M.Sc Mathematics Question Paper
Loyola College M.Sc. Mathematics April 2003 Number Theory Question Paper PDF Download
LOYOLA COLLEGE (AUTONOMOUS), CHENNAI600 034.
M.Sc. DEGREE EXAMINATION – mathematics
FourTh SEMESTER – APRIL 2003
MT 4950/ M 1055 number theory
23.04.2003
1.00 – 4.00 Max: 100 Mark
 (a) (i) If (a,m)=1, prove that a^{f}^{(m) }º1(modm). Hence deduce Fermats theorem.
(OR)
(ii) of n ³1 , prove that (8)
 (i) State and prove Wilson’s theorem
(ii) Salve the congruence x^{2 }+ x +7 º 0 (mod 189)
(OR)
 (iii) Solve x^{2 }+ x +7 º 0(mod 7^{3})
(iv) Reduce the congruence 4x^{2 }+2x +1 º0 (mod5) to the form x^{2 }º a (mod p)
hence find the solutions (17)
 a) (i) Let P be an odd prime with (a, p) = 1. Consider he least nonnegative
residues module p of the integers a, 2a, 3a, ….
If n denotes the number of these residues that exceed p/2, then
prove that the Legendre symbol
(OR)
(ii) Find the value of the Legendre symbol
(iii) Find the highest power of 7 that divides 1000! (8)
 b) (i) If p is an odd prime and (a,2p) = 1 then prove that the
Legendre symbol
(ii) Define the Jacobi symbol
(OR)
(iii) If f(n) is a multiplicative function and if then prove that
F(n) in multiplicative.
(iv) Define the Moebius function m(n) and prove that inversion formula that
if for every positive integer n
then F(n) . (17)
 (a) (i) Find all the integral solutions of the equation ax + by =c if they
exist, where a, b, c and integers.
(OR)
(ii) Show that there exists at least one positive solution of ax + by = c if
g = (a, b) satisfies the condition gc and gc >ab. (8)
(b) (i) Prove that all solutions of 3x +5y =1 can be written in the
form x = 2 + 5t, y = 1 3t.
(ii) Define a primitive solution of x^{2 }+ y^{2 }= z^{2}. Prove that the positive
primitive solutions of x^{2 }+ y^{2 }= z^{2} with y even are give by x = r^{2 }= s^{2},
y = 2rs, z = r^{2 }+ s^{2}, where r and s are arbitrary integers of opposite
parity with r > s >0 and (r,s) =1.
(OR)
(ii) Prove that every positive integer is a sum of four squares of integers. (17)
 (a) (i) If P(n) is the partition function, with the usual notation
prove that p_{m}^{(n)} = p_{m1}^{(n) }+ P_{n}^{(nm) }if n ³ m>1.
(ii) using the graph of a partition, prove that the usual notation prove that
the number of partitions of n into m summands is the same as the
number of partitions of n having largest sum m and m.
(OR)
(iii) State Euler’s formula and use it prove that Euler’s identify for
any positive integer n. (8)
(b) (i) If n ³ 0 then prove that
(OR)
(ii) of
^{ }
 (iii) Prove that for o £ x<1, the series (17)
Loyola College M.Sc. Mathematics April 2003 Mechanies – II Question Paper PDF Download
LOYOLA COLLEGE (AUTONOMOUS), CHENNAI600 034.
M.Sc. DEGREE EXAMINATION – mathematics
FourTh SEMESTER – APRIL 2003
MT 4801/ M 1026 mechanies – II
16.04.2003
1.00 – 4.00 Max: 100 Marks
Answer ALL the questions
 a) Explain the term ‘ABERRATION’. Also derive the relativistic formula
for aberration in the form (8)
(OR)
 Show that the operator is an invariant for
Lorentz transformation. (8)
 a) State ‘ETHER’ Hypothesis. Explain the Michelson Morley experiment
and give the conclusion. (17)
(OR)
 b) Show that Lorentz transformations forma group . (17)
 a) Obtain the transformation formula for mass in the form (8)
(OR)
 If a body of mass m disintegrates while at rest in to two parts of rest masses m_{1} and m_{2}, show that the energies E_{1 }and E_{2 }of the parts are
given by (17)
 a) Derive the equation E = m C^{2}, Deduce that p^{2 }– is an invariant under
Lorentz transformation. (17)
(OR)
 Obtain the transformation formula for force components in the
(17)
 a) Explain ‘contravariant vectors’, covariant vectors,
‘contravariant tensors’ and ‘covariant tensors’. (8)
(OR)
 If a vector has components, on cartesion coordinates then
the components in polar coordinates are and if the components be
then the polar coordinates components are (8)
 a) Define fundamental tensors and show that gmg is a Covariant tensor of
rank two. Also transform ds^{2 }= dx^{2 }+ dy^{2 }+ dz^{2} in polar and cylindrical
coordinates. (17)
(OR)
 b) Define Christoffel’s 3index symbols of the first and second kind. Also
calculate christoffel’s symbols corresponding to the metric
ds^{2 }= dr^{2 }+ r^{2}dq^{2} + r^{2}sin^{2}q df^{2}. (17)
 a) Define ‘Energy Tensor’. Show that the equationfor m = 4
gives the equation of continuity in Hydrodynamics. (8)
(OR)
 Obtain isotropic polar coordinates and Cartesian coordinates.
Also Deduce that the velocity of light at distance r_{1 }from the origin is
(8)
 a) Obtain the schwarzchild line element in the neighbourhood of an attracting particle
in the from (17)
(OR)
 b) Derive the differential equation to the planetary orbits in the
form . (17)
Loyola College M.Sc. Mathematics April 2003 Graph Theory Question Paper PDF Download
LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034.
M.Sc. DEGREE EXAMINATION – Mathematics
FourTH SEMESTER – APRIL 2003
MT 4951 / M 1056 – GRAPH THEORY
23.04.2003
1.00 – 4.00 Max : 100 Marks
Answer ALL questions. Each question carries 25 marks.
 (a) Obtain a necessary and sufficient condition for a sequence (d_{1}, d_{2,} …,d_{n}) to be graphic.
(OR)
(b) Prove that a graph is bipartite if and only if it contains no odd cycle. (8 marks)
(c) (i) Prove, with usual notation, that .
(ii) Find the shortest distance from the vertex u_{o} to all other vertices in the following
graph.
(OR)
(d) (i) Prove that a vertex v of a tree G is a cut vertex of G if and only if d (v)>1. Deduce
that every nontrivial loopless connected graph has at least two vertices that are not cut
vertices.
(ii)State Cayley’s formula and find the number of distinct spanning trees of the following
graph. (8+9 marks)
 (a) Define connectivity and edgeconnectivity of a graph. Prove, with usual notation, that
.
(OR)
 If G is 2connected, then prove that any two vertices of G lie on a common cycle.
(8 marks)
 (i) Prove that a non empty connected graph is Eulerian if and only if it has no vertex of
odd degree.
(ii) Obtain Chvatal’s sufficient condition for a simple graph to be Hamiltonian.
(OR)
 (i) Let G be a simple graph and let u and v be nonadjacent vertices in G such that
 n. Prove that G is Hamiltonian is Hamiltonian.
(ii) Describe Chinese postman problem. State Fleury’s algorithm. Find an optimal
four in the following graph.
