# LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034.

B.Sc. DEGREE EXAMINATION – MATHEMATICS

FourTH SEMESTER – APRIL 2003

## ST 4201 / sTA 201  –  MATHEMATICAL STATISTICS

28.04.2003

9.00 – 12.00                                                                                                     Max : 100 Marks

### PART – A                                       (10´ 2=20 marks)

Answer ALL the questions.

1. Two dice are thrown. What is the probability that the sum of the numbers on the two dice is eight?
2. The probability that a customer will get a plumbing contract is and the probability that he will get an electric contract is 4/9. If the probability of getting at least one is 4/5,determine the probability that he will get both.
3. Consider 2 events A and B such that and . Verify whether the given statement is true (or) false. .
4. Define i)  independent events and ii)  mutually exclusive events.
5. State any four properties of a distribution function.
6. The random variable X has the following probability function
 X = x 0 1 2 3 4 5 6 7 P (X=x) 0 k 2k 2k 3k k2 2k2 7k2+k

Find k.

1. Let f (x) =

0    ;   else where

Find E(X).

1. Let X ~ B (2, p) and Y~B (4, p). If P , find P.
2. Define consistent estimator.
3. State Neyman – Pearson lemma.

### Answer any FIVE questions.

1. A candidate is selected for three posts. For the first post three are three candidates, for the

second there are 4 and for the third there are 2. What are the chances of his getting

1. i) at least one post and  ii)  exactly one post?
2. Three boxes contain 1 white, 2 red, 3 green ; 2 white, 3 red, 1 green and 3 white, 1 red, 2 green balls. A box is chosen at random and from it 2 balls are drawn at random. The balls so drawn happen to be white and red. What is the probability that they have come from the second box?
3. Find the conditional probability of getting five heads given that there are at least four heads, if a fair coin is tossed at random five independent times.
4. Derive the mean and variance of hyper-geometric distribution.
5. Let X be a random variable having the p.d.f

f(x) =

Find the m.g.f. of X and hence obtain the mean and variance of X.

1. If X is B(n,p), show that E= p and E.
2. Let X be  N(m,s2).  i)  Find b so that
3. ii) If P (X < 89) =0.90 and P(X < 94) =0.95, find m and s2.
4. If X and Y are independent gamma variates with parameters m and n respectively,

Show that  ~ .

### PART – C                                         (2´20=40 marks)

Answer any TWO questions

1. If the random variables x1 and x2 have the joint  p.d.f

f  (x1 ,x2) =

i ) find the conditional mean of X1 given  X2 and  ii)  the  correlation coefficient

between  X1 and X2.

1. a)  Find all the odd and even order  moments of Normal distribution.
2. Let (X,Y) have a bivariate normal distribution. Show that each marginal distribution

in normal.

1. a) Derive the p.d.f of F- variate with (n1,n2) d.f.
2. Find the g.f of exponential distribution.
• a) Let X1, X2, …. Xn  be a  random sample of size n from N (q,1) . Show that the sample

mean is an unbiased estimator of the parameter q.

1. Write a short note on:
2. i) null hypothesis ii) type I and type II errors iii)    standard error
3. iv) one -sided and two -sided tests.

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# LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034.

B.Sc. DEGREE EXAMINATION – MATHEMATICS

SECOND SEMESTER – APRIL 2003

## MT 2500 / MAT 501  –  ALGEBRA ANALYTICAL GEOMETRY, cALCULUS – II

23.04.2003

9.00 – 12.00                                                                                                    Max : 100 Marks

### PART – A                                       (10´ 2=20 marks)

Answer ALL questions. Each question carries TWO marks.

1. Prove that
2. Evaluate
3. State cauchy’s root test for convergence of a given series.
4. Show that
5. If Y= show that
6. Solve
7. Solve where
8. Evaluate
9. Find the equation to the plane through the point (3,4,5) and parallel to the plane
10. Find the equation of the sphere with centre (-1, 2, -3) and radius 3 units.

### Answer any FIVE questions. Each question carries EIGHT marks.

1. If n =   prove that   .
2. Evaluate .
3. Sum the series .
4. Sum the series .
5. Solve .
6. Solve .
7. Test for convergence of the series .
8. Find the perpendicular distance from P (3, 9, -1) to the line .

### PART – C                                         (2´20=40 marks)

Answer any TWO questions. Each question carries TWENTY marks.

