Draw graphs of kinetic and potential energies of simple harmonic motion as function of displacement.
State the theorem of parallel axes of moment of inertia.
State the Kepler’s laws of planetary motion.
What is gravitational red shift?
Define Poisson’s ratio.
List any two phenomena which bring out the surface tension of water.
Distinguish holonomic constraints from non-holonomic constraints.
Two photons travel in opposite direction. Find the relative velocity of one with respect to the other.
What are beats?
How is a stationary wave formed?

PART B (4 x 7 ½ m = 30 m)

ANSWER ANY FOUR QUESTIONS

Prove that a small oscillation of a liquid in a U-tube is simple harmonic.
(a)State and explain the principle of equivalence of general theory of relativity. (b)Explain any one experiment in support of general theory of gravitation.
Explain the torsional oscillation method of determining rigidity modulus of the material of a wire.
(a) Write down the Lorentz transformation equations (b) Determine the velocity of a particle whose kinetic energy is equal to its rest mass energy.
Determine the fundamental frequency and the first overtones of (i) open pipe and (ii) closed pipe

PART C (4 x 12 ½ m = 50 m)

ANSWRER ANY FOUR QUESTIONS

(a) Find the moment of inertia of a thin rod about an axis perpendicular to its length and passing through one of its ends.(b)Obtain an expression for acceleration of a body rolling down an inclined plane
(a) Determine the time period of an earth satellite which is orbiting near the surface of earth (b) Estimate the height of a geostationary satellite (c) What is escape velocity? Find an expression for it.
Obtain an expression for the volume rate of flow of a liquid through a capillary tube by Poiseuilles method.
Solve the problems of (i) Atwood’s machine and (ii) simple pendulum by Lagrangian dynamics
What is Doppler effect? Find expression for the apparent frequency in the case of (i) source moving towards stationary observer and (ii) observer moving towards stationary source.

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Loyola College B.Sc. Mathematics April 2008 Physics For Mathematics – II Question Paper PDF Download

October 26, 2017 by 01 prakash

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

FG 15

FOURTH SEMESTER – APRIL 2008

PH 4206 / 4200 – PHYSICS FOR MATHEMATICS – II

Date : 24/04/2008 Dept. No. Max. : 100 Marks

Time : 9:00 – 12:00

PART – A

Answer all questions. (10 x 2 = 20 Marks)

Convert the given decimal into binary number 56.253₁₀=X₂
State Demorgan’s law.
State Pauli’s exclusion principle.
Define Photoelectric effect.
What is meant by Binding energy?
Define Nuclear Fission.
Does the sound waves travel through vacuum? Give reason.
Define the term acoustics and ultrasonic waves.
Define a perfectly Black body.

10.State De-Broglie’s wavelength.

PART-B

Answer any four questions. (4 x 7.5 = 30 Marks)

11.Reduce the following Boolean function with the help of a karnaugh map

F=S(1,2,3,5,7,8,9,10,11,12,14).

12.Describe the production of X-rays with a neat diagram.

13.Discuss the similarities between an atomic nucleus and a liquid drop.

14.Derive Newton’s formula for velocity of sound in a medium.

15.Why the classical theory fails? Give reasons.

PART-C

Answer any four questions. (4 x 12.5 = 50 Marks)

16.Draw a neat circuit diagram of JK flip-flop and explain it’s working with Truth table.

Describe in detail, the Continuous and Characteristic X-ray spectra

18.Discuss the general properties of a nuclei and explain the

Binding energy/nucleon versus A curve.

19.State Reverberation time. Derive Sabine’s formula for it.

20.Derive Schrodinger’s time dependent and time independent wave equations.

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

October 26, 2017 by 01 prakash

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

XZ 21

FIFTH SEMESTER – APRIL 2008

MT 5504 – OPERATIONS RESEARCH

Date : 06-05-08 Dept. No. Max. : 100 Marks

Time : 1:00 – 4:00

PART-A

Answer all the questions (10 ´ 2 = 20)

1) What are the characteristics of operations research ?

2) Explain the terms slack and surplus variables.

3) What is meant by pure strategy ?

4) What is the value of the game ?

5) What is price break ?

6) What is an assignment problem?

7) Define traveling salesman problem.

8) What are the rules to be adopted in drawing the network ?

9) Write the difference between PERT and CPM ?

10) Define critical path.

PART- B

Answer any two questions (5 ´ 8 = 40)

11) A company produces two types of pens, say A and B. Pen A is a superior quality and pen B is lower quality. Profits on pen A and pen B are Rs.5 and Rs.3 per pen respectively. Raw materials required for each pen A is twice as that of pen B. the supply of raw materials is sufficient only for 1000 pens of B per day. Pen A requires a special clip and only 400 clips are available per day. For pen B only 700 clips are available per day. Prepare a mathematical model so that the company can make maximum profit.

12) Solve by graphical method

Maximize Z = 5x + 8y

Subject to the constraints

2x + y £ 8

x + 3y £ 10 where x, y ³ 0

13) Solve the LPP by simplex method

Maximize Z = 20x + 30y

Subject to 3x + 2y £ 36

5x + 2y £ 50

2x + 6y £ 60 where x, y ³ 0.

14) Solve by dominance property the following game

player B

1 7 2

player A 6 2 7

6 1 6

15) Four jobs can be processed on four different machines, one job on one machine.

Resulting times in minutes vary with assignments are given bellow. Find the

optimum assignment of jobs to machines and the corresponding time.

Machines

15 13 14 17

11 12 15 13

13 12 10 11

15 17 14 16

Jobs

16)

Determine the following: a path, a loop, a directed loop, a tree and a spanning

tree for the following network

17) Find the shortest route for the following network.

