## Loyola College B.Sc. Mathematics April 2012 Real Analysis Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

FIFTH SEMESTER – APRIL 2012

# MT 5505/MT 5501 – REAL ANALYSIS

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

Time : 9:00 – 12:00

PART – A

Answer ALL questions:                                                                                                        (10 x 2 = 20)

1. State and prove the triangular inequality.

1. Prove that the sets Z and N are similar.

1. Prove that the union of an arbitrary collection of closed sets is not necessarily closed.

1. Prove that every neighbourhood of an accumulation point of a subset E of a metric space contains infinitely many points of the set E.

1. Show that every convergent sequence is a Cauchy sequence.

1. Define the term “complete metric space” with an example.

1. State Rolle’s theorem.

1. Prove that every function defined and monotonic on is of bounded variation on .

1. State the linearity property of Riemann-Stieltjes integral.

1. State the conditions under which Riemann-Stieltjes integral reduces to Riemann integral.

PART – B

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

1. State and prove Cauchy-Schwartz inequality.

1. Prove that the interval is uncountable.

1. State and prove the Heine-Borel theorem.

1. State and prove the intermediate value theorem for continuous functions.

1. Let and be metric spaces and . If  is compact and  is continuous on , prove that  is uniformly continuous on .

1. State and prove the intermediate value theorem for derivatives.

1. Suppose on . Prove that on  and that

.

1. a) Let be a real sequence. Prove that (a) converges to L if and only if

(b)  diverges to  if and only if .

PART – C

Answer ANY TWO questions:                                                                                             (2 x 20 = 40)

 19. (a) Prove that every subset of a countable set is countable. (b) Prove that countable union of countable sets is countable. (c) State and prove Minkowski’s inequality.                                                            (8+7+5) 20. (a) Prove that the only sets in R that are both open and closed are the empty set and the set R itself. (b) Let E be a subset of a metric space . Show that the closure  of E is the smallest closed set containing E. (c) Prove that a closed subset of a compact metric space is compact.                     (4+8+8) 21. (a) Let  and  be metric spaces and . Prove that  is continuous on X if and only if  is open in X for every open set G in Y. (b) Explain the classification of discontinuities of real-valued functions with examples. (12+8) 22. (a) State and prove Lagrange’s mean value theorem. (b) Suppose  on  and  for every  that is monotonic on . Prove that  must be constant on . (c) Prove that a bounded monotonic sequence of real numbers is convergent.       (8+4+8)

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## Loyola College B.Sc. Mathematics April 2012 Physics For Mathematics – I Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

THIRD SEMESTER – APRIL 2012

# PH 3104/3100 – PHYSICS FOR MATHEMATICS – I

Date : 28-04-2012              Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

PART – A

1. What is the range of a Projectile?
2. Define holonomic and non-holonomic constraints.
3. What do you understand by parking orbit?
4. State Newton’s law of gravitation.
5. State Hooke’s law of elasticity.
6. Distinguish between cohesive and adhesive forces.
7. Define common mode rejection ratio(CMRR) of operational amplifier.
8. State the postulates of special theory of relativity.
9. Define the term ‘frame of reference’.
10. State Einstein mass-energy relation.

PART – B

Answer any FOUR questions:                                                                                                  (4×7.5 = 30)

1. a) Explain velocity-time graph for an object.
2. b) Apply Lagrange’s equation of motion for Atwood’s machine. (4+3.5)
3. a) State kepler’s laws of planetary motion. (4.5)
b) Write a note on gravitational redshift.                  (3)
4. Derive an expression for the work done per unit volume in stretching a wire.
5. On the basis of Lorentz transformation, derive an expression for length contraction.
6. Draw a circuit diagram to solve the equations x+y=2; x-y=1using Operational amplifier.

PART – C

Answer any FOUR questions:                                                                                             (4×12.5 =50)

1. Obtain an expression for i) Resultant velocity of the projectile at any instant
ii) Maximum height reached by the projectile iii) Time taken to attain maximum height iv) Time of flight and v) Horizontal range. (5×2.5).
2. Describe Boy’s experiment to determine Universal gravitational constant G with a neat diagram.
3. With a neat sketch explain how will you determine the surface tension of a given liquid by capillary rise method
4. Explain Op-amp as i) adder ii) subractor and iii) integrator. (4.5+4+4)
5. Describe Michelson-Morley experiment. Discuss the result obtained.

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## Loyola College B.Sc. Mathematics April 2012 Operations Research Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

FIFTH SEMESTER – APRIL 2012

# MT 5507/MT 5504 – OPERATIONS RESEARCH

Date : 30-04-2012              Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

PART  – A

1. A firm plans to purchase at least 200 quintals of scrap containing high quality metal X and low quality metal Y. It decides that the scrap to be purchased must contain at least 100 quintals of X and not more than 35 quintals of Y. The firm can purchase the scrap from two suppliers (A and B) in unlimited quantities. The percentage of X and Y metals in terms of weight in the scrap supplied by A and B is given below:
 Metals Supplier A Supplier B X 25% 75% Y 10% 20%

The price of A’s scrap is Rs.200 per quintal and that of B is Rs. 400 per quintal. The firm wants to determine the quantities that it should buy from the two suppliers so that total cost is minimized. Construct a LP model.

1. Give at least four different contrasting properties of primal and dual of general LP problems
2. What is meant by unbalanced transportation problem?
3. Name four different methods to solve an assignment problem.
4. Consider the game with the following payoff table:
 Player B Player A B1 B2 A1 2 6 A2 -2 λ

Show that the game is strictly determinable, whatever λ may be.

1. When no saddle point is found in a payoff matrix of a game, how do we find the value of the game?
2. Give an example of weighted graph which is not a tree. Find two different minimal spanning trees of the weighted graph you constructed.
3. Why is in PERT network each activity time assume a Beta-distribution?
4. What are the basic information required for an efficient control of inventory?
5. What is Economic Order Quantity (EOQ) concept?

PART  – B

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

1. Using Simplex Method, solve the following Problem and give your comments on the solution: Maximize Z= 6x1+4x2, Subject to: x1+x2£  5, x2≥ 8, x1, x2≥ 0.
2. Prove that dual of the dual is the primal.

 Tasks Clerks A B C D 1 4 7 5 6 2 – 8 7 4 3 3 – 5 3 4 6 6 4 2
1. Consider a problem of assigning four clerks to four tasks. The time (hours) required to complete the task is given below:

Clerk 2 cannot be assigned to task A and clerk 3 cannot be assigned to task B. Find all the optimum assignment schedules.

1. A company has three production facilities S1, S2 and S3 with production capacity of 7, 9 and 18 units (in 100s) per week of a product, respectively. These units are to be shipped to four warehouses D1, D2, D3 and D4 with requirement of 5, 6, 7 and 14 units (in 100s) per week, respectively. The transportation costs (in rupees) pre unit between factories to warehouses are given in the table below:
 D1 D2 D3 D4 Capacity S1 19 30 50 10 7 S2 70 30 40 60 9 S3 40 8 70 20 18 Demand 5 8 7 14 34

Use North-West Corner Method to find an initial basic feasible solution to the transportation problem.

