B.Sc Mathematics Question Paper

Loyola College B.Sc. Mathematics Nov 2008 Real Analysis Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

AB 10

 

FIFTH SEMESTER – November 2008

MT 5505 – REAL ANALYSIS

 

 

 

Date : 03-11-08                     Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

SECTION – A

Answer  ALL  Questions                                                                                           (10 x 2=20 marks)

  1. Define similar sets with an example.
  2. If a and b are any real numbers such that

 

  1. Define a metric space.
  2. Give an example of a set E in which every interior point of E is also an accumulation point of E but

not conversely.

  1. Define a convergent sequence.
  2. State intermediate value theorem for continuous functions.
  3. Define Open ball and Closure of set E.
  4. When is a sequence {an} said to be Monotonic increasing and decreasing?
  5. State the linearity property of Riemann- Stieltjes integral.
  6. Define limit superior of a real sequence.

SECTION – B

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

 

  1. State and prove Minkowski’s inequality.
  2. If n is any positive integer, then prove that Nn is countably infinite.
  3. Let(X, d ) be a metric space. Then prove that
  4. i) the union of an arbitrary collection of open sets in X is open in X.
  5. ii) the intersection of an arbitrary collection of closed sets in X is closed in
  6. Prove that a closed subset of a compact metric space is compact.
  7. Prove that every compact subset of a metric space is complete.
  8. Let f : ( X, d ) → Rk be continuous on X. If X is compact, then prove that f is

bounded on X.

  1. State and prove Rolle’s theorem.
  2. Let {an} be a real sequence. Then prove that

(i) {an} converges to l  if and only if  lim inf an = lim sup an = l

(ii) {an} diverges to + ∞ if and only if  lim inf an = + ∞

SECTION C

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

 

  1. (a) Prove that the set R is uncountable. (10 marks)

(b) State and prove Cauchy –Schwartz inequality.                                                          (10 marks)

 

  1. (a) State and prove Bolzano- Weierstrass theorem. (18 marks)

(b) Give an example of a metric space in which a closed ball

is not the  closure of the open ball B(a ; r ).                                                                 (2 marks)

 

  1. (a) Let ( X, d1) and (Y, d2) be metric spaces and f : X→Y. If x0X, then prove that f is

continuous at x0 if and only if for every sequence {xn} in X that converges to x0,

the sequence { f (xn ) } converges to f (x0).                                                                  (12marks)

(b) Prove that Euclidean space k is complete.                                                                ( 8 marks)

 

 

  1. b) State and prove Taylor’s theorem.                                                                  (10 marks)

 

 

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Loyola College B.Sc. Mathematics Nov 2008 Mechanics – II Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

AB 18

 

FIFTH SEMESTER – November 2008

MT 5500 – MECHANICS – II

 

 

 

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

Time : 9:00 – 12:00

            PART-A

Answer ALL the questions:                                                                                                (10 x 2 =20 marks)

 

  1. Define the centre of mass of a rigid body.
  2. Where does the centre of gravity lie on a uniform solid tetrahedron?
  3. State the principle of virtual work.
  4. The maximum velocity of a particle moving in a simple harmonic motion is 2ft/sec and its period is Find its amplitude.
  5. If the distance x of a point moving on a straight line from a fixed origin on it and its velocity V are connected by , show that the motion is simple harmonic.
  6. Find the length of a seconds pendulum.
  7. If the angular velocity of a particle moving in a plane curve about a fixed origin is constant, show that its transverse acceleration varies as radial velocity.
  8. Define central orbit and central force.
  9. State the perpendicular axes theorem.
  10. State D’ Alemberts’ principle.

 

PART-B

Answer any FIVE questions:                                                                                                            (5 x 8 = 40 marks)

 

  1. Find the centre of gravity of a uniform solid hemisphere of radius .
  2. A uniform circular lamina of radius 3a and centre O has a hole in the form of an equilateral triangle of side 2a with one vertex at O. Prove that the distance of the centre of gravity from

       O is   .

  1. Derive the Cartesian equation of a catenary.
  2. Show that the resultant of two simple harmonic motions of the same period in the same straight line is also a simple harmonic motion. Find the amplitude and epoch.
  3. Obtain the radial and transverse components of velocity and acceleration in polar coordinates.
  4. Find the law of force towards the pole when the central orbit is the curve .
  5. If ℓ1 be the length of an imperfectly adjusted seconds pendulum which gains n seconds per hour and ℓ2 , the length of one which loses n seconds per hour in the same place, show that the true length of the seconds pendulum is    .
  6. Find the moment of inertia of a truncated cone about its axis, radius of its ends being a and b.

 

 

PART-C

Answer any TWO questions:                                                                                     (2 x 20 = 40 marks)

 

 

  1. a) Four equal rods, each of length a, are joined to form a rhombus ABCD and the points

B and D are joined by a string of length ℓ. The system is placed in a vertical plane with A resting on a horizontal plane and AC vertical. Prove that the tension in the string is  where is the weight of the rod.

  1. b) Find the centre of gravity of the area enclosed by the parabolas and  .                                                                                  (10+10)

 

  1. a) A point moves with uniform speed v along the cardioids  Show that
  1. its angular velocity  about pole is
  2. radial component of acceleration is constant,
  • magnitude of resultant is .
  1. b) Derive the pedal equation of the central orbit.                                     (10+10)

 

  1.  a) State and prove the parallel axes theorem concerning moment of inertia.

 

  1. b) A solid sphere is rolling down a plane, inclined to the horizon at an angle α and rough enough to prevent any sliding. Find its acceleration.                         (10+10)

 

  1. a)  A particle is performing simple harmonic motion of period T about a centre O and it

passes through the position P (where op = b) with velocity v in the direction op. Prove that the time that elapses before it returns to P is  .

 

  1. b) Two particles of masses M and M1 respectively are attached to the lower end of an elastic

       string whose upper end is fixed and are hung at rest. M1 falls off. Show that the distance of    

       M from the upper end of the string at time t is   where a is the unstretched   

       length of the string; b and c are the distance by which it would be stretched when

        supporting M and M1, respectively.                                                                                (10+10)

 

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Loyola College B.Sc. Mathematics Nov 2008 Operations Research Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

   B.Sc. DEGREE EXAMINATION – MATHEMATICS

AB 21

 

FIFTH SEMESTER – November 2008

MT 5504 – OPERATIONS RESEARCH

 

 

 

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

Time : 9:00 – 12:00

SECTION A

Answer all the questions. (10 X 2 = 20)

 

  1. Define feasible solution of a general Linear Programming Problem.
  2. Give the symmetric form of a LPP.
  3. State the necessary and sufficient condition for a Transportation problem to have a feasible solution.
  4. How can you convert a maximization assignment problem to a minimization problem?
  5. What is a two person zero sum game.
  6. Define EOQ.
  7. Define total float of an activity.
  8. What is payoff matrix?
  9. What is a project? List the 3 main phases of a project.
  10. What are the different types of inventory?

 

SECTION B

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

 

  1. Solve the following L.P.P by the Graphical method:

Max Z = 3x1 + 2x2 subject to

-2x1 + x2 ≤ 1,

x1≤ 2,

x1 + x2 ≤ 3 and x1, x2 ≥ 0.

