## Loyola College B.Sc. Mathematics April 2011 Real Analysis Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

FIFTH SEMESTER – APRIL 2011

# MT 5505/MT 5501 – REAL ANALYSIS

Date : 11-04-2011              Dept. No.                                                    Max. : 100 Marks

Time : 9:00 – 12:00

SECTION  A

Answer ALL questions.                                    (10 x 2 = 20)

1. State the least upper bound axiom.

1. Prove that any infinite set contains a countable subset.

1. Prove that the intersection of an arbitrary collection of open sets need not be open.

1. Distinguish between adherent and accumulation points.

1. Prove that any polynomial function is continuous at each point in .

1. Give an example of a continuous function which is not uniformly continuous.

1. State Rolle’s theorem.

1. If a real-valued function has a derivative at , prove that is continuous at .

1. Give an example of a sequence of real numbers whose limit inferior and limit superior exist, but the sequence is not convergent.

1. Give an example of a function which is not Riemann-Stieltjes integrable.

SECTION  B

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

1. State and prove Cauchy-Schwartz inequality.

1. Prove that the Cantor set is uncountable.

1. Prove that a subset E of a metric space is closed if and only if it contains all its adherent points.

1. Prove that a closed subset of a complete metric space is also complete.

1. State and prove Lagrange’s mean value theorem.

1. If a real-valued function is monotonic on , prove that the set of discontinuities of is countable.

1. If a real-valued function is continuous on , and if exists and is bounded in , prove that  is of bounded variation on .

1. State and prove integration by parts formula concerning Riemann-Stieltjes integration.

SECTION  C

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

 19. (a) Prove that the set of rational numbers is not order-complete. (b) Prove that the set of all rational numbers is countable. (c) State and prove Minkowski’s inequality.                                                          (10+5+5) 20. (a) Prove that every bounded and infinite subset of  has at least one accumulation point. (b) State and prove the Heine-Borel theorem.                                                       (16+4) 21. (a) Let  and  be metric spaces and . Show that  is continuous at  if and only if for every sequence  in X that converges to , the sequence  converges to . (b) Prove that a continuous function defined on a compact metric space is uniformly continuous.                                                                                                        (10 + 10) 22. (a) State and prove Taylor’s theorem. (b) Prove that a monotonic sequence of real numbers is convergent if and only if it is bounded.                                                                                                                (12+8)

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## Loyola College B.Sc. Mathematics April 2011 Modern Algebra Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

FOURTH SEMESTER – APRIL 2011

# MT 4502 – MODERN ALGEBRA

Date : 07-04-2011              Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

SECTION-A                                             (10X2=20)              Answer ALL the questions.

1. Let R be the set of all numbers. Define * by x*y=xy+1 for all x,y in R. Show that  is commutative but not associative.
2. Define a partially ordered set and give an example.
3. Show that the intersection of two normal subgroups is again a normal subgroup.
4. Give an example of an abelian group which is not cyclic.
5. Let G be the group of non-zero real numbers under multiplication. and f:GG  be defined

by f(x)=x for all xG. Is this map a homomorphism of G into G?  Justify.

1. If f is a homomorphism of a group G into a group G’ then prove that kernel of f is a

normal subgroup of G.

1. Prove that an element a in a Euclidean ring R is a unit if d(a)=d(1).

8 Let Z be the ring of integers. Give all the maximal ideals of  Z.

1. Show that every field is a principal ideal domain.
2. Find all the units in Z[i]={x +iy/x,y Z}

SECTION-B                                                             (5X8=40)

1. Prove that a non-empty subset H of a group G is a subgroup of G if and only if HH=H and H=H-1.
2. Let H be a subgroup of a group G. Then prove that any two left coset in G are either identical or have

no element in common.

1. Show that a subgroup N of a group G is a normal subgroup of G iff every left coset of N in G is a

right coset of N in G.

1. Prove that any group is isomorphic to a group of permutations.
2. Prove that an ideal of the Euclidean ring R is a maximal ideal of R if and only if it is generated by a

prime element of R.

1. Show that Qis a field under the usual addition and multiplication.
2. Let R be an Euclidean ring. Then prove that any two elements a and b in R have a greatest common

divisor   d   which can be expressed by  a + b.

