Loyola College M.Sc. Mathematics Nov 2008 Mathematics For Computer Applications Question Paper PDF Download

November 15, 2017 by 01 prakash

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

AB 31

M.C.A. DEGREE EXAMINATION – COMPUTER APPLICATION

FIRST SEMESTER – November 2008

MT 1902 – MATHEMATICS FOR COMPUTER APPLICATIONS

Date : 11-11-08 Dept. No. Max. : 100 Marks

Time : 1:00 – 4:00

Part A (Answer ALL questions) 2 x 10 = 20

Define Lattice homomorphism between two lattices.
With usual notations prove that (i)(ii) .
Define context free grammar.
What is the difference between deterministic finite automata and non-deterministic finite automata?
Let G = (N, T, P, S), where N = {S}, T = {a}, P: {S → SS, S → a}. Check whether G is ambiguous or unambiguous.
Give a deterministic finite automata accepting the set of all strings over {0, 1} containing 3 consecutive 0’s.
If R and S be two relations defined by and , then find

RS, RR and R.

Let and ,. Write the matrix of

of R and sketch its graph.

Define ring with an example.
State Kuratowski’s theorem.

Part B (Answer ALL questions) 5 x 8 = 40

(a) Show that De Morgan’s laws given by and hold in a

complemented, distributive lattice.

(OR)

(b) Let be a lattice. For any prove the following distributive inequalities:

) and .

(a) Show that L(G) = is accepted by the grammar G = (N, T, P, S) where N = {S,A} T = {a, b}, P consists of the following productions: S → aSA, S → aZA, Z → bZB, Z → bB, BA → AB, AB → Ab, bB → bb, bA→ ba.

(OR)

(b) Let the grammar G = ({S,A}, {a, b}, P, S) where P consists of S →aAS, S → a, A → SbA , A → SS, A → ba. For the string aabbaa find a

(i) leftmost derivation

(ii) rightmost derivation

(iii) derivation tree.

(a) (i) Define deterministic finite state automata.

(ii) Draw the state diagram for the deterministic finite state automata, M = where Q =, Σ ={a, b}, F = and δ is defined as follows:

δ	a	b

Check whether the string bbabab is accepted by M. (3+5)

(OR)

(b) Given an non-deterministic finite automaton which accepts L. Prove that there exists a deterministic finite automaton that accepts L.

(a) (i) Write short on Hasse diagram.

(ii) Let and relation be such that if x divides y. Draw the

Hasse diagram of . (4+4)

(OR)

(b) (i) Show that n³+2n is divisible by 3 using principle of mathematical induction.

(ii) If the permutations of the elements of {1,2,3,4,5} be given by

, then find

α ^-1^-1. (4+4)

(a) Prove that there is a one- to-one correspondence between any two left cosets of H in G.

(OR)

(b) (i) If G is a graph in which the degree of every vertex is atlest two, then prove that G

contains a cycle.

(ii) Prove that the kernel of a homomorphism g from a group to is a subgroup

of . (4+4)

Part C (Answer ANY TWO questions) 2 x 20 = 40

16.(a) Let G be (p,q)graph, then prove that the following statements are equivalent:

(i) G is a tree. (ii) Every two vertices of G are joined by a unique path (iii) G is connected

and (iv) G is acyclic and p = q+1.

(b) Let H be a subgroup of G. Then prove that any two left cosets of H in G are either

identical or have no element in common. (14+6)

(a) Let be a Boolean Algebra. Define the operations + and · on the elements of B by,

. Show that is a boolean ring with identity 1.

(b) Prove that every chain is a distributive lattice. (15+5)

(a) If G = (N, T, P, S) where N = {S, A,B}, T = {a,b}, and P consists of the following rules:

S → aB, S → bA, A → a, A → aS, A → bAA, B →b, B → bS, B → aBB. Then prove the following:

S w iff w consists of an equal number of a’s and b’s
A w iff w has one more a than it has b’s.
B w iff w has one more b than if has a’s

(b) State and prove pumping lemma. (10+10)

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Loyola College M.Sc. Mathematics Nov 2008 Differential Geometry Question Paper PDF Download

November 15, 2017 by 01 prakash

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

AB 29

M.Sc. DEGREE EXAMINATION – MATHEMATICS

FIRST SEMESTER – November 2008

MT 1807 – DIFFERENTIAL GEOMETRY

Date : 11-11-08 Dept. No. Max. : 100 Marks

Time : 1:00 – 4:00

Answer ALL the questions

I a) Prove that the curvature is the rate of change of angle of contingency with respect to

arc length.

(or)

b) Show that the necessary and sufficient condition for a curve to be a straight line is that

for all points. [5]

c) (1) Find the centre and radius of an osculating circle.

(2) Derive the formula for torsion of a curve in terms of the parameter u. [8+7]

(or)

d) Derive the Serret-Frenet Express them in terms of Darboux vector. [15]

II a) Show that the circle , has three point contact at the

origin with a paraboloid with

(or)

b) Derive the necessary and sufficient condition for a space curve to be a helix. [5]

c) If two single valued continuous functions and of the real variable are given then prove that there exists one and only one space curve determined uniquely except for its position in space, for which s is the arc length, k is the curvature and is the torsion.

(or)

d) Find the intrinsic equation of the curve [15]

III a) Derive the equation satisfying the principal curvature at a point on the space curve.

(or)

b) Prove that the metric is always positive. [5]

c) Prove that is a necessary and sufficient condition for a surface to be

developable.

(or)

d) Define developable. Derive polar and rectifying developables associated with a

space curve. [15]

IV a) State and prove Meusnier Theorem.

(or)

b) Prove that the necessary and sufficient condition for the lines of curvature to be

parametric curves is that [5]

c) (1) Derive the equation satisfying the principal curvature at point on a surface.

(2) How can you find whether the given equation represent a curve or a surface?

(3) Define oblique and normal section. [9+2+4]

(or)

d) (1) Define geodesic. State the necessary and sufficient condition that the curve

be a geodesic .

(2) Show that the curves are geodesics on a surface with metric

. [5+10]

V a) Prove that the Gaussian curvature of a space curve is bending invariant.

(or)

b) Show that sphere is the only surface in which all points are umbilics. [5]

c) Derive the partial differential equation of surface theory. Also state Hilbert

Theorem.

(or)

d) State the fundamental theorem of Surface Theory and demonstrate it with an

example. [15]

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Loyola College M.Sc. Mathematics Nov 2008 Computer Algorithms Question Paper PDF Download

November 15, 2017 by 01 prakash

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

AB 30

M.Sc. DEGREE EXAMINATION – MATHEMATICS

FIRST SEMESTER – November 2008

MT 1808 – COMPUTER ALGORITHMS

Date : 13-11-08 Dept. No. Max. : 100 Marks

Time : 1:00 – 4:00

Answer ALL questions

a) Ifthen prove that

b) Explain sequential labeling of the nodes of a binary tree. Give an example of a complete binary tree which is not full. (5)
c) Give an algorithm to form a heap of n elements inserting one element at a time. Simulate it on A(1:8) = (34, 67, 12, 55, 80, 71, 99, 10).

d) Give an algorithm to form a heap of n elements using algorithm ADJUST. Simulate it on A(1:7) = (66, 83, 14, 90, 45, 33, 81). (15)
a) Give the control abstraction for divide and conquer strategy.

b) Explain the problem ‘Optimal Merge Pattern’. (5)

c) State algorithm BinSearch.

Simulate it on A(1:12) = (12, 23, 33, 45, 56, 61, 66, 70, 78, 89, 90, 95) when (i) x=23, (ii) x=57 and (iii) x=95. Draw the binary decision tree when n = 12.

d) State algorithm QUICKSORT.