18) A manufacturer purchases 10,000 items per year for sales in his shop. The unit cost is Rs.10/year, the holding cost is 80 paise/month and cost of reordering is Rs.200. Determine economic lot size, optimal total yearly cost, number of orders per year and time between orders if shortages are not allowed.

PART- C

Answer any Two questions (2 ´ 20 = 40)

19) Solve the LPP by Big M- method, Minimize Z = 4x₁ + x₂

subject to 3x₁ + x₂ = 3

4x₁ + 3x₂ ³ 6

x₁ + 2x₂ £ 4 where x₁, x₂ ³ 0.

20) The following table gives the activities in a construction project.

Activity 1-2 1-3 2-3 2-4 3-4 4-5

Duration 20 25 10 12 6 10

Draw the network for the project. Find the critical path and project duration.

Find the total float for each activity.

21) (a) Solve the LPP by Dual Simplex method ,

Minimize Z = 3x + 2y

subject to 7x + 2y ³ 30

5x + 4y ³ 20

2x + 8y ³ 16 where x, y ³ 0

(b) Find the value of the following game by graphical method

( 10 + 10 )

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Loyola College B.Sc. Mathematics April 2008 Modern Algebra Question Paper PDF Download

October 26, 2017 by 01 prakash

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

XZ 13

FOURTH SEMESTER – APRIL 2008

MT 4502 / 4500 – MODERN ALGEBRA

Date : 26/04/2008 Dept. No. Max. : 100 Marks

Time : 9:00 – 12:00

PART – A

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

Define an equivalence relation on a set.
Define a binary operation on a set.
Define a cyclic group.
Define a quotient group of a group.
Define an isomorphism.
Define a permutation group.
Define a division ring.
Define a field.
Define an integral domain.
What is a Gaussian integer?

PART – B

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

If G is a group, then prove that

for every
for all

Prove that anon – empty subset H of a group G is a subgroup of G if and only if

(i)

(ii)

If H is a subgroup of a group G, then prove that any two left Cosets of H in G either are identical or have no element in common.
If H is a subgroup of index 2 in a group G, prove that H is a normal subgroup.
If is a homomorphism of a group G into a group , prove that

(i) , the identity element of G¹

(ii) for all

Show that the additive group G of integers is isomorphic to the multiplicative group
Prove that the intersection of two subrings of a ring R is a subring of R.
Find all the units in Z(i).

PART – C

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

State and prove the Fundamental theorem of arithmetic.
a) State and prove Lagrange’s theorem.

b) Show that every subgroup of an abelian group is normal. (14+6)

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

b) Define an endomorphism an epimorphism and an automorphism.

State and prove unique factorization theorem.

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

October 26, 2017 by 01 prakash

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

XZ 18

FIFTH SEMESTER – APRIL 2008

MT 5500 – MECHANICS – II

Date : 28-04-08 Dept. No. Max. : 100 Marks

Time : 1:00 – 4:00

PART – A

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

Define centre of mass.
Define centre of gravity of a compound body.
State the principle of virtual work.
Define suspension bridge.
Define amplitude.
Define periodic time of S.H.M.
Write down the differential equation of the central orbit in p-r coordinates.
State the theorem of parallel axis.
Define equimomental system.
State D’Alemberts principle.

PART – B

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

A piece of uniform wire is bent in the shape of an isosceles triangle sides are ‘a’ ‘a’ and ‘b’. Show that the distance of C.G from the base of the triangle is .
A regular hexagon is composed of six equal heavy rods freely jointed together and two opposite angles are connected by 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 .
A particle moves in S.H.M 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 motion is .
The velocity of a particle along and perpendicular to radius vector from a fixed origin are and components of acceleration are and .
An elliptic lamina of semi axes a and b swings about a horizontal axis through one of the foci in a vertical plane. Find the length of the S.E.P.
Find the resultant of two simple harmonic motions of the same period in the same straight line.
Find the centre of gravity of a uniform solid right circular cone.
Find the components of the velocity and acceleration along radial and transverse directions.

PART – C

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

a) Find the centre of gravity of the area enclosed by the parabolas and .

b) A uniform chain, of length , is to be suspended from two points A and B, in the same horizontal line so that either terminal tension is n times that at the lowest point. Show that the span AB must be . …….(10+10)

a) Define catenary and derive the equation of the catenary.

b) Four equal rods, each of length a, are jointed to form a rhombus ABCD and the points B and D are joined by a string of length . The system is placed in a vertical plane with A resting on a horizontal pane and AC vertical. Prove that the tension in the string is where W is the weight of each rod. ……(10+10)

a) Obtain the differential equation of a central orbit in the form .

b) Show that the M.I about the x-axis of the parabola bounded by the latus rectum supposing the density at each point to vary as cube of the abcissa where M is the mass of the lamina. ……(10+10)