1. Show that every finite integral domain is a field.

SECTION-C                                                       (2X20=40)

1. a) If H and K are finite subgroups of a group G then prove that  o(HK)= o(H)o( K)/o(H
2. b) Prove that every subgroup of a cyclic group is cyclic.                            (12+8)
3. a) Prove that there is a one-one correspondence between any two left cosets of a subgroup

H in G and thereby prove the Lagrange’s theorem.

1. b) State and prove Euler’s theorem and Fermat’s theorem.                                                         (10+10)
2. a) State and prove Fundamental homomorphism theorem for groups.
3. b) Let R be a commutative ring with unit element whose only ideals are (0) and R itself.

Prove that R is a field.                                                                                                               (12+8)

1. a) State and prove unique factorization theorem.
2. b) Let R be the ring of all real valued functions on the closed interval [0,1].

Let M={f R/   f(1/2)=0}. Show that M is a maximal ideal of R.                                            (10+10)

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## Loyola College B.Sc. Mathematics April 2011 Mechanics – II Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

SIXTH SEMESTER – APRIL 2011

# MT 6604/MT 5500 – MECHANICS – II

Date : 07-04-2011              Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

PART – A

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

1. What is the Centre of Gravity of a compound body?
2. Where does the C.G of a uniform hollow right circular cone lie?
3. Define virtual work.
4. What is common catenary?
5. Show that frequency is the reciprocal of the periodic time in a simple harmonic motion.
1. If the maximum velocity of a particle moving in a simple harmonic motion is

2ft/sec and its period is 1/5 sec, prove that the amplitude is  feet.

1. What is the pr equation of a parabola and an ellipse?
2. What are the radial and transverse components of acceleration?
3. Define moment of inertia?
4. Explain the conservation of angular momentum.

PART –B

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

1. A homogenous solid is formed of a hemisphere of radius r soldered to a right circular cylinder of

the same radius. If h be the height of the cylinder, show that  the center of gravity of the solid from

the common base is .

1. Find the center of gravity of a uniform trapezium lamina.
2. A uniform rod AB of length 2a  with one end A against a smooth vertical  wall being supported by

a string of length 2l, attached to the other end of the rod  B and to a point C of the wall vertically

above A.   Show that if the rod rests  inclined to the wall at an angle q, then  cos2 q =.

1. Derive the intrinsic equation of the common catenary.
2. A second pendulum is in a lift which is ascending with uniform acceleration . Find the number of seconds it will gain per hour. Calculate the loss if

the lift were descending with an acceleration of .

1. Show that the composition of two simple harmonic motions of the same period

along two perpendicular lines is an ellipse.

1. Prove that the areal velocity of a particle describing a central orbit is constant.

Also show that its linear velocity varies inversely as the perpendicular distance

from the centre upon the tangent at P.

1. Show that the Moment of inertia of a truncated cone about its axis, the radii of its

ends being a and b, (a<b) is .

PART –C

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

1. (a) Find the centre of gravity of the area in the first quadrant bounded by the co-

ordinate axes and the curve .

(b) AB and AC are two uniform rods of length 2a and 2b respectively. If

, prove that the distance from A of the Centre of gravity of two the

rods is                                                                              (10 + 10)

1. (a) Show that the length of a chain whose ends are tied together and hanging over

a circular pulley of radius a, so as to be in contact with two thirds of the

circumference of the pulley is a    .

(b) Derive the expression for velocity and acceleration of a particle moving on a

curve.                                                                                                                                (10 + 10)

1. (a) A particle P describes the orbit under a central force. Find the

law of force.

(b) The law of force is  and a particle is projected from an apse at a distance

Find the orbit when the velocity of projection is  .                                                (10 + 10)

1. (a) State and prove Parallel axis theorem.