 (a) Prove that a matching M in G is a maximum matching if and only if G contains no
Maugmenting path.
(OR)
(b) Prove , with usual notation, that n if (8 marks)
 (i) State and prove Hall’s theorem.
(ii) Find an optimal matching in the graph given by the following matrix.
(OR)
 (i) Define an independent set. Prove with usual notation that n.
(ii) Prove that in a bipartite graph the number of edges in a maximum matching is the
same as the number of vertices in a minimum covering. (8+9 marks)
 (a) State and prove Brook’s theorem.
(OR)
(b) Let G be a kcritical graph with a 2vertex cut {u, v}. Prove that .
(8 marks)
(c) State and prove kuratowski’s theorem.
(OR)
 (i) Obtain Euler’s formula and deduce that k_{5} is non
(ii) State and prove the fivecolour theorem. (9+8 marks)
Loyola College M.Sc. Mathematics April 2003 Computer Algorithms Question Paper PDF Download
LOYOLA COLLEGE (AUTONOMOUS), CHENNAI600 034.
M.Sc. DEGREE EXAMINATION – mathematics
FourTh SEMESTER – APRIL 2003
MT 4952/ M 1057 computer Algorithms
26.04.2003
1.00 – 4.00 Max: 100 Mark
Answer ALL questions. Each question carries 25 marks
I a) (i) Give procedure SEARCH to search for an element x in an array A(1:n)
and to return k if a(k) = x and zero otherwise.
(ii) Give a recursive produce to find GCD of 2 accepted numbers. (OR)
 b) Give a procedure to create a leap of n elements, inserting one item, at a time. (8)
 c) (i) Discuss: Analyzing algorithms in general
(ii) Explain the conditional statements and loop structures in SPARKS. (OR)
 d) Give HEAPSORT to sort numbers in an array. Simulate it on
A(1:6) = (14,17,25,12,13,7) (17)
II a) Give procedure BINSRCH and simulate it on
A(1:7) = (45, 70, 82, 90, 95, 100, 110)when x = 46.(OR)
 b) Give procedure MAXMIN and find its best, worst and average lese number
of comparisons when n is a power of 2. (8)
 c) Give procedure MERGESORT
If the time for merging operation is proportional to n, then find the computing
time ד (n) for MERGESORT. When n = 2^{k}, prove that ד(n) =O(n log n) (OR)
 Give procedure SELECT to find the k^{th} smallest element in an array simulate it on
A(1:8) = (14, 12, 61, 60, 17, 20, 6, 10 ) to find the 3^{rd} smallest element. (17)
III a) Explain the problem of optimal storage on rapes. With usual notation.
if l_{1} £ l_{2} £ ..£ l_{n} then prove that the ordering i_{j} = j_{i} £ n minimizes
outing over all possible permutations of the i_{j} . (OR)
 State a greedy algorithm to generate shortest paths from a given vertex to
all other vertices in a graph. (8)
 Give line procedure due to krushkal to find a minimum sparring tree of a graph.
Prove also that Kruskal’s algorithm generates a minimum cost sparring
tree for every corrected undirected graph G. (OR)
 State procedure GREEDY With used notation, if
p_{1}w_{1}³ p_{2}w_{2} ³ …³ p_{n}w_{n} , then prove that GREEDY KNAPSACK generates
an optimal solution to the given instance of the knapsack problem. (17)
IV a) Explain sum of subsets problem and give 2 different formulations for the same.
(OR)
 b) Explain in detail low backtracking works on the 4 queen problem. (8)
 c) (i) Give recursive backtracking algorithm for sum of subsets problem
(ii) State procedure MCOLO RING to find all mcolorings in a graph.
(OR)
 d) Define a hamiltonian cycle.Give an example of (i) a hamiltonian graph.
(ii) Give algorithm HAMITONIANJ to generate all hamiltonian cycles in a graph (17)
Loyola College M.Sc. Mathematics April 2003 Complex Analysis – II Question Paper PDF Download
LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034.
M.Sc. DEGREE EXAMINATION – mathematics
SECOND SEMESTER – APRIL 2003
MT 2802 / M 823 – COMPLEX ANALYSIS – II
24.04.2003
1.00 – 4.00 Max : 100 Marks
Answer ALL questions.
 (a) Show that . (8)
 Let Re Z_{n} >0 for all Prove that Converges to a non zero number if and only if the series (8)
 (i) Obtain the Gauss formula for the Gamma function and show that
éé
(ii) If  z  and p ³ o then prove that (7+2+8)
OR
 (i) Let f be a real valued function defined on such that for all x > 0.
Suppose f(x) statistics the following properties.  log f (x) is convex,
 f (x+1) = xf (x) for all x,
 f (1) =1. Then show that f (x) = [(x) for all x.
 State and prove Euler’s Theorem.
II (a) State and prove first version of Maximum Principle for Harmonic Functions. (8)
OR
(b) Let be a path from a to b and let be an analytic
continuation along There is a number Î>0 such that if is any path
from a to b with for all t, and if is any
contribution along with [go]_{a} = [fo]_{a }: than prove that [g_{1}]_{b} = [f_{1}]_{b} . (8)
 (i) Define Poisson kernal and prove the four properties of Poisson kernal.
(ii) Stating the required conditions, solve the Dirichlet’s problem for the unit disk. (6+11)
OR
(d) State the prove Harnack’s inequality and hence prove Hernack’s theorem. (5+12)
III a. State and prove Poisson – Jenson formula. (8)
OR
 State and prove Little Picard’s theorem. (8)
 Define order and genus of an entire function and prove that if f is an entire function
of finite genus then f is of finite order l £ +1. (5+12)
OR
 State and prove Bloch’s theorem. (17)
IV a. Prove that a discrete module consists of ether of Zero alone, of the integral multiples
nco of a single complex number ¹ o or 1 or of all linear combinations n_{1 }w_{1} + h_{2 }w_{2}
with integral coefficient of two numbers w_{1}, w_{2} with non real ration . (8)
OR
 Show that the zeroes a_{1}, a_{2 }….. a_{n} and poles b_{1}, b_{2} ….b_{n} of an elliptic function satisfy (modm)
 (i) Define weierslvass p function. Derive the differential equation satisfied by the
weierslvass p –function.
(ii) Show that p (z +u) + p(z) + p (u) = (2+8+7)
OR
 Show that ℒ(z) is an odd function and prove that ℒ(z) = p(z), also derive the le gendre relation.
Loyola College M.Sc. Mathematics Nov 2003 Mathematical Statistics – I Question Paper PDF Download
LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034
M.Sc., DEGREE EXAMINATION – MATHEMATICS
THIRD SEMESTER – NOVEMBER 2003
ST 3951 / S 972 – MATHEMATICAL STATISTICS – I
10.11.2003 Max:100 marks
1.00 – 4.00
SECTIONA
Answer ALL questions. (10×2=20 marks)
 Illustrate that pairwise independence does not imply mutual independence of random events.
 Prove that every distribution function is continuous atleast from the left.
 Show that if the probability of a random event equals zero, it does not follow that this event is impossible. Similarly prove that if the probability of a random event equals one, its does not follow that this event is sure.
 Define truncated distribution of a random variable X and given an example.
 Give two examples of random variables for which expected value does not exist.
 Define convergence in law of a sequence of random variables and give an example.
