B.Sc Mathematics Question Paper

Loyola College B.Sc. Mathematics April 2009 Modern Algebra Question Paper PDF Download

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       LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

ZA 16

FOURTH SEMESTER – April 2009

MT 4502 / 4500 – MODERN ALGEBRA

 

 

 

Date & Time: 24/04/2009 / 9:00 – 12:00  Dept. No.                                                   Max. : 100 Marks

 

 

PART – A

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

1.)     Define an equivalent relation on set A.

2.)     Define a partially ordered set.

3.)     Define a cyclic group.

4.)     If G is a group of order n and a Î G, show that an = e.

5.)     Define an automorphism of a group.

6.)     Define the alternating group of degree n.

7.)     Define a ring.

8.)     Define a field.

9.)     Define an ordered integral domain.

10.)   Define a maximal ideal of a ring.

PART – B

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

11.)   If G is a group, prove that

a.) the identity element of G is unique

b.) every a Î G has a unique inverse in G.

12.)   Show that the union of two subgroups of G is a subgroup of G if and only if one is contained in the other.

13.)   Prove that every subgroup of a cyclic group is cyclic.

14.)   If a and b are elements of a group and a5 = e, b4 = e, ab = ba3, prove that

(i)  a2b = ba        and               (ii) ab3 = b3a2.

15.)   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.

16.)   If f is a homomorphism of a ring R into a ring R′, then prove that the kernel of f is an ideal
of R.

17.)   Prove that the intersection of two subrings of a ring is a subring.

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

PART – C

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

19.)   State and prove the fundamental theorem of arithmetic.

20.) (a) State and prove Lagrange’s theorem.

(b) Let H be a subgroup of index 2 in a group G.  Prove that H is a normal subgroup.  (15+ 5)

21.) (a) State and prove the fundamental theorem of homomorphism on groups

(b) Define an integral domain and a division ring                                                            (14+ 6)

22.)   State and prove unique factorization theorem.

 

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Loyola College B.Sc. Mathematics April 2009 Mechanics – II Question Paper PDF Download

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 LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

ZA 37

SIXTH SEMESTER – April 2009

MT 6604 – MECHANICS – II

 

 

 

Date & Time: 21/04/2009 / 9:00 – 12:00 Dept. No.                                                Max. : 100 Marks

 

 

PART – A

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

 

  1. Define centre of mass.
  2. Write down the formula for the C.G of a rigid body.
  3. State the principle of virtual work.
  4. Define suspension bridges.
  5. Define Simple Harmonic motion.
  6. Define second pendulum.
  7. Write down the components of the acceleration of a particle in polar coordinates.
  8. Define central orbit.
  9. State the theorem of perpendicular axes.
  10. State D’Alembert’s principle.

PART – B

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

 

  1. A piece of uniform wire is bent in the shape of an isosceles triangle whose sides are a, a and b. Show that the distance of the C.G from the base of the triangle is .
  2. Find the C.G of a uniform solid right circular cone.
  3. A regular hexagon is composed of six equal heavy rods freely jointed together and two opposite angles are connected by a string which is horizontal, one rod being in contact with a horizontal plane; at the middle point of the opposite rod a weight W’ is placed. If W be the weight of each rod, show that tension in the string is .
  4. Find the equation of the catenary in the Cartesian farm.
  5. A particle moves in SHM in a straight line. In the first second, after starting from rest, it travels a distance ‘a’ and in the next second, it travels a distance ‘b’ in the same direction. Prove that the amplitude of the motion is .
  6. Determine the maximum speed with which a car can turn round a level curve of radius 100 meters without slipping given that the coefficient of friction between the tyres and the track is 0.3.
  7. Derive the Pedal equation or p-r equation of a central orbit.
  8. Find the M.I of a hollow sphere.

 

 

                                                                   

                                                                 

PART – C

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

 

  1. (a) Find the C.G of the area enclosed by the parabolas .

(b) ABCDEF is a regular hexagon formed of light rods smoothly jointed at their ends with a

diagonal rod AD. Four equal forces ‘P’ act inwards at the middle points of the rods AB, CD, DE, FA  and at right angles to the respective sides. Find the stress in the diagonal AD and state whether it is a tension or a thrust.                                                  (10+10)

  1. (a) A string of length 2l hangs over two small smooth pegs in the same horizontal level. Show

that if h is the sag in the middle, the length of either part of the string that hangs vertically is

.

(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 the theorem of parallel axis.

(b) Find the M.I of the square lamina about a diagonal of length l.   (10+10)

  1. (a) A particle acted on by a central attractive force is projected with a velocity at an

angle of with its initial distance ‘a’ from the centre of force. Show that the path is the equiangular spiral .