Simulate it on (1:9) = (33, 67, 78, 31, 90, 51, 29, 66, 84). (15)

a) Give the control abstraction for greedy method.

b) Explain clearly the job sequencing problem with deadlines. (5)

c) Give algorithm GREEDYKNAPSACK. If p₁/w₁≥ p₂/w₂≥ … p_n/w_n then prove that GREEDYKNAPSACK generates an optimal solution to the given instance of the knapsack problem.

d) Explain the optimal storage on tapes problem with an example. With usual notations prove that if then the ordering minimizes over all possible values of the (15)

a) Explain the inorder and the preorder tree traversals with examples.

b) Explain the breadth first search traversal with an example. (5)
c) Explain two different formulations of the sum of subsets problem. Give a recursive backtracking algorithm for sum of subsets problem.

d) Explain in detail the 4-queens problem. Give a backtracking algorithm to solve the n-queens problem. (15)
a) What is Satisfiability problem? State Cook’s Theorem.

b) Define a nondeterministic algorithm with an example. (5)
c) Explain maximum clique problem with an example. Prove that CNF-satisfiability α clique decision problem.

d) Explain the node cover decision problem with an example. Prove that the problem is NP-complete. (15)

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Loyola College M.Sc. Mathematics Nov 2008 Linear Algebra Question Paper PDF Download

November 15, 2017 by 01 prakash

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

AB 26

M.Sc. DEGREE EXAMINATION – MATHEMATICS

FIRST SEMESTER – November 2008

MT 1804 – LINEAR ALGEBRA

Date : 04-11-08 Dept. No. Max. : 100 Marks

Time : 1:00 – 4:00

Answer ALL the questions.

a) i) Prove that the similar matrices have the same characteristic polynomial.

ii) Let T be a linear operator on which is represented in the standard ordered basis by the matrix . Prove that T is not diagonalizable. (5)
b) i) State and prove Cayley-Hamilton theorem.

ii) Let V be a finite dimensional vector space over F and T a linear operator on V.Then prove that T is diagonalizable if and only if the minimal polynomial for T has the form p=where are distinct elements of F. (15)

a) i) Let V be a finite dimensional vector space. Let be subspaces such that with for. Then prove that are independent subspaces.

ii) Let W be an invariant subspace for T. Then prove that the minimal polynomial for divides the minimal polynomial for T. (5)
b) i) State and prove Primary Decomposition theorem.

ii) If, then prove that there exist k linear operators on V such that
Each is a projection.
.

The range of is.

iii) Prove that if are k linear operators which satisfy conditions 1, 2 and 3 of the above and if let be range of then. (8+7)

a) i) Let T be a linear operator on a vector space V and W a proper T-admissible subspace of V. Prove that W and Cyclic subspace Z(a;T) are independent.

ii) Let T be a linear operator on which is represented in the standard ordered basis by the matrix . Prove that T has no cyclic vector. What is the T-cyclic subspace generated by the vector (1,-1, 3)? (5)
b) i) Let a be any non-zero vector in V and let be the T-annihilator of . Prove the following statements:
The degree of is equal to the dimension of the cyclic subspace Z(a;T).
If the degree of is k, then the vectors a, Ta, ,… form the basis for Z(a;T).
If U is the linear operator on Z(a;T) induced by T, then the minimal polynomial for U is .

ii) Let T be a linear operator on a finite dimensional vector space V and let

be a proper T-admissible subspace of V. Prove that there exist non-zero vectors in V with respective T-annihilators such that V=and divides, k=2, 3…r. (15)

a) i) Define a positive matrix. Verify that the matrix is positive.

ii) Let V be a complex vector space and f a form on V such that f () is real for every. Then prove that f is hermitian. (5)
b) i) Let F be the field of real numbers or complex numbers. Let A be an nxn matrix over F. Then prove that the function g defined by is a positive form on the space if and only if there exists an invertible nxn matrix P with entries in F such that.
ii) State and prove Principle Axis theorem. (6+9)

iii) State and prove Spectral theorem and hence prove if, then for. (15)

a) i) Define a bilinear form on a vector space over a field. Let m and n be positive integers and F a field. Let V be the vector space of all mxn matrices over F and A be a fixed mxm matrix over F. If, prove that is a bilinear form.

ii) State and prove polarization identity for symmetric bilinear form f. (5)
b) i) Let V be a finite dimensional vector space over the field of complex numbers. Let f be a symmetric bilinear form on V which has rank r. Then prove that there is an ordered basis for V such that the matrix of f in the ordered basis B is diagonal and f () =1, j=1,…,r. Furthermore prove that the number of basis vectors for which =1 is independent of the choice of basis.

ii) If f is a non-zero skew-symmetric bilinear form on a finite dimensional vector space V then prove that there exist a finite sequence of pairs of vectors,with the following properties:

1) f ()=1, j=1,2,,…,k.

2) f ()=f ()=f ()=0,ij.

3) If is the two dimensional subspace spanned by and, then V=where is orthogonal to all and and the restriction of f to is the zero form. (15)

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Loyola College M.Sc. Mathematics Nov 2008 Mathematical Methods In Biology Question Paper PDF Download

November 15, 2017 by 01 prakash

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

AB 37

M.Sc. DEGREE EXAMINATION – MATHEMATICS

THIRD SEMESTER – November 2008

MT 3875 – MATHEMATICAL METHODS IN BIOLOGY

Date : 10-11-08 Dept. No. Max. : 100 Marks

Time : 9:00 – 12:00

Answer ALL Questions:

(a) (i) Write and explain any five basic SQLPLUS command.

(or)

(ii) Write a short note on LIKE command using an example. (5)

(b) (i) Create a table named “STUDENT_LIST” with student_id, student_name, department, address_ty. What are the different methods involved in inserting a record in the student_list table?

(ii) What are the advantages of chi-square test? (12 + 3)

(or)

(iii) Briefly explain normalization using an example. (15)

(a) (i) A sample analysis of examination results of 200 MBA’s was made. It was found that

46 students had failed, 68 secured a third division, 62 secured a second division and the rest were placed in first division. Are these figures commensurate with the general

examination result which is in the ratio 4: 3: 2: 1 for various categories respectively?

(or)

(ii) Let where,, and d is defined as:

d	0	1

Draw the state diagram and verify if 11010101 is in L (M). (5)

(b) (i) Write a short note on combining logic.

(ii) Briefly explain the different types sampling. (7 + 8)

(or)

(iii) The following figures show the distribution of digits in number chosen at random from a telephone directory:

Digits	0	1	2	3	4	5	6	7	8	9
Frequency	966	1007	1000	1023	1075	900	1107	1005	964	953

Test whether the digits may be taken to occur equally frequently in the directory.

(iv) Write notes on Centromere, and RNA. (8 + 7)

(a) (i) Write an account on the aims and tasks of Bioinformatics.

(or)

(ii) Define Gene, Chromosome, DNA and Genome. (5)

(b) (i) Explain generalized and specialized data bases.

(ii) Write notes on FASTA and BLAST. (7 + 8)

(or)

(iii) Discuss the application of bioinformatics.

(iv) Briefly describe Lavenshtein and Hamming distance. (10 + 5)

(a) (i) Explain the different approaches in phylogenetic analysis.

(or)

(ii) What is called optical alignment? (5)

(b) (i) Discuss the important features of algorithms.

(ii) Differentiate Homologs, Orthologs, Paralogs and Xenologs. (7 + 8)

(or)

(iii) Explain Global and Local alignment.

(iv) Briefly explain scoring mutations, Deletions and Substitutions. (8 + 7)

(a) (i) Write a note on nucleic acid sequence data bases.

(or)

(ii) Explain the responsibilities of DBMS. (5)

(b) (i) Form an essay on the methods of phylogenetic tree construction.

(ii) What are clade, Taxon and Node? (10 + 5)

(or)

(iii) Briefly explain Word of k-tuple.

(iv) Explain Dot Matrix.

(v) What are high scoring and low scoring matches? (5 + 5 + 5)

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Loyola College M.Sc. Mathematics Nov 2008 Mathematical Social Sciences Question Paper PDF Download

November 15, 2017 by 01 prakash

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

AB 36

P.G. DEGREE EXAMINATION – COMMON PAPER

THIRD SEMESTER – November 2008

MT 3925 – MATHEMATICAL SOCIAL SCIENCES

Date : 12-11-08 Dept. No. Max. : 100 Marks

Time : 9:00 – 12:00

Answer ALL Questions.