 

  1. Find the initial basic feasible solution for the following transportation problem by least cost method.

To                          Supply

From          1          2          1          4          30

3          3          2          1          50

4          2          5          9          20

Demand          20        40        30        10

 

  1. Solve the following 2 X 2 game:

B

A         5          1

4          0

 

  1. Construct the network for the project whose activities are given below and compute the total, free and independent float of each activity and hence determine the critical path and the project duration:

 

Activity: 0-1    1-2       1-3       2-4       2-5       3-4       3-6       4-7       5-7       6-7

 

Duration:3       8          12        6          3          3          8          5          3          8

 

  1. For an item, the production is instantaneous. The storage cost of one item is Re. one per month and the set up cost is 25 per run. If the demand is 200 units per month, find the optimum quantity to be produced per set-up and hence determine the total cost of storage and set-up per month.

 

  1. A commodity is to be supplied at a constant rate of 200 units per day. Supplies for any amounts can be had at any required time, but each ordering costs Rs. 50. Cost of holding the commodity in inventory is Rs. 2.00 per unit per day while the delay in the supply of the items induces a penalty of Rs.10 per unit per delay of one day. Formulate the average cost function of this situation and find the optimal policy (q,t) where t is the reorder cycle period and q is the inventory level after reorder. What should be the best policy if the penalty cost becomes infinite?

 

  1. The processing time in hours for the jobs when allocated to the different machines is indicated below. Assign the machines for the jobs so that the total processing time is minimum.
Machines
 

 

Jobs

 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 the following:

Maximize

Z=15x1 + 6x2+ 9x3 + 2x4

Subject to 2x1 + x2+ 5x3+6x4 ≤ 20

3x1+x2+3x3+25x4 ≤ 24

7x1 + x4 ≤70

x1, x2, x3, x4 ≥ 0.

SECTION C

Answer any two questions: (2 X 20 = 40)

19a) Use Penalty method to solve Z=2x1 + x2+ x3

Subject to 4x1 + 6x2+ 3x3 ≤ 8

3x1– 6x2– 4x3 ≤ 1

2x1 + 3x2 – 5x3  ≥ 4

x1, x2, x3 ≥ 0.

 

19b) A person wants to decide the constituents of a diet which will fulfill his daily requirements of essential nutrition at the minimum cost. The choice is to be made from four different types of foods. The yields per unit of these foods are given in the following table:

Food type Yield/unit Cost / unit (Rs)
Proteins Fats Carbohydrates
1 3 2 6 45
2 4 2 4 40
3 8 7 7 85
4 6 5 4 65
Minimum requirement 800 200 700

Formulate the linear programming model for the [problem.

 

20a) Solve the following Transportation problem to minimize the total cost of transportation:

 

Destination
 

origin

 

 D1 D2 D3 D4 supply
O1 14 56 48 27 70
O2 82 35 21 81 47
O3              99 31 71 63 93
Demand 70 35 45 60 210

 

 

20b) Solve the following Travelling salesman problem:

A B C D
 

 

From

 A – 46 16 40
B 41 – 50 40
C 82 32 – 60
D 40 40 36 –

 

21a) Three time estimates (in months) of all activities of a project are given below:

Time in months
Activity a m b
1-2 0.8 1.0 1.2
2-3 3.7 5.6 9.9
2-4 6.2 6.6 15.4
3-4 2.1 2.7 6.1
4-5 0.8 3.4 3.6
5-6 0.9 1.0 1.1

 

Find the expected duration and standard deviation of each activity,

Construct the project network,

Determine the critical path, expected project length and expected variance of the project length.

 

21b) For the pay-off matrix given below, decide optimum strategies for A and B

B

  • 2

A         1          200      80

2          110      170

 

22a) Explain the purchase inventory model with n-price breaks.

 

22b) Find the optimal order quantity for a product for which the price break is as follows:

Quantity                                  Unit cost

0 ≤ Q1 < 50                             Rs. 10

50 ≤Q2<100                            Rs. 9

100 ≤Q3                                   Rs. 8

The monthly demand for the product is 200 units, the cost of the storage is 25% of the unit cost and ordering cost is Rs. 20 per order.

 

 

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

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

   B.Sc. DEGREE EXAMINATION – MATHEMATICS

QB 02

 

FIFTH SEMESTER – November 2008

MT 5502 – LINEAR ALGEBRA

 

 

 

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

Time : 9:00 – 12:00

 

SECTION – A

Answer all questions:                                                                        (10 x 2 = 20 marks)

 

  1. Show that union of two subspaces of V need not be a subspace of V.
  2. If the set {v1, v2, ….vm} is a linearly independent set of vectors of a vector space V then prove that any non-empty set of this set is linearly independent.
  3. Define basis of a vector space and give an example.
  4. Show that  defined by  is a vector space homomorphism.
  5. Define an inner product space and give an example.
  6. Define an Eigen value and an Eigen vector.
  7. If {vi} is an orthonormal set, then prove that the vectors in {vi} are linearly independent.
  8. If A and B are Hermitian, Show that AB_BA is skew-Hermitian.
  9. Let R3 be the inner product space over R under the standard inner product. Normalize .
  10. Prove that the product of two invertible linear transforms on V is itself an invertible linear transformation on V.

SECTION – B

 

Answer any FIVE questions:                                                           (5 x 8 = 40 marks)

 

  1. Show that } is a basis of the Vector space F[x] of all polynomials of degree at most n.
  2. If A and B are subspaces of a vector space V over F, prove that (A+B) / B A / AB.
  3. State and prove Schwarz’s inequality.
  4. Prove that  is invertible if and only if the constant term of the minimal polynomial for T is not zero.
  5. If dim V=n and , then prove that T can have atmost n distinct eigen values.
  6. Let V=R3 and suppose that  is the matrix of  relative to the standard basis (1,0,0), (0,1,0), (0,0,1). Find the matrix relative to ,  & .
  7. Let A be an mxn matrix over a field F, and let r be its rank. Then prove that is equal to the size of the largest non-singular square submatrix of A.
  8. If  in V, then prove that T is unitary.

 

SECTION – C

Answer any TWO questions:                                                           (2 x 20 = 40 marks)

 

  1. a)   If V is a vector space of finite dimension that is the direct sum of its subspaces U and

W,  then prove that .

  1. Find the Co-ordinate vector of (2, -1, 6) of R3 relative to the basis .                                                                   (15+5)
  2. If then prove that is of dimension m2.
  3. a) State and prove Gram-Schmidt ortho normalisation theorem.
  4. Normalize in C3, relative to the standard inner product. (15+5)
  5. a) Prove that is invertible if and only if the constant term in the minimal polynomial for T is not zero.
  6. b) The linear transformation T on V is unitary if and only if it takes an orthonormal basis of V onto an orthonormal basis of V. (14+6)

 

 

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Loyola College B.Sc. Mathematics Nov 2008 Mechanics – I Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

   B.Sc. DEGREE EXAMINATION – MATHEMATICS

AB 11

 

FIFTH SEMESTER – November 2008

MT 5506 – MECHANICS – I

 

 

 

Date : 05-11-08                     Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

PART – A

Answer ALL the questions.                                                                         (10 X 2 = 20)

  1. When two forces of equal magnitudes are inclined at the angle 2α, their resultant is twice as great as when they are inclined at an angle 2β. Prove that Cos α = 2 Cos β.
  1. State the triangle law of forces.
  2. State Newton’s laws of motion.
  3. Define angle of friction.
  1. If T is the time of flight, R the horizontal range and α, the angle of projection, show that gT2=2R tan α.
  2. State Newton’s experimental law on impacts.
  3. Define the limiting velocity in a resisting medium.
  4. Define a couple and the moment of a couple.
  5. A particle is moving with uniform acceleration in a straight line, velocity u at A and V at B. Find the velocity at the midpoint of AB.
  6. Define relative angular velocity.

