Download DISTANCE EDUCATION M.Sc. (Mathematics) DEGREE

DE-7482
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DISTANCE EDUCATION
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M.Sc. (Mathematics) DEGREE EXAMINATION,
DECEMBER 2009.
ALGEBRA
Time : Three hours
Maximum : 100 marks
Answer any FIVE questions.
All questions carry equal marks.
1. (a)
Suppose that H is a subgroup of G such that wherever H a  H b then
aH  bH . Prove that gHg1  H for all g  G .
(b)
Give an example of a group G and a subgroup H such that N ( H )  C ( H ) .
Is there any containing relation between N (H ) and C (H ) ?
2. (a)
Prove that every group is isomorphic to a subgroup of A(S ) for some
appropriate S.
(b)
3. (a)
(b)
4. (a)
(b)
5. (a)
(b)
6. (a)
Prove that every permutation is the product of its cycles.
If
R
is
a
commutative
ring
with
unit
element
and
M is an ideal of R, then prove that M is a maximal ideal of R if and only if
R/M is a field.
If R is a unique factorization domain then so is R[x ] .
Let R be a commutative ring with unit element whose only ideals are (0)
and R itself. Prove that R is a field.
Let D be an integral domain a, b  D . Suppose that a n  bn and a m  bm
for two relative prime positive integers m and n. Prove that a  b .
Prove that the intersection of two subspaces of V is a subspace of V.
Prove that if V is a finite dimensional vector space, then it contains a
finite set v1 , v2 , ... vn of linearly independent elements whose linear span
is V.
Prove that their exists a subspace W of V, invariant under T, such that
V  V1  W .
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(b)
7. (a)
(b)
8. (a)
If S and T are nilpotent linear transformations which commute, prove
that ST and S  T are nilpotent linear transformations.
Prove that if L is a finite extension of F and K is a subfield of L which
contains F , then [ K : F ] /[ L : F ]
If L is an algebraic extension of K and if K is an algebraic extension of F
then prove that L is an algebraic extension of F.
If the field F has pm elements then prove that F is the splitting field of
m
the polynomial x p  x .
(b)
If the finite field F has pm elements then prove the every a  F
m
satisfies a p  a .
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DISTANCE EDUCATION
M.Sc. (Mathematics) DEGREE EXAMINATION,
DECEMBER 2009.
REAL ANALYSIS
Time : Three hours
Maximum : 100 marks
Answer any FIVE questions.
All questions carry equal marks.
1. (a)
(b)
2. (a)
Prove that compact subsets of metric spaces are closed.
Prove that a set E is open if and only if its complement is closed.
Let f be a continuous mapping of a compact metric space X into a
metric space Y . Then prove that f is uniformly continuous on X .
(b)
3. (a)
(b)
State and prove the cantor intersection theorem.
State and prove the mean value theorem for derivatives.
State and prove the inverse function theorem.
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4. (a)
(b)
5. (a)
(b)
6. (a)
Explain the terms Zeroderivative and local extrema.
Explain the directional derivative and the total derivative.
If f is continuous on a, b then prove that f  R  on a, b .
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State and prove the fundamental theorem of calculus.
Prove that if
f
and
1
p
g
are complex functions in
R   , then
1
q
b
 b p
 
q

fg d   f d   g d  .
 a
  a

a
b

(b)


Prove that if f  R  on
b
 f d
a, b
and if
f x   M
on
a, b
then
 M  b   a  .
a
7. Let
f
and
g
be
measurable
real-valued
function
defined
2
on X , let F be real and continuous on R and put hx   F  f x , g x 
x  X  . Then prove that h is measurable. In particular f  g and fg are
measurable.
8. State and prove Bounded Convergence theorem.
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DE-7484
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DISTANCE EDUCATION
M.Sc. (Mathematics) DEGREE EXAMINATION,
DECEMBER 2009.
DIFFERENTIAL EQUATIONS AND NUMERICAL METHODS
Time : Three hours
Maximum : 100 marks
Answer any FIVE questions.
All questions carry equal marks.
1. (a)
(b)
State and prove the Existence Theorem.
Write a theorem on a formula for the Wronkskian and prove it.
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2. Derive the Legendre equation.
3. Find the general curves of the equations
dx
dy
dz
.