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

October 26, 2017 by 01 prakash

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034 LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034B.Sc. DEGREE EXAMINATION – MATHEMATICSFOURTH SEMESTER – APRIL 2008MT 4501 – MECHANICS – I
Date : 29/04/2008 Dept. No. Max. : 100 Marks Time : 9:00 – 12:00 PART – A Answer ALL questions.: (10 x 2 = 20)1. Define coplanar forces.2. Distinguish between internal and external forces.3. Define a couple.4. Define moment or torque of a force .5. Define dynamical friction.6. Define angular velocity.7. A body of mass ‘m’ is carried in a lift moving with downwards acceleration f. Find the pressure on the lift.8. Define cone of friction.9. Define angle of projection.10. Define elastic body.
PART – BAnswer any FIVE questions. (5 x 8 = 40)11. Two forces acting on a particle are such that if the direction of one of them is reversed, the direction of the resultant is turned through a right angle. Prove that the forces must be equal in magnititude.12. Prove that the sum of any two coplanar forces about any point in the plane of forces equals the moment of the resultant about that point.13. 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’ a part. If two pw and qw are suspended from P and Q respectively, show that when the reactions at R and S are equal, the distance PR is given by 14. A system of forces in the plane of is equivalent to a single force at A1 acting along the internal bisector of the angle BAC and a couple of moment G1. If the moments of the system about B and C are respectively G2 and G3, prove that .15. 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 w.r.t the other is zero, prove that the line joining them subtends at the centre an angle whose cosine is .16. Two particles of masses m1 and m2 (m1>m2) are connected by means of light inextensible string passing over a light, smooth, fixed pulley. Discuss the motion.17. If t be the time in which a projectile reaches a point P in its path and t1, the time from P till it reaches the horizontal plane through the point of projection, show that the height of P above the horizontal plane is gtt1.18. 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 as (1+e): (1-e).
PART – CAnswer any TWO questions. (2 x 20 = 40)19. a) State and prove Lami’s theorem.b) Three equal strings of no sensible weight are knotted together to form an equilateral and a weight W is suspended from A. If the triangle and the weight be supported with BC 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+10)
20. a) A ladder which stands on a horizontal ground leaning against a vertical wall is so loaded that its centre of gravity is at the distances a and b from the lower and upper ends respectively. Show that if the ladder is in limiting equilibrium, its indination to the horizontal is given by where and 1 are the coefficients of friction between the ladder and the ground and the wall respectively.b) A particle is projected upwards under the action of gravity in a resisting medium where the resistance varies as the square of the velocity. Discuss the motion. (10+10)
21. a) Obtain the equation of the path of a projectile in Cartesian form.b) A particle is projected so as to clear two walls, first of height a at a distance b from the point of projection and the second of height b at a distance a from the point of projection. Show that the range on the horizontal plane is and the angle of projection exceeds . (10+10)
22. a) Two smooth spheres m1 and m2 moving with velocities u1 and u2 respectively in the direction of line of centres impinge directly. Discuss the motion of each mass after impact, given that e is the coefficient of restitution.b) A body, sliding down a smooth inclined plane, is observed to cover equal distances, each equal to a, in consecutive intervals of time t1 and t2. Show that the indination of plane to the horizon is . (10+10)

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Loyola College B.Sc. Mathematics April 2008 Mathematics For Physics Question Paper PDF Download

October 26, 2017 by 01 prakash

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

XZ 3

THIRD SEMESTER – APRIL 2008

MT 3102 / 3100 – MATHEMATICS FOR PHYSICS

Date : 07/05/2008 Dept. No. Max. : 100 Marks

Time : 1:00 – 4:00

SECTION A

Answer ALL questions. (10 x 2 = 20)

Write the Leibnitz’s formula for the nth derivative of a product uv.
Prove that the subtangent to the curve y=a^x is of constant length.
Prove that =
Find L[e^-2tsin2t]
If y = log ( 1+x ).then find D²y
Expand tan 7q in terms of tanq
Prove that the matrix is orthogonal
If tan = tan h then show that cosx coshx = 1
Find the A.M. of the following frequency distribution.

x : 1 2 3 4 5 6 7

f : 5 9 12 17 14 10 6

Write the general formula in Poisson’s distribution.

SECTION B

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

11.If y=sin^-1x, prove that ( 1-x² )y₂–xy₁=o and (1-x²)y_n+2-(2n+1)xy_n+1-n²y_n=o

12.Find the length of the subtangent, subnormal, tangent and normal at the point (a,a) on the

cissoid y² =

Sum to infinity the series:-

Verify Cayley Hamilton theorem for the matrix

A =

If sin () = tan ( x + iy) , Show that
If sin (A + iB) = x + iy ,

Prove that (i) (ii)

Find L^-1
Ten coins are tossed simultaneously. Find the probability of getting at least seven heads.

SECTION C

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

(a) Prove that 1 +

(b) Find the mean and standard deviation for the following table, giving the age

distribution of 542 members.

Age in years

20-30

30-40

40-50

50-60

60-70

70-80

80-90

No. of members

132

153

140

20.(a) Prove that 64cos⁶q – 80 cos⁴q + 24 cos²q – 1

(b) Expand sin³q cos⁴q in terms of sines of multiples of angles. (10 + 10)

21.a)Find the maxima and minima of x⁵-5x⁴+5x³+10

b)Find the length of the subtangent and subnormal at the point `t’ of the curve

x = a(cost + t sint),

y = a(sint-tcost) (10 + 10)

a)Solve the equation , given that y =when t = 0.

b)Find L^-1 (15 + 5)

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Loyola College B.Sc. Mathematics April 2008 Mathematical Statistics Question Paper PDF Download

October 26, 2017 by 01 prakash

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMAT., PHYSICS & CHEMIST.

NO 15

FOURTH SEMESTER – APRIL 2008

ST 4206 / 4201 – MATHEMATICAL STATISTICS

Date : 28/04/2008 Dept. No. Max. : 100 Marks

Time : 9:00 – 12:00

PART – A 10×2 = 20 marks

Answer all questions

Define conditional probability
Define independent and mutually exclusive events
State the definition of random variable.
Let X have the pdf f(x) = 1/3, -1<x<2, find E(X)
State any two cases where poisson distribution can be applied.
Define the p.d.f. of continuous uniform distribution.
State any two applications of t-test
State Neyman – Pearson Lemma.
Define Maximum Likelihood Estimator.
Define Null and Alternative hypothesis.

