## Loyola College B.Sc. Mathematics April 2007 Real Analysis Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc.

 CV 13

DEGREE EXAMINATION – MATHEMATICS

FIFTH SEMESTER – APRIL 2007

MT 5501REAL ANALYSIS

Date & Time: 28/04/2007 / 1:00 – 4:00          Dept. No.                                                     Max. : 100 Marks

Answer all the questions:                                                                           10 x 2 = 10

1. Define an inductive set with an example.

1. Prove that every positive integer n (except 1) is either a prime or a product of primes.

1. State and prove Euler’s theorem for real numbers.

1. Define a Metric space.

1. State Cantor’s intersection theorem for closed sets.

1. Define an interior point and an open set.

7.Give an example of a uniformly continuous function.

1. Define a Cauchy sequence.

1. Suppose f and g are defined on (a, b) and are both differentiable at c Î (a, b), then prove

that the function fg is also differentiable at c.

1. Define total variation of a function f on .

Answer any five questions:                                                                                         5 x 8=40

1. Prove that the set R of all real numbers is uncountable.

1. State and prove Bolzano-Weirstass theorem for R.

1. Prove that every compact subset of a metric space is complete.

1. Let (X, d1) and (Y, d2) be metric spaces and f: X Y be continuous on X. If X is compact, then prove that f (X) is a compact subset of Y.

1. Let (X, d1) and (Y, d2) be metric spaces and f: X Y be continuous on X. Then show that a map f: X Y is continuous on X if and only if f -1 (G) is open in X for every open set G in Y.

16    Prove that in a metric space (X, d)

( i ) Arbitrary union of open sets in X is open in X

( ii) Arbitrary intersection of closed sets in X  is closed in X.

1. Let f: R and f have a local maximum or a local minimum at a point c.

Then prove that f ’(c) = 0.

1. Let f be of bounded variation onand xÎ (a, b) Define V:  R as   follows:

V (a) = 0

V (x) =Vf , a <  x ≤ b.

Then show that the functions V and V – f are both increasing functions on.

Answer any two questions:                                                                                                      2 x 20 = 40

19   State and prove Intermediate value theorem for continuous functions.

1.   State and prove Lagrange’s theorem for a function f :  R

21.(a) Suppose c Î (a ,b) and two of the three integrals f da ,f da , and f da

exists. Then show that the third also exists andf da =f da +f da.

(b) When do we say f is Riemann-Stieltjes integrable?

1. (a) State and prove Unique factorization theorem for real numbers.

(b) If F is a countable family of countable sets then show that  is also countable.

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## Loyola College B.Sc. Mathematics April 2007 Physics For Mathematics Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

 SC 06

THIRD SEMESTER – APRIL 2007

# PH 3100 – PHYSICS FOR MATHEMATICS

Date & Time: 28/04/2007 / 9:00 – 12:00 Dept. No.                                            Max. : 100 Marks

PART A

Answer ALL questions:                                                                 10 x 2 = 20 marks

1. Define Simple Harmonic motion.
2. A heavy fly wheel of moment of inertia 0.3 kg m2 is mounted on a horizontal axle of radius 0.01 m and negligible mass compared with the flywheel. Neglecting friction, find the angular acceleration if a force of 40 N is applied tangentially to the axle.
3. State any two Kepler’s laws.
4. Define Gravitational Potential.
5. Explain the terms stress and strain.
6. Define Surface tension of a liquid.
7. State Einstein mass-energy relation.
8. Name the transformation under which Maxwell’s equations are invariant.
9. Explain Doppler Effect.
10. A tuning fork A of frequency 384 Hz gives 6 beats per second when sounded with another tuning fork B. On loading B with a little wax, the number of beats per second becomes 4.  What is the frequency of B?

PART B

Answer any FOUR questions:                                                      4 x 7.5 = 30 marks

1. Derive expressions for the periods of oscillation of the mass suspended when two springs are connected (i) in series (ii) in parallel.
2. Obtain an expression for the escape velocity of a body from the surface of the earth. Calculate its value.
3. Derive an expression for the work done in stretching a wire.
4. Obtain the time dilation formula of special theory of relativity.
5. Two trains traveling in opposite directions at 100 km/hr each, cross each other while one of them is whistling. If the frequency of the note is 800 Hz find the apparent pitch as heard by an observer in the other train: (i) before the trains cross each other (ii) after the train have crossed each other.