xz
y z  y2
4. (a)
(b)
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Find the steady state temperature at any point of a square plate whose
two adjacent sides are kept at the constant temperature 100C.
Find the solution of
2
2 y
2  y
such that


t 2
x 2
(i)
y = 0 when x  0 or  for all values of t .
(ii)
y
 0 when t  0 for all values of x.
t
(iii)
y  sin x from x  0 to  / 2
 when t = 0.
 0 from x   / 2 to  
5. Use the Newton's method to approximate positive solution of the system of
equations x 2  y 2  z 2  1; 2x 2  y2  4z  0; 3x 2  4 y  z 2  0 starting with
the initial approximation.
6. (a)
(b)
State and prove Chebyshev theorem.
Derive the sequence of Legendre polynomials.
1
7. Explain Gauss-Legendre integration method and evaluate I 
dx
1 x
2
using
0
three point formula.
8. Derive Euler's method on differential equation and solve numerically the initial
value problem u'  2tu 2 , u(0)  1 with h = 0.2, 0.1 and 0.05 on [0, 1]
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DISTANCE EDUCATION
M.Sc. DEGREE EXAMINATION, DECEMBER 2009.
Mathematics
OPERATIONS RESEARCH
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Time : Three hours
Maximum : 100 marks
Answer any FIVE questions.
All questions carry equal marks.
1. (a)
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Use Simplex method to solve the LPP :
Maximize Z  4 x1  x 2  3x 3  4 x 4
subject to,
 4x1  6x 2  5x 3  4x 4  20
 3x1  2x 2  4x 3  x 4  10
 8x1  3x 2  3x 3  2x 4  20
and
(b)
x1 , x 2 , x 3 , x 4  0 .
Construct the dual of the problem
Minimize Z  3x1  2x 2  4 x 3
subject to,
3 x1  5 x 2  4 x 3  7
6 x1  x 2  3 x 3  4
7x1  2x 2  x 3  10
x1  2 x 2  5 x 3  3
4 x1  7 x 2  2 x 3  2
and x1 , x 2 , x 3  0 .
2. (a)
Use
dual
simplex
method
to
solve
LP problem :
Minimize Z  3x1  2x 2  x 3  4 x 4
subject to
2x1  4x 2  5x 3  x 4  10
3x1  x 2  7x 3  2x 4  2
5x1  2x 2  x 3  6x 4  15
and
(b)
x1 , x 2 , x 3 , x 4  0 .
Use revised simplex method to solve the LPP :
Maximize Z  2x1  x 2
subject to
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the
following
3x1  4 x 2  6
6x1  x 2  3 and
x1 , x 2  0 .
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3. (a)
The following matrix gives the pay-off of different strategies S1 , S2 and
S 3 against conditions N1 , N 2 , N 3 and N 4 .
N1
N2
N3
N4
Rs.
Rs.
Rs.
Rs.
S1
4,000
100
6,000
18,000
S2
20,000
5,000
400
0
S3
20,000
15,000
–2,000
1,000
Indicate the decision taken under the following approach :
(i)
Pessimistic
(ii)
Optimistic
(iii) Equal probability
(b)
(iv)
Regret
(v)
Hurwicz criterion, his degree of optimism being 0.7.
An
ice-cream
retailer
buys
ice-cream
at
a
cost
of
Rs. 5 per cup and sells it for Rs. 8 per cup; any remaining unsold at the
end of the day can be disposed of at a salvage price of Rs. 2 per cup. Past
sales have ranged between 15 and 18 cups per day; there is no reason to
believe that sales volume will take on any other magnitude in future.
Find the EMV if the sale history has the following probabilities.
Market size : 15
Probability :
4. (a)
16
17
18
0.1 0.2 0.4 0.3
Use graphical method, solve the rectangular game whose pay-off matrix
for Player A is
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 2  1 5  2 6
 2 4  3 1 0 .