PART – B

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

  1. The angle between two forces of magnitudes P + Q and P – Q is 2α and the resultant of the forces makes an angle  f with the bisection of the angle between the forces. Show that p tan f = Q tan α.
  2. State and prove Lami’s theorem.
  3. Discuss the motion of a particle moving along a straight line with uniform acceleration f.
  4. Two like parallel forces P and Q (P > Q) act at A and B respectively. If the magnitudes of the forces are interchanged, show that the point of application of the resultant on AB will be displaced through the distance.
  5. Three balls of masses m1, m2, m3 respectively for which е is given are lying in a straight line. m1 is projected with a given velocity so as to impinge on m2 which in turn impinges on m3. If each impinging ball after impact is reduced to rest, prove that m22 = m1m3.
  6. Two particles of masses m1 and m2 (m1>m2) are connected by means of a light in extensible string that passes over a light, smooth, fixed pulley. Discuss the motion.
  7. Two smooth spheres of masses m1 and m2 moving with velocities u1 and u2 respectively in the direction of line of centres impinge directly. Discuss the motion of each mass after impact, given that e is the coefficient of restitution.

 

 

  1. Show that the velocity with which a particle must be projected down a smooth inclined plane of length  and height h so that the time of decent shall be the same as taken by another particle in falling freely through a distance equal to the height of the plane is.

SECTION – C

Answer any Two questions.                                                                   (2 X 20 = 40)

  1. (a) Three equal strings of no sensible weight are knotted together to form an equilateral  and a weight W is suspended from A. If the triangle and the weight be supported with BC horizontally by means of two strings at B and C each at an angle 1350 with BC, show that the tension in BC is .

(b) A weight is supported on a smooth plane inclined at an angle α with the horizon, by a string inclined to the vertical at the angle β. If the inclination of the plane is increased to ٧ and the inclination of the string with the vertical is unaltered, the tension in the string is doubled in supporting the weight. Prove that        (10+10)

  1. (a) State and prove Varignon’s theorem.

(b) Two rough particles connected by a light string rest on an inclined plane. If their weights and corresponding coefficients of friction are w1, w2 and μ1, μ2 respectively and μ1> tan α > μ2 where α is the inclination of the place with the horizon, prove that , if both particles are on the point of moving down the plane.

  1. (a) A particle is projected vertically upwards with the velocity of and after t seconds, another particle is projected upwards from the same point with the same velocity. Prove that the particles will meet at a height  after a time  seconds from rest.

(b) Discuss the motion of a particle falling under gravity in a medium whose resistance

varies as the square of the velocity.

  1. Show that the path of a projectile is a parabola. Also show that the speed of a projectile at any point on its path equals the speed of a particle acquired by it in falling from the directrix to that point.

 

 

 

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Loyola College B.Sc. Mathematics Nov 2008 Graphs, Diff. Equ., Matrices & Fourier Series Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

AB 02

 

   B.Sc. DEGREE EXAMINATION – MATHEMATICS

FIRST SEMESTER – November 2008

MT 1501 – GRAPHS, DIFF. EQU., MATRICES & FOURIER SERIES

 

 

 

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

Time : 1:00 – 4:00

PART – A        (10 × 2 = 20)

Answer ALL the questions

 

  1. What are linear functions?

 

  1. Find the slope of the line x = -2y–7.

 

  1. Write the normal equations of y = ax+b.

 

  1. Reduce y = aebx to normal form.

 

  1. Find the particular integral of yn+2-4yn+1+3yn = 2n

 

  1. Solve yn+2 -5yn+1 + 6yn = 0.
  2. Find the eigen values of and hence the eigen values of A2
  3. Define a quadratic form.

 

  1. Write down the Fourier series expansion of an even function f(x) = x in .

 

  1. Find the Fourier coefficient a0 if f(x) = x in the range 0 to π.

 

 

PART-B        (5 × 8 = 40) 

Answer any FIVE questions

 

  1. (a) Graph the function f(x) = x2+4x. (4+4)

 

(b) If the cost in rupees to produce x kilograms of a milk product is given by c(x) = 500-3x+2x2,

then find

(i) Total cost for 9 kilograms.

(ii) Marginal cost.

 

  1. Find the minimum average cost if the cost function is given by C(x) = 36x-10x2+2x3. Find also

the marginal cost at that point.

 

  1. Using the method of least squares fit a straight line to the following data.
x 3 4 4 6 8
y 4 5 6 8 10

 

  1. Solve the difference equation, yn+2 -3yn+1+2yn = 5n+2n

 

  1. (a) Verify Cayley – Hamilton Theorem for .

 

(b) What are the Eigen values of a triangular matrix?                                                         (6+2)

 

  1. (a) Find the characteristic vectors of . (6+2)

(b) Write down the matrix form corresponding to the quadratic form x2+2y2+3z2+4xy+8yz+6zx.

 

  1. Find the Fourier series expansion of ex in the range –π to π .
  2. Show that in , x(π-x) = .

 

PART-C        (2 × 10 = 20)      

Answer any TWO questions

 

  1. (a) Draw the graph and find the equilibrium price (y) and quantity (x) for the demand and supply

curves given below                                                                                             (10+10)

2y = 16-x

y2 = 4(x-y).

 

(b) Fit a curve of the form y = a+bx+cx2 to the following data

x 0 1 2 3 4
y 1 1.8 1.3 2.5 6.3

 

  1. (a) Solve yn+2+yn+1+yn =  n2+n+1.                                                                               (12+8)

 

(b) Solve  yn+2-2yn+1+yn  =  n2-2n

 

  1. (a) Find a Fourier series expansion for the function f(x) = x2 in and deduce that .

 

(b) Obtain the half range sine series for the function f(x) = cosx  .            (10+10)

 

  1. Diagnolise .

 

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

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

 

AB 22

 

FIFTH SEMESTER – November 2008

MT 5400 – GRAPH THEORY

 

 

 

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

Time : 9:00 – 12:00

PART- A

                                                                                                                                                                                 

Answer ALL the questions. Each question carries 2 marks.                                (10 x 2 = 20 marks)

 

  1. Show that every cubic graph has an even number of vertices.
  2. Let G = (V, E) be a (p, q) graph. Let and . Find the number of vertices and edges in G – v and G – e.
  3. Define a walk and a path.
  4. What is a connected graph?
  5. Give an example of a disconnected graph with 4 components.
  6. Draw all non-isomorphic trees on 6 vertices.
  7. Define an Eulerian trail and a Hamiltonian cycle.
  8. What is a cut edge? Give an example.
  9. Determine the chromatic number of Kn.
  10. Define a planar graph and give an example of a non-planar graph.

 

 

PART –  B

 

Answer any FIVE questions. Each question carries EIGHT marks.                   (5 x 8 = 40 marks)

 

  1. (a). Show that in any group of two or more people there are always two with exactly

the same number of friends inside the group.

(b). 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.                                                           (4+4)

 

  1. Let G1 be a (p1, q1)-graph and G2 a (p2, q2)-graph. Show that G1 + G2 is a (p1 + p2, q1 + q2 + p1p2) – graph and G1 x G2 is a (p1 p2, q1p2 + q2p1) – graph.