(a) Define a research problem.

(or)

What are the essential characteristics of sampling? (5 marks)

Explain briefly all the sources of hypothesis.

(or)

Explain the utilities of Social Science research. Explain the difficulties in social science research.

(15 marks)

(a) (i) Give an example of a graph which has a cut vertex but does not have a bridge.

(ii) Find G₁ + G₂ and G₁ x G₂ if G₁ = and G₂ =

(2 + 3 marks)

(or)

(b) Define (i) closed walk, (ii) trial, and (iii) path. Give examples. (5 marks)

(ii) Find adjacency matrix and incidence matrix for the following graph.

(iii) In the following graph, find x – y walk, x – y trial, x – y path and length of the x – y walk, trial and path.

(2 + 7 + 6 marks)

(or)

(d) (i) Define directed graph, indegree and outdegree of a directed graph. Give an example.

(ii) Prove that, in a graph G every x – y walk contains an x – y path.

(iii)If G contains no odd degree vertices then prove that G contains no bridges.

(2 + 7 + 6 marks)

III. (a) Calculate the mean for the following distribution.

Cost of production: 2 – 6 6 – 10 10 – 14 14 – 18 18 – 22 22 – 26 26 – 30 30 – 34

No. of Holdings : 1 9 21 47 52 36 19 3

(or)

(b) Certain crosses of pea gave 5321 yellow and 1804 green seeds. The expectation is 25 percent of green seeds on a Mendelian hypothesis. Can the divergence from the expected value have arisen from the fluctuation of simple sampling only? (5 marks)

Series A 192 288 236 229 184 260 348 291 330 243

Series B 83 87 93 109 124 126 126 101 102 108

(or)

(d) Find the mean, median and mode for the following data.

Marks 0 – 4 4 – 8 8 – 12 12 – 16 16 – 20 20 – 24 24 – 28

No. of students 10 12 18 7 5 3 2

(15 marks)

(a) Write L-formula, P-formula, L-P formula, M.E formula, Walsch Price index formula for the

construction of index numbers.

(or)

(b) Construct quantity index numbers taking 1967 as the base:

Commodity Average Price Production

1967 1968 1969 1970

A 1.00 62 65 66 90

B 1.50 138 120 110 80

C 0.25 500 540 580 800

D 2.25 10 10 10 10

(5 marks)

(or)

(d) Prepare price and quantity index numbers for 1972 with 1961 as base year from the following data by using (i) Laspeyre’s, (ii) Paasche’s, (iii) Marshall-Edgeworth, and (iv) Fisher’s method.

Article I Article II Article III Article IV

Year Price Quantity Price Quantity Price Quantity Price Quantity

1961 5.00 5 7.75 6 9.63 4 12.50 9

1972 6.50 7 8.80 10 7.75 6 12.75 9

(15 marks)

(a) Explain briefly scaling individual test items in terms of difficulty.

(or)

(b) The fifth grade norms for a reading examination are mean = 60, s.d = 10, and for an arithmetic examination mean = 26, s.d = 4. Ram scores 55 on the reading test and 24 on the arithmetic test. Compute his -scores. In which test is he better? (5 marks)

(c) (i) Five problems are solved by 15%, 34%, 50%, 62% and 80% respectively of a large unselected group. If the zero point of ability in this test is taken to be at – 3, what is the –value of each problem as measured from this point?

(ii) A test is administered on 400 pupils. It gives mean 60 and standard deviation 12. Complete the following table of equivalent raw scores.

Raw Score : 84 78 72 66 60 54 48 42 36

– score : __ __ 1 __ 0 __ __ __ __

Standard score: __ __ __ __ __ __ 40 __ __

(10 + 5 marks)

(or)

(d) (i) Given three test items 1, 2 and 3 are passed by 50%, 40% and 30% respectively of a large group, on the assumption of normality of the distribution, what percentage of this group must pass test item 4 in order for this to be as much more difficult than 3, as 2 is more difficult than 1?

(ii) Write short notes on Intelligence Quotient. (10 + 5 marks)

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Loyola College M.Sc. Mathematics Nov 2008 Algorithmic Graph Theory Question Paper PDF Download

November 15, 2017 by 01 prakash

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

AB 35

M.Sc. DEGREE EXAMINATION – MATHEMATICS

THIRD SEMESTER – November 2008

MT 3806 – ALGORITHMIC GRAPH THEORY

Date : 15-11-08 Dept. No. Max. : 100 Marks

Time : 1:00 – 4:00

Answer all questions. Each question carries 20 marks.

1(a)(i). Prove that an edge of a graph is a cut edge if and only if it is contained in no cycle

of the graph.

(OR)

(ii). Prove that an interval graph satisfies the triangulated graph property. Give an

example to show that not triangulated graph is an interval graph. (5)

(b) (i). Define connectivity and edge-connectivity with examples.

(ii). Prove that a graph G with is 2-connected if and only if any two vertices of

G are connected by at least two internally-disjoint paths. (5+10)

(OR)

(iii). State the depth-first search algorithm and simulate it on the following graph by

selecting the vertex a.

(15)

2(a)(i). Define a simplicial vertex. Obtain a perfect elimination scheme for the following

graph.

(OR)

(ii). Define a vertex separator and a minimum vertex separator. Obtain a minimum

vertex separator for the following graph.

(5)

(b)(i). Let G be an undirected graph. Then prove that the following statements are

equivalent.

(1). G is triangulated.

(2). Every minimal vertex separator induces a complete subgraph in G.

(ii). State the Lexicographic breadth first search algorithm and apply it to the following

graph and obtain the perfect elimination scheme.

(OR)

(iii). Suppose that there is a tree T whose vertex set is the set of maximal cliques of a graph G such that each induced subgraph : v ε V is connected, where K_v consists of those maximal cliques which contain v. Prove that G is the intersection graph of a family of subtrees of a tree and hence deduce that G is triangulated.

(iv). Prove that a family of subtrees of a tree satisfies the Helly property. (10+5)

(a)(i). Define covering and edge-coverings in a graph with examples.

(OR)

(ii). Let G be a split graph with the vertex set partitioned into a stable set S and a

clique K. If |S| = α(G) and |K| = ω(G) – 1, then prove that there exists an x ε S

such that K +{x} is a clique. (5)

(b)(i). Let G be an undirected graph. Prove that the following statements are equivalent.

(1). G is a split graph

(2). G and are triangulated graphs

(3). G contains no induced subgraph isomorphic to 2K₂, C₄ or C₅. (15)

(OR)

(ii). Define a graphic sequence. Check whether the sequence (8, 8, 6, 5, 5, 4, 3, 3, 2)

is graphic.

(iii). Obtain a necessary and sufficient condition for a sequence to be graphic. (5+10)

(a)(i). Define a vertex colouring and a critical graph. Prove that in a k-critical graph

the minimum degree of vertices is at least k – 1.

(OR)

(ii). What is a permutation labeling? Illustrate with an example. (5)

(b)(i). Prove that an undirected graph G is a permutation graph if and only if G and

are comparability graphs. (15)

(OR)

(ii). Define a permutation graph. Draw the permutation graph corresponding to the

permutation [5,2,1,7,6,8,3,4].

(iii). Let G be an undirected graph. Prove with usual notations that a bijection L

from V to {1, 2, 3 … n} is a permutation labeling if and only if the mapping

, is an injection. (5+10)

(a)(i). Define a planar graph. Prove that the complete graph on 5 vertices is non-planar.

(OR)

(ii). Obtain an interval representation of the interval graph given below.

(5)

(b)(i). Let G be an undirected graph. Then prove that the following statements are

equivalent.

(1). G is an interval graph

(2). G contains no chordless 4-cycle and its complement is a

comparability graph.

(3). The maximal cliques can be linearly ordered such that, for every

vertex v of G the maximal cliques containing v occurs consecutively.