 Show that if the moment of order k of a random variable X exists, then
where a > 0.
 State the theorem of Bochner, giving necessary and sufficient conditions for a function to be a characteristic function.
 The characteristic function of the random variable X is given by = exp Find the density function of this random variable.
 State Lindeberg – Levy Central Limit
SECTIONB
Answer any FIVE questions. (5×8=40 marks)
 Let {An}, n =1, 2, ….., be a non increasing sequence of events and let A be their product. Then show that P (A) = .
 Show that the conditional probability satisfies the axioms of the theory of probability.
 a) State and prove Bayes
 b) Illustrate the application of Bayes
 State and prove a necessary and sufficient condition for the independence of the random variables X and Y of the discrete type.
 If a random variable has a symmetric distribution and its expected value exists, then show that this expected value equals the center of symmetry. Hence show that for a symmetric distribution the central moments of odd orders (if they exist) are equal to zero.
 a) If not all the moments exist, then show that those moments that do exists fail to determine the distribution function F (x).
 b) Define convergence in r^{th} mean and given an example.
 The random variables X and Y have the joint density given by
f (x,y) = . Compute the coefficient of correlation.
 Define the t, chisquare and F distributions.
SECTIONC
Answer any TWO questions. (2×20=40 marks)
 a) State and prove Lapunov inequality concerning absolute moments. (10)
 b) Show that the expected value of the product of an arbitrary finite number of
independent random variables, whose expected values exist, equals the product of the
expected values of these variables. (4)
 c) Show that the covariance of two independent random variables equals zero. Is the
converse true? Justify your answer. (6)
 a) State and prove Levy Inversion Theorem concerning the determination of the
distribution function by the characteristic function. (14)
 b) Prove that the probability function of the Poisson distribution can be obtained as the
limit of a sequence of probability functions of the binomial distribution. (6)
 a) Show that for n ³ 2, the binomial distribution can be obtained from the zeroone
distribution. (4)
 b) Examine the additive property for Gamma random variables. (6)
 c) State and prove Bernoulli’s weak law of large numbers. (10)
 a) State and prove the Chebyshev Show that in the class of random variables
whose second order moment exists, one cannot obtain a better inequality than the
Chebyshev inequality. (10)
 b) If the ^{th }moment of a random variable exists, then show that can be expressed
in terms of the ^{th }derivative of the characteristic function of this random variable
at t = 0. (8)
 c) Define the strong law of large numbers. (2)
Loyola College M.Sc. Mathematics Nov 2003 Functional Analysis Question Paper PDF Download
LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034.
 Sc. DEGREE EXAMINATION – MATHEMATICS
IV SEMESTER – APRIL 2004
MT 4800/M 1025 – FUNCTIONAL ANALYSIS
Date : Max. Marks : 100 Marks
Hours : 3 hours
Answer All questions:
 a) Show that every vector space has a Hamel basis
(Or)
If , prove that the hull space Z(f) has deficiency O or 1 in X. Conversely, show that if Z is a subspace of X of deficiency O or 1, then there is an such that
Z = Z(f). (8)
 b) (i) Let X and Y be normed linear spaces and let B(X, Y) denote the set of all bounded linear transformations from X into Y. Then prove that B(X, Y) is a normed linear space.
(ii) Let X and Y be normed linear spaces and let T : XY be a linear transformation. Prove that T is bounded if and only if T is continuous. (9 + 8)
(Or)
State and prove the Hahn – Banach Theorem (real version) ( 17 )
 a) Let X and Y be Banach spaces and let T be a linear transformation of X into Y. Prove that if the graph of T is closed, then T is bounded.
(Or)
State and prove FRiesz Lemma (8)
 State and prove the uniform boundedness theorem. Give an example to show that the uniform boundedness principle is not true for every normed vector space.
(Or)
If X and Y are Banach spaces and if T is a continuous linear transformation of X onto Y, then prove that T is an open mapping. (17)
 a) State and prove Bessel’s inequality
(Or)
If T is an operator on X, then show that (Tx, x) = 0
 b) i) If x^{‘} is a bound linear functional on a Hilbert space X, prove that there is a unique
such that x^{‘}(x) = (x,
 ii) If M and N are closed linear subspaces of a Hilbert space H and if P and Q are projections on M and N, then show that PQ = O QP = O. (9 + 8)
(Or)
Prove that two Hilbert spaces are isomorphic iff they have the same dimension. (17)
4) a) Define a topological divisor of Zero. Let S be the set of singular elements in a Banach algebra . Prove that the set of all topological divisors of Zero is a subset of S.
(Or)
Let A be a Banach algebra and Then prove that the spectrum of x, is nonempty.
 State and prove the Spectral Theorem.
(Or)
Define the spectral radius of an element x in a Banach Algebra A. In the usual notation, prove that . (17)
Loyola College M.Sc. Mathematics April 2006 Topology Question Paper PDF Download
LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
M.Sc. DEGREE EXAMINATION – MATHEMATICS

THIRD SEMESTER – APRIL 2006
MT 3800 – TOPOLOGY
Date & Time : 28042006/1.004.00 P.M. Dept. No. Max. : 100 Marks
Answer ALL questions. All questions carry equal marks.
 a) i) Let X be a metric space with metric d. Show that d_{1} defined by is also a metric on X. Give an example of a pseudo metric which is not a metric.
(or)
 ii) In any metric space X, show that each open sphere is an open set. Prove that any union of open sets in X is open. (8)
 b) i) Let X be a complete metric space and let Y be a subspace of X. Prove that Y is complete iff it is closed.
 ii) State and prove Cantor’s Intersection Theorem.
iii) If is a sequence of nowhere dense sets in a complete metric space X, prove that there exists a point in X which is not any of the s. (6+6+5)
 iv) Let X and Y be metric spaces and f be a mapping of X into Y. Then prove that f is continuous iff is open in X whenever G is open in Y.
 v) Prove that the set C(X,R) of all bounded continuous real functions defined on a metric space X is a Banech space with respect to point wise addition and scalar multiplication and the norm defined by . (6+11)
 a) i) Show that every separable metric space is second countable.
(or)
 ii) Prove that the product of any nonempty class of compact spaces is compact.
(8)
 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+5)
(or)
 State and prove Lindelof’s Theorem.
 v) Let X be any nonempty set, and let S be an arbitrary class of subsets of X. Show that S can serve as an open subbase for a topology on X. (6+11)
III. a) i) Prove that a metric space is compact iff it is complete and totally bounded.
(or)
 ii) Prove that every compact Hausdorff space is normal. (8)
 b) i) In a sequentially compact metric space, prove that every open cover has a Lebesque number.
 ii) Show that every sequentially compact metric space is totally bounded.
iii) Prove that every sequentially compact metric space is compact. (9+4+4)
(or)
 b) iv) In a Hausdorff space, show that any point and disjoint compact subspace can be separated by open sets.
 v) Show that every compact subspace of a Hausdorff space is closed.
 vi) Prove that a 1–1 mapping of a compact space on to a Hausdorff space is homeomorphism. (7+5+5)
 a) i) Prove that any continuous image of a connected space is connected.
(or)
 ii) Let X be a T_{1} Prove that X is normal iff each neighbourhood of a closed set F contains the closure of some neighbourhood of F. (8)
 b) i) State and prove the Urysohn Imbedding Therorem.