 

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

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

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

(b). Find the composition of the following graphs.

 

(4+4)

 

 

 

 

 

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

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

(4+4)

  1. Let v be a vertex of a connected graph. Then prove that the following statements are equivalent:
    1. v is a cut-point 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 points u and w distinct from v such that v is on every (u, w)-

path.

 

  1. 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.

 

  1. State and prove the five-colour theorem.

 

PART – C

 

Answer any TWO questions. Each question carries 20 marks.                            (2 x 20 = 40 marks)

 

  1. (a). 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.

(b). Show that an edge e of a graph G is a cut edge if and only if it is not contained in any cycle

of G.                                                                                                                                (15+5)

 

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

are of even length.

(b). Let G be graph with with p ≥ 3 and. Then prove that G is Hamiltonian.                      (10+10)

  1. (a). If G is Hamiltonian, prove that for every non-empty proper subset S of V, the

number of components of G \ S , namely, ω(G \ S) ≤ | S |.

 

(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)

 

  1. Let G be a (p, q)-graph. Prove that the following statements are equivalent.
  2. G is a tree.
  3. Any two vertices of G are joined by a unique path.
  4. G is connected and p = q + 1.
  5. G is acyclic and p = q + 1.

(20)

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

AB 25

 

   B.Sc. DEGREE EXAMINATION – MATHEMATICS

FIFTH SEMESTER – November 2008

MT 5404 – FORMAL LANGUAGES AND AUTOMATA

 

 

 

Date : 14-11-08                     Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

PART-A

Answer ALL questions.                                                                                    (10 x2 = 20)

 

1)   If T = {0,1} then write any four elements of T +.

2)   Define a regular grammar.

3)   Give an example for a context-free grammar.

4)   Prove that every regular language context-free language .

5)   Show that the family of PSL is closed under star.

6)   Define a derivation tree.

7)   Prove that G =  is ambiguous.

8)   Define a language accepted by a deterministic finite automation

9)   Define a nondeterministic finite automation.

10) Construct a DFA accepting all strings in (0 + 1)* ending with 1.

 

PART-B

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

 

11) Let G be a grammar with  SAaS / SS/a  ,     A SbA / SS /ba .  For the string

aabbaaa  find   i)  a leftmost derivation

  1. ii) a right most derivation

iii)  a derivation tree

12) Write a note on Chomskian hierarchy.

13) Define concatenation of two languages. Also prove that  the family of CSL is closed

under concatenation.

14) Write a CNF grammar for the language

15) Prove that  is not a context free language.

16) Let G be a grammar with productions S S + S / SS / a / b  .Find a

derivation tree for the string  ab + ab

17) Construct a finite  automation that accepts  exactly those input  strings of 0’s and1’s

that end in 101.

18) Construct a finite automation which can  test wether a given positive integer is

divisible by 2.

 

PART-C

Answer ANY TWO questions                                                                   (2 x 20 = 40 )

 

19) Let L be any context-free language. Prove that there exist constants p and q  such

that if  z  is a  word in L  with  then z can be written as   where

such that for each integer  is in L.

 

20) i) Write a note on Chomsky normal form.

  1. ii) Find a grammar in CNF grammar for the language

(8 + 12)

21) Let L =consisting of an equal number of a’s and b’s}.Write a CFG

to generate L

 

22) i) Write an ambiguous and unambiguous grammar for

  1. ii) Let L be a set accepted by a nondeterministic finite automation. Then prove

that there exists a deterministic finite automation that accepts L.           (8 + 12)

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

AB 24

 

   B.Sc. DEGREE EXAMINATION – MATHEMATICS

FIFTH SEMESTER – November 2008

MT 5402 – COMBINATORICS

 

 

 

Date : 14-11-08                     Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

SECTION-A

 

ANSWER ALL THE QUESTIONS:                                                                 (10×2 = 20)

 

  1. Define Stirling number of second kind. Find
  2. How many ways can one move from (0,0,0 ) to (a,b,c).
  3. Find per .
  4. Find the cycle of length 1, length 3 and length 4 for the permutation                                                     1 2 3 4 5 6 7 8  3 2 5 1 4 8 6 7 .
  5. Define Ordinary generating function.
  6. Define Ferrers graph.
  7. Find the Rook polynomial for the chess board C in the diagram below,

 

     

 

 

 

 

 

  1. Define Euler’s function.
  2. Define derangement using Sieve’s formula.
  3. Define cycle index of a permutation group.

 

SECTION-B

ANSWER ANY FIVE QUESTIONS:                                                                 (5×8 = 40)

 

  1. An executive attending a week long business seminar has five suits of different colours. On Mondays, she does not wear blue or green, on Tuesdays, red or green, on Wednesdays, blue, white or yellow, on Fridays, white. How many ways can she dress without repeating a colour during the seminar?

 

  1. Prove that the element f of R [t ] given by f(t) = has an inverse in R[t] if and only if has an inverse in R.

 

  1. Prove that the cardinality of the set of permutations of m symbols taken n at a time is m(m-1) (m-2)……..(m-n+1).

 

  1. State and prove Multinomial theorem .

 

  1. (i) An examination paper with 10 questions consists of 6 questions in algebra and 4 questions in geometry. At least 1 question from each section is to be attempted. In how many ways can this be done?

(ii) Find the number of ways of selecting 4 letters from the word EXAMINATION.       (5+3)

 

 

 

  1. (i) Define exponential generating function.

(ii) How many permutations can be formed from the two symbols and  with the added condition that  may occur at most twice and  may occur at most once. Illustrate the problem using exponential generating function.

(3+5)

  1. If the number of permutation in of type (λ) = , then show that .

 

  1. Describe the following Symmetric functions with an example

(a) Monomial (b) Complete homogenous (c) Elementary.

 

SECTION-C

ANSWER ANY TWO QUESTIONS                                                                (2×20 = 40)

 

  1. (i) If there exists a bijection between the set of n-letter words with distinct letters out of an alphabet of m letters and the set of n-tuples on m letters, without repetitions. Show that the cardinality of each of these sets is

m(m-1)(m-2)…(m-n+1).

(ii) Define increasing and strictly increasing of a word. Also prove that there  exists a bijection between the set of increasing words of length n on m ordered letters, and the set of distributions on n non-distinct objects into m distinct boxes.

(10+10)

 

  1. (i) In how many ways can a total of 16 be obtained by rolling 4 dice once?

(ii) Define a partition for the integer n. Give the recurrence formula for . Tabulate the values of for n, m = 1,2,3…7                                         (8+12)

 

  1. (i) State and prove Generalized Inclusion and Exclusion principle.

(ii) With proper illustrations describe the problem of Fibonacci.

(10+10)

 

  1. State and prove Burnside’s lemma.

 

 

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AB 16

 

   B.Sc. DEGREE EXAMINATION – MATHEMATICS

FIFTH SEMESTER – November 2008

MT 5407 – FORMAL LANGUAGES AND AUTOMATA

 

 

 

Date : 14-11-08                     Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

SECTION A

Answer ALL the questions:                                                                                       (102=20)

 

  1. Define a context sensitive grammar.
  2. Show that the language L ={is accepted by the grammar

G= ({S},{a,b,c}P,S) where P consists of the productions S → , S → bc.