PART C

Answer any FOUR questions:                                                    4 x 12.5 = 50 marks

1. An object rolls without slipping down a smooth inclined plane find its acceleration.  Compare the acceleration of a solid cylinder with that of a hollow cylinder of the same mass.
2. State Newton’s law of gravitation.  With a neat diagram describe Boy’s experiment to determine G.  (2.5 + 10)
3. Define coefficient of viscosity and derive Poiseuille’s formula for the flow of a liquid through a capillary tube.
4. State the postulates of the Special Theory of Relativity.   On the basis of Lorentz transformation, derive an expression for length contraction.
5. State the laws of vibration of strings and describe experiments to verify the law concerning (i) length (ii) tension and (iii) linear density.

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## Loyola College B.Sc. Mathematics April 2007 Operations Research Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

 CV 16

B.Sc.  DEGREE EXAMINATION –MATHEMATICS

FIFTH SEMESTER – APRIL 2007

MT 5504 – OPERATIONS RESEARCH

Date & Time: 03/05/2007 / 1:00 – 4:00          Dept. No.                                                     Max. : 100 Marks

SECTION –A

## Answer All:                                                                                     2 x 10 = 20

1. Define Operations Research.
2. What are the three methods to find Initial Basic Feasible solution in Transportation problem ?
3. Solve the Transportation problem by Least Cost Method.
 A1 A2 A3 Supply B1 4 6 2 5 B2 3 1 5 15 B3 4 5 3 15 Demand 10 10 10

1. Solve the game:
 2 1 4 1 4 3 2 2 6

1. Define Unbalanced situation in Transportation problem.
2. Define Feasible Solution.
3. Solve the Assignment problem
 3 7 5 4 7 2 5 4 6

1. What is Dummy activity in Network problem ?
2. Define Optimistic Time Estimate.
3. Define Economic Order Quantity.

SECTION –B

## Answer any five:                                                                              5x 8 = 40

1. Find the Initial Basic Feasible solution in Transportation problem using i) North West Corner Rule ii) Least Cost Method.
 A1 A2 A3 A4 Supply B1 4 2 1 3 20 B2 8 4 2 4 20 B3 1 2 3 4 30 B4 5 2 4 6 20 Demand 10 30 10 40

1. Using Graphical method solve the Linear Programming Problem

Max z = 2x1+4x2    subject to the constraints

2x1+4x≤ 5 ,       2x1+4x2 ≤ 4,         x1, x2 ≥ 0.

1. Solve the Assignment problem

 M1 M2 M3 M4 M5 J1 9 22 58 11 19 J2 43 78 72 50 63 J3 41 28 91 37 45 J4 74 42 27 49 39 J5 36 11 57 22 25

1. Solve using Matrix Oddment method

 -1 2 1 1 -2 2 3 4 -3

1. Define critical path and draw the Network diagram for

Activity:   A    B   C    D    E   F    G    H     I      J       K

Immediate predecessor:    –     –    –     A    B   B    C    D    E    H,I    F,G

1. Solve using Dominance property
 1 7 3 4 5 6 4 5 7 2 0 3

1. Solve the Transportation problem
 A1 A2 A3 A4 Supply B1 1 2 1 4 30 B2 3 3 2 1 50 B3 4 2 5 9 20 Demand 20 40 30 10

1. The probability distribution of monthly sales of certain item is as follows:

Number of items:    0         1          2        3         4          5          6

P(d)    :   0.02    0.05    0.30    0.27    0.20     0.10     0.06

The cost of carrying inventory  is Rs 10 per unit per month . Find the

shortage cost for one item for one unit of time.