(or)
 ii) State and prove the Weierstrass Approximation Theorem. (17)
Loyola College M.Sc. Mathematics April 2006 Probability Theory And Stochastic Processes Question Paper PDF Download
LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
M.Sc. DEGREE EXAMINATION – MATHEMATICS
SECOND SEMESTER – APRIL 2006
ST 2902 – PROBABILITY THEORY AND STOCHASTIC PROCESSES
Date & Time : 28042006/9.0012.00 Dept. No. Max. : 100 Marks
PART – A
Answer ALL the questions (10 ´ 2 = 20)
 Define probability by classical method.
 Give an example for a discrete probability distribution.
 Define an induced probability space.
 State the properties of a distribution function.
 Define the distributed function of a continuous random variable.
 Write the formula to find the conditional mean and variance of Y given X = x.
 What do you mean by a Markov matrix? Give an example
 Write a note on onedimensional random walk.
 Define (i) recurrence of a state (ii) periodicity of a state
 Define renewal function.
PART – B
Answer any FIVE questions. (5 ´ 8 = 40)
 State and prove Boole’s inequality.
 Explain multinomial distribution with an example.
 Given the dF
F(x) = 0 , x < – 1
= , 1
= 1 , 1
compute (a) P(1/2 < X 1/2) (b) P(X = 0) (c) P(X = 1) (d) P (2 < X 3).
 Let X have the pdf f(x) = 2x, 0 < x < 1, zero elsewhere. Find the dF and p.d.f. of Y = X^{2}.
 (a) When is a Markov process called a Markov chain?
(b) Show that communication is an equivalence relation. (2 + 6)
 A Markov chain on states {0,1,2,3,4,5} has t.p.m.
Find the equivalence classes.
 Find the periodicity of the various states for a Markov chain with t.p.m.
 Derive the differential equations for a pure birth process clearly stating the postulates.
PART – C
Answer any TWO questions. (2 ´ 20 = 40)
 (a) The probabilities that the independent events A,B and C will occur are ¼, ½ , ¼ respectively.
What is the probability that at least one of the three events will occur?
 Find the mean and variance of the distribution that has the dF
F(x) = 0 , x < 0
= x/8 , 0 £ x < 2
= x^{2}/16 , 2 £ x < 4
= 1 , 4 £ x (5 + 15)
 If X_{1} and X_{2} have the joint p.d.f.
f(x_{1},x_{2)} =
find (i) marginal pdf of X_{1 }and X_{2.}
_{ }(ii) conditional pdf of X_{2} given X_{1} = x_{1} and X_{1} given X_{2} = x_{2}.
(iii) find the conditional mean and variance of X_{2} given X_{1 }= x_{1} and
X_{1} given X_{2} = x_{2. }(4 + 4 + 12)_{ }
 Derive a Poisson process clearly stating the postulates.
 Derive the backward and forward Kolmogorov differential equations for a
birth and death process clearly stating the postulates.
Loyola College M.Sc. Mathematics April 2006 Mathematical Statistics – II Question Paper PDF Download
LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
M.Sc. DEGREE EXAMINATION – MATHEMATICS

FOURTH SEMESTER – APRIL 2006
ST 4900 – MATHEMATICAL STATISTICS – II
(Also equivalent to ST 4953)
Date & Time : 03052006/9.0012.00 Dept. No. Max. : 100 Marks
SECTIONA (10 ´ 2 = 20)
Answer ALL questions. Each question carries 2 marks.
 Let T have a tdistribution with 10 degrees of freedom. Find P(T > 2.228).
 Find the variance of S^{2} = (1/n) ∑ ( x_{i} – x )^{2} , when X_{1}, X_{2},…., X_{n} is a random sample from N(µ , σ^{2} ).
 How do you obtain the joint p.d.f. of any two order statistics Y_{r} and Y_{s} when Y_{r} < Y_{s} ?
 What do you understand by a sufficient statistic for a parameter?
 Define: UMVUE.
 State RaoCramer Inequality.
 Distinguish between randomized and nonrandomized tests.
 Illustrate graphically, the meaning of UMPT of level α test.
 Define a renewal process.
 When do you say that a stochastic matrix is regular?
SECTIONB (8 x 5 = 40)
Answer any 5 questions. Each question carries 8 marks.
 Let and S^{2} be the mean and the variance of a random sample of size 25 from a distribution N (3, 100). Evaluate P (0 < < 6, 55.2 < S^{2} < 145.6).
 Derive the central Fdistribution with (r_{1}, r_{2}) degrees of freedom.
 Let Y_{1} < Y_{2} < Y_{3} be the order statistics of a random sample of size 3 from the uniform distribution having p.d.f.
f(x; θ ) = 1/θ, 0 < x < θ, 0 < θ < ∞, zero elsewhere. Show that 4Y_{1}, 2Y_{2} and (4/3)Y_{3} are all unbiased estimators of θ. Find the variance of (4/3)Y_{3}.  If az^{2} + bz + c = 0 for more than two values of z, then show that a = b = c = 0. Use this result to show that the family{ B(2, p): 0 < p < 1} is complete.
 State and prove LehmannScheffe’s theorem.
 Let X have a p.d.f. of the form f(x; θ) = θ x^{θ1} , 0 <x < 1, θ =1,2, zero elsewhere. To test H_{0} : θ =1 against H_{1}: θ =2, use a random sample X_{1}, X_{2} of size n = 2 and define the critical region to be C = { (x_{1}, x_{2}) : ¾ ≤ x_{1 }x_{2 }}. Find the power function of the test.
 Prove or disprove: “UMPT of level α always exists for all types of testing problems”. Justify your answer.
 A certain genetic model suggests that the probabilities of a particular trinomial distribution are, respectively, p_{1 =}p^{2}, p_{2} = 2p (1p), and p_{3} = (1p)^{2} , where 0 < p < 1. If X_{1, }X_{2}, X_{3} represent the respective frequencies in ‘n’ independent trials, explain how we could check on the adequacy of the genetic model.
SECTIONC ( 20 ´ 2 = 40 )
Answer any 2 questions. Each question carries 20 marks.
 a) State and prove Factorization theorem. (12)
 b) Given the p.d. f. f(x; θ) = 1 / ( π [1 + ( x – θ)^{2} ) , ∞ < x < ∞ , ∞ < θ < ∞. Show that the RaoCramer lower bound is 2/n, where n is the size of a random sample from this Cauchy distribution. (8)
 a) State and prove the sufficiency part of NeymanPearson theorem. (12)
 b) Let X_{1}, X_{2},…, X_{n} denote a random sample from a distribution having the p.d.f.
f(x; p) = p^{x} (1p)^{1x} , x = 0,1, zero, elsewhere. Show that C = { (x_{1}, …,x_{n}) : Σ x_{i} ≤ k }is a best critical region for testing H_{0}: p = ½ against H_{1}: p = 1/3. Use the central limit theorem to find n and k so that approximately the level of the test is 0.05 and the power of the test is 0.9. (8)
 a) Derive the likelihood ratio test for testing H_{0}: θ_{1}=0, θ_{2} > 0 against
H_{1}: θ_{1} ≠ 0, θ_{2} >0 when a random sample of size n is drawn from N(θ_{1} , θ_{2} ). (12)  b) By giving suitable examples, distinguish between unpaired and paired ttests. (8)
 a) Show that the Markov chain is completely determined by the transition matrix and the initial distribution. (8)
 b) Give an example of a random walk with an absorbing barrier. (4)
 c) Explain in detail the properties of a Poisson process. (8)
Loyola College M.Sc. Mathematics April 2006 Mathematical Statistics – I Question Paper PDF Download
LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
M.Sc. DEGREE EXAMINATION – MATHEMATICS

THIRD SEMESTER – APRIL 2006
ST 3951 – MATHEMATICAL STATISTICS – I
Date & Time : 27042006/1.004.00 P.M. Dept. No. Max. : 100 Marks
SECTIONA (10 ´ 2 = 20)
Answer ALL questions. Each question carries 2 marks.