  1. Define concatenation of two languages.
  2. Check whether the families of Phrase structure language and Context sensitive language are closed under intersection. Justify your answer.
  3. Let G = (N, T, P, S), where N = {S,A}, T = {a, b} and P consists of the rules S → aAb,

S → abSb, S → a, A → bS, A → aAAb. Check whether the grammar is ambiguous or unambiguous.

  1. Define leftmost derivation.
  2. Define Chomsky normal form.
  3. Define Self embedding property.
  4. What is the difference between deterministic finite automaton and non-deterministic finite automaton?
  5. Draw the state diagram for the non-deterministic finite state automaton M = (K, I, , , F) where K =   , I = {0,1}, F = {}  and δ is defined as follows,

 

       δ      0      1
 

 

 

SECTION B

Answer any FIVE questions:                                                                                     (5=40)

 

  1. (a) Give a context sensitive grammar to generate L = .

(b) Let L = . Show that L is generated by the regular grammar             G = (N, T, P, S) where N = {S, A}, T = {a, b} and P consists of the rules S → aS,            S → aA, A →bA, A →b.

  1. (a) Define regular grammar.

(b) Let L =   Show that a Context free grammar generating L is given by

G = (N. T, P, S) where  N = {S,A}, T = {a, b} and P consists of the following productions: S → aSb, S →aAb,    A →bA, A →b.

  1. Prove that the families of Phrase structure language, Context sensitive language, Context free language and Regular language are closed under union and product.
  2. Prove that the families of Phrase structure language, Context sensitive language, Context free language and Regular language are closed under inverse homomorphism. Prove also that the family of Context free language is not closed under intersection.
  3. Show that the grammar G =where = { S, ,V,

= { They, are, flying, planes}, = { S → () ,  → They,  (V) (,

V → are, , A →flying, V → (Aux) (P), Aux → are, → N, P → flying} and S is the start symbol, generates the language consisting of the sentence { They are flying planes}.

  1. Write a grammar in Chomsky normal form to generate L= .
  2. Let L be accepted by a non- deterministic finite automata. Then show that there exists a deterministic finite automaton that accepts L.
  3. Draw the state diagram for the following finite state automaton, M = (K, I, , , F) where K = {}, I = {a, b, c}, F = { is defined as follows,

 

        a      b      c
       Φ

 

SECTION C

Answer any TWO questions:                                                                                     (220=40)

 

  1. (a) Let L = . Prove that a context free grammar generating L is

G  =  ( {S,A}, {a, b}, P, S) where P consists of the rules S → aSb, S → aAb, A → bAa,

A → ba.

(b) Define Kleene closure and reflection of the language L.

(c) Prove that the families of Phrase structure language, Context sensitive language, Context free language and Regular language are closed under reflection.

  1. (a) Let G = (N, T, P,S) where  N = {S}, T = {a}, P = { S → SS, S → a}. Show that the grammar is ambiguous.

(b) Given a context free grammar  G = (N, T, P,S), prove that there exists an equivalent context free grammar such that for each non-terminal A S in G,

=  is infinite.

(c) Write the construction of a grammar in Greibach normal form.

  1. State and prove u-v theorem and illustrate it with example.
  2. Given a finite state automaton M, prove that there exists a right-linear grammar G such that L(G) is set accepted by M and illustrate it with an example for non-deterministic finite state automata.

 

 

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AB 14

 

   B.Sc. DEGREE EXAMINATION – MATHEMATICS

FIFTH SEMESTER – November 2008

MT 5405 – FLUID DYNAMICS

 

 

 

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

Time : 9:00 – 12:00

 

SECTION A

 

Answer ALL questions:                                                                                 (10 ´ 2 = 20)

 

  1. Define a steady flow.
  2. Define pathlines.
  3. What is the condition if the rigid surface in contact with the fluid motion is at rest?
  4. Determine pressure, if the velocity field qr = 0, , qz = 0 satisfies the equation of motion where A, B are constants.
  5. Find the stream function y, if j = A(x2 – y2) represents a possible fluid motion.
  6. What is the complex potential of sinks a1, a2 …… an with strength m1, m2 …… mn situated at the points z1, z2 …… zn respectively?
  7. Define a two-dimensional doublet.
  8. Define vortex tube.
  9. Find the vorticity components of a fluid motion, if the velocity components are

u = Ay2 + By + C, v = 0, w = 0.

  1. Define the term camber.

 

SECTION B

 

Answer any FIVE questions:                                                                         (5 ´ 8 = 40)

 

  1. The velocity in a 3-dimensional flow field for an incompressible fluid is . Determine the equation of streamlines passing through the point (1, 1, 1).
  2. Derive the equation of continuity.
  3. Draw and explain the working of a Venturi tube.
  4. Prove that for the complex potential the streamlines and equipotentials are circles.
  5. Obtain the complex potential due to the image of a doublet with respect to a plane.
  6. Show that the velocity vector is everywhere tangent to the lines in the XY-plane along which y(x, y) = a constant.
  7. Let , (A, B, C are constants) be the velocity vector of a fluid motion. Find the equation of vortex lines.
  8. Discuss the structure of an aerofoil.

 

 

 

 

 

SECTION C

Answer any TWO questions:                                                                         (2 ´ 20 = 40)

 

  1. a) For a two-dimensional flow the velocities at a point in a fluid may be expressed in the Eulerian coordinates by u = x + y + 2t and v = 2y + t. Determine the Lagrange coordinates as functions of the initial positions , and the time t.
  2. b) If the velocity of an incompressible fluid at the point (x, y, z) is given by where . Prove that the fluid motion is possible and the velocity potential is .                                                                                                                   (10 + 10)

 

  1. Derive the Euler’s equation of motion and deduce the Bernoulli’s equation of motion.

 

  1. a) Obtain the complex potential due to the image of a source with respect to a circle.
  2. b) The particle velocity for a fluid motion referred to rectangular axes is given by the components, where A is a constant. Find the pressure associated with this velocity field. (12 + 8)

 

  1. a) Show the motion specified by , (k being a constant) is an irrotational flow.
  2. b) State and prove the theorem of Kutta-Joukowski.                             (5 + 15)

 

 

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   B.Sc. DEGREE EXAMINATION – MATHEMATICS

AB 17

 

FIFTH SEMESTER – November 2008

MT 5408 – GRAPH THEORY

 

 

 

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

Time : 9:00 – 12:00

PART – A

ANSWER ALL QUESTIONS                                                                          (10 x 2 = 20)

 

  1. Define a cubic graph.
  2. Prove that .
  3. Give an example for an isomorphism between two graphs.
  4. Define a self complementary graph and give an example.
  5. Define walk and and path.
  6. Define an eulerian graph.
  7. Prove that every Hamiltonian graph is 2-connected.
  8. Define the centre of a tree.
  9. If G is k-critical, then prove that .
  10. Write the chromatic number for .

 

PART – B

ANSWER ANY FIVE QUESTIONS                                                                 (5 x 8= 40)

  1. a) Prove that in any graph G , the number of points of odd degree is even.
  2. b) Prove that any self complementary graph has 4n or 4n + 1    (4 + 4)

 

  1. a) Let G be a k regular bigraph with bipartition (V1,V2) and k>0. Prove that

.

  1. b) Prove that a closed walk of odd length contains a cycle.                   (4 + 4)

 

  1. Let x be a line of a connected graph G. Then prove that the following are

equivalent

  1. x is a bridge of G.
  2. ii) There exists a partition of V into subsets U and W such that for each

uU and  wW, the line x is on every  u – w  path.

iii) There exist two points u , w  such that the line x is on every  u –  w path.