(P.T.O)

SECTION –C

## Answer any two:                                                                              2x 20 = 40

1. Solve the following Linear Programming Problem using Simplex method

Max z = 3x1+2x2    subject to the constraints

x1+2x≤ 6 ,       2x1+x2 ≤ 8,       -x1+x2 ≤ 1,         x2 ≤ 2,        x1, x2 ≥ 0.         (20)

20 a) Solve the Transportation problem to maximize the profit

 A1 A2 A3 A4 Supply B1 40 25 22 33 100 B2 44 35 30 30 30 B3 38 38 38 30 70 Demand 40 20 60 30

1. b) Solve the following traveling sales man problem

 A B C D E A – 3 6 2 3 B 3 – 5 2 3 C 6 5 – 6 4 D 2 2 6 – 6 E 3 3 4 6 –

(10+10)

21 a) Solve the game graphically

 1 0 4 -1 -1 1 2 5

1. b) 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. Find

1. i) Economic Order Quantity ii) Time between two consecutive orders

iii) Number of orders per year             iv) The optimal total cost                      (10+10)

22 a) Draw the Network diagram ,the Critical path ,the project duration and the total float

for the following activities

Activity:    1-2       2-3      3-4        3-7       4-5        4-7       5-6        6-7

Duration:     3          4          4           4           2          2           3           2

1. b) What is the probability that the project will be completed in 27 days? Draw the

network diagram also .

Activity:    1-2       1-3      1-4        2-5      2-6         3-6        4-7       5-7        6-7

T0          :      3          2          6           2          5            3           3           1           2

Tm         :      6          5         12          5         11           6           9           4           5

Tp          :      15       14        30          8         17          15         27          7           8

(10+10)

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## Loyola College B.Sc. Mathematics April 2007 Graph Theory Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

 CV 17

B.Sc.  DEGREE EXAMINATION –MATHEMATICS

FIFTH SEMESTER – APRIL 2007

MT 5400GRAPH THEORY

Date & Time: 03/05/2007 / 9:00 – 12:00        Dept. No.                                                     Max. : 100 Marks

Part A

Answer all the questions. Each question carries 2 marks.

1. Show that Kpv = Kp – 1.
2. Give an example of a self-complementary graph.
3. Write down the incidence and adjacency matrices of the following graph.

1. Give an example of a disconnected graph with three components each of which is

3-regular.

1. Give an example of a non-Eulerian graph which is Hamiltonian.
2. For what values of m and n is Km,n Eulerian?
3. Draw all non-isomorphic trees on 5 vertices.
4. Give an example of a closed walk of even length which does not contain a cycle.
5. Define a planar graph and give an example of a non-planar graph.
6. Define the chromatic number of a graph.

Part B

Answer any 5 questions. Each question carries 8 marks.

1. (a). Prove that in any graph G, the sum of degrees of the vertices is twice the number

of edges. Deduce that the number of vertices of odd degree in any graph is even.

(b). Draw the eleven non-isomorphic graphs on 4 vertices.                           (4+4)

1. (a). Let G be a (p, q)-graph all of whose vertices have degree k or k + 1. If G has t

vertices of degree k then show that t = p(k+1)-2q.

(b). Define isomorphism of graphs. If two graphs have the same number of

vertices and same number of edges are they isomorphic? Justify your answer.                                                                                                                           (4+4)

1. (a). Define product and composition of two graphs. Illustrate with examples.

(b). Prove that any self – complementary graph has 4n or 4n+1 vertices.       (4+4)

1. (a). Prove that a closed walk of odd length contains a cycle.

(b). Prove that a graph with p vertices and  is connected.              (4+4)

1. (a). Show that if G is disconnected then GC is connected.

(b). Determine the centre of the following graph.

1. Prove that a graph G with at least two points is bipartite if and only if all its cycles

are of even length.

1. Let v be a vertex of a connected graph. Then prove that the following statements are equivalent:
1. v is a cut-vertex of G.
2. There exists a partition of V – {v} into subsets U and W such that for each

uU and  wW, the point v is on every (u, w) – path.

1. There exist two vertices u and w distinct from v such that v is on every (u, w)-

path.

1. Let G be graph with p ≥ 3 and. Then prove that G is Hamiltonian.

Part C

Answer any 2 questions. Each question carries 20 marks.

1. Prove that the maximum number of edges among all graphs with p vertices with no triangles is [p2 / 4], where [x] denotes the greatest integer not exceeding the real number x.                                     (20)
2. (a).Prove that every connected graph has a spanning tree.

(b).Prove that the following statements are equivalent for a connected graph G.