1. a) Evaluate I =                                                                                           (10)
2. b) Find the reduction formulae for In =                                                  (10)
3. a) Solve by variation of parameter method.                                         (10)
4. b) Solve (10)
5. a) Sum the series (10)

1. b) Sum the series (10)
2. a) Show that the series  is convergent when k is greater than unity and

divergent when k is equal to or less than unity.                                                           (10)

1. Find the equation of the sphere which passes through the circle

and touch the plane            (10)

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# B.Sc. DEGREE EXAMINATION – MATHS / CHEMISTRY

II SEMESTER – NOVEMBER 2003

### PH 203 / 205 / 403 – GENERAL PHYSICS  II

08.11.2003                                                                                                                              100 Marks

## 1.00 – 4.00

PART – A

Answer All questions                                                                         (10 x 2 = 20 marks)

1. What is a Zone plate?
2. Give the geometry of a Nicol prism
3. Define specific rotatory power of an optically active substance
4. State Gauss’s law in differential form
5. Three capacitors of capacitance values 1 mF, 2 mF and 3 mF are arranged in series. What is the effective capacitance?
6. Define the ampere, the unit of current.
7. Distinguish between amplitude and frequency modulations.
8. What are the charge carriers in semiconductor devices?
9. Give the truth table of the NAND gate
10. List any four properties of X-rays.

#### PART – B

Answer any FOUR questions                                                        (4 x 7 ½  = 30 marks)

1. Prove the rectilinear propagation of light by Fresnel’s theory of half-period zones
2. Derive an expression for the loss of energy on sharing of charges between two capacitors.
3. Find the magnetic field at any point due to an infinitely long wire carrying current.
4. State and prove De Morgan’s theorems.
5. Discuss the theory and production of X-rays with a neat diagram.

#### PART – C

Answer any FOUR questions                                                      (4 x 12 ½ = 50 marks)

1. Explain the theory of production and of analysis of different types of polarized beams.
2. Using Gauss’s law, determine the intensity of electric field due to (i) a charged sphere and (ii) a line charge.
3. Derive an expression for the intensity of magnetic field along the axis of a current carrying circular coil.
4. Explain the working of a two-stage RC coupled amplifier with a circuit diagram. Also explain the frequency response of the amplifiers.
5. Discuss with necessary theory the working of Bragg spectrometer.

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# B.Sc. DEGREE EXAMINATION  –  MATHEMATICS

Fourth  SEMESTER  – NOVEMBER 2003

# ST 4201/STA 201 MATHEMATICAL  STATISTICS

14.11.2003                                                                                        Max: 100 Marks

9.00 – 12.00

SECTION A

Answer ALL the questions.                                                      (10 ´ 2 = 20 Marks)

1. Define an event and probability of an event.
2. If A and B any two events, show that P (AÇBC) = P(A) – P(AÇB).
3. State Baye’s theorem.
4. Define Random variable and p.d.f of a random variable.
5. State the properties of distribution function.
6. Define marginal and conditional p.d.fs.
7. Examine the validity of the given Statement “X is a Binomial variate with

mean 10 and S.D  4”.

1. Find the d.f of exponential distribution.
2. Define consistent estimator.

SECTION B                          (5 ´ 8 = 40 Marks)

Answer any FIVE questions.

1. An urn contains 6 red, 4 white and 5 black balls.  4 balls are drawn at random.

Find the probability that the sample contains at least one ball of each colour.

1. Three persons A,B and C are simultaneously shooting. Probability of A hit the

target is  ;  that for B is    and for C is  Find   i)  the probability that

exactly one of them will hit the target ii) the probability that at least one of them

will hit the target.

1. Let the random variable X have the p.d.f

Find P( ½ < X <  ¾) and    ii) P ( – ½ < X< ½).

1. Find the median and mode of the distribution

.

1. Find the m.g.f of Poisson distribution and hence obtain its mean and variance.

1. If X and Y are two independent Gamma variates with parameters m and g

respectively,  then show that    Z =  ~ b (m,g).

1. Find the m.g.f of Normal distribution.
2. Show that the conditional mean of Y given X is linear in X in the case of bivariate normal distribution.

## SECTION – C

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

1. Let X1and X2 be random variables having the joint p.d.f

Show that the conditional means are

(10+10)

1. If f (X,Y) has a trinomial distribution, show that the correlations between

X and Y is   .