(b) A square lamina of side 2a rotates in a vertical plane about a horizontal axis passing through

one of the vertices and perpendicular to its plane and a weight equal to that of lamina is placed at the opposite vertex. Find the length of S.E.P.                        (10+10)

 

 

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Loyola College B.Sc. Mathematics April 2009 Mechanics – I Question Paper PDF Download

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      LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

ZA 15

FOURTH SEMESTER – April 2009

MT 4501 – MECHANICS – I

 

 

 

Date & Time: 02/05/2009 / 9:00 – 12:00 Dept. No.                                                   Max. : 100 Marks

 

 

 

PART – A

Answer ALL the questions:                                                                            10 x 2 = 20     

 

  1. When do you say a concurrent system of forces is in equilibrium?
  2. State the converse of Lami’s theorem.
  3. Define moment of a force.
  4. When do you say two unlike parallel forces form a couple?
  5. Write down the components of the acceleration of a particle in the tangential and normal directions.
  6. A ship is steaming north at and a man walks across its deek in a direction due west at 8 k.m.p.h. Find its resultant velocity in space.
  7. State the principle of conservation of momentum.
  8. Define coefficient of restitution.
  9. Define (i) trajectory and (ii) horizontal range
  10. Define limiting velocity.

 

PART – B

Answer any FIVE questions:                                                             5 x 8 = 40

 

  1. State and prove Lami’s theorem.
  2. A weight W hangs by string and is drawn aside by horizontal force until the string makes an angles 60o with the vertical. Find the horizontal force and the tension in the string.
  3. Find the resultant of two unlike parallel forces.
  4. State the laws of friction.
  5. A ship sails north-east at 15 kmph and to a passenger on board, the wind appears to blow from north with velocity of Find the true velocity of the wind.
  6. A particle is dropped from the top of a tower and describes during the last second of its fall (9/25) of the height of the tower. Find the height of the tower.
  7. A particle projected from the top of a wall 50 m. high, at an angle of 30o above the horizon, strikes the level ground through the foot of the wall at an angle of 45o. Show that the angle of depression of the point of striking the ground from the point of projection is .

 

 

 

 

  1. A particle is projected from a point in a smooth fixed horizontal plane with velocity is at an elevation a. Show that the particle ceases to rebound from the plane at the end of time and that the total horizontal distance described in this period is .

PART – C

Answer any TWO questions:                                                              2 x 20 = 40

 

  1. a) Three equal strings of no sensible wei0ght are knotted together to form an equilateral and a weight W is suspended from A. If the triangle and the weight be supported with BC horizontal by means of two strings at B and C each at an angle 135o with BC, show that the tension in BC is .
  2. b) The top of a pole is held by means of four horizontal wires which enert the following tensions: 20 lbs weight due North, 30 lbs weight due East, 40 lbs weight due South West and 50 lbs weight due South East. Find the magnitude and direction of the resultant pull on the post.
  3. a) Discuss the equilibrium of a particle on a rough inclined plane acted on by an external force.
  4. b) A uniform ladder rests at the angle 45o with its upper extremity against a rough vertical wall and its lower extremity on the ground. If be the coefficients of friction between the ladder and the ground and the wall respectively, show that the least horizontal force which will move the lower extremity towards the wall is where W is the weight of the ladder.
  5. Obtain the equation of the path of a projectile in Cartesian form. Also determine the time of flight, greatest height and horizontal range.
  6. Two smooth spheres of masses m1 and m2 moving with velocities u1 and u2 imping directly. Obtain
  7. the motion after impact
  8. the impulse imparted to each other due to impact
  • the change is kinetic energy due to impact.

 

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Loyola College B.Sc. Mathematics April 2009 Mechanics – I Question Paper PDF Download

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  LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

ZA 30

FIFTH SEMESTER – April 2009

MT 5506 – MECHANICS – I

 

 

 

Date & Time: 24/04/2009 / 1:00 – 4:00  Dept. No.                                                   Max. : 100 Marks

 

 

PART – A

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

 

  1. Define coplanar forces.
  2. State the law of parallelogram of forces.
  3. Define couple.
  4. State the theorem on “equilibrium of three coplanar forces”.
  5. State the principle of conservation of linear momentum.
  6. Define angular velocity.
  7. State Newton’s laws of motion.
  8. Define horizontal range of a projectile.
  9. Define impulsive force.
  10. Define the coefficient of elasticity.

PART – B

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

 

  1. State and prove Lami’s theorem.
  2. A weight bar hangs by a string and is drawn aside by a horizontal force until the strings makes an angle 60o with the vertical.  Find the horizontal force and the tension in the string.
  3. A straight rod PQ of length ‘2a’ and weight ‘W’ rests on smooth horizontal pegs R and S at the same level at a distance ‘a’ apart.  If two weights p w and q w are suspended from P and Q respectively, show that, when the reactions at R and S are equal the distance PR is given by .
  4. Two rough particles connected by a light string rest on an inclined plane.  If their weights and corresponding coefficients of friction are W1, W2 and µ1, µ2 respectively and , where α is the inclination of the plane with the horizon, prove that , if both particles are on the point of moving down the plane.
  5. A and B describe concentric circles of radii ‘a’ and ‘b’ with speeds u and v, the motion being the same way round.  If the angular velocity of either with respect to the other is zero, prove that the line joining them subtends at the centre an angle whose cosine 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 is projected so as to graze the tops of 2 walls, each of height 20 feet, at distances of 30 ft and 170 ft respectively, from the point of projection.  Find the angle of projection and the highest point reached in the flight.
  8. A ball A impinges directly on an 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 is to that of A is (1+e): (1-e).