 Give an example for a nondecreasing sequence of sets.
 Distinguish between experiment and random experiment.
 Let f (x) = x/15, x=1,2,3,4,5
0, otherwise.
Find the median of the above distribution.
 Let f(x) = (4x) / 16, 2 < x < 2 ,zero elsewhere, be the p.d.f. of X.
If Y = X , compute P(Y ≤ 1).
 Give an example of a random variable in which mean doesn’t exist.
 Prove that E(E(X / Y)) = E(X).
 Define Hyper Geometric distribution.
 Define the characteristic function of a multidimensional random vector.
p p p
 If X_{n} → X and Y_{n} → Y, then show that X_{n} + Y_{n } → X +Y.
 State LindebergFeller theorem.
SECTIONB (8 x 5 = 40)
Answer any 5 questions. Each question carries 8 marks.
 Let f(x) = ½, 1 < x < 1, zero elsewhere, be the p.d.f. of X. Find the distribution
function and the p.d.f. of Y = X^{2}.
 State and prove Chebyshev’s inequality.
 If X_{1} and X_{2} are discrete random variables having the joint p.m.f.
f(x_{1},x_{2}) = ( x_{1} + 2 x_{2} ) / 18, (x_{1}, x_{2}) = (1,1), (1,2), (2,1), (2,2), zero elsewhere, determine the conditional mean and variance of X_{2}, given X_{1} =x_{1}, for x_{1} = 1 or 2.
Also, compute E[ 3X_{1} – 2 X_{2} ].
 State and prove any two properties of MGF.
 Stating the conditions, show that binomial distribution tends to Poisson
distribution.
 Obtain the central moments of N (µ, σ2).
 Let X ~ G (n_{1}, α) and Y ~ G (n_{2}, α) be independent. Find the distribution of X/Y.
 Explain in detail the difference between WLLN and SLLN.
SECTIONC ( 20 x 2 = 40 )
Answer any 2 questions. Each question carries 20 marks.
 a) State and prove Bayes’ theorem. (10)
 b) Bowl I contains 3 red chips and 7 blue chips. Bowl II contains 6 red chips and 4 blue chips. A bowl is selected at random and then 1 chip is drawn from this bowl. Compute the probability that this chip is red. Also, relative to the hypothesis that the chip is red, find the conditional probability that is drawn from bowl II. (10)
 a) Find the mean and variance of the random variable X having the distribution function:
F(x) = 0, x < 0,
( x/4), 0≤x <1,
(x^{2} /4), 1≤x<2,
1 , x≥ 2. (10)
 b) Let X have the uniform distribution over the interval ( π/2 , π/2). Find the
distribution of Y = tan X. (10)
 a) State and prove Kolmogorov’s strong law of large numbers. (12)
 b) State and prove BorelCantelli lemma. (8)
 a) Examine if central limit theorem (using Lyapounov’s condition) holds for the
following sequence of independent variates:
P X_{k} = ± 2^{k} = 2^{(2k + 1)} , P X_{k} = 0 = 1 – 2^{–2k } (8)
 b) State and prove LindebergLevy central limit theorem. (12)
Loyola College M.Sc. Mathematics April 2006 Functional Analysis Question Paper PDF Download
LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
M.Sc. DEGREE EXAMINATION – MATHEMATICS

FOURTH SEMESTER – APRIL 2006
MT 4800 – FUNCTIONAL ANALYSIS
Date & Time : 20042006/FORENOON Dept. No. Max. : 100 Marks
ANSWER ALL QUESTIONS
 a) i) Show that every vector space has a Hamel basis
(or)
 ii) If f Î X^{*}, prove that the null space Z(f) has deficiency 0 or 1 in X. Conversely, show that if Z is a subspace of X of deficiency 0 or 1, then there is an f Î X^{*} such that Z=Z(f).
 b) iii) Show that every element of X/Y contains exactly one element of Z, where Y and Z are complementary subspaces of X.
 iv) Let X and Y be normed linear spaces and let B(X, Y) denote the set of all bounded linear transformations from X into Y. Then prove that B(X,Y) is a normed linear space.
(or)
 v) Let X be a real vector space, p be a real valued function on X such that P(x+y) £ p(x) + p(y) and p(ax) = a p(x) ” x,y Î X and a ³ 0, and let Y be a subspace of X. If f is a linear functional on Y and f(x) £ p(x) ” x Î Y, prove that there is a linear functional F on X such that F(x)=f(x) ” x Î Y and F(x) £ p(x) ” x Î
 a) i) If x ¹ 0 is an element of a real normed linear space X, then show that there exists an x Î x¢ such that x¢(x) = x and x¢ = 1.
(or)
 ii) Let X and Y be Banach spaces and let T be a linear transformation of X into Y. Prove that if the graph of T is closed, then T is bounded. (8)
(or)
 b) iii) State and prove the uniform boundedness theorem.
 iv) Give an example to show that uniform boundedness principle is not for every normed vector space. (10+7)
(or)
 v) Let X and Y be Banach spaces and if T is a continuous linear transformation of X onto Y, then prove that T is an open mapping. (17)
 a) i) State and prove the Riesz – Representation Theorem.
(or)
 ii) If M and N are closed linear subspaces of a Hilbert space X and if P and Q are projections on M and N, then show that M ^N Û PQ = 0 Û QP=0 (8)
 b) iii) If T is an operator on a Hilbert space X, then prove that T is normal iff its real and imaginary parts commute.
 iv) Prove that how Hilbert spaces are isomorphic iff they have the same dimension. (7+10)
(or)
 v) If P is a projection on a closed linear space M of a Hilbert space X, prove that M is invariant under T Û TP =PTP
 vi) If P_{1}, P_{2}, … P_{n} are projections on closed linear subspaces M_{1}, M_{2}, … M_{n} on X, then prove that P= P_{1} + P_{2} + …+P_{n} is a projection iff the P_{i} are pairwise orthogonal and in the case P is a projection on M=M_{1}+M_{2}+…+M_{n}. (5+12)
 a) i) Prove that every element x in a Banech algebra A for which x–1 < 1, is regular, and the inverse of such an element is given by .
(or)
 ii) Let A be a Banech algebra and x Î Then prove that the spectrum of x, s(x), is nonempty. (8)
 b) iii) Let G be the set of regular elements in A and S be the set of singular elements in A. Prove that G is an open set and therefore S is a closed set.
 iv) Show that the mapping x à x^{–1} of G into G is continuous and is therefore a homeomorphism. (5+12)
(or)
 v) State and prove the Spectral Theorem. (17)
Loyola College M.Sc. Mathematics Nov 2006 Topology Question Paper PDF Download
LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034 M.Sc. DEGREE EXAMINATION – MATHEMATICS

THIRD SEMESTER – NOV 2006
MT 3803 – TOPOLOGY
(Also equivalent to MT 3800)
Date & Time : 25102006/9.0012.00 Dept. No. Max. : 100 Marks
Answer all the questions.