PART – B 5×8 = 40 marks

Answer any five questions

State and prove Additional theorem of probability.
An um contains 6 white, 4 red and 9 black balls. If 3 balls are drawn at random, find the probability that (i) two of the balls drawn are white (ii) one is of each colour (iii) none is red (iv) atleast one is white.
Let X and Y be two r.v’s each taking three values – 1,0 & 1 and having the joint probability distribution

X Y	-1	0	1
-1	0	.1	.1
0	.2	.2	.2
1	0	.1	.1

(i) Show that X and Y have different expectations

(ii) Prove that X and Y are uncorrelated

(iii) Find Var (X) and Var (Y).

From a bag containing 3 white and 5 black balls, 4 balls are transferred into an empty bag. From this bag a ball is drawn and is found to be white . what is the probability that out of four balls transferred ,3 are white and 1 is black ?
Derive the MGF of Poisson distribution and hence obtain its mean and variance.
Let the random variable X have the marginal density

f₁(x)=1,-1/2<x<1/2 and let the conditional density of Y given X=x be

f(y│x)= 1, x<y<x+1, -1/2<x<0

= 1, -x<y<1-x , 0<x<1/2 . Show that X and Y are uncorrelated.

If X and Y are independent gamma variates with parameters µ and v respectively, show that the variables u=X+Y,Z=X/(X+Y) are independent and U is gamma variate with parameter (µ+ v ) and Z is a β₁ (µ, v ) variate
Define the following (i) Unbiased ness (ii) Consistency (iii) Efficiency of an estimator

PART – C 2×20 = 40 marks

Answer any two questions

a) State and prove Baye’s theorem

b) Derive the recurrence relation satisfied by the central moments of the

Poisson distribution.

a) Suppose that two – dimensional continuous random variable (x,y) has joint d.f. given by f(x,y) = 6x² y, o<x<1, o<y<1,

= 0, otherwise

Find (i) P(0<X<3/4,1/3<Y<2),,(ii) P(X +Y <1) (iii) P(X>Y) (iv) P(X<1/Y<2)

b) State and prove Chebyshev’s inequality.

a) Discuss the properties of normal distribution.(8)

b) The mean yield for one – acre plot is 662 kgs with s.d. of 32 kgs. Assuming normal diet, how many one – acre plots in a batch of 1000 plots would you expect to have yield, i) over700 kgs ii) below 150 kgs,

iii) what is the lowest yield of the best 100 plots?(12)

a) Derive the probability density function of t-distribution with n degrees of freedom

b) In random sampling from a normal population N(µ, s²), find the estimators of the parameters by the method of moments.

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

October 26, 2017 by 01 prakash

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

XZ 20

FIFTH SEMESTER – APRIL 2008

MT 5502 – LINEAR ALGEBRA

Date : 03/05/2008 Dept. No. Max. : 100 Marks

Time : 1:00 – 4:00

SECTION – A

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

Illustrate by an example that union of two subspaces need not a subspace of a vector space.
Give an example of a linearly dependent set of vectors in R² over R.
“Every linearly independent set of vectors is a basis.” True or False. Justify.
Show that kernel of a homomorphism in a vector space is a subspace.
If V is an inner product space show that where a and b are scalars.
For T є A(v)define eigen value and eigen vector of T.
Find the trace of
Is the following matrix skew symmetric?
If T є A(v)is a Hermitian show that all its eigen values are real.
When do you say that a linear transformation T on V is unitary?

SECTION – B

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

Let V be a vector space of dimension n and be libearly independent vectors in V. Show that there exists n-r new vectors in V such that is a basis of V.
If V is a vector space of finite dimension that is the direct sum of its subspace U and W show that
Verify that defined by T(a,b)=(a-b, b-a, -a) for a, b є R is a vector space homomorphism. Find the rank and nullity of T.
State and prove triangular inequality.
Show the T є A(V)is invertible if and only if the constant term of the minimal polynomial for T is not zero.
If V has dimension n andT є A(V), show that the rank of T is equal to the rank of the corresponding matrix m(T) in Fn.

Check the consistency of the following set of equations.

x₁+2x₂+x₃+11

4x₁+6x₂+5x₃=8

2x₁+2x₂+3x₃=19

If for all v є V, show that T is unitary.

SECTION – C

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

(a) Find whether the vectors (2, 1, 1, 1), (1, 3, 1, -2) and (1, 2, -1, 3) in R⁴ are linearly dependent or independent.

(b) If U and V are vector spaces of dimension m and n respectively over F, show that the vector space Hom(U,V) is of dimension mn.

a) Apply Gram Schmidt orthonormalization process to obtain an orthonormal basis for the subspace of R⁴ generated by the vectors. (1, 1, 0, 1), (1, -2, 0, 0),

(1, 0, -1, -2).

b) If are distinct eigen values of T є A(V) and if v₁, v₂, …, v_n are eigen vectors of T belonging respectively, show that v₁,v₂,…,v_n are linearly independent over F.

a) Let V be a vector space of dimension n over F and let T є A(V). If m₁(T) and m₂(T) are the matrices of T relative to two bases {v₁, v₂, …, v_n} and {w₁,w₂,…,w_n} of V respectively, show that there is an invertible matrix C in F_n such that m₂(T)=Cm₁(T)C^-1.

b) Show that any square matrix A can be expressed uniquely as the sum of a symmetric matrix and a skew-symmetric matrix.
c) If A and B are Hermitian show that AB+BA is Hermitian and AB-BA is skew Hermitian.

a) Find the rank of

b) If T є A(V)show that T* є A(V)and show that

(i)

(ii)

(iii) (T)^* = T^*

(iv) (T^*)^* = T

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

October 26, 2017 by 01 prakash

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

AK 22

FIFTH SEMESTER – APRIL 2008

MT 5400 – GRAPH THEORY

Date : 05/05/2008 Dept. No. Max. : 100 Marks

Time : 9:00 – 12:00

SECTION A

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

Define a graph.
Give an example for the following

(i) Bipartite graph (ii) Regular graph

Draw a graph with five vertices and find its complement.
Define a walk and a path.
What is a cut point? Give an example.
Give an example of a graph, which is Eulerian but non-Hamiltonian.
State Fleury’s algorithm.
Define a planar graph and give an example of a non-planar graph.
Define the center of the tree.
Define the chromatic number of a graph.