1. i)    Derive  the p.d.f of ‘t’ distribution with ‘n’ d.f
2. ii) Find all odd order moments of Normal distribution.                       (15+5)
3. i) Derive the p.d.f of ‘F’ variate with (n1,n2) d.f                                     (14)

1. ii) Define   i)   Null and alternative Hypotheses                                      (2)
2. ii) Type I and Type II errors.                                                (2)

and         iii)   critical region                                                                   (2)

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## Loyola College B.Sc. Mathematics Nov 2003 Algebra, Anal. Geometry, Calculus & Trigonometry Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

B.Sc., DEGREE EXAMINATION – MATHEMATICS

# MT – 1500/MAT 500 – ALGEBRA, ANAL. GEOMETRY, CALCULUS & TRIGONOMETRY

01.11.2004                                                                                                           Max:100 marks

1.00 – 4.00 p.m.

SECTION – A

Answer ALL Questions.                                                                                (10 x 2 = 20 marks)

1. If y = sin (ax + b), find yn.
2. Show that in the parabola y2 = 4ax, the subnormal is constant.
3. Prove that cos h2x = cos h2x + sin h2
4. Write the formula for the radius of curvature in polar co-ordinates.
5. Find the centre of the curvature xy = c2 at (c, c).
6. Prove that .
7. Form a rational cubic equation which shall have for roots 1, 3 – .
8. Solve the equation 2x3 – 7x2 + 4x + 3 = 0 given 1+is a root.
9. What is the equation of the chord of the parabola y2 = 4ax having (x, y) as mid – point?
10. Define conjugate diameters.

SECTION – B

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

1. Find the nth derivative of cosx cos2x cos3x.
2. In the curve xm yn = am+n , show that the subtangent at any point varies as the abscissa of the point.
3. Prove that the radius of curvature at any point of the cycloid

x = a (q + sin q) and  y = a  (1 – cos q) is 4 a cos .

1. Find the p-r equation of the curve rm = am sin m q.
2. Find the value of a,b,c such that .
3. Solve the equation

6x6 – 35x5 + 56x4 – 56x2 + 35x – 6 = 0.

1. If the sum of two roots of the equation x4 + px3 + qx2 + rx + s = 0 equals the sum of the other two, prove that p3 + 8r = 4pq.
2. Show that in a conic, the semi latus rectum is the harmonic mean between the segments of a focal chord.

SECTION -C

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

1. a) If y = , prove that

(1 – x2) y2 – xy1 – a2y = 0.

Hence show that (1 – x2) yn+2 – (2n +1) xyn+1 – (m2 + a2) yn = 0.                     (10)

1. Find the angle of intersection of the cardioid r = a (1 + cos q) and r = b (1 – cos q).

(10)

1. a) Prove that  = 64 cos6 q – 112 cos4q + 56 cos2q –                                       (12)

1. b) Show that (8)
2. a) If  a + b + c + d = 0, show that

.                               (12)

1. b) Show that the roots of the equation x3 + px2 + qx + r = 0 are in Arithmetical

progression if 2 p3 – 9pq + 27r = 0.                                                                             (8)

1. a) Prove that the tangent to a rectangular hyperbola terminated by its asymptotes is

bisected at the point of contact and encloses a triangle of constant area.                     (8)

1. b) P and Q are extremities of two conjugate diameters of the ellipse and S is

a focus.  Prove that PQ2 – (SP – SQ)2 = 2b2.                                                              (12)

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## Loyola College B.Sc. Mathematics April 2006 Mathematical Statistics Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS, PHYSICS & CHEMISTRY

 AC 10

FOURTH SEMESTER – APRIL 2006

# ST 4201 – MATHEMATICAL STATISTICS

(Also equivalent to STA  201)

Date & Time : 22-04-2006/9.00-12.00         Dept. No.                                                       Max. : 100 Marks

Part A

Answer all the questions.

1. Define conditional probability of the event A given that the event B has happened.
2. If A1 and A2 are independent events with P(A1) = 0.6 and  P(A2) = 0.3, find     P(A1 U A2), and P(A1 U A2c)
3. State any two properties of a distribution function.
4. Define the covariance of any two random variables X and Y. What happens when they are independent?
5. The M.G.F of a random variable is  [(2/3) + (1/3) et]5 . Write the mean and variance.
6. Define a random sample.
7. Explain the likelihood function.
8. Let X have the p.d.f. f(x) =1/3, -1<x<2, zero elsewhere. Find the M.G.F.
9. Define measures of skewness and kurtosis through moments.
10. Define a sampling distribution.