1. A soft drink company calculated the market share of two products against its major competitor having three products and found out the impact of additional advertisement in any one of its products against the other
 Company B Company A B1 B2 B3 A1 6 7 15 A2 20 12 10

What is the best strategy for the company as well as the competitor? What is the payoff obtained by the company and the competitor in the long run? Use graphical method to obtain the solution.

1. Solve the following game after reducing it to a 2×2 game
 Player B Player A B1 B2 B3 A1 1 7 2 A2 6 2 7 A3 5 1 6
1. An assembly is to be made from two parts X and Y. Both parts must be turned on a lathe and Y must be polished whereas X need not be polished. The sequence of activities together with their predecessors is given below.
 Activity Description Predecessor  Activity A Open work order – B Get material for X A C Get material for Y A D Turn X on lathe B E Turn Y on lathe B, C F Polish Y E G Assemble X and Y D, F H Pack G

Draw a network diagram of activities for the project.

1. The production department for a company requires 3600 kg of raw material for manufacturing a particular item per year. It has been estimated that the cost of placing an order is Rs. 36 and the cost of carrying inventory is 25% of the investment in the inventories. The price is Rs. 10 per kg. Calculate the optimal lot size, optimal order cycle time, minimum yearly variable inventory cost and minimum yearly total inventory cost.

PART  – C

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

1. (a) Use graphical method and solve the LPP: Maximize Z= 15x1+10x2, Subject to: 4x1+6x2£  360,

3x1£  180, x2≥ 8, 5x2£  200, x1, x2≥ 0.                                                                         (7 marks)

(b) Use simplex method to solve the LPP: Maximize Z= 3x1+5x2+4x3, Subject to:  2x1+3x2£  8,

2x2+5x3£  10, 3x1+2x2+4x3£  15, x1, x2, x3≥ 0.                                                           (13 marks)

1. (a) ABC limited has three production shops supplying a product to five warehouses. The cost of        production varies from shop to shop and cost of transportation form one shop to a warehouse also varies. Each shop has a specific production capacity and each warehouse has certain amount of requirement. The costs of transportation are given below:
 Warehouse Shop I II III IV V Supply A 6 4 4 7 5 100 B 5 6 7 4 8 125 C 3 4 6 3 4 175 Demand 60 80 85 105 70 400

The cost of manufacturing the product at different production shops is given by:

 Shop Variable Cost CCost Fixed Cost A 14 7000 B 16 4000 C 15 5000

Find the optimal quantity to be supplied from each shop to different warehouses at minimum total cost.                                                                                                                           (10 marks)

(b) Two firms A and B make smart phones and tablets. Firm A can make either 150 smart phones in a

week or an equal number of tablets, and make a profit of Rs. 400 per smart phone and Rs. 300 per

tablet. Firm B can, on the other hand, make either 300 smart phones, or 150 smart phones and 150

tablets, or 300 tablets per week. It also has the same profit margin on the two products as A. Each

week there is a market of 150 smart phones and 300 tablets and the manufacturers would share

market in the proportion in which they manufacture a particular product. Write the payoff matrix

of A per week. Obtain graphically A’s and B’s optimal strategies and value of the game.

(10 marks)

1. A small project is composed of 7 activities whose time estimates are listed in the table below. Activities are identified by their beginning(i) and ending(j) node numbers.
 Activity Estimated Duration (weeks) (i–j) Optimistic Most Likely Pessimistic 1-2 1 1 7 1-3 1 4 7 1-4 2 2 8 2-5 1 1 1 3-5 2 5 14 4-6 2 5 8 5-6 3 6 15
• Draw the network of the activities in the project (5 marks)
• Find the expected duration and variance for each activity. (5 marks)
• What are the expected project length and critical path? (5 marks)
• Calculate the standard deviation of the project length.                         (5 marks)

1. (a) The annual demand of a product is 10,000 units. Each unit costs Rs.100 if orders are placed in

quantities below 200 units but for orders of 200 or above the price is Rs. 95. The annual inventory

holding costs is 10% of the value of the item and the ordering cost is Rs. 5 per order. Find the

economic lot size.                                                                                                       (8 marks)

(b) A dealer supplies you the following information with regard to a product dealt in by her: Annual

demand=10000 units; Ordering Cost=Rs 10/order; Price=Rs.20/unit; Inventory carrying cost=20%

of the value of the inventory per year.  The dealer is considering the possibility of allowing some

backorder to occur. She has estimated that the annual cost of backordering will be 25% of the

value of inventory.

• What should be the optimum number of units of the product she should buy in one lot?
• What quantity of the product should be allowed to be backordered, if any?
• What would be the maximum quantity of inventory at any time of the year?
• Would you recommend her to allow backordering? If so, what would be the annual cost saving by adopting the policy of backordering?                                                      (12 marks)

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## Loyola College B.Sc. Mathematics April 2012 Numerical Methods Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034B.Sc. DEGREE EXAMINATION – MATHEMATICSSIXTH SEMESTER – APRIL 2012MT 6605 – NUMERICAL METHODS
Date : 20-04-2012 Dept. No.         Max. : 100 Marks                 Time : 1:00 – 4:00
PART – A Answer ALL questions: (10 X 2 = 20 marks)
1. Explain Cramer’s rule.2. Distinguish between Gauss Elimination and Gauss Seidel methods.3. State the condition for convergence in Newton Raphson method.4. What do you mean by transcendental equation?5. Write Newton forward interpolation formula.6. Write any two properties of divided differences.7. Write Stirling’s formula using central difference notation.8. Write the derivatives using Newton’s forward difference formula.9. Define numerical integration.10. Write Simpson’s 1/3rd and 3/8th rule.
PART – B
Answer any FIVE questions: (5 X 8 = 40 marks)
11. Solve by Gauss elimination method: 12. Find the real root of  correct to three significant figures using Regula falsi method.13. Write a C program to  interpolate using Newton’s forward interpolation formula.14. Use Lagrange’s formula to find the value of y at x = 6 from the following data: x = 3, 7, 9, 10 and the corresponding value of y = 168, 120, 72, 63.15. Using the following data, find f’(5) by Newton’s divided difference formula:             : 0 2 3 4 7 9                   : 4 26 58 112 466 92216. Derive Laplace Everett’s formula.17. Apply Simpson’s rule to evaluate  to two decimal places by dividing the range into 4 equal parts.18. Solve  with the initial condition x = 0, y = 0 using Euler’s modified formula.

PART – C
Answer any TWO questions:                (2 X 20 = 40 marks)
19. (a)  Solve by Gauss Seidel method:
(b)  Find by Newton’s method the root of the equation  , which is approximately 2, correct        to three places of decimals.
20. (a) Given   : 0 1 2 5             : 2 3 12 14    find the cubic function of x using Newton’s          divided difference formula.
(b) Using Newton’s formula find the value of f(1.5) from the following data: :    0    1    2    3   4        : 858.3 869.6 880.9 892.3 903.6
21. (a) Use Stirling’s formula to find f(1.63) given  : 1.50    1.60      1.70        1.80       1.90       : 17.609   20.412   23.045   25.527    27.875            (b)Given  X: 0 4 8 12Y: 143 158 177 199 calculate  y5 by Bessel’s formula.
22. (a) Apply the fourth order Runge–Kutta method, to find an approximate value of   when   = 0.2      given that            (b) Write a C program to find the value of  using Simson’s 1/3 rule.

