karnatakastateopenuniversity.in M.Sc Mathematics Model Question Paper : Karnataka State Open University

Organisation : Karnataka State Open University
Exam : M.Sc.
Document Type : Model Question Paper
Category or Subject : Mathematics

Website : https://ksoumysuru.ac.in/
Download Sample Question Paper :
Math 4.1 : https://www.pdfquestion.in/uploads/9476-Math%204.1.pdf
Math 4.2 : https://www.pdfquestion.in/uploads/9476-Math%204.2.pdf
I Sem MSc 2015 : https://www.pdfquestion.in/uploads/9476-I-Sem-MSc.pdf
IV Sem MSc 2016 : https://www.pdfquestion.in/uploads/9476-IV-Sem-MSc.pdf
III Sem MSc 2015 : https://www.pdfquestion.in/uploads/9476-II%20Sem-MSc.pdf
II Sem MSc 2016 : https://www.pdfquestion.in/uploads/9476-III-Sem-Msc.pdf

Karnataka M.Sc Mathematics Question Paper

Download Question Paper of M.Sc. Mathematics Sample Question Paper is now available in the official website of Karnataka State Open University.

Related : Karnataka State Open University Physics Model Question Paper : www.pdfquestion.in/9472.html

Instructions

Number Theory :
Time : 3 Hours
Max. Marks : 80
Note :
1) Answer any five questions.
2) All questions carry equal marks.

Model Questions

1. a) If a and b are any two integers, not both of them are zero, then show that there exists integers x and y such that gcd (a, b) = ax + by.
b) State and prove the fundamental theorem of arithmetic.
c) Explain the prime number theorem.

2. a) Show that for each positive integer n1 n = n/d @ , the sum being extended over all positive divisors of n.
b) Explain the Hill Cipher with an example.
c) Show that Dirichlet multiplication is commulative and associative.

3. a) Let p be an odd prime and gcd (a, p) = 1. Then show that ‘a’ is a quadratic residue of p if and only if a(p – 1)/2 1(mod p).
b) State and prove the Gauss lemma for quadratic residue. Evaluate n of Gauss lemma for (11/23).

4. a) Define the finite continued fraction. Show that every rational number can be written as a finite simple continued fraction.
b) Solve each linear diophantine equation using continued fraction
i) 12x + 13y = 14
ii) 28x + 91y = 119.
c) If the number x has a periodic simple continued fraction expansion, then show that x is a quadratic irrational.

Graph Theory And Algorithms :
1. a) State and prove Handshaking property. Hence prove that there is no graph with odd number of odd degree vertices.
b) Show that every u.v. walk in graph contains a u-v path.
c) Define a bipartite graph. Show that a graph is bipartite if and only if all its cycler are even.

2. a) Define the following with an example
i) composition product
ii) normal produt
iii) tensor product

3. a) If G has a Hamiltonian cycle, then show that for each non-empty set S V(G),the graph G – S has atmost |S| components.
b) Explain the following
i) Seating problem
ii) Travelling salesman problem
c) Show that a connected graph is a tree if and only if it is minimally connected

4. a) Show that any graph G is connected if and only if it has a spanning tree.
b) Show that every cut-set in a connected graph G must contain at least one branch of every spanning tree of G.
c) Define the following with an example
i) Block graph
ii) Cut-vertex graph

Physics :
1. (a) Define electromagnetic potentials. Express Maxwell’s equations in terms of the electromagnetic potentials. (10)
(b) Show that the gauge transforms A’ and ? ‘ satisfy the Lorentz condition if and only if the gauge functions satisfy the wave equation. (5)
OR
2. (a) Obtain an expression for Lienard-Wiechert potentials of a moving point charge. (10)
(b) Show that under Lorentz gauge E and B also satisfy the wave equation. (5)

3 .(a) Deduce the Abraham-Lorentz formula for radiation reaction and explain its significance. (10)
(b) Show that the expression of power radiated by an oscillating electric dipole leads to Larmor’s formula.
OR
4 .(a) Derive expression for the power radiated by an oscillating electric dipole. (10)
(b) Write a note on Pinch effect. (5)

5. (a) Deduce an expression for Poynting vector and discuss the importance of Poynting theorem. (10)
(b) Starting from Fresnel’s equations obtain Brewster’s law. (5)
OR
6 .(a) Obtain an expression for Clausius- Mossotti equation for electric fields in solids. (10)
(b) Write a note on retardation plates. (5)

7. (a) Give the theory of multiple reflections from a plane parallel film. (10)
(b) Note down the conditions for sustainable interference. (5)
OR
8. (a) Give a detailed description of diffraction at a circular aperture. (10)
(b) Discuss Fresnel’s diffraction in brief. (5)