1. G is Eulerian.
2. Every vertex of G has even degree.
3. The set of edges of G can be partitioned into cycles. (5+15)
4. Let G be a (p, q)-graph. Prove that the following statements are equivalent.
5. G is a tree.
6. Any two vertices of G are joined by a unique path.
7. G is connected and p = q + 1.
8. G is acyclic and p = q + 1. (20)

1. (a).Let G be a connected plane graph with V, E and F as the sets of vertices, edges

and faces respectively. Then prove that | V | – | E | + | F | = 2.

(b). State and prove the five-colour theorem.                                                 (10+10)

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## Loyola College B.Sc. Mathematics April 2007 Formal Languages And Automata Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

 CV 19

B.Sc.  DEGREE EXAMINATION –MATHEMATICS

FIFTH SEMESTER – APRIL 2007

MT 5404FORMAL LANGUAGES AND AUTOMATA

Date & Time: 04/05/2007 / 1:00 – 4:00          Dept. No.                                                     Max. : 100 Marks

PART A

Answer all questions. Each question carries two marks.                                                                         10×2=20

1. Define context – free grammar and give an example.
2. Write a grammar to generate all palindromes over {a,b}
3. Show that every regular language is a context free language.
4. Define the concatenation of two languages and give an example.
5. Show that the PSL is closed under reflection.
6. Define an unambiguous grammar.
7. Show that SSS, Sa, Sb is ambiguous.
8. Construct a finite automation that accepts exactly those input strings of 0’s and 1’s that end in 11.
9. Construct a DFA to test whether a given positive integer is divisible by 2.
10. Define a non – deterministic finite automation.

PART B

Answer any five questions. Each question carries 8 marks.                                                          5×8=40

1. Prove that PSL is closed under union.
2. Let G be a grammar with SAaS|SS|a, ASbA|SS|ba. For the string aabbaaa find
1. a left most derivation
2. a right most derivation
3. Construct a grammar G for the language L(G) = {anbam / n, m 1}
4. Discuss about the Chomskian hierarchy
5. Prove that L={ai / i is a prime} is not a CFL.
6. Give an ambiguous and an unambiguous grammar to generate L={anbn / n1}.
7. Construct a DFA to test whether a given positive integer is divisible by 3.
8. Give a deterministic finite automation accepting the set of all strings over {0,1} with three consecutive 0’s.

PART C

Answer any two questions. Each question carries 20 marks.                                                            2×20=40

1. a) Write a note on the construction of CNF
2. b) Find a grammar in CNF equivalent to a grammar whose productions are

SaAbB,  AaA|a,  BbB|b                                                           (5+15)

1. State and prove uvwxy theorem.
 b
 a
1. Construct a DFA for the NDFA given below

a                           b

1. a) Construct a DFA accepting all strings over {0,1} having even number of 0’s.
2. b) Let G = ( N, T, P, S), N = {S, A},   T = {a,b},   P = { SaA,  A bS, Ab}.

Find L(G) and also construct an NDFA accepting L(G)         (10+10)

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## Loyola College B.Sc. Mathematics April 2007 Fluid Dynamics Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

 CV 18

B.Sc.  DEGREE EXAMINATION –MATHEMATICS

FIFTH SEMESTER – APRIL 2007

MT 5401FLUID DYNAMICS

Date & Time: 03/05/2007 / 9:00 – 12:00        Dept. No.                                                     Max. : 100 Marks

SECTION A

Answer ALL Questions.                     (10 x 2 = 20)

1. Define Lagrangian method of fluid motion.
2. State the components of acceleration in Cartesian coordinates?
3. What is the equation of continuity for (i) a homogeneous steady flow of fluid, (ii) a non-homogeneous incompressible flow of fluid.
4. Show that u = a+ by – cz, v = d – bx + ez, w = f + cx – ey are the velocity components of a possible liquid motion.
5. Write down the boundary condition when a liquid is in contact with a rigid surface.
6. Write down the stream function in terms of fluid velocity.
7. If = A(x2 – y2) represents a possible flow phenomena, determine the stream function.
8. State the Bernoulli’s equation for a steady irrotational flow?
9. What is the complex potential of sources at a1, a2, ….,an with strengths m1, m2,…,mn respectively?
10. Describe the shape of an aerofoil.

SECTION B

Answer ANY FIVE Questions.         (5 x 8 = 40)

1. (a) Define a streamline. Derive the differential equation of streamline.