(OR)

(iii). Prove that an undirected graph G is a circular arc graph if and only if its

vertices can be circularly indexed v₁.v₂ …v_n so that for all i and j

(15 marks)

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Loyola College M.Sc. Mathematics Nov 2008 Analytic Number Theory Question Paper PDF Download

November 15, 2017 by 01 prakash

AB 34

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – MATHEMATICS

THIRD SEMESTER – November 2008

MT 3805 – ANALYTIC NUMBER THEORY

Date : 07-11-08 Dept. No. Max. : 100 Marks

Time : 9:00 – 12:00

ANSWER ALL QUESTIONS

I a) Prove that if the integer has r distinct odd prime factors , then

[OR]

b) Prove that the identity function is completely multiplicative. (5)

c) i) Write the relation between and .

. ii) Let the arithmetic function be multiplicative. Then prove that is completely

multiplicative if, and only if for all . . (7 + 8)

[OR]

d) Prove that the set of all arithmetic functions f with f(1)0 forms an abelian group

under Dirichlet product (15)

II a) Write a note on the test for divisibility by 11.

[OR]

b) Write a product formula for . (5)

c) If f has a continuous derivative on the interval . . (15)

[OR]

d) If prove that
i) .
ii) (7 + 8)

III a) If , then prove that the linear congruence has exactly one

solution.

[OR]

b) State and prove Wolstenholme’s (5)

c) i) State and prove Lagrange’s theorem for polynomial congruences.
ii) Let . Prove that is composite. (7 + 8)

[OR]

d) i) Solve the congruence (mod 120).
ii) Write any two properties of residue classes. (7 + 8)

IV a) Write a note on quadratic residues and give an example.

[OR]

b) Prove that Legendre’s symbol is a completely multiplicative function. (5)

c) State and prove Gauss’ lemma. Also derive the value of m defined in Gauss

lemma.

[OR]

d) Determine those odd primes p for which 3 is a quadratic residue and those for

which it is a nonresidue. (15)

V a) State and prove reciprocity law.

[OR]

b) Write a note on partitions. (5)

c) Derive a generating function for

[OR]

d) State and prove Euler’s pentagonal-number theorem (15)

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Loyola College M.Sc. Mathematics April 2009 Topology Question Paper PDF Download

November 15, 2017 by 01 prakash

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – MATHEMATICS

ZA 60

THIRD SEMESTER – April 2009

MT 3803 / 3800 – TOPOLOGY

Date & Time: 16/04/2009 / 1:00 – 4:00 Dept. No. Max. : 100 Marks

Answer ALL questions. All questions carry equal marks.

01.(a) (i) Let X be a non-empty set, and let d be a real function of ordered pairs of
elements of X which satisfies the following two conditions:

d(x, y) = 0 Û x = y
d(x, y) £ d(x, z) + d(y, z)

Show that d is a metric on X.

(OR)

(ii) In any metric space, show that

any union of open sets in X is open
any finite intersection of open sets in X is open. (5)

(b) (i) If a convergent sequence in a metric space has infinitely many distinct points,
then prove that its limit is a limit point of the set of points of the sequence.

(ii) State and prove Cantor’s Intersection Theorem.

(iii) State and prove Baire’s Theorem. (5 + 5 + 5)

(OR)

(iv) Proving the necessary lemmas, establish that the set Rⁿ of all n-tuples
x = (x,₁, x₂, …,x_n) of real numbers is a real Banach space with respect to
coordinatewise addition and scalar multiplication and the norm
defined by (15)

II.(a) (i) Show that every separable metric space is second countable.

(OR)

(ii) If f and g are continuous real functions defined on a topological space X,
prove that fg is continuous. (5)

(b) (i) Show that any continuous image of a compact space is compact.

(ii) Prove that any closed subspace of a compact space is compact.

(iii) Give an example to show that a compact subspace of a compact space need not
be closed. (6 + 5 + 4)

(OR)

(iv) Prove that a topological space is compact, if every subbasic open cover has a
finite subcover. (15)

III.(a) (i) Prove that a metric space is sequentially compact Û it has the
Bozano-Weierstrass property.

(OR)

(ii) Show that a metric space is compact Û it is complete and totally bounded.

(5)

(b)(i) State and prove Lebesgue’s covering Lemma.

(ii) Prove that every sequentially compact metric space is compact (10 + 5)

(OR)

(iii) If X is a compact metric space, then prove that a closed subspace of C(X, R) is
compact Û it is bounded and equicontinuous.

(15)

IV.(a) (i) Prove that the product of any non-empty class of Hausdorff spaces is a
Hausdorff space.

(OR)

(ii) Show that every compact space is normal. (5)

(b)(i) State and prove the Tietze Extension Theorem.

(OR)

(ii) State and prove the Urysohn Imbedding Theorem (15)

(a)(i) Prove that any continuous image of a connected space is connected.

(OR)

(ii) Let X be a topological space. If {A_i} is a non-empty class of connected
subspaces of X such that Ç A_i is non-empty, prove that A = È A_i is also a
connected subspace of X. (5)

(b)(i) Prove that a subspace of the real line R is connected Û it is an interval.

(ii) Let X be an arbitrary topological space. Show that each point in X is contained
in exactly one component of X. (9 + 6)

(OR)

(iii) State and prove the Weierstrass Approximation Theorem. (15)

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Loyola College M.Sc. Mathematics April 2009 Probability Theory And Stochastic Processes Question Paper PDF Download

November 15, 2017 by 01 prakash

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

ZA 59

M.Sc. DEGREE EXAMINATION – MATHEMATICS

SECOND SEMESTER – April 2009

MT 2961 – PROBABILITY THEORY AND STOCHASTIC PROCESSES

Date & Time: 02/05/2009 / 1:00 – 4:00 Dept. No. Max. : 100 Marks

PART-A

Answer all questions (10 x 2 = 20)

Find the value of k such that, 0<x<1, zero, elsewhere represents the pdf of a random variable.
Find the mgf of a random variable X with probability mass function, x = 1,2,3,…
If A and B are two events and show that .
Define convergence in distribution of a sequence of random variables.
Define a stochastic process. What are the different ways of classifying a stochastic process?
Let X and Y be two independent random variables with N(10,4) and N(15,5) respectively. What is the distribution of 2X+3Y?
State Bonferronni’s inequality.
Define the period of a state. When do you it is aperiodic?
Patients arrive at a clinic according to Poisson process with mean rate of 2 per minute. What is the probability that no customer will arrive during a 2 minute interval?
The joint pdf of two random variables is, 0<<<1, zero, elsewhere. Obtain the conditional pdf of given .

PART – B

Answer any five questions ( 5 x 8 = 40)

Show that the distribution function satisfies, , and right continuity.
Derive the mgf of the normal distribution. Hence obtain mean and variance.
Show that binomial distribution tends to Poisson distribution under some conditions, to be stated.
_{Let X have a pdf f(x) = , 0 < x < ∞ and Y be another independent random variable with pdf g(y) = ,0 < y < ∞ .obtain the pdf of U= .}
_{Let {} be a Markov chain with states 1,2 and 3 and transition probability matrix}

_{If the initial distribution is (},,), find

iii)

_{Derive the expression for in a pure birth process with X(0) =0}
_{Let X and Y have the joint pdf,zero, elsewhere. Find E[Y|x].}
_{Prove that E(XY) = E(X) E(Y) when the random variables are continuous and independent. Is the converse true? Justify.}

_{PART – C}

_{Answer any two questions ( 2 x 20 = 40)}

a) State and prove Baye’s theorem.
b) Give an example to show that pair wise independence does not imply

independence of three events.

c). State and prove Boole’s Inequality. ( 8 + 6 + 6)

a) State and prove Markov inequality. Deduce Chebyshev’s inequality.

b). Show that almost sure convergence implies convergence in probability. Is

the converse true? Justify.

c). State and prove central limit theorem for a sequence of i.i.d random variables. ( 5 + 5 + 10 )

(a). State the postulates of Poisson process and derive an expression for .