(b)
Explain
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(i)
Two person zero-sum games
(ii)
Pure strategies
(iii) Mixed strategies and
(iv)
5. (a)
(b)
Saddle point.
What are the steps involve in PERT network?
Calculate the total float for each activity.
Activity :
1-2 1-3 1-5 2-3 2-4 3-4 3-5 3-6 4-6 5-6
Duration :
(in weeks)
8
7
12
4
10
3
5
10
7
4
Find the critical path.
6. (a)
Show that for ( M / M / 1) :  / FCFS  model, the probability distribution
of queue length is  n 1    ,    /   1 and n  0 , find the average
number of customers in the system.
(b)
Cars arrive at a petrol pump, having one petrol unit, in Poisson fashion
with
an
average
of
10 cars per hour. The service time is distributed exponentially with a
mean of 3 minutes. Find
(i)
Average number of cars in the system
(ii)
Average waiting time in the queue
(iii) Average queue length
(iv)
The probability that the number of cars in the system is 2.
7. A cosmetics manufacturing company is interested in selecting the advertising
media for its product and the frequency advertising in each media. Data
collected from the past experience is given below :
Expected sales (in thousand rupees) :
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Frequency per week
TV
Radio News paper
1
220
150
100
2
275
250
175
3
325
300
225
4
350
320
250
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The cost of advertising in news paper is Rs. 500 per appearance, Rs. 1,000 in
radio and Rs. 2,000 in TV. The budget provides Rs. 4,500 per week for
advertisement. Use dynamic programming to determine the optimal mode of
advertising.
8. (a)
The annual demand for an item is 3200 units. The unit cost is Rs. 6 and
inventory carrying charges 25% per annum. If the cost of one
procurement
is
Rs. 150, determine
(i)
Economic order quantity
(ii)
Time between two consecutive orders
(iii) Number of orders per year
(iv)
(b)
The optimal total cost.
Find the optimal order quantity for a product for which the price-breaks
are as follows :
Quantity
Purchasing Cost
(Per unit)
0  Q1  100
Rs. 20
100  Q2  200
Rs. 18
200  Q3
Rs. 16
The monthly demand for the product is 400 units, the storage cost is 20%
of the unit cost of the product and the cost of ordering is Rs. 25 per
month.
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DE–7486
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DISTANCE EDUCATION
M.Sc. (Maths) DEGREE EXAMINATION, DECEMBER 2009.
MATHEMATICAL STATISTICS
Time : Three hours
Maximum : 100 marks
Answer any FIVE questions.
(5  20 = 100)
1. (a)
If
2

f x, y    3
the
joint
x  2 y , for

0
probability
density
of
X
and
Y
is
given
by
0  x  1, 0  y  1
, elsewhere
Find the conditional mean and conditional variance of X given Y  1 .
2
(b)
Derive mean and variance of Beta distribution.
2. (a)
(b)
(i)
State the properties of Distribution function.
(ii)
Let X have a Gamma Distribution with parameters  and  . Show
2
that P x  2     .
e
Let
X
and Y
have bivariate normal distribution with parameters
1  3,  2  1,  1  16,  2 2  25 and   3 . Determine the following.
5
2
(i)
P 3 Y  8 
(ii)
P 3  Y  8 X  7
(iii)
P  3  X  3
(iv)
P  3  X  3 Y  4  .
3. (a)
(b)
Derive ‘t’-distribution.
State and prove central limit theorem.
4. (a)
Let X 1 , X 2 , ... X n denote a random sample from a distance that has a
probability density function F x ;  ,    . Show that the statistic
Y1  U1 X 1 , X 2 , ... X n  is a sufficient statistic for  if and only if we can
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find two non negative functions, K1 and K 2 such that F x1 ;   .
F x2 ;   … F xn ;    K1 U1 X1 , X 2 , ..X n ;   . K 2 X 1 , X 2 ... X n  where for
every fixed value of Y1  U 1 X 1 , X 2 , ... X n  , K 2  ( X 1 , X 2 , ..., X n ) does not
depends upon  .
(b)
State and prove Rao-Cramer Inequality.
5. (a)
Determine, on the basis of the sample data shown in table, whether the
true proportion of shoppers. Favouring detergent A over detergent B is
the same in all 3 cities.
Detergent A Detergent B
U.K.
232
168
400
Japan
260
240
500
India
197
203
400
Use 0.05 level of significance.
(b)
State and prove Neyman-Pearson Lemma.
6. (a)
(b)
An oil company claims that less than 20% of all cars owners have not
tried its gasoline. Test this claim at the 0.01 level of significance, if a
random check reveals that 22 of 200 car owners have not tried the oil
company’s gasoline.
Find the critical region of the likelihood ratio test for testing the null
hypothesis H 0 :   0 against the composite alternative H1 :   0 on the
basis of a random sample of size n from a normal population with the
known variance  2 .
7. (a)
A company appoints 4 salesman A, B, C, D and observes their sales in 3
seasons. The figures (in lakhs) are given in table.
Salesman
Season
A
B
C
Total
D
Summer
36 36 21 35
128
Winter
28 29 31 32
120
Monsoon 26 28 29 29
112
Total
360
90 93 81 96
Carry out an analysis of variance.
(b)
Prove that :
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k
n
 xij  x... 
2
n 
i 1 j 1
k
 x
 x ... 
2
i
i 1
k
n
 xij  xi...
2
where x i is the mean of
i 1 j 1
the observations from the i th population and x .. is the mean of all nk
observations.
8. (a)
A Machine is set to deliver packets of a given weight. 10 samples of size 5
each were recorded. Below are given relevant data :
Sample :
 