(b) Find the lengths of the simple equivalent pendulum, for the following:

1. i) Circular wire ii) Circular disc. (10 + 10)

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## Loyola College B.Sc. Mathematics April 2011 Numerical Methods Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

SIXTH SEMESTER – APRIL 2011

# MT 6605 – NUMERICAL METHODS

Date : 09-04-2011              Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

PART – A

Answer ALL questions.                                                                                                 (10 ´ 2 = 20)

1. What is the condition of convergence for solving a system of linear equations by iteration procedure?
2. What do you mean by partial pivoting?
3. Explain the method of successive approximation.
4. What is the order of convergence in regula falsi method?
5. Write a short note on interpolation.
6. Write the Gregory-Newton backward interpolation formula.
7. State the relation between Bessel’s and Laplace-Everett’s formulae.
8. Write Bessel’s central difference interpolation formula.
9. What is the order of error in Simpson’s 1/3
10. Using Euler’s method, Solve y¢ = x + y, given y(0) = 1 for x = 0.2

PART – B

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

1. Using Gauss elimination method, solve the system

10x + y + z = 12,        2x + 10y + z = 13,      2x + 2y + 10z = 14

1. Find an approximate root of x log10 x – 1.2 = 0 by regula falsi method.
2. Find a real root of the equation cos x = 3x – 1 correct to 3 decimal places.
3. Find a polynomial which takes the following values and hence compute yx at x = 2, 12

x:         1          3          5          7          9          11

yx:        3          14        19        21        23        28

1. Obtain Newton’s divided difference formula for unequal intervals.
2. The population of a certain town (as obtained from census data) is shown in the following table. Find the rate of growth of the population in the year 1981.

Year:                           1951                1961                1971                1981                1991

Population:                 19.96               36.65               58.81               77.21               94.61

(in thousands)

1. Evaluate using (i) Simpson’s 1/3 rule and (ii) Simpson’s 3/8
2. Using Modified Euler method, find y(0.1), y(0.2) given

PART – C

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

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

10x – 5y – 2z = 3,                   4x – 10y + 3z = –3,                 x + 6y + 10z = –3

(b)  Find the positive root of f(x) = 2x2 – 3x – 6 = 0 by Newton-Raphson method correct to 3 decimal places.                                                                                                                                  ( 12 + 8)

1. (a) Using Lagrange’s formula of interpolation find y(9.5) given

x:         7          8          9          10

y:         3          1          1          9

(b)  The population of as town is a follows

Year     x:                                 1941       1951       1961          1971       1981     1991

Population in lakhs y:               20           24           29              36           46         51

(10 + 10)

1. The following table gives the values of the probability integral for certain values of x.  Find the values of this integral when x = 0.5437 using (i) Stirling’s formula (ii) Bessel’s formula and (iii) Laplace-Everett’s formula.

x:                           0.51                 0.52                 0.53                0.54                  0.55

y = f(x):           0.5292437       0.5378987       0.5464641       0.5549392       0.5633233

x:                           0.56                 0.57

y = f(x):           0.5716157       0.5798158

1. (a) Develop a C-program to implement Trapezoidal rule.

(b)  Using Runge-Kutta method of fourth order, solve given y(0) = 1 at
x = 0.2, 0.4                                                                                                               (8 + 12)

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## Loyola College B.Sc. Mathematics April 2011 Math 1 Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034.

1. Sc., DEGREE EXAMINATION – Mathematics

FIFTH SEMESTER

PART – A

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

1. Give any two application areas of a linear programming problem.
2. Define iso – profit and iso – cost lines.
3. Give the mathematical model of a transportation
4. What is opportunity cost in an assignment problem?
5. Define value of a game. When is a game said to be fair?
6. Give two examples of situations where game theory is used.
7. Define activity variance and project variance.
8. Define critical path of a network.
9. What is buffer inventory?
10. Define carrying cost of inventory.

PART – B

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

1. a) How do you find the dual of a linear programming problem.
2. Find the dual of the LPP : Minimize subject to the constraints , , , , , .
3. Obtain an initial feasible solution to the following transportation problem using the least cost rule –
 D1 D2 D3 Availability S1 1 2 6 7 S2 8 4 2 12 S3 3 7 5 11 Demand 10 10 10

1. Players A and B play a game in which each has three coins, a 5p,10p and a 20p. Each selects a coin without the knowledge of the other ’s choice. If the sum of the coins is an odd amount, then A wins B’s coin. But, if the sum is even, then B wins A’s coin. Find the best strategy for each player and the values of the game.