SECTION B

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

(a) Show that every cubic graph has an even number of vertices.

(b) Prove that .

(a) Let G be a k-regular bigraph with bipartition and . Prove that

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

(a) In a graph, prove that any u – v walk contains a u – v path.

(b) Show that a closed walk of odd length contains a cycle.

(a) Prove that a graph G with p vertices and is connected.

(b) If G is not connected then show that is connected.

Prove that a graph G with at least two points is bipartite if and only if all its cycles are of even length.
Show that the following statements are equivalent for a connected graph G.

G is Eulerian.
Every vertex of G has even degree.
The set of edges of G can be partitioned into cycles.

If G is a graph with p vertices, and , then prove that G is Hamiltonian.
(a) Prove that is non-planar.

(b) State and prove Euler’s theorem.

SECTION C

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

Prove that the maximum number of edges among all p point graphs with no triangles is , where [x] denotes the greatest integer not exceeding the real number x.

Let G be a connected graph with at least three vertices. Prove the following:

G is a block if and only if any two vertices of G lie on a common cycle.
Any two vertices of G lie on a common cycle if and only if any vertex and any edge of G lie on a common cycle.

Let G be a (p, q) graph. Show that the following statements are equivalent.

G is a tree.
Every two vertices of G are joined by a unique path.
G is connected and p = q + 1.
G is acyclic and p = q + 1.

(a) State and prove five colour theorem.

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

G is 2-colourable.
G is bipartite.
Every cycle of G has even length.

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Loyola College B.Sc. Mathematics April 2008 Fluid Dynamics Question Paper PDF Download

October 26, 2017 by 01 prakash

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

XZ 15

FIFTH SEMESTER – APRIL 2008

MT 5401 – FLUID DYNAMICS

Date : 05/05/2008 Dept. No. Max. : 100 Marks

Time : 9:00 – 12:00

PART – A

Answer all: 10 x 2 = 20

1.Define steady and unsteady flow.

2.Write the Euler’s equation of motion in terms of spherical polar co ordinates.

3.What are stagnation points?

4.What are the applications of Pitot tube?

5.Explain the term complex potential.

6 Define Source and Sink.

7.State a fundamental property of vortex.

8.Define a vorticity vector.

9.Prove that flow is irrotational for

10.What is lift of an Aerofoil ?

PART – B

Answer any five: 5 x 8 = 40

The velocity in a three dimensional flow field for an incompressible fluid is

given by . Determine the equation of streamlines passing

through the point (1,1,1).

Derive the relationship between Eulerian and Lagrangian points in space.
Show that the velocity potential satisfies the Laplace

equation. Also determine the stream lines.

Derive the Euler’s equation of motion.

Briefly explain Pitot tube.

Stream is rushing from a boiler through a conical pipe, the diameters of the ends of which are D and d. If V and v be the corresponding velocities of the stream and if the motion be supposed to be that of divergence from the vertex of the cone prove that where k is the pressure divided by density.

(P.T.O)

Find the vorticity of the fluid motion in spherical polar coordinates

, and .

Define aerofoil and discuss about its structure.

PART –C

Answer any two: 2 x 20 = 40

19a) A mass of fluid is in motion so that the lines of motion lie on the surface of

coaxial cylinders. Show that the equation of continuity is

b) For a 2-D flow the velocities at a point in a fluid may be expressed in the

Eulerian co ordinate by u= 2x+2y+3t and v = x+ y+ , determine them in

Lagrangian.

20 a) If the velocity of an incompressible fluid at the point (x ,y, z) is given by

, prove that the liquid motion is possible and that the

velocity potential is . Also determine the stream lines.

Draw and explain Venturi tube.

21 a) What arrangements of source and sinks will give rise to the function

? Draw a rough sketch of the stream lines.

b) The particle velocity for a fluid motion referred to rectangular axes is given by the components , where A is a constant. Show that this is a possible motion of an incompressible fluid under nobody force in an infinite fixed rigid tube. Also determine the pressure associated with this velocity field where .

a) Derive Joukowski transformation.

b) State and prove Kutta Jowkowski theorem.

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Loyola College B.Sc. Mathematics April 2008 Fin. Accounts & Fin. Stat. Analysis Question Paper PDF Download

October 26, 2017 by 01 prakash

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

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

RO 13

FOURTH SEMESTER – APRIL 2008

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

Date : 24/04/2008 Dept. No. Max. : 100 Marks

Time : 9:00 – 12:00

SECTION A

Answer all questions: 10 x 2 =20 marks

What is Overdraft?
How does one treat recovery of bad debts in accounts?
Differentiate between Cash discount and Trade discount?
a. Only __________transactions are recorded in the Single Column Cash Book

A ___________is a short explanation of the transaction.

What is a voucher?
Who are non- profit seeking concerns? Give examples
List any two advantages of Subsidary books?
Why is a Trial balanace prepared?
Ascertain purchases from the following figures:

Cost of goods sold Rs. 80,700

Opening Stock Rs. 5,800

Closing stock Rs. 6,000

Calculate the Capital fund by preparing Balance sheet:

Rs. Rs

Cash 6,000 Bank balance 35,000

Subscription outstanding 2,000 Furniture 20,000

Building 1,00,000 Salaries outstanding 4,000

SECTION B

Answer any five questions: 5 x 8 = 40 marks

What are Assets? Discuss the various types of assets?
Write a brief note on the importance of ratio analysis.
Briefly explain the concepts which form the backbone of Accounting.