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## Loyola College B.Sc. Mathematics April 2012 Modern Algebra Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

FOURTH SEMESTER – APRIL 2012

# MT 4502 – MODERN ALGEBRA

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

Time : 1:00 – 4:00

PART – A

ANSWER ALL THE QUESTIONS:                                                                     (10 x 2 = 20 marks)

1. Define an equivalence relation.
2. Show that if every element of the group G is its own inverse then G is abelian.
3. Show that every cyclic group is abelian.
4. Define a normal subgroup of a group
5. Define the kernel of a homomorphism of a group.
6. Express (1,3,5) (5,4,3,2) (5,6,7,8) as a product of disjoint cycles.
7. If A is an ideal of a ring R with unity and 1A show that A=R.
8. If F is a field show that its only ideals are and F itself.
9. Define a maximal ideal.
10. Show that every field is a principal ideal domain.

PART – B

ANSWER ANY FIVE QUESTIONS:                                                                    (5 x 8 = 40 marks)

1. Show that a non empty subset H of a group G is a subgroup of G if and only if a,bH implies that ab-1
2. Show that every subgroup of a cyclic group is cyclic.
3. State and prove Langrange’s theorem.
4. Show that a subgroup N of a group G is a normal subgroup of G if and only if every left coset of N in G is a right coset of N in G.
5. State and prove Cayley’s theorem.
6. Show that every finite integral domain is a field.
7. Show that every Euclidean ring is a principal ideal domain.
8. Let R be a Euclidean ring. Show that any two elements a and b in R have a greatest common divisor d which can be expressed in the form d=λa+mb for l, m in R.

PART –C

ANSWER ANY TWO QUESTIONS:                                                                    (2 X 20 = 40 Marks)

1. (i) If H and K are finite subgroup of a group G, show that

(ii) Show that a group G cannot be the union of two proper subgroups.                               (12+8)

1. (i) If G is a group and N is a normal subgroup of G, show that G/N, the set of all distinct left

cosets of N in G is also a group.

(ii) If H is the only subgroup of order o (H) in the group G, show that H is normal in G.       (12+8)

1. (i) If H and N are subgroups of a group G and suppose that N is normal in G, show that is

isomorphic to .

(ii) If R is a commutative ring with unit element whose only ideals are (O) and R itself, show that

R is a field.                                                                                                                        (10+10)

1. (i) Show that an ideal of the Euclidean ring R is a maximal ideal if and only if it is generated by a

prime element of R.

(ii) Show that the characteristic of an integral is either zero or a prime number.                    (14+6)

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## Loyola College B.Sc. Mathematics April 2012 Mechanics – II Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

SIXTH SEMESTER – APRIL 2012

# MT 6604/MT 5500 – MECHANICS – II

Date : 18-04-2012              Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

PART – A

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

1. State the conditions for non-existence of centre of gravity.
2. Mention the differences between center of gravity and centre of mass.
3. Define the work done by the tension in an elastic string.
4. Define Suspension bridge.
5. A particle executing simple harmonic motion makes 100 complete oscillations per minute and its maximum speed is 15 ft/sec. What is the length of its path and maximum acceleration?
6. Define Centripetal force.
7. Define Central Orbit.
8. An insect crawl at a constant rate u along the spoke of a earth wheel of radius a starting from centre, the wheel moving with velocity v. Find the accelerations along and perpendicular to the spoke.
9. State Parallel axis theorem.
10. Write down the formula for Moment of Inertia of a solid sphere and hollow sphere.

PART –B

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

1. Find the Centre of gravity of a uniform circular angle.
2. Derive the intrinsic equation of the catenary.
3. A particle executing simple harmonic motion in a straight line has velocities 8,7,4 at three points distant one foot from each other. Find the period.
4. ABCD is a trapezium in which AB and CD are parallel and of lengths a and b. Prove that the distance of the centre of mass from AB is where h is the distance between

AB and CD.

1. A particle describes the orbit rn=Acos-Bsin under a central force, the pole being the centre. Find the law of force.
2. Derive the p-r equation of a central orbit.
3. Find the moment of Inertia of a thin uniform parabolic lamina bounded by the parabola y2=4a(hx) about the y-axis.
4. State and prove D’Alemberts Principle.

PART –C

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

1. a) Find the centre of gravity of a hollow hemisphere.
2. b) Find the centre of gravity of the arc of the cardiod r = a(1+cosθ) lying above the initial

line.                                                                                                                                                   (8+12)

1. State and prove Principle of virtual work for a system of coplanar forces acting on a body.

1. a) Find the resultant of two simple harmonic motions of the same period in the same

straight line.

1. b) The speed v of a particle moving along the x-axis is given by

Show that the motion is simple harmonic with centre at x = 4b and amplitude 2b.

Find the time from x = 5b to x = 6b.                                                                                      (12+8)

1. a) If the law of acceleration is and the particle is projected from an apse

at a distance c with velocity , prove that the equation of the orbit is

1. b) Find the moment of inertia of an elliptic lamina.            (10 + 10)

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## Loyola College B.Sc. Mathematics April 2012 Mechanics – I Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

FIFTH SEMESTER – APRIL 2012

# MT 5506/MT 4501 – MECHANICS – I

Date : 27-04-2012              Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

PART – A

ANSWER ALL QUESTIONS:                                                                                (10 X 2 = 20 marks)

1. State the law of parallelogram of forces.
2. What is the magnitude and direction of the resultant of two forces and when and  are equal in magnitude and the angle between them is 60°?
3. Define torque of a force.
4. State any two laws of friction.
5. Define angular velocity.
6. State the triangle law of velocities.
7. State the principal of conservation of linear momentum.
8. A body of mass 50 gm is acted upon by a constant force F=100 dynes. Find the time required to move the body through a distance of 25 cms from rest.
9. Define time of flight of a projectile.
10. State Newton’s experimental law on impact.

PART – B

ANSWER ANY FIVE QUESTIONS:                                                              (5 X 8 = 40 marks)

1. State and prove Lami’s theorem.
2. A uniform plane lamina in the form of a rhombus one of whose angles is 120° is supported by two forces of magnitudes P and Q applied at the center in the directions of the diagonals so that one side is horizontal. Show that if P>Q, then P2=3Q
3. State and prove varignon’s theorem on moments.
4. Two unlike parallel forces P and Q (P>Q) act at A and B respectively. Show that if the direction of P be reversed, the resultant is displaced through the distance .
5. A and B describe circles of radii ‘a’ and ‘b’ with speeds u and v, the motion being the same way round. If the angular velocity of either w.r.t. the other is zero, prove that the line joining them subtends at the center an angle whose cosine is
6. A train goes from one station to another moving during the first part of the journey with uniform acceleration ‘’ when the steam is shut off and brakes are applied, it moves with uniform retardation ‘’. if ‘a’ be the distance between the two stations, show that the time taken is .
7. If V1 and V2 be the velocities at the ends of a focal chord of a projectile’s path and ‘u’ the horizontal component of the velocity, show that .
8. A ball impinges directly on exactly equal and similar ball B lying on a smooth horizontal table. If e is the coefficient of restitution, prove that after impact, the velocity of B to that A is (1+e) : (1-e).