PART – C

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

 

  1. a) A uniform plane lamina in the form of a rhombus, one of whose angles is 120o is supported by two forces of magnitudes P and Q applied at the centre in the directions of the diagonals so that one side is horizontal.  Show that is P>Q, then P2 = 3Q2.
  1. b) Three equal strings of no sensible weight are knotted together to form an equilateral ∆ 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 .
  2. a) State and prove Varignon’s theorem on moments.
  3. b) A mass ‘m1’ hanging vertically pulls a mass ‘m2’ vertically up a rough inclined plane of inclination α by a light inextensible string passing over a smooth, light pulley at the top of the plane, the portion of the string between m2 and the pulley being parallel to the line of greatest slope. Find the acceleration of the system and the tension in the string.

 

  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 pulley. Discuss the motion.
  2. b) A particle of mass m is projected vertically under gravity, the resistance of air being ‘mk’ times the velocity. Show that the greatest height altained by the particle is where V is the terminal velocity of the particle and λV is its initial velocity.

 

  1. a) A particle is projected up an inclined plane of inclination b. Find the range on the inclined plane.
  2. b) 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 .

 

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Loyola College B.Sc. Mathematics April 2009 Mathematical Statistics Question Paper PDF Download

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       LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

YB 15

FOURTH SEMESTER – April 2009

ST 4206/ ST 4201 – MATHEMATICAL STATISTICS

 

 

 

Date & Time: 27/04/2009 / 1:00 – 4:00  Dept. No.                                                        Max. : 100 Marks

 

 

                                                                        PART – A 

                                                     

ANSWER ALL THE QUESTIONS                                                                          (10 x 2 = 20)

 

  1. Define Mutually Exclusive Events with examples.
  2. State multiplication law of probability
  3. What is the chance that a leap year selected at random will contain 53 sundays?
  4. Define probability generating function of a random variable.
  5. If X is a random variable and a and b are constants, then show that

E (aX + b) = a E(X) + b provided all the expectations exist.

  1. Derive the moment generating function of Poisson distribution.
  2. Define Beta Distribution of First kind.
  3. Define Regression.
  4. Ten unbiased coins are tossed simultaneously. Find the probability of getting at least seven heads.
  5. Define Most Powerful test.

 

PART – B

ANSWER ANY FIVE QUESTIONS                                                                          (5 x 8 = 40)

  1. Four cards are drawn at random from a pack of 52 cards. Find the probability that
  2. These are a king, a queen, a jack and an ace.
  3. Two are kings and two are queens
  4. Two are black and two are red.
  5. There are two cards of hearts and two cards of diamonds.

 

  1. The contents of urns I, II and III are as follows:

Urn I   : 1 White, 2 Black and 3 Red balls

Urn II  : 2 White, 1 Black and 1 Red balls, and

Urn III : 4 White, 5 Black and 3 Red balls

One urn is chosen at random and two balls drawn from it. They happen to be white and              red. What is the probability that they come from Urns I, II or III?

  1. Let X be a Continuous random variable with probability density function

 

a). Determine the constant a

b). Compute P( X ≤ 1.5 )

 

  1. State and prove Chebyshev’s Inequality.

 

  1. Derive the Mean and Variance of Binomial Distribution.

 

  1. The joint probability distribution of two random variables X and Y is given by

P ( X = 0, Y = 1) =  , P ( X = 1, Y = -1) = and P ( X = 1, Y = 1) =. Find

  1. Marginal distributions of X and Y and
  2. Conditional probability distribution of X given Y = 1.

 

  1. The following figures show the distribution of digits in numbers chosen at random

from a telephone directory:

Digits 0 1 2 3 4 5 6 7 8 9 Total
Frequency 1026 1107 997 966 1075 933 1107 972 964 853 10,000

 

Test whether the digits may be taken to occur with equal frequency in the directory.

  1. Define
    1. Null Hypothesis
    2. Alternative Hypothesis
  • Level of Significance
  1. Two types of Errors                                                     ( 2 + 2 + 2 + 2 )

 

PART – C

ANSWER ANY TWO QUESTIONS                                                                        (20 x 2 = 40)

  1. (a). State and Prove Baye’s Theorem . (10)

 

(b). Three groups of Children contain respectively 3 girls and 1 boy, 2 girls and 2 boys,

and 1 girl and 3 boys. One child is selected at random from each group. Show that

the chance that the three selected consist of 1 girl and 2 boys is  .               (10)

 

  1. (a). In four tosses of a coin, let X be the number of heads. Tabulate the 16 possible

outcomes with the corresponding values of X. By simple counting, derive the

probability distribution of X and hence calculate the expected value of X        (10)

 

 

 

(b). A random variable X has the following probability density function:

 

x 0 1 2 3 4 5 6 7
p(x) 0 k 2k 2k 3k k2 2k2 7k2+k

 

  1. Find k
  2. Evaluate
  • If , find the minimum value of a.
  1. Determine the distribution function of X (10)

 

  1. (a). State any five properties of Normal Distribution (8)

 

(b). 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 :

  1. no defective and
  2. at least two defectives                                                                             (12)

 

  1. (a). If , find
    1. Var (X)
    2. Var (Y)
  • r (X,Y) (10)

(b). The mean weekly sales of soap bar in departmental stores was 146.3 bars per store.