## Part B

Answer any five questions.

1. Stat and prove Bayes theorem.
2. Derive the mean and variance of Gamma distribution.
3. Let the random variables X and Y have the joint pdf

x + y, 0<x<1, 0<y<1

f(x, y) =

0, otherwise.

Find the correlation coefficient.

1. A bowl contains 16 chips of which 6 are red, 7 are white and 3 are blue. If 4 chips are taken at random and without replacement, find the probability that
1. All the 4 are red.
2. None of the 4 is red.
• There is atleast one of each colour.
1. State and prove the addition theorem for three events A, B and C. What happens when they are mutually exclusive?
2. Derive the mgf of Poisson distribution. And hence prove the additive property of the Poisson distribution.
3. Let X1 and X2 denote a random sample of size 2 from a distribution that is       N(m, s2). Let Y1 = X1 + X2 , Y2 = X1 – X2.  Find the joint pdf of Y1 and Y2 and show that Y1 and Y2 are independent.
4. Define the cumulative distribution function F(x) of a random variable X and mention the properties of it.

# Part C

Answer any two questions.

1. a) Derive the recurrence relation for the central moments of Binomial distribution. Obtain the first four moments.
2. b) Show that Binomial distribution tends to poisson distribution under certain conditions.           (10 +10 = 20)
3. a) Discuss the properties of Normal distribution
4. b) In a distribution exactly normal, 10.03% of the items are under 25 kilogram weight and 89.97 % of the items are under 70 kilogram weight. What are the mean and standard deviation of the distribution?                                                                                                      (10 +10 = 20)
5. Let f(x, y) = 8xy, 0<x<y<1; f(x, y) = 0 elsewhere. Find
6. a) E(Y/X = x),    b). Var( Y/X = x).
7. b) If X and Y are independent Gamma variates with parameters m and v respectively, show that the variables U = X + Y, Z = X / (X + Y) are independent and that U  is a g( m + v) variate and Z is a b1(m, v) variate.                                                                                      (10 +10 = 20)
8. a) Derive the pdf of t-distribution.
9. b) Obtain the Maximum Likelihood Estimators of m and s2 for Normal distribution.         (10 + 10 = 20)

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## Loyola College B.Sc. Mathematics Nov 2006 Physics For Mathematics Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034                                B.Sc. DEGREE EXAMINATION – MATHEMATICS

 AC 03

THIRD SEMESTER – NOV 2006

# PH 3100 – PHYSICS FOR MATHEMATICS

(Also equivalent to PHY 100)

Date & Time : 28-10-2006/9.00-12.00     Dept. No.                                                       Max. : 100 Marks

PART A

Answer ALL questions:                                                                 10 x 2 = 20 marks

1. State Principle of Conservation of angular momentum.
2. Give an expression for angular acceleration.
3. State Newton’s law of Gravitation.
4. What is parking orbit?
5. Define Poisson’s ratio.
6. Explain the term viscosity of a fluid.
7. State the fundamental postulates of the special theory of relativity.
8. Explain the term ‘frame of reference’.
9. What are beats?  How are they produced?
10. The driver of a car moving towards a factory with velocity 30 m/s sounds the horn with a frequency of 240 Hz.   Find the apparent frequency of sound heard by the watchman of the factory.

PART B

Answer any FOUR questions:                                                      4 x 7.5 = 30 marks

1. Prove that the oscillation of a liquid in a U-tube is simple harmonic.
2. Using Newton’s law of gravitation calculate (a) mass and density of earth (b) mass of sun [given G = 6.67 x 10-11 Nm2 Kg-2; Radius of  earth = 6.38 x 106 m and distance of earth from centre of the Sun = 1.5 x 1011
3. Obtain Stoke’s law for the motion of body in a viscous medium from dimensional considerations. Also determine the Coefficient of viscosity of a liquid from Stoke’s formula.
4. On the basis of Lorentz transformation, derive an expression for length contraction.
5. Explain Doppler Effect. Derive a general expression for  the apparent frequency of a note when both the source and the listener are in motion.