Moon Ltd had a balance of Rs.75,000 in its Profit & Loss account on 1^st April 2006. During 2006-2007, its profit before tax amounted to

Rs. 7,62,500. The income tax provision for the year amounted to be

Rs. 3,47,500. 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 Appropriate account.

Statement of the financial position of Mr. Ram is given below:

Liabilities 2006 2007 Assets 2006 2007

Creditors 2,900 2,500 Cash 4,000 3,000

Capital 73,900 61,500 Debtors 2,000 1,700

Stock 800 1,300

Building 10,000 8,000

Land 60,000 50,000

——— ——— ———- ———

76,800 64,000 76,800 64,000

Additional information:

(i) There were no drawings

(ii) There was no purchase or sales of either building or other fixed assets.

Prepare a Statement of cash flow.

Journalise the following transactions in the books of Mr. Hari for the month of January 2008

Rs.

1 Hari started business with capital 2,50,000

5 Purchased goods from Shankar 24,000

7 Purchased goods for cash 4,000

8 Purchased chairs 1,500

10 Proprietors drawings 1,000

11 Paid rent to the landlord , Mr. Ram 1,700

13 Received interest 1,000

15 Sold goods to Suresh 4,000

20 Paid to Shankar Rs.23,500 and discount received Rs.500

22 Paid salary to the Manager, Mr. Arun 2,000

Prepare a bank reconciliation statement from the following:

Rs.

Bank balance as per Cash Book 18,500

Cheques deposited but not collected by the bank 6,000

Cheques issued but not presented for payment 4,500

Bank charges debited in the pass book 150

Dividend collected by bank not entered in the cash book 1,200

Insurance premium paid by the bank not entered in the

Cash book 350

M/s Sathish & Co furnishes the following information for the month of January 2008. Prepare a Cash book with Cash and Bank columns:

Jan Rs.

1 Balance in Hand 1,000

Bank Overdraft 3,000

3 Issued a cheque to Sharma 500

5 Received a cheque from Mr. Charles and

Deposited into bank 1,000

7 Drew from bank for office use 500

9 Paid Life insurance premium in cash 250

15 Bank collected dividend 500

SECTION C

Answer any five questions: 2 x 10 =40 marks

The following is the summary of cash transactions of Chennai Literary Club for the year ended 31^st December 2007

Receipts Rs. Payments Rs.

To Balance from last year 3,190 By Rent & Rates 1,680

To Entrance fees 2,250 By Wages 2,450

To Subscription 16,000 By Lighting 720

To Donations 1,650 By Lecture fees 4,350

To Life membership fee 2,500 By Books 2,130

To Interest 140 By Office expenses 4,500

To Sale of furniture By Placed on 6%

(Book value Rs. 500) 720 fixed deposit 8,000

By Cash at bank 2,420

By Cash in hand 200

——– ——–

26,450 26,450

——– ——–

At the beginning of the year, the Club possessed Books worth Rs.20,000 and Furniture worth Rs.8,500. Furniture worth Rs. 500 was sold on 1.1.07 for

Rs. 720 as shown in the above cash summary. Ordinary Subscription in arrears at the beginning of the year amounted to Rs. 350 and at the end of the year Rs. 450.

Prepare Income and Expenditure Account of the Club for the year ended 31^st December, 2007 and a Balance sheet at that date after writing off Depreciation at 5% per annum on Furniture and 10% per annum on Books.

Entrance fees and Donations to be capitalized.

From the following Trial Balance of M/s Ram & Sons, Prepare Trading and Profit and Loss Account for the year ending on 31^st December 2007 and the Balance Sheet as on that date

Dr. Cr.

Purchases and Sales 21,750 35,000

Discount 1,300

Wages 6,500

Salaries 2,000

Travelling expenses 400

Commission 425

Carriage inwards 275

Administration expenses 105

Trade expenses 600

Interest 250

Building 5,000

Furniture 200

Debtors 4,250

Capital 13,000

Cash 7,045

Creditors 2,100

——— ———

50,100 50,100

Additional information:-

Stock on 31^st December 2007, was Rs. 6,000. Depreciate building by 20%. Create a provisions for bad debts at 10% on debtors. Outstanding wages

Rs. 475.

Using the following data, complete the balance sheet below:

Gross profit ratio 20%

Current ratio 1.8

Stock turnover ratio 4times

Debt collection period 20 days

(360 days year)

Long-term debt to equity 40%

Total assets turnover ratio

(on sales) 0.3 times

Credit sales to total sales 80%

Gross profit Rs. 1,08,000

Share capital & Reserves and Surplus Rs. 12,00,000

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Loyola College B.Sc. Mathematics April 2008 Complex Analysis Question Paper PDF Download

October 26, 2017 by 01 prakash

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

XZ 22

SIXTH SEMESTER – APRIL 2008

MT 6600 – COMPLEX ANALYSIS

Date : 16/04/2008 Dept. No. Max. : 100 Marks

Time : 9:00 – 12:00

SECTION – A

Answer ALL questions: (10 x 2 = 20)

Prove that for any two complex numbers and ,

Verify Canchy – Riemann equations for the function

Show that is harmonic.
Define Möbius transformation.
State Cauchy – Goursat theorem.
State Cauchy’s Integral formula.
Evaluate where C is the circle .
Find the Taylor’s series expansion about z=0.
Obtain the Laurent’s series for in .
Find the residue of at .

SECTION – B

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

Prove that the function

is not differentiable at the origin, but Canchy-Riemann equations are satisfied there.

Show that the function is harmonic and find the corresponding analytic function.
Show that the transformation maps the circle onto a straight line .
Evaluate if C is the positively oriented circle
State and prove Liouville’s theorem.
State and prove Residue theorem.
State and prove Taylor’s theorem.
Find the residues of at its poles.