After an advertising campaign the mean weekly sales in 22 stores for a typical week

increased to 153.7 and showed a standards deviation of 17.2. Was the advertising

campaign successful?                                                                                           (10)

 

 

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Loyola College B.Sc. Mathematics April 2009 Linear Algebra Question Paper PDF Download

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          LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

ZA 32

FIFTH SEMESTER – April 2009

MT 5508 / 5502 – LINEAR ALGEBRA

 

 

 

Date & Time: 06/05/2009 / 1:00 – 4:00            Dept. No.                                               Max. : 100 Marks

 

 

SECTION – A

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

  1. Let V be a vector space.  Prove that (– a)v = a(– v) = – (av) for any a Î F and vÎ V.
  2. Prove that any non-empty subset of linearly independent vectors is linearly independent.
  3. Show that the vectors (1, 0, -1), (2, 1, 3), (–1, 0, 0) and (1, 0, 2) are linearly dependent in R3.
  4. Verify that the map T : R2 ® R2 defined by T(x, y) = (2x + y, y) is a homomorphism.
  5. If V is an inner product space, then prove that
  6. Define eigenvalue and eigenvector of a linear transformation.
  7. If l Î F is an eigenvalue of T Î A(V), prove that l is a root of the minimal polynomial of T over F.
  8. For A, B Î Fn, prove that tr (A + B) = tr (A) + tr (B).
  9. If A and B are Hermitian, show that AB + BA is Hermitian.
  10. Prove that the eigenvalues of a unitary transformation of T are all of absolute value 1.

 

SECTION – B

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

  1. Prove that the union of two subspaces of a vector space V over F is a subspace of V if and only if one is contained in the other.
  2. 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.
  3. Let U and V be vector spaces over a field F, and suppose that U has finite dimension n. Let {u1, u2, . . . un} be a basis of U and let v1, v2, . . . vn be arbitrary vectors in V. Prove that there exists a unique homomorphism T: U ® V such that T(u1) = v1, T(u2) = v2, . . . , T(un) = vn.
  4. Define an orthonormal set.  If {w1, w2, . . . wn} is an orthonormal set in an inner product space V, prove that .
  5. Prove that T Î A(V) is invertible if and only if the constant term of the minimal polynomial for T is not zero.

 

  1. Let V = R3 and suppose that  is the matrix of T Î 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) and w3 = (1, 2, 1).
  2. Show that any square matrix can be expressed uniquely as the sum of a symmetric matrix and a skew-symmetric matrix.
  3. Investigate for what values of l, m, the system of equations  x1 + x2 + x3 = 6,
     x1 + 2x2 + 3x3 = 10,   x1 + 2x2 + l x3 = m  over a rational field has (a) no solution  (b) a unique solution  (c) an infinite number of solutions.

 

SECTION – C

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

  1. (a) Let T : U ® V be a homomorphism of two vector spaces over F and suppose that U has finite dimension. Then prove that

                              Dim U = nullity T + rank of T

(b)  If W1 and W2 are subspaces of a finite dimensional vector space V, prove that dim (W1 + W2) = dim W1 + dim W2 – dim (W1 Ç W2)

  1. (a) Prove that T Î A(V) is singular if and only if there exists an element v ¹ 0 in V such that T(v) = 0.

(b)  Prove that every finite-dimensional inner product space V has an orthonormal set as a basis.    (8+12)

  1. (a) If T be the linear transformation on R3 defined by
    T(a1, a2, a3) = ( a1 + a2 + a3, – a1 – a2 – 4a3, 2a1 – a3), find the matrix of T relative to the basis v1 = (1, 1, 1), v2 = (0, 1, 1), v3 = (1, 0, 1).

(b)  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 Ç W2 = (0).                                                                    (8+12)

  1. (a) If T Î A(V), then prove that T* Î A(V).  Also prove that

(i) (S+T)* = S* + T*     (ii)  (ST)* = T*S*     (iii)       (iv)  (T*)* = T

(b)  Prove 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.

 

 

 

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Loyola College B.Sc. Mathematics April 2009 Graphs, Diff. Equ., Matrices & Fourier Series Question Paper PDF Download

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      LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

ZA 02

FIRST SEMESTER – April 2009

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

 

 

 

Date & Time: 22/04/2009 / 1:00 – 4:00    Dept. No.                                                       Max. : 100 Marks

 

 

SECTION – A

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

  1. Write the range and domain of
  2. Check whether each of the following defines a function:

(i)  y = -3x + 8                              (ii) x2 + y2 = 1

  1. State the principle of least square.
  2. Reduce into linear form: y = a xn, where a and n are constants.
  3. Define  (i) general solution, (ii) particular solution of a difference equation.
  4. Solve 16 yx+2 – 8 yx+1 + yx = 0.
  5. Find the eigenvalues of A5 when.
  6. Find the sum and product of eigenvalues of the matrix.
  7. State the Dirichlet conditions for Fourier series.
  8. Find the Fourier constants a0 and an for the function f(x) = x3 in (-p < 0 < p).

 

SECTION – B

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

  1. (a)  A company sold 500 tool kits in 2000 and 20000 tool kits in 2005.  Assuming that sales are approximated by a linear function, express the company’s sales S as a function of linear time t.

(b)  A company has fixed cost of Rs. 8250 and a marginal cost of Rs. 450 for each item produced.  Express the cost C as a function of the number x of items produced and evaluate the function at x = 20 and at x = 50.