01.(a)(i) Let X be a metric space with metric d. Show that d_{1}. defined by
d_{1}(x,y) =
is also a metric on X.
(OR)
(ii) Let X and Y be metric spaces and f be a mapping of X into Y.
Show that f ^{–}^{1}(G) is open in X whenever G is open in Y.
(b)(i) Let X be a metric space. Prove that any arbitrary union of open sets in X is
open and any finite intersection of open sets in X is open.
(ii) Give an example to show that any arbitrary intersection of open sets in X
need not be open.
(iii) In any metric space X, prove that each closed sphere is a closed set.(6+4+5)
(OR)
(iv) If a convergent sequence in a metric space has infinitely many distinct
points, prove that its limit is a limit point of the set of points of the
sequence.
(v) State and prove Cantor’s Intersection Theorem.
(vi) If {A_{n}} is a sequence of nowhere dense sets in a complete metric space X,
show that there exists a point in X which is not in any of the A_{n}’s. (4+6+5)
02.(a) (i) Prove that every separable metric space is second countable.
(OR)
(ii) Let X be a non–empty set, and let there be give a “closure” operation
which assigns to each subset A of X a subset of X in such a manner
that (1) = , (2) A Í , (3) , and (4) =.
If a “closed” set A is defined to be one for which A = , show that the
class of all complements of such sets is a topology on X whole closure
operation is precisely that initially given.
 (i) Show that any closed subspace of a compact space is compact.
(ii) Give an example to show that a proper subspace of a compact space need
not be closed.
(iii) Prove that any continuous image of a compact space is compact. (5+4+6)
(OR)
(iv) Let C(X đ) be the set of all bounded continuous real functions defined
on a topological space X. Show that (1) C (X đ) is a real Banach space
with respect to pointwise addition and multiplication and the norm
defined by = sup; (2) If multiplication is defined pointwise
C(X, R) is a commutative real algebra with identity in which
£ and = 1.
03.(a) (i) State and prove Tychonoff’s Theorem.
(OR)
(ii) Show that a metric space is compact Û it is complete and totally
bounded.
(b) (i) Prove that in a sequentially compact space, every open cover has a
Lesbesgue number.
(ii) Show that every sequentially compact metric space is totally bounded.(9+6)
(OR)
(iii) State and prove Ascoli’s Theorem.
04.(a)(i) Show that every subspace of Hausdorff is also a Hausdorff.
(OR)
(ii) Prove that every compact Haurdolff space is normal.
(b)(i) Let X be a T_{1} – space.
Show that X is a normal Û each neighbourhood of a closed set F contains
the closure of some neighbourhood of F.
(ii) State and prove Uryjohn’s Lemma. (6+9)
(OR)
(iii) If X is a second countable normal space, show that there exists a
homeomorphism f of X onto a subspace of R^{¥}_{.}
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.
(b)(i) Let X be a topological space and A be a connected subspace of X. If B is a
subspace of X such that A Í B Í , show that B is connected.
(ii) If X is an arbitrary topological space, then prove the following:
(1) each point in X is contained in exactly one component of X;
 each connected subspace of X is contained in a component of X;
 a connected subspace of X which is both open and closed is a component of X. (6+9)
(OR)
(iii) State and prove the Weierstrass Approximation Theorem.
Loyola College M.Sc. Mathematics Nov 2006 Ordinary Differential Equations Question Paper PDF Download
LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034 M.Sc. DEGREE EXAMINATION – MATHEMATICS

FIRST SEMESTER – NOV 2006
MT 1806 – ORDINARY DIFFERENTIAL EQUATIONS
Date & Time : 31102006/1.004.00 Dept. No. Max. : 100 Marks
ANSWER ALL QUESTIONS
 (a) If the Wronskian of 2 functions x_{1}(t) and x_{2}(t) on I is nonzero for at
least one point of the interval I, show that x_{1}(t) and x_{2}(t) are linearly
independent on I.
OR
Consider the Differential Equation x” + λ^{2} x = 0, prove that
A cos λx + B sin λx is also a solution of the Differential equation.
(5 Marks)
(b) State and prove the method of variation of parameters.
OR
By the method of variation of parameters solve x”’ − x’ = t. (15 Marks)
 (a) Obtain the indicial form of the equation
2x^{2} (d^{2}y/dx^{2 }) + (dy/dx) + y = 0
OR
Obtain the indicial form of the Bessel’s differential equation. (5 Marks)
(b) Solve the differential equation using Frobenius Method ,
x^{2} (d^{2}y/dx^{2})^{ }+ x q(x) (dy/dx) + r(x) y = 0 and discuss about their
solutions when it’s roots differ by an integer .
OR
Solve the Legendre’s equation,
(1 – x^{2}) (d^{2}y/dx^{2})^{ }– 2x (dy/dx) + L(L+1)y = 0. (15 Marks)
III. (a) Prove that ∫^{+1}_{1} P_{n}(x) dx = 2 if n = 0 and
∫^{+1}_{1} P_{n}(x) dx = 0 if n ≥ 1
OR
Show that Hypergeometric function does not change if the parameter α and
β are interchanged, keeping γ fixed. (5 Marks)
(b) Obtain Rodrigue’s Formula and hence find P_{0}(x), P_{1}(x), P_{2}(x) & P_{3}(x).
OR
Show that P_{n}(x) = _{2}F_{1}[n, n+1; 1; (1x)/2] (15 Marks)
IV.(a) Considering an Initial Value Problem x’ = x, x(0) = 1, t ≥ 0, find x_{n}(t).
OR
Find the eigen value and eigen function of x” + λ x = 0, 0 < t ≤ (5 Marks)
(b) State and prove Picard’s Boundary Value Problem.
OR
State Green’s Function. Show that x(t) is a solution of L(x) + f(t) = 0 if and
only if x(t) = ∫^{b}_{a} G(t,s) f(s) ds. (15 Marks)
V.(a) Discuss the fundamental Theorem on the stability of the equilibrium of
the system x’ = f(t, x).
OR
Obtain the condition for the null solution of the system x’ = A(t) x is
asymptotically stable. (5 Marks)
(b) Study the stability of a linear system by Lyapunov’s direct method.
OR
Study the stability of a nonlinear system by Lyapunov’s direct method.
(15 Marks)
Loyola College M.Sc. Mathematics Nov 2006 Mathematical Methods In Biology Question Paper PDF Download
LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
M.Sc. DEGREE EXAMINATION – MATHEMATICS

THIRD SEMESTER – NOV 2006
MT 3875 – MATHEMATICAL METHODS IN BIOLOGY
Date & Time : 06112006/9.0012.00 Dept. No. Max. : 100 Marks
I a) Draw the state diagram for M = { (q_{0,}q_{1,}q_{2,}q_{3}), {0,1}, δ,q_{0},{q_{0}} }
δ  0  1 
q_{0}
q_{1} q_{2} q_{3} 
q_{0},q_{1}
q_{3} _{ _} _{ }q_{3} 
q_{0,} q_{2}
_{ _} _{ }q_{3} _{ }q_{3} 
(or)
 b) Why do we need to install a program from web ? (5)
 c) How do you generate Data base? Explain with an example .