1
2
3
4
5
6
7
8
9
10
Mean X :
15 17 15 18 17 14 18 15 17 16
Range (R) :
7
7
4
9
8
7
12
4
11
5
Calculate the values for the central line and the control limits of mean chart
and then comment on the state of control.
(Conversion factors : n  5, A2  0.58 , D3  0 , D4  2.11 ).
(b)
(i)
Explain – Control charts.
(ii)
Explain : X -chart and R-Chart and C-Chart.
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DISTANCE EDUCATION
M.Sc. DEGREE EXAMINATION, DECEMBER 2009.
Mathematics
COMPLEX ANALYSIS
Time : Three hours
Maximum : 100 marks
Answer any FIVE questions.
All questions carry equal marks.
1. (a)
Discuss the Riemann Sphere and prove that any circle on the sphere
corresponds to a circle or straight line in the z-plane.
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(b)
Prove that f ( z )  |xy| is not analytic at the origin eventhough CauchyRiemann equations are satisfied at the origin.
2. (a)
If


converges, prove that f ( z ) 
an

a
nz
n
tends to f(1) as z
0
0
approaches 1 in such a way that |1  z | (1| z |) remains bounded.
(b)
3. (a)
(b)
Discuss the mapping w  e z .
State and prove Cauchy's theorem for a circular disc.
Evaluate

C
4. (a)
(b)
5. (a)
z2  1
dz where C is the circle | z  1| 1 .
z2  1
State Liouville's theorem. Using this, prove Fundamental Theorem on
Algebra.
State and prove Schwarz Lemma.
Let f (z ) be analytic in a region  and  be a simple closed contour
described in the positive sense in  which does not pass through any of
zeros and poles of f (z ) . If  ~ 0(mod  )then prove that
1
2 i
f ' (z)
 f ( z ) dz  N  P
where N and P are the number of zeros and the
r
number of poles respectively, in the interior of  .
(b)
Using Rouche's theorem, show that the equation z 5  15 z  1  0 has one
root in the disc | z |  3 / 2 and four roots in the annulus 3 / 2  | z |  2.