  1. Using the method of least square fit a straight line to the following data.

x:               0               1              2                3               4

y:               1             1.8           3.3            4.5             6.3

 

 

  1. The data in the following table will fit a formula of the type y = a + bx + cx2.  Find the formula by the method of group averages

x:           87.5         84.0          77.8            63.7          66.7            36.9

y:           292          283            270            235            197            181

  1. Solve:    yn+2 – 4 yn+1 + 4yn = 2n + 3n + p.
  2. Find the inverse of the matrix  using Cayley-Hamilton theorem.
  3. Find the eigenvalues and eigenvectors of the matrix .
  4. If  expand f(x) as a sine series in the interval (0, p)
  5. Express f(x) = | x |, -p < x < p as a Fourier series and hence deduce that

SECTION – C

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

  1. (a) Given the following total revenue R(x) = 600x – 5x2 and total cost
    C(x) = 100 x + 10,500  (i) express p as a function of x, (ii) determine the maximum level of profit by finding the vertex of p(x) and (iii) find the x intercepts and draw a rough sketch of the graph.

(b)  The temperature q of a vessel of cooling water and the time t in minutes since the beginning of observation are connected by a relation of the form
q = a ebt + c.  Their tabulated values are given below

t:             0          1         2           3          5             7          10         15        20

q:         52.2     48.8     46.0      43.5      39.7        36.5      33.0     28.7    26.0

Find the best values you can for a, b, c.

  1. Solve the following equation.
  • Dux + D2ux = cos x
  • (E2 – 5E + 6) yn = 4n (n2 – n + 5)
  1. (a)  Expand f(x) = x (2p  – x) as a Fourier series in (0, 2p)

(b)  Obtain the Fourier series for the function f(x) = x2, –p £ x £ p and from it deduce that

  1. Page: 2

    Reduce the quadratic form  in to a canonical form by the method of orthogonal reduction.

 

 

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Loyola College B.Sc. Mathematics April 2009 Fin. Accounts & Fin. Stat. Analysis Question Paper PDF Download

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            LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc., B.A. DEGREE EXAMINATION – MATHEMATICS&HISTORY,ECONOMICS

KP 14

FOURTH SEMESTER – April 2009

CO 4205 / 4201 – FIN. ACCOUNTS & FIN. STAT. ANALYSIS

 

 

 

Date & Time: 27/04/2009 / 9:00 – 12:00  Dept. No.                                                Max. : 100 Marks

 

 

SECTION A

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

  1. List the various types of cash book?
  2. What is Bank Reconciliation Statement?
  3. Journalise the following:
  • Goods worth Rs.1,000 taken by the proprietor for personal use

(ii) Paid Advertisement expenses to Anil & Co.,   Rs.1,500

  1. Classify the following under Personal, Real, Nominal account
  • Outstanding salary (b)  Goodwill  (c ) Loss by fire  (d)  Purchases
  1. Fill in the blanks:
  • Purchases book is also known as ___________.
  • Journal is the book of _________ entry.
  1. Ascertain the amount of profit/loss on sale of machinery:

Cost of Machine  Rs. 1,00,000; Accumulated Depreciation on machinery Rs. 30,000 ;  Sale value of machinery Rs. 85,000

  1. Find out Earning per share from the following:

Net profit after tax Rs. 2,00,000

10% Preference Share capital Rs. 4,00,000

Equity share capital (Rs.10 each)  Rs. 1,00,000

  1. Calculate the amount of subscription to be credited to Income and expenditure account for the year 2008:

Subscriptions received during 2008                                        Rs.24,000

Subscriptions received in advance for 2009                           Rs.  1,600

Subscriptions outstanding in the beginning of  2008             Rs.  2,000

Subscriptions outstanding at the close of  2008                     Rs.  1,500

  1. What are the different modes of expressing ‘Ratios’?
  2. Give adjusting entry for the following:
  • Depreciation @10% on furniture of Rs. 5,000
  • Wages outstanding Rs. 200

SECTION B

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

  1. Who are the users of accounting information and why do the users need the accounting information?
  2. Explain any four accounting concepts.
  3. What is ‘Petty cash book’?  What are its advantages?
  4. Prepare a bank reconciliation statement from the following data as on 31-12-2008

Rs.

  • Balance as per cash book 25,000
  • Cheques issued but not presented for payment 18,000
  • Cheques deposited in bank but not collected 20,000
  • Bank paid insurance premium   8,000
  • Direct deposit by a customer   5,000
  • Interest on investment collected by bank 12,000
  • Bank charges   1,000
  1. Enter the following transactions in Rohan’s cash book with discount and cash column:

Date                           Particulars                                                Rs.

Jan  1               Cash balance                                                   18,500

3               Cash sales                                                        33,000

7               Paid David                                                      15,850

Discount allowed by him                                    150

13               Sold goods to Mohan on credit                      19,200

15               Cash withdrawn for personal expenses            2,400

16               Purchased goods form Charles on credit        14,300

22               Paid into bank                                                 22,750

25               Cash received from Mohan                             19,000

Allowed him discount                                                     200

 

  1. A Ltd., provides the following accounting data for the year ending 31st March 2008 and request you to ascertain (a) Gross profit ratio (b) Net profit ratio

(c) Operating ratio (c) Operating profit ratio

Rs.