PART – C

ANSWER ANY TWO QUESTIONS:                                                               (2 X 20 = 40 marks)

1. a) Two weights P and Q are  suspended from a fixed point O by strings OA and OB and are kept apart by a

light rod AB. If the strings OA and OB make angle a and b with the rod, show that the angle q which

the  rod makes with the vertical is given by

1. b) 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 displaces though the

distance

1. a) A non uniform rod AD rests on two supports B and C at the same level where AB=BC=CD. If a weight p

is hung from A or a weight q is hung from D, the rod just tills. Show that weight of the rod is p+q and

that the centre of gravity of the rod divides AD in the ratio 2p+q : p+2q.

1. b) A uniform ladder rests at the angle 45° with its upper extremity against a rough vertical wall and its

lower extremity on the ground. If m and m1 be the coefficients of friction between the ladder and the

ground and the wall respectively, show then the least extremity towards the wall is

where W is the weight of the ladder.

1. a) Two particles of masses m1 and m2 (m1>m2) are connected by means of a light inextensible string

passing  over a light, smooth, fixed pully. Discuss the motion.

1. b) A particles of mass m is projected vertically under gravity, the resistance of air being ‘mk’ times

velocity.  Show that the greatest height attained by particle is where V is the

termind velocity of the particle and λV is its initial velocity.

1. a) Derive the equation of the projectile in the from .
2. b) Two smooth spheres of masses m1 and m2 moving with velocities u1 and u2 respectively in the direction

of the line of centers impinge directly. Discuss the motion.                                                         (10 + 10)

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## Loyola College B.Sc. Mathematics April 2012 Mathematics For Computer Science Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

SECOND SEMESTER – APRIL 2012

# MT 2100 – MATHEMATICS FOR COMPUTER SCIENCE

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

Time : 9:00 – 12:00

PART A

Answer ALL the questions:                                                                                  10×2 = 20

1. Define symmetric matrix with an example.
2. Prove that.
3. Remove the fractional coefficients from the equation
4. Find the partial differential coefficients of .
5. Evaluate.
6. Evaluate
7. Solve the equation = 0.
8. Derive the partial differential equation by eliminating the arbitrary constants from .
9. Find an iterative formula to , where N is a positive integer.
10. Write Simpson’s

PART B

Answer any FIVE questions:                                                                                          5×8 = 40

1. Show that the equations  are consistent and solve them.
2. Prove that
3. Find the condition that the roots of the equation may be in geometric progression.
4. Integrate with respect to x.
5. (i) Evaluate

(ii) Prove that                                                                     (4 + 4)

1. Solve the equation
2. Solve (i) (ii)                                                         (4 + 4)
3. Determine the root of correct to three decimals using, Regula Falsi method.

PART C

Answer any TWO questions:                                                                                       2×20 = 40

1. (i) Find all the characteristic roots and the associated characteristic vectors of the matrix

A =.

(ii) If  then prove that               (14+6)

1. (i) Solve the equation

(ii)  If  , prove that .                                            (14+6)

1. (i) Integrate with respect to x.

(ii) Solve                                                             (6+14)

1. (i) Solve

(ii) Evaluate  using trapezoidal rule and Simpson’s rule.                                    (8+12)

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## Loyola College B.Sc. Mathematics April 2012 Mathematical Statistics Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHS & PHYSICS

FOURTH SEMESTER – APRIL 2012

# ST 4206/4201 – MATHEMATICAL STATISTICS

Date : 19-04-2012              Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

SECTION A

Answer all the questions.                                                                                          10 X 2 = 20

1. Define random experiment.
2. What are independent events?
3. If Var(X) = 4, find Var(3X + 8).
4. Define continuous uniform distribution and write its mean and variance.
1. State any two applications of t-test.
1. Write down the mean and variance of Binomial distribution.
2. Define exponential distribution.
3. Write any two properties of regression coefficients.
4. What is null hypothesis?
5. Define critical region.

SECTION B

Answer any five questions.                                                                                       5 X 8 = 40

1. An urn contains 6 white, 4 red and 9 black balls. If 3 balls are drawn at random, find the probability that
2. Two of the balls drawn are white.
3. One ball of each colour is drawn.
• None is red.
1. At least one is white
2. If p1 =P(A), p2 =P(B) and p3 =P(AÇB), (p1, p2,p3 >0); express the following in terms of p1, p2 and p3. ,  P(A/B),  and
3. A random variable X has the following probability function:

x              :               0              1              2              3              4              5              6              7

p(x)        :               0              k              2k           2k           3k           k2            2k2          7k2 + k

Find k, evaluate P(X<6).

1. State any four properties of Distribution function.
2. Find mean and variance of Poisson distribution.
3. Derive the rth order moments of Rectangular distribution and hence find standard deviation.
4. Obtain the line of regression of Y on X for the following data:

X:            65           66           67           67           68           69           70           72

Y:            67           68           65           68           72           72           69           71

1. What are the steps involved in solving testing of hypothesis problem?

(PTO)

SECTION C

Answer any two questions.                                                                                       2 X 20 = 40

1. a) State and prove addition theorem of probability.
2. b) Sixty percent of the employees of XYZ Corporation are college graduates. Of these, ten are in sales. What is the probability that
3. An employee selected at random is in sales?
4. An employee selected at random is neither in sales nor a college graduate? (10+5+5)
5. The joint probability density function of a two-dimensional random variable (X,Y) is given by:
6. Verify that whether f(x,y) is a joint p.d.f.
7. Find the marginal density functions of X and of Y
8. Find the conditional density function of Y given X=x and conditional density function of X given Y=y.
9. Check for independence of X and Y.        (5+6+6+3)
10. a) A manufacturer, who produces medicine bottles, finds that 0.1 % of the bottles are defective. The bottles are packed in boxes containing 500 bottles. A drug manufacturer buys 100 boxes from the producer of bottles. Using Poisson distribution, find how many boxes will contain:
11. no defective.
12. atleast two defectives. (5+5)
13. Derive mean and variance of Beta distribution of first kind. (10)
14. Derive the p.d.f. of the F-statistic with (n1, n2) degrees of freedom. (20)

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## Loyola College B.Sc. Mathematics April 2012 Linear Algebra Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

FIFTH SEMESTER – APRIL 2012

# MT 5508/MT 5502 – LINEAR ALGEBRA

Date : 03-05-2012              Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

PART – A

Answer ALL questions:                                                                                          (10 x2 =20 marks)

1. If V  is a vector over a field F, Show that (-a)v = a(-v)= -(av), for a  F, v
2. Is the Union of two subspaces is a subspace?
3. Show that the vectors (1,0,-1), (2,1,3),(-1,0,0) and (1,0,1) are linearly dependent in .
4. Determine the following mapping is a vector space homomorphism: T :   by T(a,b)=ab.
5. Define inner product space.
6. Define orthonormal set in an inner product space.
7. Prove that is orthogonal.
8. For A,B Fn and then prove that tr (A+B) = tr A + tr B.
9. Define Hermitian and skew-Hermitian.
10. Find the rank of the matrix over field of rational numbers.