1. Draw the network for the following set of activities:

Activity :                         A     B         C         D         E          F          G         H         I

Immediate predecessor:  –       –           –           A       B,C        A         C      D,E,F      D

1. A company operating 50 weeks in a year is concerned about its stocks of copper cable. This costs Rs. 240 a meter and there is a demand for 8,000 meters a week. Each replenishment costs Rs. 1,050 for administration and Rs. 1,650 for delivery, while holding costs are estimated at 25 per cent of value held a year. Assuming no shortages are allowed, what is the optimal inventory policy for the company?
2. What are the major assumptions and limitations of a LPP? Discuss in detail.
3. The assignment costs of four operators to four machines are given in the following table:
 I II III IV A 10 5 13 15 B 3 9 18 3 C 10 7 3 2 D 5 11 9 7

Find the optimal assignment.

1. A factory requires 1,500 units of an item per month, each costing Rs. 27. The cost per order is Rs. 150 and the inventory carrying charges working out to 20 per cent of the average inventory. Find the economic order quantity and the number of orders per year. Would you accept a 2 per cent discount on a minimum supply quantity of 1,200 units? Compare the total costs in both the cases.

PART – C

Answer any TWO questions.                                                                                     (2 X 20= 40 marks)

1. a) Solve by simplex method: Maximize , , , , ,

1. b) Explain the graphical method of solving a LPP.

1. a) What is meant by unbalanced transportation problem? Explain the method of solving

such a problem.

1.  b) Solve the travelling salesman problem with the following cost matrix:
 City A City B City C City D City A ∞ 46 16 40 City B 41 ∞ 50 40 City C 82 32 ∞ 60 City D 40 40 36 ∞

1. a) Solve the following game using graphical method:
 B1 B2 A1 1 -3 A2 3 5 A3 -1 6 A4 4 1 A5 2 2 A6 -5 0
1. b) A project has the following data:
 Activity A      B      C      D      E     F      G     H to tm tp 4      8      4      1      2      4      10      18 5      12    5      3      2      5      14      20 6      16   12     5      2      6      18      34

A < C; B < D; A, D < E; B < F; C, E, F < G; G < H. (i) Draw the network, (ii) Find the critical path and the expected time of completion of the project, (iii) What is the probability that the project would completed in 60 days?

1. a) Explain the EOQ model with constant demand and variable order cycle time.

1. b) A contractor has to supply 10,000 bearings per day to an automobile manufacturer. He

can produce 25, 000 bearings per day. The holding cost is Rs. 2 per year and the set –

up cost is Rs. 180. How frequently should the production run be made?

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## Loyola College B.Sc. Mathematics April 2011 Mechanics – I Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

FIFTH SEMESTER – APRIL 2011

# MT 5506/MT 4501 – MECHANICS – I

Date : 18-04-2011              Dept. No.                                                    Max. : 100 Marks

Time : 9:00 – 12:00

PART – A

Answer ALL Questions                                                                                           (10 x 2 = 20 marks)

1. Define parallelogram of forces.
2. What are coplanar forces?
3. State the theorem on polygon of forces.
4. Define moment of a force.
5. State newton’s laws of motion.
6. Define angle of friction.
7. Define relative angular velocity.
8. State the principle of conservation of linear momentum.
9. Write down the “horizontal range” for projectile (with usual notations).
10. Define Newton’s experimental laws on impact.

PART – B

Answer any FIVE Questions                                                                                                 (5 x 8 = 40 marks)

1. State and prove Lami’s Theorem.
2. A weight W hangs by a string and is drawn aside by a horizontal force until the string makes an angle 60o with the vertical.  Find the horizontal force and tension in the string.
3. Find the resultant of two like parallel forces.
4. A uniform rod AB of length 2a and weight W is resting on two pegs C and D in the same level at a distance d apart.  The greatest weights that can be placed at A and B without tilting the rod are W1 and W2 respectively.  Show that .
5. A lift a ascends with constant acceleration f, then with constant velocity and finally stops under constant retardation f.  If the total height ascended is h and total time occupied is t, show that time during which the lift is ascending with constant velocity is .
6. Show that when masses P and Q are connected by a string over the edge of a table, the tension is the same whether P hangs and Q is on the table or Q hangs and P is on the table.
7. A particle projected upwards under the action of gravity in a resisting medium where the resistance varies as the square of the velocity.  Discuss the motion.
8. Two perfectly elastic smooth spheres of masses m and 3 m are moving with equal moments in the same st.line and in the same direction.  Show that the smaller sphere reduced to rest after it strikes the other.