Sales                                                                            20,00,000

Gross profit                                                                   8,00,000

Office expenses                                                               60,000

Selling expenses                                                               40,000

Finance expenses                                                             30,000

Loss on sale of plant                                                          4,000

Interest received on investments                                       5,000

Net profit                                                                      6,71,000

 

  1. Naresh Ltd., had a balance of Rs. 80,000 in its Profit & Loss appropriation account on 1st  April 2007. During 2007-2008, its profit before tax amounted to Rs. 7,62,500.  The income tax provision of the year amounted to be 347,500.  The company decided to transfer Rs. 60,000 to General Reserve, Rs,. 87,500 to sinking fund for redemption of debentures to pay a dividend for the financial year at the rate of 10%.   The company’s share capital consisted of 3,00,000 shares of Rs.10 each.  Draw up the Profit & Loss Appropriation Account.
  2. The following balances were extracted from the ledger of Ram on 31st March 2008.  You are required to prepare a trial balance as on that date in proper form:

Rs.                                                       Rs.

Drawings                                 6,000              Salaries                        9,500

Capital                                     24,000             Sales returns                1,000

Sundry creditors                     43,000             Purchase returns          1,100

Bills payable                              4,000             Travelling expenses     4,600

Loan from Karthik                  10,000             Commission paid           100

Furniture                                   4,500             Trading expenses        2,500

Opening  stock                        47,000             Discount earned          4,000

Cash in hand                                900             Rent                            2,000

Cash at bank                           12,500             Bank overdraft            6,000

Tax                                            3,500             Purchases               1,26,000

Sales                                     1,28,000

 

SECTION C

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

  1. The following Trial Balance is extracted from the books of Mr. Rahul on 31-12-2008

Particulars                                           Debit               Credit

Rs.                   Rs.

Furniture                                                                       640

Motor vehicles                                                            6,250

Buildings                                                                     7,500

Capital                                                                                                 12,500

Bad debts                                                                       125

Sundry Debtors and Creditors                                                3,800                 2,500

Stock on 1-1-08                                                          3,460

Purchases and Sales                                                    5,475               15,450

Bank overdraft                                                                                      3,050

Sales and Purchases returns                                          200                     125

Advertising                                                                   450

Interest                                                                                      118

Commission                                                                                             375

Cash                                                                              650

Taxes and insurance                                                    1,250

General expenses                                                           782

Salaries                                                                        3,300               ______

34,000               34,000

Adjustments:

Stock on 31-12-2008 was Rs. 3,250

Depreciate Buildings @ 5% Furniture @ 10% and Motor vehicle @ 20%

Salaries Rs. 300 amd taxes Rs. 120 are outstanding

Insurance prepaid for Rs. 100

Write off further Rs. 100 as bad debts

Prepare Trading and Profit and Loss account for the year ending 31-12-2008 and

a Balance Sheet as on that date.

 

20  The following is the Receipts and Payments A/c of Delhi football association for the first year ending 31st Dec 2008

Receipts & Payments Account

 

Receipts                              Rs.                Payments                             Rs.

To Donation                                  50,000       By Pavilions office              40,000

To Reserve Fund (Life

Membership and entrance fees)    4,000       By expenses in connection

To Receipts from  football                              with matches                            900

Matches                                      8,000       By furniture                           2,100

Revenue Receipts:                                           By Investments at cost         16,000

To Subscriptions                             5,200       Revenue Payments:

To  locker Rents                                   50       By Salaries                             1,800

To interest on securities                     240       By Wages                                 600

To sundries                                         350       By Insurance                             350

By Telephone                            250

By Electricity                            110

By Sundry expenses                 210

                  By Balance on hand               5,520

67,840                                                    67,840

Additional information:

  • Subscriptions outstanding for 2008 are Rs. 250
  • Salaries unpaid for 2008 are Rs. 170
  • Wages unpaid for 2008 are Rs. 90
  • Outstanding bill the sundry expenses is Rs. 40
  • Donations received have to be capitalized.

Prepare from the details given above and Income and Expenditure A/c for the year ended 31-12-2008 and the Balance Sheet of the Association as on that date.

 

21.Following are the summarized Balance Sheets of Arul Ltd as on 31st Dec 2007 and 2008

Balance Sheets

Liabilities    2007

Rs.

     2008

Rs.

 Assets     2007

Rs.

     2008

Rs.
Share capital

General reserve

P&L A/c

Bank Loan

Sundry creditors

Provision for taxation

 1,00,000

 

50,000

30,500

70,000

 

50,000

 

32,000

 

 1,50,000

 

60,000

30,000

–

 

37,200

 

35,000

 Land & Building

Plant & Machinery

Stock

Debtors

Cash

Bank

Goodwill

  

1,00,000

 

1,00,000

50,000

75,000

500

2,000

5,000

  

90,000

 

1,19,000

24,000

63,200

1,000

15,000
3,32,500 3,12,200 3,32,500 3,12,200

 

Additional information:

During the year ended 31st December 2008

  • Dividend of Rs. 23,000 was paid
  • Depreciation written off on building Rs. 10,000, Machinery Rs. 14,000
  • Income tax paid during the year Rs. 28,000

Prepare a statement of cash flow for the year ended 31 Dec 2008.