PART C

Answer any FOUR questions:                                                    4 x 12.5 = 50 marks

1. Explain simple harmonic motion and discuss its characteristics.  Derive Simple Harmonic equations by calculus.
2. What is stationary satellite? Define escape velocity.  Show that the escape velocity from the surface of the earth is equal to 11 km/s.  Distinguish between orbital velocity and escape velocity.
3. Define Young’s modulus, modulus of rigidity and Poisson’s ratio.  Show that the bulk modulus of elasticity K = Y / [3(1-2σ)].
4. Describe Michelson-Morley experiment.  Discuss the results obtained.
5. What is resonance?  Explain the resonance column find the velocity of sound in air.

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## Loyola College B.Sc. Mathematics Nov 2006 Mechanics-II Question Paper PDF Download

LOYOLA COLLEGE  ( AUTONOMOUS ) , CHENNAI – 600 034

# V SEMESTER – NOVEMBER 2006

Date       :                                                                                                                  Max  : 100 Marks

### Duration:                                                                                                                  Hours: 3 hours

————————————————————————————————————————-

SUB.CODE:MT5500                                                                                                                           SUB.NAME : MECHANICS-II

——————————————————————————————————————————————————————

Answer  ALL  the questions and each question carries 2 marks                     [  10 X 2  = 20  ]

01.State the cases of  non existence of center of gravity

02.State the forces which can be ignored in forming the equation of virtual work.

03.Define Neutral equilibrium with an example

04.Define Span of a Catenary

05.A particle is performing S.H.M. between points A and B. If the period of oscillation is

2p, show that the velocity at any point is a mean proportional between AP and BP.

06.Define Apse

07.If the angular velocity of a particle moving in a plane curve about a fixed origin is

constant, show that its transverse acceleration varies as radial velocity.

08.Find the M.I of a thin uniform rod.

09.Define radius of gyration.

10.State D’Alembert’s principle.

Answer any FIVE of the following                                                               [  5 X 8  = 40  ]

1. A uniform solid right circular cylinder of height l and base radius r is sharpened at

one end like pencil. If the height of the resulting conical part be h, find the distance

through which the C.G is displaced, it being assumed that there is no shortening of the

cylinder.

12.Find the C.G. of a uniform hollow right circular cone.

13.A uniform chain, of length l, 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

14.A uniform string hangs under gravity and it is such that the weight of each element of

it is proportional to the projection of it on a horizontal line. To determine the shape of

the string.

15.Show that the composition of 2 simple harmonic motions of the same period along 2

perpendicular lines is an ellipse.

16.A particle executing S.H.M in a straight line has velocities 8,7,4 at three points distant

one foot each other. Find the period.

17.Derive the radial and transverse components of velocity and acceleration.

1. A circular disc of radius 5cms. Weighing 100 gms. is rotating about a tangent at the

rate of 6   turns per second. Find the frictional couple which will bring it to rest in one

minute

Answer any TWO of the following                                                                [  2 X 20  = 40  ]

19.i.Discuss the stability of a body rolling over a fixed body

ii.A body consisting of a cone and a hemisphere on the same base rests on a rough

horizontal table. Show that the greatest height of the cone so that the equilibrium may

be stable is  times the radius of the sphere.

20.i.State and prove the principle of virtual work for a system of coplanar forces acting on

a rigid body.

ii.A solid hemisphere is supported by a string fixed to a point on the rim and to a point

on a smooth vertical wall with which the curved surface of the hemisphere is in

contact. If  and are the inclination of the string and the plane base of the

hemisphere to the vertical, prove that

21.A point moves with uniform speed v along the curve r = a (1+ cosq ). Show that

1. Its angular velocity w about pole is
2. Radial component of acceleration is constant and equal to numerically

iii. Magnitude of resultant acceleration is

22.i.State and prove the theorem of parallel axes

1. Find the moment of inertia of a hollow sphere.

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## Loyola College B.Sc. Mathematics Nov 2006 Graphs, Diff. Equ., Matrices & Fourier Series Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034  B.Sc. DEGREE EXAMINATION – MATHEMATICS

 AA 02

FIRST SEMESTER – NOV 2006

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

Date & Time : 03-11-2006/1.00-4.00           Dept. No.                                                       Max. : 100 Marks

SECTION A

Answer ALL Questions.                                                                            (10 x 2 = 20)

1. A firm producing poultry feeds finds that the total cost C(x) of producing x units is given by C(x) = 20x + 100. Management plans to charge \$24 per unit for the feed. How many units must be sold for the firm to break even?
2. Find the equation of the line passing through (2, 9) and (2, -9).
3. Find the domain and range of the function f(x) = .
4. Find the axis and vertex of the parabola y = x2 – 2x + 3.
5. Reduce y = axn to the linear law.
6. Solve the difference equation yx+2 – 8yx+1 + 15yx = 0.
7. State Cayley Hamilton theorem.
8. Find the determinant value of a matrix given its eigen values are 1, 2 and 3.
9. Define periodic function. Give an example.
10. Show that = 0, when n 0.