PART – C

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

1. a) Three equal strings of no sensible weights are knotted together to form an equilateral triangle

ABC and a weight W is suspended from A.  If the triangle and the weight be supported with

BC horizontal by means of two strings at B and C each at an angle 135o with BC.  Show that

the tension in BC is .                                                                                           (10)

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

displaced through the distance  . AB.                                                                    (10)

1. a) A system of forces in the plane of ABC is equivalent to a single force at A; acting along the

internal bisector of the angle BAC and a couple of moment G.  If the moments of the system

about B and C are respectively G2 and G3 prove that (b+c) G1 = b G2 + c G2.

1. b) A ladder which stands on a horizontal ground leaning against a vertical wall is so loaded that

its centre of gravity is at the distance a and b from the lower and the upper ends respectively.

Show that if the ladder is in limiting equilibrium, its inclination  to the horizontal is given by

where m and m1 are the coefficients of friction between the ladder and the

ground and the wall respectively.

1. a) A body, sliding down a smooth inclined plane is observed to cover equal distances, each equal

to a, in consecutive intervals of time t1, t2.  Show that the inclination of plane to the horizon is

.                                                                                                             (8)

1. b) Discuss the motion of two particles connected by a string.                                                 (12)
2. a) 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.                                       (10)

1. b) Show that the path of projectile in a parabola.                                                                     (10)

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## Loyola College B.Sc. Mathematics April 2011 Linear Algebra Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

FIFTH SEMESTER – APRIL 2011

# MT 5508/MT 5502 – LINEAR ALGEBRA

Date : 20-04-2011              Dept. No.                                                  Max. : 100 Marks

Time : 1:00 – 4:00

SECTION – A

Answer ALL questions                                                                                         (10 X 2 = 20 Marks)

1. Define a vector space over a field F.
2. Prove that R is not a vector space over C.
3. Define the kernel of a linear transformation.
4. Prove that in V3(R), the vectors (1, 2, 1), (2,1,0) and (1, -1, 2) are linearly independent.
5. Define an inner product space.
6. State the triangle inequality for inner product space.
7. Define an orthonormal set in an inner product space.
8. Prove that (A+B)T = AT + BT where A and B are two m X n matrices.
9. Define an invertible matrix.
10. Define Hermitian and unitary linear transformations.

SECTION – B

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

1. Prove that any set containing a linearly dependent set is also linearly dependent.
2. Let V be a vector space over a field F. Then prove that S = {v1, v2, . . ., vn} is a basis for V if and only of every element of V can be expressed as a linear combination of elements of S.
3. Prove that T : R2→R2 defined by T(a, b) = (a+b, a) is a vector space homomrphism.
4. Prove that T Є A(V) is invertible if and only of T maps V onto V.
5. Let T Є A(V) and λ Є F. Then prove that λ is an eigenvalue of T if and only if λI-T is singular.
6. Show that any square matrix can be expressed uniquely as the sum of a symmetric and a skew – symmetric matrix.
7. Show that the system of equations

X+2y+z=11

4x+6y+5z=8

2x+2y+3z=19 is inconsistent.

1. If TЄA(V) is Hermitian, then prove that all its eigen values are real.

SECTION – C

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

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

W, then prove that dim V = dim U + dim W.

1. b) If A and B are subspaces of a vector space V over F, prove that (A+B)/BA/A

(10 x 10)

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

Hom (U,V) is of  dimension mn.

1. a) Apply the Gram – Schmidt orthonormalization process to the vectors (1,0,1), (1,3,1)

and (3,2,1) to obtain an orthonormal basis for R3.