 

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Loyola College B.Sc. Mathematics April 2009 Data Structure & Algorithm Question Paper PDF Download

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     LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

ZA 35

SIXTH SEMESTER – April 2009

MT 6602 – DATA STRUCTURE & ALGORITHM

 

 

 

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

 

 

PART – A ( 10 ´ 2 = 20)

Answer ALL questions

  1. Define atomic data.
  2. What is pseudocode?
  3. What is sequential search?
  4. List any two variations of sequential search.
  5. Explain (i) random list (ii) ordered list.
  6. What is parsing?
  7. Define Fibonacci series recursively.
  8. Define complete binary tree
  9. What is sort efficiency?
  10. Explain selection sort.

 

PART – B ( 5 ´ 8 = 40)

Answer any FIVE questions

  1. Write a short note on (i) ADT operators and (ii) ADT structure.
  2. Write the algorithm for probability search.
  3. Explain the four operations associated with linear list.
  4. Write an algorithm to convert a decimal number to a binary number.
  5. Explain the queue operations (i) Enqueue (ii) Dequeue (iii) Queue Front and
    (iv) Queue Rear.
  6. Write a program to find factorial of a given number recursively.
  7. What is heap sort? Write its algorithm.
  8. Explain the straight insertion sort with example.

 

PART – C (2 ´ 20 = 40)

Answer any TWO questions

  1. What is binary search? Write its algorithm.
  2. What is backtracking? Explain eight queens problem and write its algorithm.
  3. How do you implement queues in arrays? Write the complete algorithm.
  4. Write the algorithms of Babble sort and Quick sort and also discuss their efficiency.

 

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

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

ZA 36

B.Sc. DEGREE EXAMINATION – MATHEMATICS

SIXTH SEMESTER – April 2009

MT 6603/MT 6600 – COMPLEX ANALYSIS

 

 

 

Date & Time: 18/04/2009 / 9:00 – 12:00        Dept. No.                                                          Max. : 100 Marks

 

 

PART – A

Answer ALL questions:                                                                               (10 x 2 = 20)  

  1. Find the absolute value of .
  2. Show that does not have a limit as .
  3. When do you say that a sequence of function converges uniformly?
  4. Using Cauchy integral formula evaluate where C is the circle .
  5. Find zeros of the function .
  6. When do you say that a point “a” is an isolated singularity of ?
  7. Calculate the residue of .
  8. State Cauchy’s residue theorem.
  9. Define a bilinear transformation.
  10. When do you say that a function is conformal at ?

PART – B

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

  1. Illustrate by an example that C.R. equations are not sufficient for differentiability at a point.
  2. State Liouville’s theorem and deduce the Fundamental theorem of algebra from it.
  3. Let be analylic in a region D bounded by two concentric circles and on the boundary. Let be any point in D. Show that

.

  1. State and prove Maximum modules theorem.
  2. Expand as a power series in the region .
  3. Evaluate where C is the square having vertices

.

 

 

  1. Using contour integration evaluate .
  2. Find the image of the circle under the map .

PART – C

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

 

  1. (a) Let nbe a function defined on the region D such that u and v and

their first order partial derivatives are continuous in D. If the first order partial derivatives of u and v satisfy the C.R. equations at , show that  is differentiable at

(b) Let be a given power series. Show that there exists a number R such that

such that

(i)  the series converges absolutely for every z with

(ii) if , the convergence uniform in .

(iii) if , the series diverges.                                                                        (10+10)

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

(b) Expand using Taylors series about the point .                                       (15+4)

  1. (a) State and prove Argument theorem.

(b) State and prove Rouche’s theorem.

(c) Prove that .                                                                                      (7+6+7)

  1. (a) Show that the transformation maps the circle into a straight

line given by .

(b) Find the bilinear transformation which maps the points respectively to

.

(c) Show that a bilinear transformation where maps the real axis into

itself if and only if a, b, c, d are real.                                                                          (6+6+8)

 

 

 

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

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

   B.Sc. DEGREE EXAMINATION – MATHEMATICS

THIRD SEMESTER – NOVEMBER 2010

MT 3502/MT 5503 – ASTRONOMY

 

 

 

Date : 02-11-10                     Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

PART – A

 

Answer ALL Questions                                                                                          (10 x 2 = 20 marks)

 

 

  1. Define diurnal motion of a star.
  2. Define ecliptic.
  3. Define Equinoxial points.
  4. What is the effect of refraction on the position of a celestial body?
  5. Define geocentric parallax.
  6. State Kepler’s laws of planetary motion.
  7. Define age of the moon.
  8. Define annular eclipse.
  9. What are inner planners?
  10. Write a note on Asteroids.

PART – B

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

 

 

  1. Define sidereal time and prove that its is equal to R.A ± hour angle of a star.
  2. Define circumpolar star and find the condition for any star to be circumpolar.
  3. Trace the changes in the length of the day for a place on earth’s equator.
  4. Compare refraction and geocentric parallax.
  5. Find the effect of refraction on R.A and declination of a star.
  6. Define sidereal month and synodic month for the moon and find the relation between them.
  7. Compare solar and lunar eclipses.
  8. Write a note on comets.