SECTION B

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

1. The marginal cost for raising a certain type of fruit fly for a laboratory study is \$12 per unit of fruit fly, while the cost to produce 100 units is \$1500.

(a) Find the cost function C(x), given that it is linear.

(b) Find the average cost per unit to produce 50 units and 500 units.(4 + 4 marks)

1. The profit P(x) from the sales of x units of pies is given by P(x) = 120x – x2. How many units of pies should be sold in order to maximize profit? What is the maximum profit? Draw the graph.
2. Graph the functions (a) y = x2 – 2x – 15 , (b) f(x) = .

(4 + 4 marks)

1. Fit a parabola y = a + bx + cx2 using method of group averages for the following data.

x          0          2          4          6          8          10

y          1          3          13        31        57        91

1. Solve the difference equation yk+2 – 5yk+1 + 6yk = 6k.
2. Find the eigen values and eigen vectors of A = .
3. Using Cayley Hamilton theorem, find A-1 if A = .
4. In (-), find the fourier series of periodicity 2for f(x) = .

SECTION C

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

1. (a) Suppose that the price and demand for an item are related by p = 150 – 6x2, where p is the price and x is the number of items demanded. The price and supply are related by p = 10x2 + 2x, where x is the supply of the item. Find the equilibrium demand and equilibrium price.

(b) Fit a straight line by the method of least squares for the following data.

x          0          5          10        15        20        25

y          12        15        17        22        24        30        (10 + 10 marks)

1. Solve the following difference equations.

(a) yn+2 – 3yn+1 + 2yn = 0, given y1 = 0, y2 = 8, y3 = -2.

(b) u(x+2) – 4u(x) = 9x2.                                                               (8 + 12 marks)

1. Expand f(x) = x2, when -< x < , in a fourier series of periodicity 2. Hence deduce that

(i) .

(ii) .

(iii) .

1. Diagonalize the matrix A = . Hence find A4.

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## Loyola College B.Sc. Mathematics Nov 2006 Graph Theory Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034  B.Sc. DEGREE EXAMINATION – MATHEMATICS

 AA 08

FIFTH SEMESTER – NOV 2006

# MT 5400 – GRAPH THEORY

(Also equivalent to MAT 400)

Date & Time : 03-11-2006/9.00-12.00         Dept. No.                                                       Max. : 100 Marks

Part A

Answer all the questions. Each question carries 2 marks.

1. Give an example of a regular graph of degree 0.
2. The only regular graph of degree 1 is K2. True or false? Justify your answer.
3. What is a self-complementary graph?
4. What is the maximum degree of any vertex in a graph on 20 vertices?
5. Show that the two graphs given below are not isomorphic.

1. Give an example of a closed walk of even length which does not contain a cycle.
2. Draw all non-isomorphic trees on 6 vertices.
3. Give an example of a graph which has a cut vertex but does not have a cut edge.
4. Define a block.
5. Give an example of a bipartite graph which is non-planar.

Part B

Answer any 5 questions. Each question carries 8 marks.

1. (a). Prove that in any graph,

(b). Draw the eleven non-isomorphic sub graphs on 4 vertices.                     (4+4)

1. (a). Define the incidence and adjacency matrices of a graph. Write down the

adjacency matrix of the following graph:

(b). Let G be a (p, q)-graph all of whose vertices have degree k or k + 1. If G

has t vertices of degree k then show that t = p(k+1)-2q.                                                                                                                                       (4 + 4)

1. Prove that the maximum number of edges among all graphs with p vertices, where p is odd, with no triangles is [p2 / 4], where [x] denotes the greatest integer not exceeding the real number x.
2. (a). Let G be a k-regular bipartite graph with bipartition (X, Y) and k > 0. Prove

that

(b). Show that if G is disconnected then GC is connected.                (4 + 4)

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

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

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

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

(b). Prove that every tree has a centre consisting of either one vertex or two

1. Let G be graph with with p ≥ 3 and, then prove that G is Hamiltonian.

Part C

Answer any 2 questions. Each question carries 20 marks.