6. (a)
Prove that

0
(b)
7. (a)
sin mx
dx   2 .
x
State and prove Hadamard's Factorization Theorem.
Show that any two bases of the same module are connected by a
unimodular transformation.
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(b)
8. (a)
Show that a nonconstant elliptic function has equally many poles as it
has zeros.
With the usual notations, derive
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3
' ( z )  4( z )  g2( z )  g3 .
(b)
Prove that
( z )
' ( z )
1
(u )
' (u )
1 =0
(u  z )  ' (u  z ) 1
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DE-7488
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DISTANCE EDUCATION
M.Sc. (Mathematics) DEGREE EXAMINATION,
DECEMBER 2009.
TOPOLOGY AND FUNCTIONAL ANALYSIS
Time : Three hours
Maximum : 100 marks
Answer any FIVE questions.
Each question carries 20 marks.
1. (a)
(b)
Let X be a topological space A be a subset of X. Prove that A  A  D(A).
Let f : X  Y be a mapping of one topological space into another. Prove
that the following are equivalent :
(i)
f is continuous
(ii)
f 1 ( F )
is
closed
in
X
whenever
F
is
closed
in Y.
(iii)
2. (a)
(b)
f ( A )  f ( A ) for every subset A of X.
State and prove Tychonoff theorem.
Prove that a metric space is sequentially compact  it has the BolzanoWeierstrass property.
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3. (a)
(b)
4. (a)
Prove that compact Hausdorff space is normal.
State and prove the Tietze Extension Theorem.
Prove that a subspace of the real line R is connected  it is an interval.
(b)
Prove that continuous image of a connected space is connected.
(c)
Let X be a locally connected space. If Y is an open subspace of X, then
prove that each component of Y is open in X.
5. (a)
Let M be a closed linear subspace of a Banach space N. If the norm of a
coset x  M in N / M is defined by x  M  inf x  m : m  M , then
show that N / M is a Banach space.
(b)
Let M be a linear subspace of a real normed linear space N, and let f be a
functional defined on M. If x 0 
 M and if M 0  M  { x0 } is a linear
subspace spanned by M and x 0 then prove that f can be extended to a
functional f0 defined on M 0 such that f0  f .
6. (a)
(b)
7. (a)
State and prove closed graph theorem.
State and prove the Uniform Boundedness Theorem.
State and prove Schwartz inequality.
(b)
If M and N are closed linear subspace of a Hilbert space H such that
M  N , then prove that the linear subspace M  N is also closed.
(c)
Let { e1 , e2 , e3 , ..., en } be a finite orthonormal set in a Hilbert space H. If x
is any vector in H, then prove that
n
 x, e
i
2
 x
2
.
i 1
8. (a)
If A1 and A2 are self-adjoint operators on H, then prove that their
product A1 A2 is self-adjoint  A1 A2  A2 A1
(b)
If N 1 and N 2 are normal operators on H with the property that either
commutes with the adjoint of the other, then prove that N1  N 2 and
N1 N 2 are normal.
(c)
If P and Q are the projections on closed linear subspace M and N of H,
then prove that M  N  PQ  0  QP  0 .
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DISTANCE EDUCATION
M.Sc. (Mathematics) DEGREE EXAMINATION,
DECEMBER 2009.
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GRAPH THEORY
Time : Three hours
Maximum : 100 marks
Answer any FIVE questions.
All questions carry equal marks.
1. (a)
Define walk, path and cycle in a graph.
(b)
If e is a link of G , prove that  G    G  e    G  e  .
(c)
If
is
a
tree,
then
prove
G
edges = (the number of vertices) – 1.
2. (a)
(b)
3. (a)
that
the
number
of
Prove that any spanning tree constructed by Kruskal’s algorithm is an
optimal tree.
In the following spanning tree T in k10 , find its associated sequence.
Prove that, A graph G with more than 2 vertices is connected if and only
if any two vertices of G are connected by atleast two internally disjoint
paths.
(b)
Prove that C G  , the closure of G is well defined.
(c)
If
is
non
Hamiltonian
simple
graph
with
G
n vertices n  3 , then prove that G is degree-majorised by some Cm, n .
4. (a)
(b)
Define Eulerian graph.
Prove
that
a
given
connected
graph
if and only if all vertices of G are of even degree.
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G
is
Eulerian
(c)
If G is a simple graph with n vertices n  3  and  
n
, prove that G is
2
Hamiltonian.
5. (a)
For
any
two
integers
and
prove
that
k2
l  2,
r k, l   r k, l  1  r k  1, l  . Also prove that, if r k, l  1 and r k  1, l 
are both even then r k, l   r k, l  1  r k  1, l  .
(b)
6. (a)
If G is a tree with n vertices, prove that  k G   kk  1
n 1
.
Let G be a simple graph with n vertices n  2  . Then prove that G is
bipartite
2-chromatic.
if
and
only
if
G
(b)
State and prove Brook’s theorem.
(c)
Give an example for a connected graph G for which  G    .
7. (a)
Obtain Euler’s formula for a connected plane graph.
(b)
Show that the graph K 5 is non-planar.
(c)
Prove that every planar graph is 5-vertex colourable.
8. (a)
(b)
is
Prove that the value of any flow in a network N is less than or equal to
the capacity of any cut in the network.
State and prove Max-flow Min-cut theorem.
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DE–7490
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DISTANCE EDUCATION
M.Sc. (Maths) DEGREE EXAMINATION, DECEMBER 2009.
PROGRAMMING IN C/C++
Time : Three hours
Maximum : 100 marks
Answer any FIVE questions.
All questions carry equal marks.
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1. (a)
(b)
2. (a)
(b)
3. (a)
(b)
4. (a)
(b)
5. (a)
Explain the following in C with example.
(i)
#define
(5)
(ii)
return.
(5)
Explain the different types of constants in C.
(10)
Write the basic concepts of Object Oriented Programming.
(10)
How do you declare and define user defined functions? Explain.
(10)
Distinguish between local and global variables with example.
(10)
Write a program to find the standard deviation and variance for ‘n’
numbers. (10)
Explain about logical and relational operators with example.
(10)
Write a program to find the NCR. of given numbers.
(10)
Explain the following with example.
(i)
break
(3)
(ii)
continue
(3)
(iii) while.
(b)
6. (a)
(b)
7. (a)
(b)
(4)
Write a program to display the given names into alphabetical order.
(10)
How do you define and declare structures? Explain with example.
(10)
Explain about the declaration and initialization of pointer variable.
(10)
Write a program to add and subtract the two matrices in 3  3 order.
(10)
Write short notes on :
(i)
# include
(2)
(ii)
gets ( )
(2)
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8. (a)
(b)
(iii) counters
(2)
(iv)
goto
(2)
(v)
fopen ( ).
(2)
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Explain about the formatted input and output operations in file.
(10)
Write a program that reads a text file and creates another file that is
identical except that every blank space is replaced by a single character
‘a’. (10)
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DE–7491
25
DISTANCE EDUCATION
M.Sc. (Mathematics) DEGREE EXAMINATION,
DECEMBER 2009.
DISCRETE AND COMBINATORIAL MATHEMATICS
Time : Three hours
Maximum : 100 marks
Answer any FIVE questions.
Each question carries 20 marks.
1. (a)
Using the generating function
yn2  yn1  6 yn  0 given y1  1 , y0  2 .
(b)
2. (a)
(b)
solve
the
differential
equation
Find the recurrence relation for the sequence AK   2K 2  1 .
Obtain the sum 12  22  32  ....  n 2 by determining the generating
function.
Solve the following recurrence relation
Dk   8 DK  1  16 DK  2  0 where D2  16 , D 3   80 .
3. (a)
Suppose a student want to make up a schedule for a seven day period
during which she will study one subject each day. She is taking four
subjects : Mathematics, Physics, Chemistry and Economics. Obtain the
number of schedules that devote atleast one day to each subject.
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(b)
Prove
(i)
(ii)
4. (a)
n n
n
n
 0    1   .....   n   2 .
   