(b). Obtain the pgf of a Poisson process. If and have

independent Poisson processes with parameters and respectively,

find. ( 6 + 5+ 9)

a). Show that the Markov chain with the transition probability matrix

is ergodic. Obtain the stationary distribution.

b). Show that communication is an equivalence relation.

c). Let be the minimum obtained while throwing a die n-times.

Obtain the transition probability matrix. ( 12 + 4 + 4)

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Loyola College M.Sc. Mathematics April 2009 Probability Theory And Stochastic Processes Question Paper PDF Download

November 15, 2017 by 01 prakash

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – MATHEMATICS

YB 39

SECOND SEMESTER – April 2009

ST 2902 – PROBABILITY THEORY AND STOCHASTIC PROCESSES

Date & Time: 02/05/2009 / 1:00 – 4:00 Dept. No. Max. : 100 Marks

PART-A

Answer all questions (10 x 2 = 20)

Determine C such that_, _–_{∞ < x < ∞ is a pdf of a random variable X.}
_{If the events A and B are independent show that and are independent.}
_{Write the MGF of a Binomial distribution with parameters n and p. Hence or otherwise find}
_{If two events A and B are such that , show that P(A) ≤ P(B)}
_{Given the joint pdf of and, f(,) = 2, 0<<<1.obtain the conditional pdf of given .}
_{Define a Markov process.}
_{Define transcient state and recurrent state.}
_{Suppose the customers arrive at a bank according to Poisson process with mean rate of 3 per minute. Find the probability of getting 4 customers in 2 minutes.}
_{If has normal distribution N(25,4) and has normal distribution N(30,9) and if and are independent find the distribution of 2 + 3.}
_{Define a renewal process.}

_PART-B

_{Answer any five questions (5 x 8 = 40)}

_{Derive the MGF of normal distribution.}
_{Show that F (-∞) = 0, F (∞) = 1and F(x) is right continuous.}
_{Show that binomial distribution tends to Poisson distribution under some conditions to be stated.}
_{Let X and Y be random variables with joint pdf f(x,y) = x+y, 0<x<1, 0<y<1, zero elsewhere. Find the correlation coefficient between X and Y}
_{Let {} be a Markov chain with states 1,2,3 and transition probability matrix}

_{with, i = 1,2,3}

_{Find i)}

_{Obtain the expression for in a pure birth process.}
_{State and prove Chapman-Kolmogorov equation on transition probability matrix.}
_{Let X have a pdf f(x) = e}^-x_x^m-1_{, 0 < x < ∞ and Y be another independent random variable with pdf g(y) = e}^-y_y^n-1_,

_{0 < y < ∞ .obtain the pdf of U= .}

_PART-C

_{Answer any two questions (2 x 20 = 40)}

_{. a) State and prove Bayes theorem.}

_{b) Suppose all n men at a party throw their hats in the centre of the room.}

_{Each man then randomly selects a hat. Find the probability that none of them will}

_{get their own hat. (10 + 10)}

_{a) let {} be an increasing sequence of events. Show that}

_{P(lim ) = limP(). Deduce the result for decreasing events.}

_{b) Each of four persons fires one shot at a target. let A}_i_{, i = 1,2,3,4 denote}

_{the event that the target is hit by person i. If Ai are independent and}

_{P() = P() = 0.7, P() =0.9, P() = 0.4. Compute the probability that}

_{All of them hit the target}
_{Exactly one hit the target}
_{no one hits the target}
_{atleast one hits the target. (12 + 8)}

_{. a) Derive the expression for in a Poisson process.}

_{b) If and have independent Poisson process with parameters and.}

_{Obtain the distribution of = γ given + = n.}

_{c) Explain Yules’s process. (10 + 5 + 5)}

_{22). a) verify whether the following Markov chain is irreducible, aperiodic and}

_recurrent

_{Obtain the stationary transition probabilities.}

_{b) State the postulates and derive the Kolmogorov forward differential}

_{equations for a birth and death process. (10 +10)}

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Loyola College M.Sc. Mathematics April 2009 Linear Algebra Question Paper PDF Download

November 15, 2017 by 01 prakash

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – MATHEMATICS

ZA 39

FIRST SEMESTER – April 2009

MT 1810 / 1804 – LINEAR ALGEBRA

Date & Time: 17/04/2009 / 1:00 – 4:00 Dept. No. Max. : 100 Marks

Answer ALL the questions.

a) i) Let T be a linear operator on a finite dimensional space V and let c be a scalar. Prove that the following statements are equivalent.
c is a characteristic value of T.
The operator (T–cI) is singular.
det (T–cI) =0.

ii) Let T be a linear operator on which is represented in the standard ordered basis by the matrix A=. Prove that T has no characteristic values in R. (5)
b) i) Let T be a linear operator on a finite dimensional vector space V. Prove that the minimal polynomial for T divides the characteristic polynomial for T.

ii) Let V be a finite dimensional vector space over F and T be a linear operator on V then prove that T is triangulable if and only if the minimal polynomial for T is a product of linear polynomials over F. (15)

a) i) Let V be a finite dimensional vector space. Let be independent subspaces such that , then prove that for.

ii) Let W be an invariant subspace for T. Then prove that the minimal polynomial for divides the minimal polynomial for T. (5)
b) i) State and prove Primary Decomposition theorem.

ii) Let T be a linear operator on a finite dimensional space V. If T is diagonalizable and if are the distinct characteristic values of T, then prove that there exist linear operators on V such that

1..

.
.
Each is a projection
The range of is the characteristic space for T associated with.

iii) If there exist k distinct scalars and k non-zero linear operators which satisfy conditions 1,2 and 3, then prove that T is diagonalizable , are the distinct characteristic values of T and conditions 4 and 5 are satisfied also. (15)

a) i) Let T be a linear operator on a vector space V and W a proper T-admissible subspace of V. Prove that W and Cyclic subspace Z(a;T) are independent.

ii) If U is a linear operator on a finite dimensional space W, then prove that U has a cyclic vector if and only if there is some ordered basis for W in which U is represented by the companion matrix of the minimal polynomial for U. (5)

b) i) ) Let a be any non-zero vector in V and let be the T-annihilator of . Prove the following statements:
The degree of is equal to the dimension of the cyclic subspace Z(a;T).
If the degree of is k, then the vectorsa, Ta, ,… form the basis for Z(a;T).
If U is the linear operator on Z(a;T) induced by T, then the minimal polynomial for U is .

ii) Let T be a linear operator on a finite dimensional vector space V and let

be a proper T-admissible subspace of V. Then prove that there exist non-zero vectors in V with respective T-annihilators such that V=and divides, k=2, 3…r. (15)

a) i) Define the matrix of a form on a real or complex vector space with respect to any ordered basis . Let f be the form ondefined by Find the matrix of f with respect to a basis {(1,-1), (1, 1)}.

ii) Let T be a linear operator on a complex finite dimensional inner product space V. Then prove that T is self-adjoint if and only if is real for every in V. (5)
b) i) Let f be the form on a finite-dimensional complex inner product space V. Then prove that there is an orthonormal basis for V in which the matrix of f is upper-triangular.
ii) Prove that for every Hermitian form f on a finite-dimensional inner product space V, there is an orthonormal basis of V in which f is represented by a diagonal matrix with real entries. (6+9)

iii) Let f be a form on a real or complex vector space V and a basis for the finite dimensional subspace W of V. Let M be the r x r matrix with entries and Wthe set of all vectors in V such that

f ()=0 for all in W. Then prove that Wis a subspace of V,={0} if and only if M is invertible and when this is the case V=W+W. (15)

a) i) Let V be a vector space over the field F. Define a bilinear form f on V and

prove that the function defined by f () =LLis bilinear.

ii) Define the quadratic form q associated with a symmetric bilinear form f and prove that . (5)
b) i) Let V be a finite dimensional vector space over the field of complex numbers.Let f be a symmetric bilinear form on V which has rank r. Then prove that there is an ordered basis for V such that the matrix of f in the ordered basis B is diagonal and f () =

ii) If f is a non-zero skew-symmetric bilinear form on a finite dimensional vector space V then prove that there exist a finite sequence of pairs of vectors,with the following properties:

1) f ()=1, j=1,2,,…,k.

