possible Jordan forms with characteristic polynomial (x 5) 3 (x 2) 3. Q.5 If N is normal and AN = NA then prove that AN * = N * A Sub. : Topology (61382) Q.1 Give an example to show that i A B i A ib Q.2 Let X, be a subspace of, c E X c E.. X and let E be a subset of X, then prove that Q.3 Prove that a topological space X, is locally connected iff every component of an open set is an open set in X. Q.4 If f is a one-one mapping of X onto X, then prove that, f is a homeomorphism iff f E f E for every E X. Q.5 Prove that an infinite Hausdorff space X contains an infinite sequence of nonempty, disjoint open sets. Sub. : Complex Analysis (61383) Q.1 4z 5 Show that the bilinear transformation w maps the unit circle into a circle of 2 4z radius unity and centre -1/2. Q.2 Prove that function f comes arbitrarily close to any complex value in every Q.3 neighborhood of an essential singularity. z e Evaluate using cauchy integral formula dz. 2 ( z 2)( z 1) z 3

Q.4 Evaluate 2 d. 13 5sin Q.5 Prove that all zeros of 0 z 5z 7 are in z 3 6 4 Sub. : Numerical Analysis (61384) Q.1 Explain Newton s Method and find its order of convergence. Q.2 Explain Birge-Vieta Method to obtain the linear factor of the polynomial. Q.3 Solve the nonlinear system of two equations x 2 2x y + 0.5 = 0, x 2 + 4y 2 4 = 0 with initial vector (2, 0.25) by using Newton s Method. Q.4 Find order of convergence of Secant Method. Q.5 Estimate cos (1.12) using polynomial interpolation with first three and then four points from the data. x 1 1.1 1.2 1.3 cosx 0.5403 0.4536 0.3624 0.2675 Q.6 Let P N be Lagrange polynomial agreeing with the function f at the N + 1 distinct points x 0, x 1, x 2,, x N. Suppose that f is at least (N+1) times differentiable. Then the error in using P N (x) to estimate f(x) is given by R N (x) = f(x) P N (x) = [L N (x) f (N+1) (ξ)]/(n+1)! Where ξ is some point in the interval I containing x 0, x 1, x 2,, x N. Q.7 Solve the differential equation Y = -2xy 2 with the initial condition Y(0) = 1 for h = 0.1 by using Eulers Modified method Q.1 Sub. : Differential Geometry (61385) a. For a curve β(s) = ( () ), (), ) where -1 < s < 1. Show that β has unit speed and compute its Frenet apparatus. b. Let be a unit speed curve with curvature k > and τ 0. If lies on sphere of centre c and radius r, show that - c = -ρ N - ρσb where ρ = and =. Q.2 a. Evaluate the 1 form = yzdx x 2 dz on the following vector fields. (i) V = yzu 1 + xzu 2 + xyu 3 (ii) W = xz (U 2 - U 3 ) + yz 2 (U 1 - U 2 ) (iii) X = V + W b. Show that an isometry F = T a.c has an inverse mapping (F) -1, which is also isometry. Find translation and orthogonal parts of (F) -1.

Q.3 a. Find the parametrization of the entire surface obtain by revolving the curves (i) c : y = cos hx around the X axis (ii) c : (x 2) 2 + y 2 = 1 around the Y axis (iii) c : z = x 2 around z - axis b. Find the equation of a tangent plane T P (M) for the surface X (u, θ) = (u cos θ, u sin θ, 2θ) at point P = X (2, ) c. Find the extreme values of the Gaussian curvature of a torus M : (x + y 4) 2 + z 2 = 4.