PART – B

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

1. Prove that a non empty subset W of a vector space V over F is a subspace of V if and only if aw1+bw2 W  , for all a,b F , w1,w2
2. If v1,v2,. . .,vn V are linearly independent , and if v V is not in their linear span, Prove that  v1,v2,. . .,vn  are linearly independent.
3. Find the coordinate vector of (2,1,-6) of R3 relative to the basis {(1,1,2),(3,-1,0),(2,0,-1)}.
4. Prove that T :   defined by T(a,b) = (a-b, b-a,-a) for all a,b is a vector space homomorphism.
5. State and Prove Schwarz inequality.
6. If is an eigen value of T  A(v), then for any polynomial f(x)  F[x] , f() is an eigen value of f(T).
7. If A and B are Hermitian , Show that AB + BA is Hermitian and AB-BA is skew-Hermitian.
8. Show that the system of equations x1+2x2+x3 = 11, 4x1+6x2+5x3 = 8,  2x1+2x2+3x3 = 19 is inconsistent.

PART – C

Answer any TWO questions:                                                                                         (2 x20 =40 marks)

1. (a) Let V be a vector space of finite dimension and let  W1 and W2 be subspaces of V such that

V = W1+W2 and dim V = dim W1+dim W2. Then  prove that V = W1W2.                                         (10 + 10)

(b) If A and B are subspaces of a vector space V over F , Prove that

1. (a) If V is a finite dimensional inner product space and w is a sub space of V, prove that .

(b) Show that is invertible if and only if the constant term of the minimal polynomial

for T is  not zero.                                                                                                              (10 + 10)

21.(a) If are distinct eigen values of   and if v1,v2,. . .,vn are eigen vectors of

T belonging to   respectively, then v1,v2,. . .,vn are linearly independent over F.

(b) If A,B Fn, where F is the complex field, then

(i) ,    (ii)  , ,    (iii) ,    (iv) .

(10 + 10)

1. (a) 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.

(b) (i) If  is skew-Hermitian , Prove that all of its eigen  values are pure imaginaries.

(ii) Prove that the eigen values of a unitary transformation are all of absolute value one.

(10 + 10)

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## Loyola College B.Sc. Mathematics April 2012 Graphs, Diff. Equ., Matrices & Fourier Series Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

FIRST SEMESTER – APRIL 2012

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

Date : 02-05-2012              Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

PART – A

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

1. Find the range of the following functions

(a) Let defined by f(x)  = x2. (b) Let be any constant function.

1. Find the equation of the line passing through (-3,4) and (1,6).
2. Write the normal equation of y = ax+b.
3. Reduce y = aebx to normal form.
4. Define Difference equation with an example.
5. Solve yx+2-4yx=0.
6. State Cayley Hamilton theorem.
7. Find the eigen value of the matrix .
8. Find the Fourier coefficient a0 for the function f(x) = ex in (0,2π).
9. Define odd and even function.

PART – B

Answer any FIVE questions:                                                                                     (5 X 8 = 40 Marks)

1. A company has a total cost function represented by the equation y = 2x3-3x2+12x, where y represents cost and x represent quantity.

(i) what is the equation for the Marginal cost function?

(ii)What is the equation for average cost function? What point average cost is at its minimum?

1. The total cost in Rs.of output x is given by C = Find
• Cost when output is 4 units.
• Average cost of output of 10 units.
• Marginal cost when output is 3 units.

1. Fit a straight line to the following data
 X: 0 5 10 15 20 25 Y: 12 15 17 22 24 30

Estimate the value of  Y corresponding to X =6.

1. Solve yx+2 – 5yx+1+6yx = x2+x+1.
2. Find the eigen vectors of the matrix A = .
3. Verify Cayley Hamilton theorem for the matrix A =
4. Expand f(x) = x (0 <x<2 π) as a Fourier series with period 2 π.

1. If f(x) = x  in the range (0,π)

=   0 in the range (π, 2π).  Find Fourier series of f(x) of periodicity 2 π.

PART – C

Answer any TWO questions:                                                                                 (2 X 20=40 Marks)

1. (a) Fit a second degree parabola by taking xi as the independent variable.
 X: 0 1 2 3 4 Y: 1 5 10 22 38

(b)  The total profit y in rupees of a drug company from the manufacture and sale of x drug bottles is

given by  . (i) How many drug bottles must the company sell to achieve the

maximum profit? (ii) What is the profit per drug bottle when this maximum is achieved?         (10 +10)

1. (a) Solve yx+2 – 7yx+1 – 8yx = x(x-1) 2x.

(b) Solve u(x+1) – au(x) = cosnx.                                                                                       (10+10)

1. (a) Find the Fourier series of period 2 π for f(x) = x2 in (o,2 π) . Deduce

(b)  Expand   in (0,2 π) as Fourier series of period 2 π.                                                 (10+10)

1. Determine the Characteristic roots and corresponding vectors for the matrix

.Hence diagonalise A.

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## Loyola College B.Sc. Mathematics April 2012 Graph Theory Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

FIFTH SEMESTER – APRIL 2012

# MT 5408 – GRAPH THEORY

Date : 27-04-2012              Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

PART A

Answer ALL the questions                                                                                          (10 x 2 =20)

1. Define a complete bipartite graph.
2. Prove that every cubic graph has an even number of points.
3. If G1 = K2 and G2 = C3 then find
• G1G2 (ii) G1 + G2
1. Define distance between any two points of a graph.
2. Define an Eulerian graph and give an example.
3. Prove that every Hamiltonian graph is 2-connected.
4. Draw all possible trees with 6 vertices.
5. Define an eccentricity of a vertex v in a connected graph G.
7. Find the chromatic number for the following graph.

PART B

Answer any FIVE questions                                                                                                    (5 x 8 =40)

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) Prove that                                                                                      (5+3)

1. If Let be a graph and  be a  graph then prove that

(i)  is a  graph.

(ii)  is a  graph.

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

(b) If a graph G is not connected then prove that the graph  is connected.                   (5+3)

1. (a) 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.

(b) Prove that every non – trivial connected graphs has atleast two points which are not cut points.                                                                                                                              (5+3)

1. If G is a graph with vertices and , then prove that G is Hamiltonian.
2. (a) Prove that every tree has a centre consisting of either one point or two adjacent points.

(b) Let T be a spanning tree of a connected graph. Let x = uv be an edge of G not in T. Then   prove that T + x contains a cycle.                                                                                  (4+4)

1. State and prove Euler’s theorem.
2. Prove that every planar graph is 5-colourable.

PART C

Answer any TWO questions                                                                                    (2 x 20 =40)

1. (a) Prove that the maximum number of lines among all p point graphs with no triangles is .