PART – C

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

 

 

  1. a) Explain the equatorial system to fix a celestial body. Bring out the merits and demerits.           (10)

 

 

  1. b) Define twilight and find the condition for twilight to last the whole night. (10)

 

  1. a) Derive Newton’s deductions from kepler’s laws.                                                                (12)
  2. b) Define equation of time and prove that it vanishes four times a year. (8)
  3. a) Derive tangent formula for refraction.
  4. b) Explain different calendars.
  5. a) Trace the changes in the elongation and phases of the moon in one lunation.
  6. b) Find the maximum number of eclipses in a year.

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Loyola College B.Sc. Mathematics April 2009 Algebra, Calculus & Vector Analysis Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE Examination – Mathematics

Third Semester – OCT/NOV 2010

MT 3501/MT 3500 – Algebra, Calculus and Vector Analysis

 

 

Date & Time:                                Dept. No.                                                    Max. : 100 Marks

 

 

PART – A

Answer ALL questions.                                                                                                  (10 ´ 2 = 20)

  1. Evaluate
  2. Find the Jacobian of the transformation x = u (1 + v) ; y = v (1 + u).
  3. Find the complete solution of z = xp + yq + p2 – q2.
  4. Solve
  5. For , find div at (1, -1, 1)
  6. State Green’s theorem.
  7. What is L(f¢¢ (t))?
  8. Compute
  9. Find the sum and number of all the divisors of 360.
  10. Define Euler’s function f(n) for a positive integer n.

 

PART – B

Answer any FIVE questions                                                                                          (5 ´ 8 = 40)

  1. Evaluate by changing the order of the integration.
  2. Express in terms Gamma functions.
  3. Solve z2( p2+q2 + 1 ) = b2
  4. Solve p2 + q2 = z2(x + y).
  5. Find
  6. Find
  7. Prove that
  8. Show that 18! + 1 is divisible by 437

 

PART – C

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

  1. (a) Evaluate  taken through the positive octant of the sphere x2 + y2 + z2 = a2.

(b)  Show that

  1. (a) Solve (p2 + q2) y = qz.

(b)  Solve (x2 – y2)p + (y2 – zx)q = z2 – xy

  1. (a) Verify Gauss divergence theorem for taken over the region bounded by the planes x = 0, x = a, y = 0 y = a, z = 0 and z = a.

(b)  State and prove Fermat’s theorem.

  1. (a) Using Laplace transform solve  given that .

(b)  Show that if n is a prime and r < n, then (n – r)!  (r – 1)! + (-1)r – 1 º 0 mod n.

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Loyola College B.Sc. Mathematics April 2009 Alg.,Anal.Geomet. Cal. & Trign. – I Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

   B.Sc. DEGREE EXAMINATION – MATHEMATICS

FIRST SEMESTER – NOVEMBER 2010

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

 

 

 

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

Time : 1:00 – 4:00

 

PART – A

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

 

  1. Find yn when
  2. Show that, in the parabola y2=4ax,  the subtangent at any point is double the abscissa.
  3. Find the radius of curvature of xy=30 at the point (3,10).
  4. Define evolutes.
  5. Form equation given that 3+2c is a root.
  6. If α,β,γ, are the roots of the equation x3+px2+qx+r=0 find the value of ∑α2.
  7. Evaluate
  8. Prove that cosh =
  9. Find the polar of the point (3,4) with respect the parabola y2=4ax.
  10. Define conormal and concyclic points.

 

PART  –  B

 

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

 

  1. Show that in the curve hy2=(x+a)3 the square of the subtangent varies as the subnormal.
  2. Find the radius of curvature at the point ‘t’ of the curve

x=a(cost+tsint); y=a(sint-tcost).

  1. Find the coordinates of the centre of curvature at given point on the curve y=x2;
  2. Solve the equation x4+2x3-5x2+6x+2=0 given that 1+is a root of it.
  3. Find the real root of the equation x3+6x-2=0 using Horner’s method.
  4. Expand sin3θ cos4θ in terms of sines of multiples of θ.
  5. If sin(θ+iφ) =tanα + isecα , prove that cos2 θ cosh2φ =3.
  6. Show that the area of the triangle formed by the two asymptotes of the rectangular hyperbola xy=c2 and the normal at (x1,y1) on the hyperbola is .

PART – C

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

 

  1. a) Prove that if y=sin(msin-1x), then (1-x2)y2-xy1+m2y=0.
  2. b) Show that the evolute of the cycloid x=a(θ – sinθ);y=a(1-cosθ) is another cycloid.

 

  1. a) solve 2x6-9x5+10x4-3x3+10x2-9x+2=0.
  2. b) If α is a root of the equation x3+x2-2x-1=0 show that α2 -2 is also a root.

 

  1. a) if u=log tan show that tanh   = tan  and θ = -i log tan
  2. b) sum to infinity the series
  3. a) Find the locus of mid points of normal chords to the ellipse
  4. b) Find the polar of the point (x1, y1) with respect to the parabola y2=4ax.

 

­­­­­­­­­

 

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

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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

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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)

Answer any FIVE questions.

  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)

Answer Any Two

  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 Numerical Methods Question Paper PDF Download

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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 Mechanics – II Question Paper PDF Download

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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 p–r 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 Math 1 Question Paper PDF Download

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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

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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

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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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