2) f ()=f ()=f ()=0,ij.

3) If is the two dimensional subspace spanned by and, then V=where is orthogonal to all and and the restriction of f to is the zero form. (15)

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Loyola College M.Sc. Mathematics April 2010 Quantum Mechanics Question Paper PDF Download

November 15, 2017 by 01 prakash

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Loyola College M.Sc. Mathematics Nov 2010 Linear Algebra Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – MATHEMATICS

FIRST SEMESTER – NOVEMBER 2010

MT 1810/ 1804 – LINEAR ALGEBRA

Date : 30-10-2011 Dept. No. Max. : 100 Marks

Time : 1:00 – 4:00

a) (i) Prove that similar matrices have the same characteristic polynomial.

(OR) (5)

(ii) Let T be the linear operator on Â³ which is represented in the standard ordered basis by

the matrix . Find the characteristic polynomial of A.

b) (i) State and prove Cayley-Hamilton theorem.

(OR) (15)

(ii) Let V be a finite dimensional vector space over F and T a linear operator on V. Then

prove that T is diagonalizable if and only if the minimal polynomial for T has the

form are distinct elements of F.

a) (i) Let T be a linear operator on a finite dimensional space V and let c be a scalar. Prove

that the following statements are equivalent.

c is a characteristic value of T.
The operator (T – cI) is singular.
det (T – cI) = 0.

(OR) (5)

(ii) Let W be an invariant subspace for T. Then prove that the minimal polynomial for T_w

divides the minimal polynomial for T.

b) (i) State and prove Primary Decomposition theorem.

(OR) (15)

(ii) Let T be a linear operator on a finite dimensional space V. If T is diagonalizable and if

c₁,…,c_k are the distinct characteristic values of T, then prove that there exist linear

operators E₁,…,E_k on V such that

T = c₁E₁+…+ c_kE_k.
I = E_j +…+ E_k.
E_iE_j = 0,i≠j.
Each E_i is a projection

III. a) (i) Let W be a proper T-admissible subspace of V. Prove that there exists a nonzero a in

V such that W Ç Z (a ; T) = {0}.

(OR) (5)

(ii) Define T_–annihilator, T_–-admissible, Projection of vector space V and Companion

matrix.

b) (i) State and prove Cyclic Decomposition theorem.

(OR) (15)

(P.T.O.)

ii) Let P be an m x m matrix with entries in the polynomial algebra F [x]. The following are

equivalent.

P is invertible
The determinant of P is a non-zero scalar polynomial.
P is row-equivalent to the m x m identity matrix.
P is a product of elementary matrices.

a) (i) Let V be a complex vector space and f be a form on V such that f (a,a) is real for

every a. Then prove that f is Hermitian. (5)

(OR)

ii) Let f  be the form on a finite-dimensional complex inner product space V. Then prove

that there is an orthonormal basis for V in which the matrix of f is upper-triangular.

b) (i) Let f be a form on a finite dimensional vector space V and let A be the matrix of f in an

ordered basis B. Then f is a positive form iff A = A^* and the principal minors of A are all

positive.

(OR) (15)

(ii) Let V be a finite-dimensional inner product space and f a form on V. Then show that there is a

unique linear operator T on V such that f(a,b) = (Ta½b) for all a, b in V, and the map f ®T is an

isomorphism of the space of forms onto L(V,V).

a) (i) Let V be a vector space over the field F. Define a bilinear form f on V and prove that the

function defined by f (a;b) = L₁ (a) L₂ (b) is bilinear.

(OR) (5)

ii) Define the quadratic form q associated with a symmetric bilinear form f and prove that

b) i) Let V be a finite dimensional vector space over the field of complex numbers. Let f be a symmetric bilinear form on V which has rank r. Then prove that there is an ordered basis

B ={b_1,b_2, … b_n} for V such that the matrix of f in the ordered basis B is diagonal and .

(OR) (15)

ii) If f is a non-zero skew-symmetric bilinear form on a finite dimensional vector space V then

prove that there exist a finite sequence of pairs of vectors, (a₁, β₁), (a₂, β₂),… (a_k, β_k) with the

following properties.

f (a_j, β_j) = 1 , j=1,2,,…,k.
f (a_i, a_j)=f(β_i, β_j)=f(a_i,β_i)=0,i≠j.
c) If W_j is the two dimensional subspace spanned by a_j and β_j, then V=W₁Å W₂Å …W_k Å W₀

where W₀ is orthogonal to all a_j and β_j and the restriction of f to W₀ is the zero form.

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Loyola College M.Sc. Mathematics April 2011 Math-I Question Paper PDF Download

November 15, 2017 by 01 prakash

PART – A

Answer all the questions (10 X 2 = 20)

Write down the n^thderivative of cos²5x.
Show that for y²=4ax, the subnormal at any point is a constant.
Give the formula for the radius of curvature in Cartesian form.
Define evolute.
If α, β , γ are the roots of x³+px²+qx+r=0 find the value of .
Give the number of positive roots of x³+2x+3=0.
Show that sin ix =i sinh x.
Evaluate
Find the polar of (3, 4) with respect to y² = 4ax.
Define an asymptote of a hyperbola.

PART – B

Answer any FIVE questions. (5 X 8 = 40)

Show that in the curve by²=(x+a)³ the square of the subtangent varies as the subnormal .
Find the radius of curvature at ‘θ’ on x = a(cos θ+ θ sin θ), y=a(sin θ – θ cos θ).
Find the p-r equation of r sin θ + a = 0.
Solve: x⁴+2x³-5x²+6x+2=0 given that (1+i) is a root.
Remove the second term from the equation x³-6x²+11x-6=0.
Separate into real and imaginary parts tanh (x+iy).
Find the locus of poles with respect to y²=4ax of tangents to x²+y²=c².
Derive the polar equation =1 + e cos θ of a conic.

PART –C

Answer any TWO questions. 2 X 20 = 40

a) If , show that (1-x²)y_n+2– (2n+1)xy_n+1 – (n²+a²)y_n=0.
b) Find the slope of =cos(θ-α) + ecos θ. (12 + 8)
a) Show that the radius of curvature at any point on r = ae^{θ cot α} is r cosec α.
b) Solve 6x⁵-x⁴-43x³+43x²+x-6 = 0. (10 +10)
a) Calculate to two places of decimals, the positive root of x³+6x-2 = 0 by Horner’s method.
b) Expand cosh⁸θ in terms of hyperbolic cosines of multiples of θ. (12 + 8)
a) Sum of infinity : …
b) If e and e₁are two extremities of hyperbola and its conjugate show that

(10 +10)

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Loyola College M.Sc. Mathematics April 2012 Topology Question Paper PDF Download