(b) Let  be a  graph then prove that .                                              (15 +5)

1. (a) Prove that a graph G with atleast two points is bipartite if and only if all its cycles are of even length.

(b) Let G be a connected graph with atleast three points then prove that G is a block if and only if any two points of G lie on a common cycle.                                                    (12+8)

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

(i) G is eulerian.

(ii) Every point of G has even degree.

(iii) The set of edges of G can be partitioned into cycles.

(b) Show that the Petersen graph is nonhamiltonian.                                                       (12+8)

1. (a) Let be a graph then prove that the following statements are equivalent

(i) G is a tree.

(ii) Every two points of G are joined by a unique path.

(iii) G is connected and .

(iv) G is acyclic and .

(b) Prove that every uniquely n – colourable graph is (n – 1) connected.             (14+6)

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## Loyola College B.Sc. Mathematics Nov 2012 Physics For Mathematics – I Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

FIRST SEMESTER – NOVEMBER 2012

# PH 1101 – PHYSICS FOR MATHEMATICS – I

Date : 03/11/2012             Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

PART – A

1. Draw distance – time and velocity – time graph for a particle moving with constant velocity.
2. What are generalized co-ordinates in Lagrangian formulation?
3. State Newton’s law of gravitation.
4. Calculate the mass of the earth from the following data. Radius of the earth = 6371 km
G=6.66 x 10-11 Nm2/kg2
5. State Hooke’s law of elasticity.
6. Define surface tension. Give its unit.
7. Draw a circuit diagram for the given function : F(A,B,C) = AB +
8. State two postulates of special theory of relativity.
9. What is meant by CMRR in op-amp?
10. The mean life of π meson is 2 x 10-8 Calculate the mean life of a meson moving with the velocity of 0.8c.

PART – B

1. Define centripetal force. Derive an expression for it.
2. Define escape velocity. Show that the escape velocity from the surface of the earth is 11km/s.
3. Derive Poiseuille’s formula for determining the coefficient of viscosity of a liquid.
4. With a neat circuit diagram and truth table, explain the working of a full adder.
5. Derive the Lorentz space-time transformation formulae.

PART – C

1. Obtain the relation between the three moduli of elasticity.
2. Setup and solve Lagrange’s equation for i) Simple Pendulum ii) Atwood’s machine
3. a) State the postulates of general theory of relativity. (4)
b) Show that the effect of gravitation is to introduce a curvature in space-time manifold. (8.5)
4. a) With a neat circuit diagram, explain the working of an op-amp inverting amplifier. (8.5)
b) Simplify using K-map : Y = F (A,B,C) = ∑ ( 1,2,3,5,7) (4)
5. Describe the Michelson –Morley experiment and explain the physical significance of negative results.

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## Loyola College B.Sc. Mathematics Nov 2012 Operations Research Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

FIFTH SEMESTER – NOVEMBER 2012

# MT 5507/MT 5504 – OPERATIONS RESEARCH

Date : 06/11/2012             Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

PART – A

Answer any ALL questions:                                                                                    (10 x 2 = 20 Marks)

1. Define the following: (i) Basic solution (ii) Basic feasible solution
2. Express the following linear programming problem into standard form:

Maximize

Subject to

3

1. What is a transportation problem?
2. Give the mathematical formulation of an assignment problem.
3. Define a pure strategy in game theory.
5. Define a spanning tree in a network.
6. Define a critical path in a network.
7. What is the Economic order quantity?
8. Differentiate the deterministic and the probabilistic demand inventory models.

PART – B

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

1. Use the graphical method to solve the following linear programming problem.

Minimize

Subject to

1. Solve the following LPP by dual simplex method.

Maximize

Subject to

1. Determine an initial basic feasible solution to the following transportation problem by Vogel Approximation Method.

Available

Requirement     6      10    12    15

1. Solve the following assignment problem.

Jobs

I         II       III

Men

1. Solve the following game graphically

1. A project consists of a series of tables labeled A, B, …, H, I with the following relationships (W < X, Y means X&Y cannot start until W is completed; X, Y < W means W cannot start until both X&Y are completed). With this notations construct the network diagram having the following constraints: A < D, E; B, D < F; C < G; B < H; F, G < I

1. Determine the critical path of the following network.

1. A particular item has a demand of quantity 9000 units/year. The cost of the one procurement is Rs.100 and the holding cost per unit is Rs.2.40 per year. The replacement is instantaneous and no shortages are allowed. Determine
• the economic lot size
• the number of orders per year
• the time between orders

PART – C

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

1. a) Find the optimal solution for the following transportation problem using MODI method.

D1              D2               D3               D4       Supply

 S1 19 30 50 10 7 S2 70 30 40 60 9 S3 40 8 70 20 18 Demand 5 8 7 14 34

1. Use the penalty (Big-M) method to solve the following LP problem.  (10)

Minimize

Subject to

1. a) Define the Total float, free float and Independent float.                               (6)

1. b) The following indicates the details of the activities of a project.

The durations are in days.                                                                                                            (14)

 Activities TO TM TP 1 – 2 4 5 6 1 – 3 8 9 11 1 – 4 6 8 12 2 – 4 2 4 6 2 – 5 3 4 6 3 – 4 2 3 4 4 – 5 3 5 8
• Draw the network
• Find the critical path
• Find the mean and standard deviation of the project completion time

1. a) Reduce the following game to game and hence find the optimum strategies and the value

of the game.                                                                                                                  (12)

Player B

 I II III IV I 3 2 4 0 II 3 4 2 4 III 4 2 4 0 IV 0 4 0 8

Player A

1. b) Solve the following unbalanced assignment problem of minimizing total time for doing all the jobs.                                        (8)
 Jobs Operators 1 2 3 4 5 1 6 2 5 2 6 2 2 5 8 7 7 3 7 8 6 9 8 4 6 2 3 4 5 5 9 3 8 9 7 6 4 7 4 6 8

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## Loyola College B.Sc. Mathematics Nov 2012 Multimedia Technologies Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc., B.A.,B.COM. DEGREE EXAMINATION – MATHS, ECO.,SOCI.,ENGL., & COMM

THIRD SEMESTER – NOVEMBER 2012

# CS 3206/CA3206 – MULTIMEDIA TECHNOLOGIES

Date : 09/11/2012             Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

Part – A

Answer ALL the questions:                                                                                                     (10 * 2 = 20)

• Define Multimedia.
• What is synchronization?
• What is meant Nyquist frequency?
• Define the term: Flat Noise.
• State the objective of JPEG.
• What are the three key parameters need to be considered for evaluating a compression system?
• What is Multimedia system service?
• Draw the diagram for Single server and Co-servers.
• Define MMV.
• What is meant by Authoring systems?

Part – B

Answer ALL the questions:                                                                                                         (5 * 8 = 40)

• a) Briefly explain the multimedia and personalized computing.

(Or)

1. b) Write a short note on QOS architecture.

• a) Give a brief account on MIDI

(Or)

1. b) Give a detail note on Three-sensor and single-sensor color video camera.

• a) Write a short note on Temporal Access control.

(Or)

1. b) Discuss briefly about temporal transformations.