 

  1. Prove that a line x of a connected graph G is a bridge if and only if x is not

on any cycle of G.

 

  • a) Prove that every tree has a centre consisting of either one point or two

adjacent  points.

  1. b) Prove that a graph G with p points and is connected.               (4 + 4)
  • Prove that every nontrivial connected graph has atleast two points which are not cutpoints.
  • If G is a graph with vertices and , then show that G is Hamiltonian.
  • State and prove five colour theorem.

 

PART – C

 

 

ANSWER ANY TWO QUESTIONS                                                               (2 x 20 = 40)

 

  • a) Prove the maximum number of lines among all p point graphs with no

triangles is .

  1. b) Let G1 be a  graph and G2 a    Then prove that is

a  graph.                                                                   (15 + 5)

 

  • a) Prove that a graph G with at least two points is bipartite if and only if its cycles

are of even length.

  1. b) State and prove Chavatal theorem. (10 + 10)

 

  • Prove that the following are equivalent for a connected graph G.
  1. G is eulerian.
  2. Every point of G has even degree.
  • The set of edges of G can be partitioned into cycles.

 

  • Let G be a tree. Then prove that following are equivalent.
  1. i) G is a tree.
  2. ii) Every two points of G are joined by a unique path.

iii)   G is connected and  p = q + 1.

  1. iv) G is acyclic and  p = q +

 

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

AB 15

 

   B.Sc. DEGREE EXAMINATION – MATHEMATICS

FIFTH SEMESTER – November 2008

MT 5406 – COMBINATORICS

 

 

 

Date : 14-11-08                     Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

SECTION A

ANSWER ALL THE QUESTIONS:                                                                                (10×2 = 20 )

 

  1. Define Binomial number.
  2. Define Bell number.
  3. Find the Ordinary generating function and Ordinary enumerator for the combinations of five symbols a, b, c, d, e.
  4. Give all the partitions for 6.
  5. Draw the Ferrers graph for λ = (764332). Also find λ.
  6. Define hλ and find h4.
  7. Prove that = pn.
  8. Find per .
  9. Define derangement and deduce the formula for it using Sieve’s formula.
  10. Define weight of a function.

 

SECTION B

ANSWER ANY FIVE QUESTONS:                                                                                (5×8 = 40 )

 

  1. Give the recurrence formula for. Tabulate the values of for n, m = 1,2, …, 6.

 

  1. Show that there exists a bijection between the following two sets:

(a) The set of n-tuples on m letters without repetition.

(b) The set of injections of an n-set into an m-set.

Prove that the cardinality of each of these sets is m (m-1) (m-2) … (m-n+1).

 

  1. Prove that the elements f of R[t] given by f =has an inverse in R[t] if and only if  has an inverse in R.

 

 

  1. Let n be a positive integer. Show that the Ordinary Enumerator,
  • for the partitions of n is F(t) = .
  • for the partitions of n into precisely m parts is .
  • for the partitions of n into parts all of which are odd is .
  1. Describe the monomial symmetric function and the elementary symmetric function with

an example.

 

  1. State and prove Generalized inclusion and exclusion principle.
  2. Define a rook polynomial. Prove with usual notation that R(t,) = t R(t, ) + R(t,).

 

  1. List 6 elements of the group of rotational symmetries of a regular hexagon and their

types.

 

SECTION C

ANSWER ANY TWO QUESTIONS:                                                                              (2×20 = 40)

 

  1. (i) Prove that number of distributions on n distinct objects into m distinct boxes with the

objects in each box arranged in a definite order is .

(ii) Define the combinatorial distribution with an example.

(15+5)

  1. Explain in detail about the power sum symmetric functions.
  2. (i) How many permutations of 1, 2, 3, 4 are there with 1 not in the 2nd position, 2 not in

the 3rd position, 3 not in the 1st or 4th position and 4 not in  the 4th position.

(ii) Prove with the usual notation that.

(10+10)

  1. State and prove the Burnside’s lemma.

 

 

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   B.Sc. DEGREE EXAMINATION – MATHEMATICS

AB 09

 

THIRD SEMESTER – November 2008

MT 3502 / MT 5503 – ASTRONOMY

 

 

 

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

Time : 9:00 – 12:00

SECTION – A

 

Answer all questions:                                                            (10 x 2 = 20 marks)

 

  1. Define: diurnal motion.
  2. What is meant by a circumpolar star?
  3. Define Astronomical refraction.
  4. Write any two uses of the Equatorial.
  5. State any one Kepler’s law.
  6. Define Equation of time.
  7. Define: ‘Synodic month’.
  8. What is meant by ‘phase of moon’?
  9. What are inner planets?
  10. Define the term ‘opposition’ as applied to a planet.

 

SECTION – B

Answer any FIVE questions:                                                           (8 x 5 = 40 marks)

 

  1. Describe the Equatorial coordinate system to fix the position of body in the celestial sphere.
  2. Find the condition that twilight may last through out night.
  3. Derive cassini’s formula for refraction, indicating the assumptions made.
  4. If the moon’s horizontal parallax is 57’ and her angular diameter is 32’, find her radius and her distance from the earth. (earth’s radius = 4000 miles).
  5. Derive Newton’s deductions from Kepler’s laws.
  6. Describe briefly the surface structure of the moon.
  7. Enumerate the chief points of difference between the lunar and solar eclipses.
  8. Define ‘stationary points’. Find the angle subtended at the sun by two planets when they are stationary as seen from each other.

 

SECTION – C

Answer any TWO questions:                                                           (2 x 20 = 40 marks)

 

  1. a) Write notes on Horizontal coordinate system to find the position of any body in the celestial sphere.
  1. b) Trace the variations in the duration of day and night during the year at a place in the north torrid zone.
  1. (a) Find the effect of parallax on the longitude and latitude of a star.

(b) Describe the sextant.

  1. (a) Discuss the different phases of moon during a lunation.

(b) Prove that the Equation of time vanishes four times a year.

  1. (a) Find the maximum number of eclipses in a year.

(b) Write a short note on ‘Comets’.

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034        LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034B.Sc. DEGREE EXAMINATION – MATHEMATICSTHIRD SEMESTER – November 2008MT 3501 – ALGEBRA, CALCULUS AND VECTOR ANALYSIS
Date : 06-11-08 Dept. No.         Max. : 100 Marks                 Time : 9:00 – 12:00                                               PART – A        (10 × 2 = 20 marks)
Answer ALL the questions
1. Evaluate   .2. What is  ?3. Find the complete integral of  q = 2yp2
4. Write down the complete integral of  z = px + qy + pq.
5. Find the constant k, so that the divergence of the vector   is zero.
6. State Gauss Divergence theorem.
7. Find L(cos23t).8. Find  .9. Find Φ(360).
10. Find the highest power of 5 in 79!