1. b) State and prove Bessel’s inequality.                                             (10 + 10)

1. a) Let V=R3 and suppose that is the matrix of T Є A(V) relative to the

standard basis V1 = (1,0,0), V2 = (0, 1, 0), V3 = (0,0,1). Find the matrix of T relative to

the basis W1 = (1,1,0),  W2 =  (1,2,0), W3 = (1,2,1).

1. b) Show that 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.                                     (10 + 10)

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## Loyola College B.Sc. Mathematics April 2011 Complex Analysis Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

SIXTH SEMESTER – APRIL 2011

# MT 6603/MT 6600 – COMPLEX ANALYSIS

Date : 05-04-2011              Dept. No.                                                 Max. : 100 Marks

Time : 9:00 – 12:00

PART – A

Answer ALL Questions                                                                                          (10 x 2 = 20 marks)

1. Express the function f(z) = z3+z+1 in the form f(z) = u(x, y) + iv(x, y).
2. Show that the radius of convergence of the series
3. Find the modulus of
4. Define removable singularity and pole for a function.
5. State Morera’s theorem.
6. State Cauchy’s Residue theorem.
7. Find the points where the mapping is conformal.
8. Calculate the residues of at z = 1, 2 and 3.
9. Find the Singular points and its nature for the function e1/z.
10. Find the fixed points of the transformation w = .

PART – B

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

1. If z1 and z2 are two complex numbers, show that
2. Given v(x,y) = find f(z) = u(x,y) + iv(x,y) such that f(z) is analytic.
3. State and Prove the fundamental theorem of algebra.
4. Obtain the Taylor’s or Laurent’s series for the function f(z) = when (i)

(ii) 1< .

1. Evaluate using Cauchy Residue theorem where C is the circle
2. State Cauchy’s theorem and Cauchy’s integral formula. Evaluate where C is positively oriented circle .
3. State and prove maximum modulus theorem.
4. Prove that the cross ratio is invariant under Mobius transformation.

PART – C

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

1. a) Prove that for the function the Cauchy – Reimann equations

are satisfied at z = 0, but f(z) is not differentiable at z = 0.

1. b) State and prove Cauchy – Hadamard’s theorem for radius of convergence. (10 + 10)

1. a) If f(z) is analytic inside and on a simple closed curve C except for a finite number of

poles inside C and has no zero on C, Prove that  where N is the

number of zeros and P is the number of poles of inside C.

1. b) Using contour integration evaluate .                                              (10 + 10)
2. a) State and Prove Taylor’s theorem.
3. b) Show that when                                   (14+6)
4. a) Let f be analytic in a region D and for Prove that  is conformal at .
5. b) Show that by means of the inversion the circle is mapped into the circle

1. c) Find the general bilinear transformation which maps the unit circle onto and

the points                                                               (10+5+5)

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## Loyola College B.Sc. Mathematics April 2011 Astronomy Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

THIRD SEMESTER – APRIL 2011

# MT 3502/MT 5503 – ASTRONOMY

Date : 25-04-2011              Dept. No.                                                  Max. : 100 Marks

Time : 1:00 – 4:00

PART – A

Answer ALL questions.                                                                                                   (10 x 2 = 20)

1. State cosine formula.
2. Define twilight.
3. Define refraction.
4. What is aberration of light?
5. State Kepler’s third law.
6. What is equation of time?
7. Define synodic period of moon?
8. Explain annular eclipse.
9. Name the different celestial objects in the solar system.
10. What is a constellation? Name any two constellations.

PART – B

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

1. Describe the equatorial system of coordinates with a neat diagram. Give the advantages and disadvantages of this system.
2. Trace the variations in the duration of day and night during a year for a place on the equator.
3. Explain Gnomon with a neat diagram.
4. Write a note on calendars.
5. Derive Newton’s deductions from Kepler’s laws.
6. Write a note on surface structure of moon.
7. Find the maximum and minimum number of eclipses possible near a node.
8. Write a note on comets.

PART C

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

1. a) Derive an expression for hour angle and azimuth of a celestial body at rising and setting.

1. b) Explain the different zones of earth with a neat diagram.