• a) Describe the goals of Multimedia system services.

(Or

1. b) Explain about Media stream protocol.

• a) Explain about the issues facing the design of multimedia information.

(Or)

1. b) With a neat sketch, explain the architecture of HyOctane and Hy-time engine.

Part – C

Answer any TWO:                                                                                                          (2 * 20 = 40)

• i) Explain the detail about uses of multimedia in geographical information systems and educational areas.(10)

1. ii) Discuss in detail about analog video artifacts.(10)

• i) List out and explain any two Video Compression techniques.(10)

1. ii) Explain in detail about the purpose of toolkits. (10)

• i) Discuss in detail about the Current state of the Industry.(10)

1. ii) Explain in detail the framework for Multimedia systems. (10)

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## Loyola College B.Sc. Mathematics Nov 2012 Modern Algebra Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

FOURTH SEMESTER – NOVEMBER 2012

MT 4502/4500 – MODERN ALGEBRA

Date: 03-11-2012                       Dept. No.                                                       Max. : 100 Marks

Time: 1.00 – 4.00

SECTION – A                                       (10 ´ 2 = 20)

1. Define partially ordered set and give an example.
2. Show that identity element of a group is unique.
3. Let and . Prove that divides .
4. Show that any group of order up to 5 is abelian.
5. Define kernel of a homomorphism.
6. State fundamental theorem of homomorphism.
7. Define a ring and give an example.
8. If is a ring with unit element , then for all , show that .
9. State unique factorization theorem.
10. If is a commutative ring with unity, prove that every maximum ideal of  is a prime ideal.

PART – B ( 5 ´ 8 = 40)

1. If is a group in which  for three consecutive integers  for all , show that  is abelian.
2. Show that a subgroup of a group  is a normal subgroup of  if and only if every left coset of  in  is a right coset of in .
3. `Prove that every group of prime order is cyclic.
4. State and prove Cayley’s theorem.
5. Show that any two finite cyclic groups of the same order are isomorphic.
6. Define a subring of a ring. Show that the intersection of two subrings of a ring  is a subring of .
7. Show that every finite integral domain is a field.
8. Show that every Euclidean ring is a principal ideal domain.

PART – C (2 ´ 20 = 40)

1. (a) State and prove the fundamental theorem of arithmetic.

(b)  Show that a nonempty subset  of a group  is a subgroup of  if and only if implies .

1. (a) State and prove the Lagrange’s theorem.

(b)   Let  be a commutative ring with unit element whose only ideals are  and  itself.  Show that  is a field.

1. (a) Determine which of the following are even permutations:
• (ii)

(b)   If  is a group, then show that , the set of automorphisms of , is also a group.

1. (a) Show that an ideal of the Euclidean ring  is a maximal ideal of  if and only if it is generated by a prime element of .

(b)   Show that , the set of all Gaussian integer, is a Euclidean ring.

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## Loyola College B.Sc. Mathematics Nov 2012 Mechanics – II Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc., DEGREE EXAMINATION – MATHEMATICS

SIXTH SEMESTER – NOVEMBER 2012

# MT 6604 / MT 5500 – MECHANICS – II

Date :5/11/2012                Dept. No.                                        Max. : 100 Marks

Time :1.00 – 4.00

PART – A

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

1. State the conditions for non-existence of centre of gravity.
2. Write down the co-ordinates of center of gravity for a solid cone.
3. Define catenary.
4. Define suspension bridge.
5. A particle executing simple harmonic motion makes 100 complete oscillations per minute and its maximum speed is 15 feet per sec. What is the length of its path and maximum acceleration?
1. Write down any two applications of simple harmonic motion.
2. A point P describes with a constant angular velocity about O the equiangular

spiral r = a eθ. O being the pole of the spiral. Obtain the radial and transverse

acceleration of P.

1. Define central orbit.
2. Define Moment of Inertia of a particle about a straight line.
3. Find the work that must be done on a uniform flywheel of mass 50 lbs and radius

6״  to increase its speed of rotation from 5 to 10 rotation per second.

PART –B

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

1. Find the center of gravity of a hollow right circular cone of height h.
2. Prove that if a dynamical system is in equilibrium, then the work done by the

applied forces in a virtual displacement is zero.

1. Discuss the motion of a particle executing two simple harmonic motions in

perpendicular directions with same period.

1. A square hole is punched out of a circular lamina of radius as its diagonal. Show

that the distance of Centre of gravity of the remainder from the centre of the circle

is  a/(4π-2).

1. Derive the pedal p-r equation of a central orbit.
2. If the law of acceleration is and the particle is projected from an

apse at a distance c with velocity , prove that the equation of the orbit is

.

1. Find the moment of Inertia of a thin uniform parabolic lamina bounded by the

parabola  and y axis about the y-axis.

1. Derive the equation of motion of a rigid body about a fixed axis.

PART –C

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

1. (a) From a solid cylinder of height h, a cone whose base coincides with the base

of the cylinder is scooped out so that the mass centre of the remaining solid

coincides with the vertex of the cone. Find the height of the cone.

(b) Find the centre of gravity of the arc of the cardiod r=a(1+cosθ) lying above

the initial line.                                                                                                             (10 + 10)

1. (a) Derive the equation to the common catenary in the form y = C cosh x/c.

(b) A chain of length 2l is to be suspended from two points  A and B in the same

horizontal level so that either terminal tension is n times that at the lowest

point. Show that the span AB must be

• + 10)
1. (a) A particle executing simple harmonic motion in a straight line has velocities

8,7,4 at three points distant one foot from each other. Find the period.

(b) Find the resultant of two simple harmonic motions of the same period in the

same straight line.                                                                                                  (10 + 10)

1. (a) State and prove Perpendicular axis theorem

(b) Show that the moment of inertia of the part of the parabola  cut off

by the double ordinate  is about the tangent at the vertex and

about its axis, M being the mass.                                                               (6 + 14)

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## Loyola College B.Sc. Mathematics Nov 2012 Mechanics – I Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

FIFTH SEMESTER – NOVEMBER 2012

# MT 5506/MT 4501 – MECHANICS – I

Date : 03/11/2012             Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

PART – A

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

1. State the conditions for equilibrium of a system of concurrent forces.
2. State the law of parallelogram of forces.
3. Define torque of a force.
4. State any two laws of friction.
5. Find the resultant of two velocities 6 mt/sec and 8 mt/sec inclined to each other at an angle

of 30.

1. Define angular velocity.
2. Define momentum.
3. State the principle of conservation of linear momentum.
4. Define range of flight for a projectile.
5. Define the coefficient of elasticity.

PART – B

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

1. State and prove Lami’s theorem.

1. A uniform plane lamina in the form of a rhombus one of whose angles is 120° is supported by two forces of magnititudes P and Q applied at the centre in the directions of the diagonals so that one side is horizontal. Show that if P > Q, then P2 = 3Q2.