PART – B         (5 × 8 = 40 marks)
Answer any FIVE questions
11. Change the order of integration and evaluate  .12. Prove that β(m,n+1 )+ β(m+1,n) = β(m,n).
13. Solve p tanx + q tany = tanz.
14. If   are irrotational, prove that
(a)   is  solenoidal.
(b)Find the unit vector normal to the surface   z = x2 + y2 – 3 at (2,-1,2).                              (4+4)15. Evaluate   by Stokes Theorem where   & C is the boundary of the        triangle with vertices (0,0,0), (1,0,0) and (1,1,0).
16. Find  (a) L(te-t sint).
(b)L(sin3t cosh2t).
17. Find  .18. (a) If N is an integer, prove that N5-N is divisible by 30.                                                            (6+2)
(b)State Fermat’s Theorem.
PART – C        (2 × 10 = 20 marks)
Answer any TWO questions
19. (a) Evaluate   over the positive octant of the sphere x2+y2+z2 = a2
(b)Establish β(m,n) =  .                                                                                 (10+10)
20. (a) Solve  .
(b) Solve by Charpit’s Method, pxy + pq + qy = yz.                                                           (10+10)
21. (a) Verify Green’s theorem for   where C is the boundary of the              region  x=0, y=0, x+y=1.
(b) Evaluate   .                                                                                         (10+10)22. (a) Using Laplace Transform, solve   given that y(0)=1, y`(0)=0..
(b) Using Wilson’s Theorem, prove that 10!+111   0 mod 143.                                               (12+8)

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

   B.Sc. DEGREE EXAMINATION – MATHEMATICS

AB 07

 

THIRD SEMESTER – November 2008

MT 3500 – ALGEBRA, CALCULUS & VECTOR ANALYSIS

 

 

 

Date : 06-11-08                     Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

SECTION-A

Answer all questions:                                                                        (10 x 2=20)

 

  1. Evaluate .
  2. If .
  3. Obtain the partial differential equation by eliminating
  4. Solve
  5. If j =j  at  (1, -2, -1).
  6. State Stoke’s theorem.
  7. Find .
  8. Find .
  9. Find the number and sum of all the divisors of 360.
  10. Find the number of integers less that 720 and prime to it.

 

 

SECTION-B

Answer any five questions:                                                              (5 x 8=40)

 

  1. Evaluate  where R is the region bounded by the curves and .
  2. Express interms of Gamma function and evaluate .
  3. Solve .
  4. Solve .
  5. Show that .
  6. (a) Find .

(b) Find .

  1. Find .
  2. Show that (18) is divisible by 437.

 

 

SECTION-C

Answer any two questions:                                                                          (2 x 20=40)    

 

  1. (a) Change the order of integration  and evaluate the integral.

(b) Evaluate  taken over the volume bounded by the plane .

(c) Evaluate .

  1. (a) Solve

(b) Find a complete integral of .

  1. Verify Gauss divergence theorem for  for the cylinderical region S given by .
  2. Solve .

 

 

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Loyola College B.Sc. Mathematics Nov 2008 Alg.,Anal.Geomet. Cal. & Trign. – I Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

AB 01

 

   B.Sc. DEGREE EXAMINATION – MATHEMATICS

FIRST SEMESTER – November 2008

MT 1500 – ALG.,ANAL.GEOMET. CAL. & TRIGN. – I

 

 

 

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

Time : 1:00 – 4:00

PART – A

 

Answer all questions:                                                                        (10 x 2 = 20 marks)

 

  1. If find .
  2. In the curve , prove that the subtangent is of constant length.
  3. Write the formula for radius of curvature in parametric form.
  4. Define evolute.
  5. If is a root of find the other root.
  6. Define a reciprocal equation.
  7. Prove that .
  8. Find .
  9. Find the equation of the chord of the parabola having the mid point at .
  10. Define a rectangular hyperbola.

 

PART – B

 

Answer any FIVE questions:                                                           (5 x 8 = 40 marks)

 

  1. Show that in the curve the subnormal varies as the cube of the ordinate.
  2. Show that the radius of curvature at any point of the catenary .
  3. If where  find the minimum value of u.
  4. Find the radius of curvature of the cardivid
  5. Solve the equation given that  is a root of it.
  6. Solve the reciprocal equation
  7. Express interms of .
  8. Derive the polar equation of a conic.

 

PART – C

 

Answer any TWO questions:                                                           (2 x 20 = 40 marks)

 

  1. a) If show that (1-
  2. b) Find the evolute of the parabola
  3. a) Solve given that it has two pairs of equal roots.
  4. b) Find the positive root of the equation correct to two places of decimals, using Horner’s method.
  5. a) Prove that 64
  6. b) Prove that
  7. a) Sum the series .
  8. b) Show that in a conic the semilatus rectum is the harmonic mean between the segments of a focal chord.

 

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Loyola College B.Sc. Mathematics April 2009 Real Analysis Question Paper PDF Download

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   LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

ZA 29

FIFTH SEMESTER – April 2009

MT 5505 / 5501 – REAL ANALYSIS

 

 

 

Date & Time: 16/04/2009 / 9:00 – 12:00       Dept. No.                                                       Max. : 100 Marks

 

 

SECTION-A

 

Answer all questions:                                                                        (10 x 2=20)

 

  1. Define an order – complete set and give an example of it.
  2. When do you say that two sets are similar?
  3. Define discrete metric space.
  4. Give an example of a perfect set in real numbers.
  5. Define open map and closed map.
  6. When do you say that a function is uniformly continuous?
  7. If a function f is differentiable at c, show that it is continuous.
  8. Define “total variation” of a function f on [a,b].
  9. Give an example of a sequence {an}whose lim inf and lim Sup exist, but the sequence is not convergent.
  10. Give an example of a function which is not Riemann Stieltjes integrable.

 

SECTION-B

 

Answer any five questions:                                                              (5 x 8=40)

 

  1. Let a,b be two integers such that (a,b)=d. Show that there exists integers  such that .
  2. Show that the set of all real numbers is uncountable.
  3. Let E be a subset of a metric space X. Show that  is the smallest closed set containing E.
  4. State and prove Heire Borel theorem.
  5. Show that in a metric space every convergent sequence is Cauchy, but not conversely.
  6. Let f, g be differentiable at Show that f  g  is differentiable at c and if is also differentiable at c.
  7. State and prove Lagrange’s mean valve theorem.
  8. Suppose {an}is a real sequence. Show that lim Supan=l if and only if for every ,
  • there exists a positive integer N such that for all and
  • given any positive integer m, there exists an integer such that .

 

 

 

 

 

 

SECTION-C

 

Answer any two questions:                                                                          (2 x 20=40)    

 

  1. (a) State and prove Minkowski’s inequality.

(b) Show that there is a rational number between any two distinct real numbers.

(c) If show that e is irrational.

  1. (a) If F is a family of open intervals that covers a closed interval [a,b], show that a finite sub family of F also covers [a,b].

(b) Let SÌRn. If every infinite subset of S has an accumulation point in S, show that S is closed and bounded.

  1.  (a) Let (X,d1), (Y, d2) be metric spaces and . Show that f is continuous at if and only if for every sequence in X that converges to  the sequence converges to .

(b) State and prove Bolzano theorem.

 

  1. (a) State and prove Taylor’s theorem.

(b) Suppose on [a,b]. Show that on [a,b] and

 

 

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Loyola College B.Sc. Mathematics April 2009 Operations Research Question Paper PDF Download

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         LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

ZA 31

FIFTH SEMESTER – April 2009

MT 5507 / 5504 – OPERATIONS RESEARCH

 

 

 

Date & Time: 17/04/2009 / 9:00 – 12:00    Dept. No.                                                       Max. : 100 Marks

 

 

SECTION A

Answer  ALL  Questions                                                                               (10 x 2 = 20)

 

  1. Write down any two uses of Operations Research.
  2. Define Slack variables.
  3. Which is the necessary and sufficient condition for the transportation problem to have

a feasible solution?