SECTION – C

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

a) If is a regular function of , prove that

b) Find the bilinear transformation that maps the points onto the points .

State and prove Cauchy’s theorem.
Expand in the regions.

ii)

iii)

Using the method of Contour integration, prove that

ii)

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Loyola College B.Sc. Mathematics April 2008 Applied Algebra Question Paper PDF Download

October 26, 2017 by 01 prakash

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

XZ 23

SIXTH SEMESTER – APRIL 2008

MT 6601 – APPLIED ALGEBRA

Date : 21/04/2008 Dept. No. Max. : 100 Marks

Time : 9:00 – 12:00

PART – A

Answer ALL questions: (10 x 2 = 20)

Construct the truth table for the statement formula Pù Q.
What is meant by well formed formula.
When is a statement A said to tautologically imply a statement B?
Define conditional statement.
State Zorn’s lemma
Define Boolean Algebra and give an example of it.
Define Mealy and More automaton.
Give an example of a semi group which is not a monoid.
Define anti homomorphism and anti isomorphism of semi groups.
Define automata homomorphism.

PART – B

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

Show that

( P®(Q®R)ÛP®(ùQÚR) Û(PÙQ) ®R

(ii) ( ù PÙ (ù QÙ R) Ú(QÙR) Ú(PÙR) ÛR

Show that

((PÚQ) Ùù ( ù PÙ(ùQÚùR)) Ú( ùPÙùR) is a tautology.

Obtain a conjunctive normal form of each of the formula as given below

(i) (PÙ(P ®Q) (ii) ù (PÚQ) D (PÙQ)

Obtain the product – of –sums canonical forms of the following formulas.

(i) (PÙQ) Ú ( ù PÚQ) Ú(PÙù Q)

(ii) (PÙQ) Ú ( ù PÙ QÙR)

Let be a lattice ordered set. If Π; show that (L,Π, ) is an algebraic lattice.
State and prove Representation theorem for Boolean Algebra.
For any semigroup (S,o), Show that there exists a set N such that (S,o) is embeddable in (N^N, o).
Show that for any monoid (S,o) there exists a semi automation whose monoid is isomorphic to (S,o).

PART – C

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

a) Show that

(i) ù (PÙQ) ® ( ù PÚ( ù PÚQ) Û (ù PÚQ)

(ii) (PÙQ) Ù ( ù PÙ ( ù PÙQ) Û (ù PÙQ)

b) Show that the truth values of the following formulas are independent of their

components.

(i) (PÙ ( P® Q) ® Q

(ii) (P® Q) D( ù PÚQ)

a) Obtain the principal disjunctive normal forms of

(i) ( ù PÚQ)

(ii) (PÙQ) Ú( ù PÙR) Ú( QÙR)

b) Show that a modular lattice is distributive if and only if none of its sub lattices is isomorphic to the diamond lattice .

a) Explain marriage semi automation.

b) Let ~₁ and ~₂ be two congruence relations on (S,o). Show that ~₁, C ~₂ if and only if is an epimorphism of (S/~_1,o) onto (S/~₂, o).
c) For any f show that there exists a semi group F which is free on B.
a) For two automation if show that is a homomorphic image of .
b) If the automation is a homomorphic image of , show that is a homomorphic image of .
c) Explain parallel composition and series composition of two automata and .

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Loyola College B.Sc. Mathematics April 2008 Algebra, Calculus And Vector Analysis Question Paper PDF Download

October 26, 2017 by 01 prakash

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

XZ 8

THIRD SEMESTER – APRIL 2008

MT 3501 – ALGEBRA, CALCULUS AND VECTOR ANALYSIS

Date : 26-04-08 Dept. No. Max. : 100 Marks

Time : 1:00 – 4:00

SECTION – A

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

Evaluate .
If find the Jacobian of x and y with respect to r and .
Solve
Find the complete solution of

Find at (2,0,1) for .
State Stoke’s theorem.
Evaluate ë (Sinh at).
Evaluate ë^-1.
Find the sum of all divisors of 360.
Compute (720).

SECTION – B

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

By the changing the order of integration evaluate

Express interms of Gamma function and evaluate .
Obtain the complete and singular solutions of .
Solve.
Find if
Evaluate (i) ë (ii) ë
Find ë^-1
Show that if x and y are both prime to the prime n, then x^n-1-y^n-1 is divisible by n. Deduce that x¹²-y¹² is divisible by 1365.

SECTION – C

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

a) Evaluate over the tetrahedron bounded by the planes and the coordinate planes.

b) Show that .
c) Using gamma function evaluate.

a) Solve

b) Solve the following by Charpit’s method

c) Solve

a) Verify Green’s theorem for where C is the region bounded by y=x and y=x^2.

b) Show that 18!+1 is divisible by 437.

a) State and prove Wilson’s theorem.

b) Solve given using Laplace

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Loyola College B.Sc. Mathematics April 2008 Algebra, Calculus & Vector Analysis Question Paper PDF Download

October 26, 2017 by 01 prakash

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

XZ 7

THIRD SEMESTER – APRIL 2008

MT 3500 – ALGEBRA, CALCULUS & VECTOR ANALYSIS

Date : 26-04-08 Dept. No. Max. : 100 Marks

Time : 1:00 – 4:00

PART – A

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

Show that G (n+1) = n G(n).
Show that
Form the partial differential equation by eliminating the arbitrary function from .
Solve:
Show that is solenoidal.
Show that curl
Find ë .
Find ë .
Define Euler’s function.
Find the number of integer, less than 600 and prime to it.

PART – B

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

Show that.
Show that é
Solve:
Find the general integral of
Find the directional derivative of xyz-xy²z³ at(1,2,-1) in the direction

If find where C is the curve y=2x² from (0,0) to (1,2).
Find ë if

for

With how many zeros does end.