(or)
 d) Comment on ‘ Internet is a powerful tool for bio informatics ’. (15)
II a) Expand HTTP and explain Motif.
(or)
 b) Define Edit graph and explain it for ANN and CAN. (5)
 c) Write notes on recurrence relation and about the correctness
of general relation
(or)
 d) Briefly describe on dynamic programming. (15)
III a) Explain briefly on calculations of edit distance using tabulation method .
(or)
 b) Construct a deterministic finite automata accepting words over {0,1}
ending with ‘111’. (5)
 c) When both i and j are strictly positive, prove that
D(i,j) = min [D(i1,j)+1, D(i,j1)+1, D(i1,j1)+t(i,j)]
(or)
d). What skills does a bioinformatician should have ? (15)
IV What do you mean by sequence alignment data ?
(or)
 Define Global alignment problem . (5)
 c) Describe the salient features of Human Genome project.
(or)
 d) Bio informatics is just a collection of Building Data bases Explain. (15)
V a) What type of questions does the bio informatics to be answered in the field of
biomaths ?
(or)
 b) Define string alignment with an example. (5)
 c) What does informatics mean to biologists ?
(or)
 d) Explain about the sequence matching of aniridia a human gene and
eyeless a fruit fly gene.
(15)
Loyola College M.Sc. Mathematics Nov 2006 Real Analysis Question Paper PDF Download
LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034 M.Sc. DEGREE EXAMINATION – MATHEMATICS

FIRST SEMESTER – NOV 2006
MT 1805 – REAL ANALYSIS
Date & Time : 28102006/1.004.00 Dept. No. Max. : 100 Marks
 a)(1) When does the RiemannStieltjes integral reduce to Riemann integral. Explain with usual notations.
OR
(2) If a < s < b, f ÎÂ (a) on [a,b] and a (x) = I (x – s), the unit step function, then prove that = f (s). (5)
b)(1) Let f be a bounded function on [a,b] having finitely many points of discontinuity on [a,b]. Let a be continuous at every point at which f is discontinuous. Prove that f ÎÂ(a). (8)
(2) Suppose f is strictly increasing continuous function that maps an interval [A.B] onto [a,b]. Suppose a is monotonically increasing on [a,b] and f ÎÂ (a) on [a,b]. Define b and g on [A,B] by b (y) = a (f (y)), g (y) = f (f (y)). Then prove that g ÎÂ (b) and . (7)
OR
(3) Let a be monotonically increasing function on [a,b] and let a¢ Î R on [a,b]. If f is a bounded real function on [a,b] then prove that f ÎÂ (a) on [a,b] Û f a¢ ÎÂ (a) on [a,b].(8)
(4) Let f ÎÂ (a) on [a,b]. For a £ x £ b, define F(x) = , then prove that F is continuous on [a,b]. Also, if f is continuous at some x_{ o} Î (a,b) then prove that F is differentiable at x_{ o }and F¢ ( x_{ o} ) = f (x_{ o }). (7)
 a) Let : [a,b] ® R ^{m} and let x Î (a,b). If the derivatives of exist at x then prove that it is unique.
OR
(2) Suppose that maps a convex open set E Í R^{n} into R^{m}, is differentiable on E and there exists a constant M such that M, ” x Î E, then prove that
ú (b) – (a)ú £ M ú b – aú , ” a, b Î E. (5)
 b) (1) Suppose E is an open set in R ^{n }; maps R into R ^{m }; is differentiable at x_{ o} Î E, maps an open set containing (E) into R ^{k} and is differentiable at f (x_{o}). Then the mapping of E into R ^{k}, defined by is differentiable at x_{o} and . (8)
(2) Suppose maps an open set EÍ Â ^{n} into Â ^{m}. Let be differentiable at x Î E, then prove that the partial derivatives (D_{j }f _{i}) (x) exist and , 1£ j £ m, where {e _{1}, e_{ }_{2, }e_{ }_{3, }…, e _{n}} and {u _{1}, u _{2, }u _{3, }…, u _{m}} are standard bases of R ^{n} and R ^{m}. (7)
(3) If X is a complete metric space and if f is a contraction of X into X, then prove that there exists one and only one x ÎX such that f (x) = x. (15)
III. a) (1) Prove: where {f _{n}} converges uniformly to a function f on E and x is a limit point of a metric space E.
OR
(2) Suppose that {f _{n}} is a sequence of functions defined on E and suppose that ½f _{n }(x)½£ M _{n}, x ÎE, n = 1,2,… Then prove that converges uniformly on E if converges. (5)
 b) (1) Suppose that K is a compact set and
* {f _{n}} is a sequence of continuous functions on K
** {f _{n}} converges point wise to a continuous function f on K
*** f _{n }(x) ³ f _{n+1 }(x), ” n ÎK, n= 1,2,… then prove that f _{n} ® f uniformly on K. (7)
(2) State and prove Cauchy criterion for uniform convergence of complex functions defined on some set E. (8)
OR
(3) State and prove StoneWeierstrass theorem. (15)
IV a) (1)Show that converges if and only if n >0.
OR
(2) Prove that G = . (5)
b)(1) Derive the relation between Beta and Gamma functions. (7)
(2) State and prove Stirling’s formula. (8)
OR
3) If f is a positive function on (0,¥) such that f (x+1) = x f (x); f (1) =1 and log f is convex then prove that f (x) = G (x). (8)
(4) If x >0 and y >0 then (7)
 a) (1)If f (x) has m continuous derivatives and no point occurs in the sequence x _{0}, x _{1}, ..,x _{n} more than (m+1) times then prove that there exists exactly one polynomial P_{n} (x) of degree £ n which agrees with f (x) at x _{0}, x _{1}, …, x _{n}.
OR
2) Show that the error estimation for sine or cosine function f in linear interpolation is given by the formula ½f(x)P(x)½£ . (5)
b)(1) Let x_{0}, x_{1}, …, x_{n} be n+1 distinct points in the domain of a function f and let P be the interpolation polynomial of degree £ n, that agrees with f at these points. Choose a point x in the domain of f and let [a,b] be any closed interval containing the points x _{0}, x _{1}, …, x _{n } and x. If f has a derivative of order n+1 in the interval [a,b], then prove that there is at least one point c in the open interval (a,b) such that where A (x) = (x – x_{0}) (x – x_{1})…(x – x _{n}). (7)
(2) Let P _{n+1} (x)= x ^{n+1} +Q(x) where Q is a polynomial of degree £ n and let maximum of ½P _{n+1} (x)½, 1 £ x £ 1. Then prove that we get the inequality . Moreover , prove that if and only if , where T _{n+1} is the Chebyshev polynomial of degree n+1. (8)
OR
3) Let f be a continuous function on [a,b] and assume that T is a polynomial of degree £ n that best approximates f on [a,b] relative to the maximum norm. Let R(x) = f (x) –T(x) denote the error in the approximation and let D = . Then prove that
(i) If D= 0 the function R is identically zero on [a,b].
(ii) If D >0, the function R has at least (n+1) changes of sign on [a,b]. (15).