November 15, 2017 by 01 prakash

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – MATHEMATICS

THIRD SEMESTER – APRIL 2012

MT 3810 – TOPOLOGY

Date : 21-04-2012 Dept. No. Max. : 100 Marks

Time : 1:00 – 4:00

Answer all questions. All questions carry equal marks. 5 x 20 = 100 marks

01)	(a)	(i)	Let X be a non-empty set and let d be a real function of ordered pairs of elements of X which satisfies the following conditions. a) b) Show that d is a metric on X. (or)
		(ii)	Let X be a metric space. Prove that a subset G of X is open it is a union of open spheres. (5)
	(b)	(i)	Let X be a metric space, and let Y be a subspace of X. Prove that Y is complete iff Y is closed.
		(ii)	State and prove Cantor’s Intersection Theorem.
		(iii)	State and prove Baire’s Theorem. (6+5+4) (or)
		(iv)	Let X and Y be metric spaces and let f be a mapping of X into Y. Prove that f is continuous at and f is continuous is open in X whenever G is open in Y. (15)
02)	(a)	(i)	Prove that every separable metric space is second countable. (or)
		(ii)	Define a topology on a non-empty set with an example. Let be a topological space and be an arbitrary subset of . Show that each neighbourhood of intersects . (5)
	(b)	(i)	Show that any continuous image of a compact space is compact.
		(ii)	Prove that any closed subspace of a compact space is compact.
		(iii)	Give an example to show that a compact subspace of a compact space need not be closed. (6+6+3) (or)
		(iv)	Show that a topological space is compact, if every subbasic open cover has a finite subcover. (15)
03)	(a)	(i)	State and prove Tychnoff’s Theorem. (or)
		(ii)	Show that a metric space is compact if it is complete and totally bounded. (5)
	(b)	(i)	Prove that in a sequentially compact space, every open cover has a Lebesgue’s number.
		(ii)	Prove that every sequentially compact metric space is totally bounded. (10+5) (or)
		(iii)	State and prove Ascoli’s Theorem. (15)
04)	(a)	(i)	Show that every subspace of Hausdorff space is also Hausdorff. (or)
		(ii)	Prove that every compact Hausdorff Space is normal. (5)
	(b)	(i)	Prove that the product of any non-empty class of Hausdorff Spaces is a Hausdorff Space.
		(ii)	Prove that every compact subspace of a Hausdorff space is closed.
		(iii)	Show that a one-to-one continuous mapping of a compact space onto a Hausdorff Space is a homeomorphism. (6+4+5) (or)
		(iv)	If X is a second countable normal space, prove that there exists a homeomorphism f of X onto a subspace of and X is therefore metrizable. (15)
05)	(a)	(i)	Prove that any continuous image of a connected space is connected. (or)
		(ii)	Show that the components of a totally disconnected space are its points. (5)
	(b)	(i)	Show that the product of any non-empty class of connected spaces is connected.
		(ii)	Let X be a compact Hausdorff Space. Show that X is totally disconnected, iff it has open base whose sets are also closed. (6+9) (or)
		(iii)	State and prove the Weierstrass Approximation Theorem. (15)

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Loyola College M.Sc. Mathematics April 2012 Theory Of Fuzzy Subsets Question Paper PDF Download

November 15, 2017 by 01 prakash

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – MATHEMATICS

FOURTH SEMESTER – APRIL 2012

MT 4961 – THEORY OF FUZZY SUBSETS

Date : 25-04-2012 Dept. No. Max. : 100 Marks

Time : 1:00 – 4:00

Answer all the questions. Each question carries 20 marks.

a)1) Find the following index of fuzzinessfor the given fuzzy subsets

={(x₁/0),(x₂/0.3), (x₃/0.7),(x₄/1), (x₅/0), (x₆/0.2),(x₇/0.6)} and

={(x₁/0.3),(x₂/1), (x₃/0.5),(x₄/0.8), (x₅/1), (x₆/0.5),(x₇/0.6)}.

a)2) Give the ordinary subset of level α for the fuzzy subset ={(x₁/0.7),(x₂/0.5), (x₃/1),(x₄/0.2), (x₅/0.6)} i) α =0.1 ii) α=0.6 iii) α=0.8 iv) 0.9 (5)

b)1) State and prove decomposition theorem for fuzzy subsets. Decompose the fuzzy subset {(x₁|0.3), (x₂|0.7), (x₃|0.5), (x ₄|0.1), (x₅|0.6)}.

b)2) Let = {(x₁/0),(x₂/0.3), (x₃/0.7),(x₄/1), (x₅/0), (x₆/0.2),(x₇/0.6)}

= {(x₁/0.3),(x₂/1), (x₃/0.5),(x₄/0.8), (x₅/1), (x₆/0.5),(x₇/0.6)} and

= {(x₁/1),(x₂/0.5), (x₃/0.5),(x₄/0.2), (x₅/0), (x₆/0.2),(x₇/0.9)}.

Calculate (6+9)

c)1) Let = {(x₁|0.2), (x₂|0.7), (x₃|1), (x ₄|0), (x₅|0.5)} and = {(x₁|0.5), (x₂|0.7), (x₃|0), (x ₄|0.5), (x₅|0.5)}.

Check whether?

c)2) List down all the ‘algebraic’ properties of fuzzy subsets. Explain in detail, giving the implications of those properties that make a difference between crisp sets and fuzzy subsets. (5+10)

II. a)1) Choosing a suitable example, explain fuzzy subsets induced by a mapping

a)2) Choosing a suitable example, explain normal and global projections. (5)

b)1) Let then prove that ; where is the strongest path existing from x to y of length k.

b)2) Define the algebraic product and sum of two fuzzy relations. Explain with examples. (5+10)

c)1) Prove that the transitive closure of any fuzzy binary relation is transitive binary relation.

c)2) For , and as given below, verify the following conditions.

(5+10)

	A	B	C	D	E		A	B	C	D	E		A	B	C	D	E
A	0	1	1	1	1	A	1	0.5	0.5	1	0.7	A	0	0.3	1	0	0.5
B	0	0	0.9	0.7	0.3	B	0	1	0.7	0.7	0	B	0.3	0.2	0	0.8	0.1
C	0	0	0	0.7	0.3	C	0	1	1	0.7	0	C	1	0	0	0.2	1
D	0	0	0	0	0.3	D	0	0.3	0.3	0	0	D	0	0.8	0.2	1	0.4
E	0	0	0	0	0	E	1	0.5	0	0.5	1	E	0.5	0.1	1	0.4	0.4

a)1) Contrast fuzzy ordinal relation with fuzzy resemblance relation. Give an example.

a)2) Consider the relation given with the membership function

Is this relation a resemblance relation? (5)

b)1) Define anti symmetric and perfect anti symmetric fuzzy binary relations. Give examples. Is it true to say that any perfect anti symmetric relation is evidently anti symmetric?

b)2) Let be a similitude relation. Let x, y, z be the elements of E. Put then prove that (7+8)

b)3) Explain the following in detail with examples: Relation of (i)preorder (ii) anti reflexive preorder (iii) similitude (iv)dissimilitude and (v)Resemblance. (15)

a)1) Explain the three fundamental problems in the process of pattern recognition.

a)2) State the fuzzy c-means algorithm as given J. Bezdek. (5)

b)1) How will you justify that fuzzy applications will yield better results in the field of pattern recognition rather than any other traditional methods.

b)2) Explain how fuzzy clustering methods are based on fuzzy equivalence relation. Given any relation, how is it possible to apply this method. (7+8)

c)1) Explain in detail with examples (i)the two fuzzy clustering methods and (ii) the two pattern recognition methods. (15)

a) Explain how fuzzy application could make a difference in the field of Economics OR Engineering. (5)
b) Explain in detail, with a suitable example, fuzzy application in the field of Medicine OR (15)

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Loyola College M.Sc. Mathematics April 2012 Relativistic Mechanics Question Paper PDF Download

November 15, 2017 by 01 prakash

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – MATHEMATICS

FOURTH SEMESTER – APRIL 2012

MT 4813 – RELATIVISTIC MECHANICS

Date : 23-04-2012 Dept. No. Max. : 100 Marks

Time : 1:00 – 4:00

Answer ALL the questions and each question carries 20 marks.

a. i.what is Relativity?

ii..Explain the meaning of absolute quantities with examples.

iii..Is relativistic energy is absolute constant?

Explain why Michelson-Morley experiment should be repeated during nights and days and

during all seasons of the year?

What do you understand by Relativistic effects?

i. What is the rest frame of a moving body?

ii .If two events are simultaneous in a reference frame S, will they also be simultaneous in another

reference frame S’ moving with constant velocity relative to S ?

iii. If velocity of light in a frame S is v = c, what will be the velocity of light in the frame S’.

What is the rest mass of a light photon?
What types of energies are included in E= mc² ?

-6 marks

c. Derive the Lorentz Transformations

d. i. Define Aberration and determine the relativistic value of Aberration and deduce its

classical value.