1. State and prove Varignon’s theorem on moments.

1. Two particles weighing 2 kg and 1 kg are placed on the equally rough slopes of a double inclined plane whose inclinations with the horizontal are 60° and 30° The particles are connected by a light string passing over a smooth pulley at the common vertex of planes.  If the heavier particle is on the point of slipping downwards, show that the coefficient of friction is

1. A particle is dropped from an aeroplane which is rising with acceleration f and t secs after this; another stone is dropped. Prove that the distance of between the stones at time t after the second stone is dropped is .
2. Two particles of masses m1 and m2 (m1 > m2) are connected by means of light inextensible string passing over a light, smooth, fixed pulley. Discuss the motion.

1. Show that when masses P and Q are connected by a string over the edge of a table, the tension is the same wheter P hangs and Q is on the table or Q hangs and P is on the table.

1. Two balls impinge directly and the interchange their velocities after impact. Show that they are perfectly elastic and of equal mass.

PART – C

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

1. a) Two strings AB and AC are knotted at A, where a weight W is attached. If the weight

hangs freely and in the position of equilibrium, with BC horizontal,

AB : BC : CA = 2 : 4 : 3, show that the tensions in the strings are

1. b) A system of forces in the plane of D ABC is equivalent to a single force at A, acting

along the internal bisector of the angle BAC and a couple of moment G1.  If the moments

of the system about B and C are respectively G2 and G3,  prove that (b+c) G1 = bG2 + cG3.

(10 + 10)

1. a) Two unlike parallel forces P and Q (P>Q) act at A and B respectively. Show that if the

direction of P be reversed, the resultant is displaced through the distance .

1. b) A particle moving in a st. line is subject to a resistance KV3 producing retardation

where v is the velocity.  Show that if v is the velocity at any time t when the distance is

s,  and  where u is the initial velocity.                              (10 + 10)

1. Derive the equation to the path of the projectile in the form

1. A particle falls under gravity in a medium where the resistance varies as the square of the velocity. Discuss the motion.

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## Loyola College B.Sc. Mathematics Nov 2012 Linear Algebra Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

FIFTH SEMESTER – NOVEMBER 2012

# MT 5508/MT 5502 – LINEAR ALGEBRA

Date : 08/11/2012             Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

PART – A

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

1. Define a vector space over a field F.
2. Show that the vectors (1,1) and (-3, 2) in R2 are linearly independent over R, the field of real numbers.
3. Define homomorphism of a vector space into itself.
4. Define rank and nullity of a vector space homomorphism T: u®
5. Define an orthonormal set.
6. Normalise in R3 relative to the standard inner product.
7. Define a skew symmetric matrix and give an example.
8. Show that is orthogonal.
9. Show that is unitary.
10. Define unitary linear transformation.

PART – B

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

1. Prove that the intersection of two subspaces of a vector space v is a subspace of V.
2. If S and T are subsets of a vector space V over F, then prove that
3. S T implies that L(S) ≤ L(T)
4. L(L(S)) = L(S)
5. L(S U T) = L(S) + L(T).
6. Determine whether the vectors (1,3,2), (1, -7, -8) and (2, 1, -1) in R3 are linearly dependent on independent over R.
7. If V is a vector space of finite dimension and W is a subspace of V, then prove that

dim V/W = dim V – dim W.

1. For any two vectors u, v in V, Prove that .
2. If and l Î F, then prove that l is an eigen value of T it and only if [l I – T] is singular.
3. Show that any square matrix can be expressed as the sum of a symmetric matrix and a skew symmetric matrix.
4. For what values of T, the system of equations over the rational field is consistent?

PART – C

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

1. a) Prove that the vector space V over F is a direct sum of two of its subspaces W1 and W2

if and only if V = W1 + W2 and W1  W2 = {0}.

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

W, than prove that dim V = dim U + dim W.                                                                   (10 + 10)

1. If U and V are vector spaces of dimension m and n respectively over F, then prove that the

vector space Hom (U, V) is of dimension mn.

1. Apply the Gram – Schmidt orthonormalization process to obtain an orthonormal basis for

the subspace of R4 generated by the vectors (1, 1, 0, 1) , (1, -2, 0, 0) and (1, 0, -1, 2).

1. a) Prove that TÎA(V) is singular if and only it there exists an element v ≠ 0 in V such that

T(v) = 0.

1. b) Prove that the linear transformation T on V is unitary of and only if it takes an

orthonormal basis of V onto an orthonormal basis of V.                                                       (10 +10)

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## Loyola College B.Sc. Mathematics Nov 2012 Graphs, Diff. Equ., Matrices & Fourier Series Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

FIRST SEMESTER – NOVEMBER 2012

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

Date : 10/11/2012               Dept. No.                                          Max. : 100 Marks

Time : 1:00 – 4:00

PART – A

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

1. Let f: be defined by.  Find the  range of the function .
2. Find the equation of the line passing through (-3,4) and (1,6).
3. Write the normal equation of y = a+bx.
4. Reduce y = aebx to normal form .
5. Define Difference equation with an example.
6. Solve yx+2 – 4yx=0.
7. State Cayley Hamilton theorem.
8. Find the eigen value of the matrix .
9. Find the Fourier coefficient a0 for the function f(x) = ex in (0,2π).
10. Define odd and even function.

PART – B

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

1. A company has a total cost function represented by the equation y = 2x3-3x2+12x, where y

represents cost and x represent quantity.  (i) what is the equation for the Marginal cost function?

(ii) What is the equation for average cost function? What point average cost is at its minimum?

1. The total cost in Rs.of output x is given by . Find  Cost when output is 4 units.

(i)   Average cost of output of 10 units.

(ii)  Marginal cost when output is 3 units.

1. Fit a straight line to the following data. Estimate the sale for 1977.
 Year: 1969 1970 1971 1972 1973 1974 1975 1976 Sales(lakhs) 38 40 65 72 69 60 87 95
1. Solve yx+2 – 4yx = 9x2.
2. Find the eigen vectors of the matrix .
3. Verify Cayley Hamilton theorem for the matrix A =
4. Expand f(x) = x (-π <x< π) as a Fourier series with period 2 π.
5. Obtain a Fourier series expansion for f(x) = – x  in the range (-π,0)

=      x   in the range [0, π).

PART – C

Answer any TWO  questions                                                                                                            (2 X 20=40 Marks)

1. (a) Fit a curve of the form y = a + bx +c x2 for the following table:
 X: 0 1 2 3 4 Y: 1 1.8 1.3 2.5 6.3

(b)  The total profit y in rupees of a drug company from the manufacture and sale of x drug

bottles is given by  .

(i) How many drug bottles must the company sell to achieve the maximum  profit?

(ii) What is the profit per drug bottle when this maximum is achieved ?                                                     (10+10)

1. (a) Solve yx+2 – 7yx+1 – 8yx = x(x-1) 2x.

(b) So lve u(x+1) – au(x) = cosnx.                                                                                                                                 (10+10)

1. (a) Find the Fourier series of period 2 π for  f(x) = x2 in (o,2 π) . Deduce

(b)  Find the Fourier (i) Cosine series (ii) Sine series for the function f(x) =  π-x  in (0, π).                        (10+10)

1. (a) Determine the Characteristic roots and corresponding vectors for the matrix

.Hence diagonalise A.

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