1. a) Derive a tangent formula for refraction. How do you find the value of coefficient of

refraction?

1. b) Derive Cassini’s formula.

1. a) Derive the condition for occurrence of a total solar eclipse.

1. b) Derive an expression for equation of time. Prove that it vanishes four times in a year.

1. a) Derive an expression for different phases of moon.

1. b) Write a note on planets.

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## Loyola College B.Sc. Mathematics April 2011 Algebra, Calculus And Vector Analysis Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

THIRD SEMESTER – APRIL 2011

# MT 3501/ MT 3500 – ALGEBRA, CALCULUS AND VECTOR ANALYSIS

Date : 12-04-2011              Dept. No.                                                    Max. : 100 Marks

Time : 1:00 – 4:00

PART – A

Answer ALL questions.                                                                                                 (10 ´ 2 = 20)

1. Evaluate
2. Find when u = x2 – y2; v = x2 + y2
3. Solve
4. Find the complete integral of z = px + qy +p2q2
5. Find grad f if f = xyz at (1, 1, 1)
6. Evaluate divergence of the vector point function
7. Find L[sin2 2t]
8. Find
9. Find the sum of all divisors of 360.
10. Find the remainder when 21000 divisible by 17.

PART – B

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

1. Change the order of integration and evaluate
2. Express in terms of Gamma functions and evaluate
3. Solve p2 + pq = z2
4. Solve xp + yq = x
5. Show that the vectoris irrotational.
6. Evaluate: (a) L[cos 4t sin 2t]                   (b) L[e-3t sin2t]
7. Find
8. Show that 18! + 1 is divisible by 437.

PART – C

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

1. (a) Evaluate where the region V is bounded by x + y+ z = a (a > 0),
x = 0; y = 0; z = 0
.

(b)  Evaluate  where R is the region in the positive quadrant for which
x + y £ 1.

(c)  Show that

1. (a) Solve (x2 + y2 + yz)p + (x2 + y2 – xz)q = z(x+y)

(b)  Find the complete integral and singular integral of p3 + q3 = 8z

1. (a) Solve y¢¢ + 2y¢ – 3y = sin t given that y(0) = y¢(0) = 0

(b)  State and prove the Weirstrass inequality.

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

(b)  Verify Green’s theorem in the XY plane for where C is the closed curve in the region bounded by y = x; y = x2.

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## Loyola College B.Sc. Mathematics April 2011 Algebra, Anal.Geo & Calculus – II Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

SECOND SEMESTER – APRIL 2011

# MT 2501/MT 2500 – ALGEBRA, ANAL.GEO & CALCULUS – II

Date : 08-04-2011              Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

PART – A

Answer ALL questions:                                                                                                     (10 x 2 = 20)

1. Evaluate .
2. Evaluate .
3. Solve: .
4. Solve: .
5. Prove that the series is convergent.
6. Test for convergency the series .
7. Find the general term in the expansion of .
8. Prove that the coefficient of in the expansion of is .
9. Find the equation of the sphere which has its centre at the point and touches the

plane  .

1. Find the distance between the parallel planes and

PART – B

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

1. Prove that .
2. If ( n being a positive integer), prove that .

Also evaluate  and .

1. Solve: .

1. Solve
2. Test for convergency and divergency the series
3. Show that the sum of the series .
4. Show that if

1. Find the equation of the plane passing through the points

.

PART – C

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

1. a) Evaluate (10 marks)
2. b) Find the area and the perimeter of the cardiod .                    (10 marks)
3. a) Solve: . (10 marks)
4. b) Discuss the convergence of the series for

positive values of .                                                                                       (10 marks)

21.a) Show that the error in taking  as  an approximation to  is

approximately equal to  when  is small.                                                   (10 marks)

1. b) show that (10 marks)

1. a) A sphere of constant radius passes through the origin and meets the axes in A, B, C.

Prove that the centroid of the triangle ABC lies on the sphere

(10 marks)

1. b) Find the shortest distance between the lines

.

Also find the equation of the line of shortest distance.                              (10 marks)

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## Loyola College B.Sc. Mathematics Nov 2011 Physics For Mathematics-I Question Paper PDF Download

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