Loyola College M.Sc. Mathematics Nov 2006 Fluid Dynamics Question Paper PDF Download
LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034 M.Sc. DEGREE EXAMINATION – MATHEMATICS

THIRD SEMESTER – NOV 2006
MT 3953 – FLUID DYNAMICS
Date & Time : 01112006/9.0012.00 Dept. No. Max. : 100 Marks
Answer ALL Questions.
I a) (i) Derive the equation of continuity in the form
[OR]
(ii)State and prove Euler’s equation of motion. (8)
 b) (i) The velocity of an incompressible fluid is given by .
Prove that the liquid motion possible and that the velocity potential is .
Also find the stream lines.
[OR]
(ii)State and prove Holemn Hortz vorticity theorem (17)
II a) (i)Show that the two dimensional flow described by the equation
is irrotational. Find the stream lines and equaipotentials.
[OR]
(ii)State and prove Milne Thomson circle theorem. (8)
 b) (i) In a two dimensional fluid motion the stream lines are
given by .Then show that where A and B are constants. Also find the velocity.
[OR]
(ii) State and prove Blasius theorem. (17)
P.T.O.
III a)(i)Write a note on Joukowskis transformation.
[OR]
(ii) State and prove Kutta and Joukowskis theorem. (8)
b)(i) Discuss the geometrical construction of an aerofoil.
[OR]
(ii) Discuss the liquid motion past a sphere. (17)
IV a) (i) Find the exact solution of a liquid past a pipe of elliptical cross section.
[OR]
(ii) Discuss the flow between two parallel plates. (8)
 b) (i) Prove that .
[OR} (ii) Derive the NavierStokes equation of motion for viscous fluid. (17)
Loyola College M.Sc. Mathematics Nov 2006 Linear Algebra Question Paper PDF Download
LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034 M.Sc. DEGREE EXAMINATION – MATHEMATICS

FIRST SEMESTER – NOV 2006
MT 1804 – LINEAR ALGEBRA
Date & Time : 26102006/1.004.00 Dept. No. Max. : 100 Marks
Answer ALL Questions.
I ) a) Let T be a linear operator on an ndimensional vector space V. Then prove that the characteristic and the minimal polynomials for T have the same roots, except for multiplicities.
[OR]
Let W be an invariant subspace for T. Then prove that the characteristic polynomial for the restriction operator divides the characteristic polynomial for T. Also prove that the minimal polynomial for divides the minimal polynomial for T. (5)
 b) State and prove CayleyHamilton theorem.
[OR]
Let V be a finitedimensional vector space V over F and let T be a linear transform on V. Then prove that T is diagonalizable if and only if the minimal polynomial for T has the form where are distinct elements of F. (15)
II )a) Let V be a finitedimensional vector space. Let be the subspaces of V and let . Then prove the following are equivalent.
 i) are independent.
 ii) For each we have = {0}.
[OR]
Let be a nonzero vector in V and let be the Tannihilator of .Then prove that
 i) If the degree ofis k, then the vectors form a basis for.
 ii) If U is the linear operator on induced by T, then the minimal polynomial for U is. (5)
 b) State and prove the primary decomposition theorem.
[OR]
Let T be a linear operator on a finitedimensional vector space V over F. If T is diagonalizable and if are the distinct characteristic values of T, then prove that there exist linear operators on V such that
(i) T;
(ii) I=;
(iii);
(iv)
(v) the range of is the characteristic space for T associated with
Conversely, if there exist k distinct scalars and k nonzero linear operators which satisfy conditions (i),(ii) and (iii), then show that T is diagonalizable, are the distinct characteristic values of T, and conditions (iv) and (v) are satisfied . (15)
III a) Write a note on the Jordon form.
[OR]
Let T be a linear operator on which is represented in the standard basis by the matrix. Find the minimal polynomial for T. (5)
 b) State and prove cyclic decomposition theorem.
[OR]
State and prove generalized CayleyHamilton theorem. (15)
IV a) Prove that a form f is Hermitian if and only if the corresponding linear operator T is self adjoint.
[OR]
If , then prove that . (5)
 b) i) State and prove Principal Axis Theorem.
 ii) Let V be a complex vector space and f a form on V such that fis real for every .Then prove that f is Hermitian. (9+6)
[OR]
Let T be a diagonalizable normal operator with spectrum S on a finitedimensional inner product space V .Suppose f is a function whose domain contains S. Then prove that f(T) is a diagonalizable normal operator with spectrum f(S) .If U is a unitary map of V onto V’ and T’=UTU, prove that S is the spectrum of T’ and f(T)= Uf(T)U . (15)
V a) Find all bilinear forms of F over F.
[OR]
Let f be a nondegenerate bilinear form on a finitedimensional vector space V.
Then prove that the set of all linear operators on V which preserve f is a group under the operation of composition. (5)
 Let V be a finitedimensional vector space V over a field of characteristic zero, and let f be a symmetric bilinear form on V. Then prove that there is an ordered basis for V in which f is represented by a diagonal matrix.
[OR]
Let V be an ndimensional vector space over a sub field of the complex numbers, and let f be a skewsymmetric bilinear form on V. Then prove that the rank r of f is even, and if r = 2k, then there is an ordered basis for V in which the matrix of f is the direct sum of the (nr) x (nr) zero matrix and k copies of the 2×2 matrix
. (15)
Loyola College M.Sc. Mathematics Nov 2006 Differential Geometry Question Paper PDF Download
LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
M.Sc. DEGREE EXAMINATION – MATHEMATICS

FIRST SEMESTER – NOV 2006
MT 1807 – DIFFERENTIAL GEOMETRY
Date & Time : 02112006/1.004.00 Dept. No. Max. : 100 Marks
Answer ALL the questions
I a) Obtain the equation of tangent at any point on the circular helix.
(or)
 b) Show that the necessary and sufficient condition for a curve to be a plane curve
is = 0. [5]
 c) Derive the equation of the osculating plane at a point on the curve of intersection of
two surfacesin terms of the parameter u. [15]
(or)
 d) Derive the SerretFrenet formulae and deduce them in terms of Darboux vector.
II a) Define involute and find the curvature of it.
(or)
 b) Prove that a curve is of constant slope if and only if the ratio of curvature to torsion
is constant . [5]
 c) State and prove the fundamental theorem for space curve. [15]
(or)
 d) Find the intrinsic equations of the curve given by
III a) What is metric? Prove that the first fundamental form is invariant under the
transformation of parameters.
(or)
 b) Derive the condition for a proper transformation from regular point. [5]
 c) Show that a necessary and sufficient condition for a surface to be developable is
that the Gaussian curvature is zero. [15]
(or)
 d) Define envelope and developable surface. Derive rectifying developable associated
with a space curve.
IV a) State and prove Meusnier Theorem.
(or)
 b) Prove that the necessary and sufficient condition that the lines of curvature may be
parametric curve is that [5]
 c) Prove that on the general surface, a necessary and sufficient condition that the curve
be a geodesic is for all values of the parameter . [15]
(or)
 d) Find the principal curvature and principal direction at any point on a surface
V a) Derive Weingarten equation. [5]
(or)
 b) Prove that in a region R of a surface of a constant positive Gaussian curvature
without umbilics, the principal curvature takes the extreme values at the boundaries.
 c) Derive Gauss equation. [15]
(or)
 d) State the fundamental theorem of Surface Theory and illustrate with an example