(b) Determine the equation of streamline for the flow given by .         (4 + 4)

1. Explain local, convective and material derivatives.
2. The velocity field at a point is . Obtain pathlines and streaklines.
3. Show that the velocity potential satisfies the Laplace equation. Also find the streamlines.
4. Derive Euler’s equation of motion for one-dimensional flow.
5. Explain how to measure the flow rate of a fluid using a Venture tube.
6. Derive the complex potential of a doublet.
7. Explain the image system of a source with regard to a plane.

SECTION C

Answer ANY TWO Questions.          (2 x 20 = 40)

1. The velocity components of a two-dimensional flow system can be given in Eulerian system by . Find the displacement of the fluid particle in the Lagrangian system.
2. (a) Show that is a possible form of a bounding surface of a liquid.

(8 + 12 marks)

1. (a) Derive Bernoulli’s equation.

(b) Explain the functions of a pitot tube with a neat diagram.                             (10 + 10 marks)

1. State and prove the theorem of Kutta and Joukowski.

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## Loyola College B.Sc. Mathematics April 2007 Astronomy Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc.

 CV 15

DEGREE EXAMINATION –MATHEMATICS

FIFTH SEMESTER – APRIL 2007

MT 5503ASTRONOMY

Date & Time: 02/05/2007 / 1:00 – 4:00          Dept. No.                                                     Max. : 100 Marks

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## Loyola College B.Sc. Mathematics April 2007 Alg.,Anal.Geomet. Cal. & Trign. – I Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034B.Sc. DEGREE EXAMINATION – MATHEMATICSFIRST SEMESTER – APRIL 2007MT 1500 – ALG.,ANAL.GEOMET. CAL. & TRIGN. – I
Date & Time: 24/04/2007 / 1:00 – 4:00 Dept. No. Max. : 100 Marks
SECTION –AAnswer all:                                                                              2 x 10 = 20
1. If y = a cos5x + b sin5x, show that  . 2. Write down the nth derivative of eax. 3. What is the formula for radius of curvature in parametric form.                  4. Find the sub tangent and the sub normal for y = 3×3. 5. If x = sin 2ө and y = cos 2ө, find  . 6. Determine the quadratic equation having 3-2i  as one of its roots. 7. Derive the relation sin ix = i sinh x. 8. Separate into real and imaginary parts for cos (x+iy). 9. Define conjugate diameters.10. Write the polar form of the conic.
SECTION –BAnswer any five:                                                                              5x 8 = 40
11. Find the nth derivative of sin2x sin4x sin6x.12. Find the angle of intersection of the cardioids r = a(1+cosө) and r = b(1-cosө).13. Find the lengths of the sub tangent and the sub normal at the point (a, a)      for the curve y = x3+ 3x+4.  14. Show that the roots of the equation x3+px2+qx+r =0 are in A.P if    2p3-9pq+27r =0.15. Solve the equation 6×5+11×4-33×3-33×2+11x+6= 0.
16. Prove that  = 7 – 56 sin2ө + 11 2 sin4ө – 64 sin6ө.                        17. Prove that 32 cos6ө = cos6ө + 6 cos4ө +15 cos2ө + 10.                    18. Prove that the product of the focal distances of a point on an ellipse is equal to the           square of the semi-diameter which is conjugate to the diameter through the point.

SECTION –CAnswer any two:                                                                              2x 20 = 40
19. State and prove Leibnitz theorem and prove that (1-x2)y2 –xy1+m2y = 0 and       (1-x2) yn+2 –(2n+1)xyn+1+(m2-n2)yn = 0  for  y = sin( msin-1x).                                                                                                                                               (P.T.O)20 a) Find the evolute of the ellipse  .     b) Find the p-r equation of rm = am sinm ө.                                                (10+10)
21 a) Find the equation whose roots are the roots of the equation          x4-x3-10×2+4x+24 = 0 increased by 2 and hence solve the equation.      b) Find the sum of the fourth power of the roots of the equation             x3-2×2+x+1 = 0.                                                                                   (10+10)                                                                                          22 a) Prove that  .
b) Prove that the tangent to a rectangular hyperbola terminated by its asymptotes          is bisected at the point of contact and encloses a triangle of constant area.                                                                                                                                                                                                                                                                                       (10+10)

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