PART – C

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

a) Evaluate

b) Evaluate over the region in the positive octant for which .

a) Find the complete integral of using charpits method.

b) If where is a constant vector and is the position vector of a point show that curl .

a) Verify Stoke’s theorem for

where S is the upper half of the sphere and C its boundary.

b) Find (i) ë^ùand (ii) ë^ù

a) Solve using Laplace transforms

given that

and at t = 0.

b) Find the highest power of 11 in .

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Loyola College B.Sc. Mathematics April 2008 Algebra, Anal.Geo & Calculus – II Question Paper PDF Download

October 26, 2017 by 01 prakash

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

XZ 54

SECOND SEMESTER – APRIL 2008

MT 2500 – ALGEBRA, ANAL.GEO & CALCULUS – II

Date : 23/04/2008 Dept. No. Max. : 100 Marks

Time : 1:00 – 4:00

PART – A

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

Evaluate
Write the value of
Is exact?
Solve
State Raabe’s test.
Define uniform convergence of a sequence.
Find the Coefficient of in the expansion of
Write down the last term in the expansion of
Write the intercept and normal forms of the equation of a plane.
Find the Centre and radius of the sphere

PART – B

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

Evaluate
Solve
Test the Convergence of
Find the sum to infinity of the series
Sum the series
If a, b, c denote three Consecutive integers, show that

The foot of the perpendicular drawn form the origin to the plane is (12,-4,-3); find the equation of the plane.
Find the equation to the sphere through the four points (0,0,0), (a,0,0), (0,b,0), (0,0,c) and determine its radius.

PART – C

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

a) Evaluate

b) Find the area of the cardioid (12+8)

a) Prove that the series

is convergent if and

b) Sum the series

Find the image of the point (1,3,4) in the plane . Hence prove that the image of the line is .
Through the circle of intersection of the sphere and the plane two spheres and are drawn to touch the place . Find the equations of the spheres.

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

October 26, 2017 by 01 prakash

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

XZ 1

FIRST SEMESTER – APRIL 2008

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

Date : 07/05/2008 Dept. No. Max. : 100 Marks

Time : 1:00 – 4:00

PART – A

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

Find the n^th derivative of e^ax.
Prove that the subtangent to the curve is of constant length.
Find the coordinates of the center of curvature of the curve at .
What is the curvature of a (i) circle (ii) straight line.
Determine the quadratic equation having (3-2i) as a root.
If are the roots of the equatim . Show that .
Prove that .
Prove that .
Find the pole of the line with respect to the parabola y²=4ax.
If are the eccentricities of a hyperbola and its conjugate,

prove that .

PART – B

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

At which point is the tangent to the curve parallel to the line
Final the angle at which the radius vector cuts the curve .
Prove that the radius of curvature at any point of the cycloid

and is .

Show that if the roots of the equation are in arithmetic progression then .
If show that .

If prove that

Find the locus of poles of chords of the parabola which subtend a right angle at the focus.
Find the equation of a rectangular hyperbola referred to its asymptotes as axes.

PART – C

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

a) If prove that and

b) Find the (p,r) –equation of the curve and hence show that the radius of curvature at any point varies as the cube of the focal distance.

a) Find the equation of the evolute of the parabola .

b) Solve .

a) Find the real root of to two places of decimals using Horner’s method.

b) Evaluate .

a) Prove that .

b) Derive the equation of the tangent at the point whose rectorial angle is on the conic .

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

October 26, 2017 by 01 prakash

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

AB 19

FIFTH SEMESTER – November 2008

MT 5501 – REAL ANALYSIS

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

Time : 9:00 – 12:00

SECTION-A

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

State principle of induction.
Show that the set Z is similar to N.
Define isolated point of a set in a metric space.
“Arbitrary intersection of open sets in open” True or False. Justify your answer.
If {x_n} is a sequence in a metricspace, show that { x_n} converges to a unique point.
Define complete metric space and give an example of a space which is not complete.
When do you say that a function has a right hand derivative at ?
Define (i) Strictly increasing function

(ii) Strictly decreasing function

When do you say that a partition is a refinement of another partition? Illustrate by an example.
Define limit superior and limit inferior of a sequence.

SECTION-B

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

Show that is an irrational number.
Show that collection of all sequences whose terms are 0 and 1 is uncountable.
Let Y be a subspace of a metric space (X,d). Show that a subset A of Y is open in Y if an only if for some open set G in X.
Show that every compact subset of a metric space is complete.
Show that every compact set is closed and bounded in a metric space.
Let be differentiable at c and g be a function such that where I is some open interval containing the range of f. If g is differentiable at f(a), show that g_of is differentiable at c and (g_of)^’(c)=g^’(f(o).f^’(c).

If f is of bounded variation on [a,b] and if f is also of bounded variation on [a,c] and [c,b] for , show that .
Show that lim inf(a_n) if and only if for .

there exists a positive integer N such that for all and
given any positive integer m, there exists such that .

SECTION-C

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

(a) State and prove Unique factrization theorem for integers.

(b) If S is an infinite set, show that S contains a countably infinite set.

(a) State and prove Heine theorem.

(b) If S, T be subsets of a metric space M,

show that (i)

(ii)

Illustrate by an example that

(a) Let X be a compact metric space and be continuous on X.

Show that is a compact subset of Y.

(b) Show that on R is continuous but not uniformly continuous.

(a) Let f be of bounded variation on [a,b] and V be the variation of f. Show that V is continuous

from the right at if and only if f is continuous from the right at c.

(b) on [a,b] and g is strictly increasing function defined on [c,d] such that

g ([c,a])=[a,b]. Let h (y) = f (g (y)) and .

Show that .

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