  1. Describe a traveling salesman problem.
  2. What is meant by mixed strategy?
  3. What is the value of the game?
  4. Define Total float
  5. Explain: Minimal Spanning tree problem?
  6. Give any two reasons for maintaining inventory
  7. Define Recorder level.

 

SECTION B

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

 

  1. Solve the following Linear Programming Problem by graphical method

Maximize  Z = 5x1 + 8x2

Subject to:

15x1 + 10x2  ≤  180

10x1 + 20x2  ≤  200

15x1 + 20x2  ≤  210

and x1, x2  ≥ 0

  1. Use simplex method to solve the following LPP

Maximize  Z = 4x1 + 10x2

Subject to:

2x1 + x2  ≤  50

2x1 + 5x2  ≤  100

2x1 + 3x2  ≤  90

and x1, x2  ≥ 0

 

  1. Find the initial basic feasible solution for the following transportation problem
D1 D2 D3 D4 Supply
S1 21 16 25 13 11
S2 17 18 14 23 13
S3 32 27 18 41 19
Demand 6 10 12 15

 

 

 

 

 

 

 

  1. Write the algorithm for solving Assignment problem.

 

  1. Solve the following game using dominance property.

 

I II III Row mini.
I 1 7 2 1
II 6 2 7 2
III 6 1 6 1
Column max. 6 7 7

 

 

 

 

 

 

  1. Draw the network for the project whose activities with their predecessor relationships

are given below:

A, C, D can start simultaneously; E >B, C;  F, G >D;  H, I > E ,F ; J >I, G ;

K > H; B > A.

 

  1. The annual demand of a product is 10,000 units, each unit cost Rs.100 if orders

placed in quantities below 200 units but for orders of 200 or above, the price is Rs.95,

the annual inventory holding cost is 10% of the value of the item, and the ordering

cost is Rs.5 per order. Find the economic lot size.

 

  1. The demand for an item in a company is 18,000 units per year and the company can

produce the item at a rate of 3000 per month. The cost of one set up is Rs. 500 and

the holding cost of one unit per month is 15 paise. The shortage cost of one unit is

Rs. 20 per month. Determine the optimum manufacturing quantity and the number

of shortage. Also determine the manufacturing time and time between setups.

 

 

SECTION C

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

  1. Solve the following Linear Programming Problem by Dual Simplex method

Minimize  Z = x1 + x2

Subject to:

2x1 + x2  ≥ 2

-x1 – x2 ≥ 1

and x1, x2  ≥ 0

 

  1. (a) Solve the following transportation problem using Least Cost Method to find the

Initial Basic Feasible solution.

 

A1 A2 A3 A4 A5 Supply
B1 4 1 2 6 9 100
B2 1 4 7 3 8 120
B3 7 2 4 7 7 120
Demand 40 20 70 90 90

.

 

 

 

(b)Solve the following traveling sales man 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

(10 + 10)

  1. (a) Solve the following game graphically
  B1 B2 B3 B4
A1 1 0 4 -1
A2 -1 1 2 5

 

(b) In a game of matching points with 2 players suppose A wins one unit value when

there are 2 heads, wins nothing when there are 2 tails, and loses  ½ unit value when

there are 1 head and 1 tail. Determine the pay-off matrix, the best strategy for each

player and the value of the game.                                                        (10+10)

 

22    (a) Draw the network, determine the critical path , 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

 

(b) ABC manufacturing company purchases 9,000 parts of a machine for its annual

requirement, ordering one month’s usage at a time. Each part costs Rs.20.

The ordering cost per order is Rs.15, and the carrying charges are 15% of the

average inventory per year.

You have been asked to suggest a more economical purchasing policy for the

company. What advice would you offer and how much would it save the company

per year?                                                                                  (10+10)

 

 

 

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Loyola College B.Sc. Mathematics April 2009 Numerical Methods Question Paper PDF Download

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        LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

ZA 38

SIXTH SEMESTER – April 2009

MT 6605 – NUMERICAL METHODS

 

 

 

Date & Time: 23/04/2009 / 9:00 – 12:00 Dept. No.                                                     Max. : 100 Marks

 

 

 

PART – A (10 ´ 2 = 20)

Answer ALL questions.

 

  1. What do you mean by partial pivoting?
  2. State Cramer’s rule.
  3. What is the order of convergence in Newton-Raphson method?
  4. Explain bisection method.
  5. What is meant by interpolation?
  6. Write the error polynomial in the Newton’s forward interpolation formula?
  7. Write the Stirling’s central difference interpolation formula.
  8. Write any two advantages of central difference interpolation formula.
  9. What is the order of error in the Trapezoidal rule?
  10. Write the formula for third order Range-Kutta method.

 

PART – B (5 ´ 8 = 40)

Answer any FIVE questions.

  1. Solve the following system of equations by Gauss elimination method

28x + 4y – z = 32,       x + 3y + 10z = 24    and     2x + 17y +4z = 35.

  1. Solve for a positive root of x3 – 4x + 1 = 0 by Regula Falsi method.
  2. Write a C program to for Lagrange’s interpolation formula.
  3. Obtain Newton’s forward interpolation formula for equal intervals.
  4. Find the first two derivatives of at x = 50 and x = 56 given the table below:

x                        :              50            51              52          53           54           55             56

:           3.6840     3.7084      3.7325    3.7563   3.7798     3.8030      3.8259

 

  1. Use Laplace-Everett’s formula to obtain f(1.15) given that f(1) = 1, f(1.1) =1.049,  f(1.2) = 1.096, f(1.3) = 1.14.
  2. Evaluate by        (i) Trapezoidal rule      (ii) Simpson’s 1/3  rule   and

(iii) Simpson’s 3/8 rule.

  1. Solve in the range 0 £ x £2 using (i) Euler’s method  (ii) improved Euler’s method

 

 

 

 

 

PART – C (2 ´ 20 = 40)

Answer any TWO questions.

  1. (a) Solve by Gauss-Seidel Method, the following system of equations.

                                    28x + 4y – z = 32,      

                                    x + 3y + 10z = 24,     

                        and      2x + 17y + 4z = 35

(b) Find the real positive root of 3x – cos x – 1 = 0 by Newton-Raphson method correct to 6 decimal places.

  1. (a) From the following table find f(x) and hence f(6) using Newton’s divided difference formula.

x          :           1             2            7             8

                        f(x)       :           1             5            5             4

  • The following table gives the value of density of saturated water for various temperatures of saturated stream.

Tempo C ( = T)            :           100      150      200      250      300

Density hg/m3 (= d)     :           958      917      865      799      712

      Find by interpolation, the densities when the temperatures are 130oC and 275oC respectively.

 

  1. (a) Using Gauss’s forward interpolation formula, find the value of log 337.5 from the following table.

x                 :              310           320            330            340          350          360

       yx = log x        :           2.4914       2.5052       2.5185      2.5315      2.5441   2.5563

  • Using Bessel’s formula, find the derivative of f(x) at x = 3.5 from the following table.

x          :           3.47     3.48     3.49     3.50     3.51     3.52     3.53

f(x)       :           0.193   0.195   0.198   0.201   0.203   0.206   0.208

  1. (a) Using Range-Kutta method of fourth order, solve for y(0.1) and y(0.2) given that                 y¢ = xy + y2, y(0) =1.

(b)  Develop a C program to implement Simpson’s 3/8 rule.

 

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