 
ws18
 2n 
n
   2   n 2 .
2
2
How many permutations of the integers from 1 to n are there in which
atleast one integer is left in its own place?
(b)
Show that the number of dearrangements of n symbols is
1 1 1

n 1 
n ! 1     ...   1

1! 2! 3!
n! 

5. (a)
Let D and R be sets and let G be a permutation group of the set D . Let
the function f1 : D  R be said to be related to a function f2 : D  R if
there is a permutation   G such that f1  f2 . Show that this relation
is an equivalence relation.
(b)
Find
the
faces
a , b, c
number
and
of
d
of
ways
the
of
painting
pyramid
with
the
two
four
colours
x and y.
6. State and prove Polyyas fundamental theorem.
7. (a)
Define
modular
and
distributive
lattices.
that in a distributive lattice L, the following are equivalent
(i)
a b  x  a b.
(ii)
x  a  x   b  x   a  b for a, b, x  L .
Prove
(b)
Define complemented lattice. Prove that in a distributive lattice, if and
element has a complement then it is unique.
(c)
Let
L
be
a
lattice
with
upper
and
lower
a

0

a
a

1
 a,
bounds 1 and 0. Prove that for any a in L, a  1  1,
,
a  0  0.
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8. (a)
(b)
(c)
Define Boolean algebra. Show by an example of a lattice which is not a
Boolean algebra.
Express the polynomial px1 , x 2 , x 3   x1  x 2 in an equivalent sum of
products canonical form in three variables x1 , x 2 and x 3 .
Represent
the
f a, b, c   a  b  c .
following
function
by
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Karnaugh
map
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