Discuss about Doppler’s effect. –14marks

02 .a. Derive the transformation formula for force.

Derive the relativistic equations of motion and energy. -6 marks

c. Discuss the concept of Minkowski space and Space-like and Time-like intervals.

i.Prove that is invariant under Lorentz transformation.

ii.If a moving particle has velocities u and u’ in the frame S and S’ respectively. Prove that

-14marks

3.a. OR

If B_ij= A _i,j– A _j,i, prove that B _ij,k +B _jk,i + B _ki,j = 0.

–6 marks

Transform ds² = dx² + dy² + dz²into ds² = dr² + r² dq² +dz² and express it in terms

Christoffel symbol.

i. State the Rigorous Quotient Law with an Illustration.
A quantity A (p,q) is such that A(p,q)B^q_s= C_pswhere B^q_s is an arbitrary tensor and C_ps is a

tensor. Show that A(p,q) is a tensor. What is its type?

-14marks

4.a. Derive Einstein’s law of gravitation in empty space.

b. Discuss about the principle of equivalence

–6marks

Derive the equation of Geodesic in the form

d. Obtain the equation of the Geodesic for the metric ds² = -e^-2kt(dx² + dy² + dz²) + dt².

–14marks

5.a. Prove that the Isotrophic polar coordinates in the form

b. Discuss about Material energy tensor.

–6marks

c. Derive the deflection of light in passing through gravitational field in the neighborhood of the

sun.

d. Derive the differential equation to the planetary orbits in the form

where .

–14marks

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Loyola College M.Sc. Mathematics April 2012 Real Analysis And Linear Algebra Question Paper PDF Download

November 15, 2017 by 01 prakash

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – MATHEMATICS

SECOND SEMESTER – APRIL 2012

MT 2906 – REAL ANALYSIS AND LINEAR ALGEBRA

Date : 26-04-2012 Dept. No. Max. : 100 Marks

Time : 9:00 – 12:00

Answer ALL Questions

(a) For any , prove that .

(OR)

(b) If is a convergent series, then prove that . (5 marks)

(ii) Find such that and find the limit of . (10+ 5 marks)

(OR)

(d) (i) State and prove Leibnitz rule.

(ii) Prove that the series is convergent. (10 + 5 marks)

(a) If the limit of f(x) as xa exists, then prove that it is unique.

(OR)

(b) State and prove the Binomial theorem for any positive integer m. (5 marks)

(ii) Prove that the limit of the product of two functions is the product of their limits. (6 + 9 marks)

(OR)

(d) (i) State and prove Chain rule for differentiation.

(ii) Use inverse function theorem to find the derivative of . (10 + 5 marks)

(a) Define the Riemann integral of a function.

(OR)

(b) If f R[a, b], then prove that | f | R[a, b]. (5 marks)

(ii) Let f(x) = x². For each n N, Let be the partition of [0, 1]. Compute and . (9 + 6 marks)

(OR)

(d) State and prove the First Fundamental theorem of Calculus. (15 marks)

(a) Prove that a system of linear equations in n unknowns f_i = 0, i = 1, 2, …, m are linearly dependent if and only if the rank r of the augmented matrix is less than the number m of equations.

(OR)

(b) Prove that a square matrix is singular if and only if its columns are linearly dependent. (5 marks)

(c) (i) If p vectors from a set of k vectors A₁, A₂, …, A_k where p < k are linearly dependent, then prove that all the vectors are linearly dependent.

(ii) Find the complete solution of the system of equations: x₁ – x₂ + 2x₃ = 1; 2x₁ + x₂ – x₃ = 2. (8 + 7 marks)

(OR)

(d) If a homogeneous linear system of m equations in n unknowns AX = 0 has rank r < n, then prove that all solutions may be written as linear combinations of n – r linearly independent solutions. Also prove that when r = n, the only solution is the dependent vector 0. (15 marks)

(a) Explain the process of reduction to diagonal form.

(OR)

(b) Write short notes on quadratic forms. Write down the matrix of the quadratic form x₁² + x₂² – 3x₃² + 2x₁x₂ – 6x₁x₃. (5 marks)

(OR)

(d) Apply Gram-Schmidt process to the vectors {1, 2, 1, 1}, {1, -1, 0,2}, {2, 0, 1, 1) to find a set of orthonormal vectors. (15 marks)

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Loyola College M.Sc. Mathematics April 2012 Partial Differential Equations Question Paper PDF Download

November 15, 2017 by 01 prakash

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – MATHEMATICS

SECOND SEMESTER – APRIL 2012

MT 2812 – PARTIAL DIFFERENTIAL EQUATIONS

Date : 21-04-2012 Dept. No. Max. : 100 Marks

Time : 9:00 – 12:00

Answer all questions. Each question carries 20 marks.

(a) Using Charpit’s method solve . (5)

(OR)

Solve. (5)

Obtain the condition for compatibility of f(x, y, z, p, q) = 0 and g(x, y, z, p, q) = 0.
Show that, are compatible and find its solution. (7 + 8)

(OR)

Determine the characteristics of z = p² – q² and find the integral surface which passes through the parabola 4z + x² = 0, y = 0. (15)

(a) If f and g are arbitrary functions show that is a solution of provided . (5)

(OR)

(b) Solve . (5)

(c) Define affine transformation and prove that the sign of the discriminate of a second of second order partial differential equation is invariant under the general affine transformation. (15)

(OR)

(d) Obtain the canonical form of the parabolic partial differential equation.

(e) Reduce to canonical form. (10 +5)

(a) Obtain the Poisson’s equation. (5)

(OR)

(b) D’Alembert’s solution of the one-dimensional wave equation. (5)

(c) Solve a two dimensional Laplace equation subject to the boundary conditions; u(x, 0), u (x, a) = 0, u (x, y) → 0 as x → ∞, where x ≥ 0 and 0 ≤ y ≤ a.

(15)

(OR)

(d) State and prove Interior Dirichlet Problem for a Circle. (15)

(a) Solve the wave equation given by , , subject to the initial conditions , , . (5)

(OR)

(b) Find the steady state temperature distribution u(x, y) in a long square bar of side p with one face maintained at constant temperature u₀and the other faces at zero temperature. (5)

0 < t < ¥ subject to the conditions u(0, t) = 0, u(l, t) = g(t), 0 < t < ¥ and u(x, 0) = 0, 0 < x < l. (15)

(OR)

(d) State and prove Helmholtz Theorem. (15)

(a) Using Fredholm determinants, find the resolvent kernel when K(x, t) = xe^t, a = 0,

b = 1. (5)

(OR)

(b) If a kernel is symmetric then prove that all its iterated kernels are also symmetric.

(5)

(OR)

(d) State and prove Hilbert theorem. (15)

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MT 1902 – MATHEMATICS FOR COMPUTER APPLICATIONS

MT 1807 – DIFFERENTIAL GEOMETRY

MT 1808 – COMPUTER ALGORITHMS

MT 1804 – LINEAR ALGEBRA

MT 3875 – MATHEMATICAL METHODS IN BIOLOGY

MT 3925 – MATHEMATICAL SOCIAL SCIENCES

MT 3806 – ALGORITHMIC GRAPH THEORY

MT 3805 – ANALYTIC NUMBER THEORY

MT 3803 / 3800 – TOPOLOGY

MT 2961 – PROBABILITY THEORY AND STOCHASTIC PROCESSES

ST 2902 – PROBABILITY THEORY AND STOCHASTIC PROCESSES

MT 1810 / 1804 – LINEAR ALGEBRA

MT 1810/ 1804 – LINEAR ALGEBRA

MT 3810 – TOPOLOGY

Show that the components of a totally disconnected space are its points. (5)

MT 4961 – THEORY OF FUZZY SUBSETS

MT 4813 – RELATIVISTIC MECHANICS

MT 2906 – REAL ANALYSIS AND LINEAR ALGEBRA

Answer ALL Questions

MT 2812 – PARTIAL DIFFERENTIAL EQUATIONS