1. Let G1 be a (p1, q1)-graph and G2 a (p2, q2)-graph. Show that
2. G1 x G2 is a (p1 p2, q1p2 + q2p1)-graph and
3. G1[G2 ] is a (p1 p2, q1p22 + q2p1)-graph.

1. Prove that the following statements are equivalent for a connected graph G.
2. G is Eulerian.
3. Every vertex of G has even degree.
4. The set of edges of G can be partitioned into cycles.

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

1. (a). Obtain Euler’s formula relating the number of vertices, edges and faces of

a plane graph.

(b). Prove that every planar graph is 5-colourable.                             (10+10)

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## Loyola College B.Sc. Mathematics Nov 2006 Formal Languages And Automata Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034                          B.Sc. DEGREE EXAMINATION – MATHEMATICS

 AA 12

FIFTH SEMESTER – NOV 2006

# MT 5404 – FORMAL LANGUAGES AND AUTOMATA

Date & Time : 06-11-2006/9.00-12.00         Dept. No.                                                       Max. : 100 Marks

PART A

Answer all questions. Each question carries two marks.                             10×2=20

• Define context – sensitive language and give an example.
• Write a grammar for the language L(G) = L={anbn / n1}.
• Show that every context – free language is a context – sensitive language.
• If L = { L={anb / n1}then find LR
• Construct a grammar to generate the set of all strings over {a,b} beginning with a.
• Define an unambiguous grammar.
• Show that the grammar SSS, Sa, Sb is ambiguous.
• Construct a DFA which can test whether a given positive integer is divisible by 5.
• Define a non-deterministic finite automation.
1. Construct a finite automation that accepts exactly those input strings of 0’s and 1’s that end in 00.

PART B

Answer any five questions. Each question carries 8 marks.                          5×8=40

1. Prove that CSL is closed under union.
2. Find a grammar generating L={anbncm/ n1, m0}
3. Write a note on Chomskian hierarchy.
4. Prove that L= {} is not a CFL.
5. Prove that PSL is closed under star.
6. Give an ambiguous and an unambiguous grammar to generate L={anbn / n1}.
7. Give a deterministic finite automation accepting the set of all strings over {0,1} with three consecutive 1’s.
8. Let G = ( N, T, P, S), N = {S, A},   T = {a,b},   P = { SaA,  A bS, Ab}. Find L(G). Also construct an NDFA accepting L(G).

PART C

Answer any two questions. Each question carries 20 marks.          2×20=40

1. a) Write a note on the construction of CNF
2. b) Write a grammar in CNF to generate L= {anbman/ n0, m1}              (5+15)

20     State and prove u-v theorem

21     Let M =  (K, I, , F) where K = {}, I = {a,b}, F = {}

.

Find the corresponding DFA.

22     i) Construct a DFA to accept all strings over {a,b} containing the substring   aabb.

1. ii) Construct a DFA accepting all strings over {0,1} having even number of 0’s.

(10+10)

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## Loyola College B.Sc. Mathematics Nov 2006 Alg.,Anal.Geomet. Cal. & Trign. – I Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034  B.Sc. DEGREE EXAMINATION – MATHEMATICS

 AA 01

FIRST SEMESTER – NOV 2006

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

(Also equivalent to MAT 500)

Date & Time : 01-11-2006/1.00-4.00           Dept. No.                                                       Max. : 100 Marks

SECTION –A

## Answer all:                                                                                    2 x 10 = 20

1. If y = A emx+B e-mx , show that y2 = m2y.
2. Write down the nth derivative of .
3. If x = at2 and y = 2at , find .
4. Prove that the sub tangent to the curve y = ax is of constant length.
5. Determine the quadratic equation having 1+ as one of its roots.
6. Calculate the sum of the cubes of the roots of equation x4+2x+3 = 0.
7. Prove that cos ix = cosh x.
8. Separate sin (x+iy) into real and imaginary parts.
9. Define conjugate diameters.
10. Write the angle between the asymptotes of the hyperbola

SECTION –B

## Answer any five:                                                                              5x 8 = 40

1. Find the nth derivative of .
2. Find the angle at which the radius vector cuts the curve .
3. Show that the parabolas and intersect at right

angles.

14 Solve the equation 6x4-13x3-35x2-x+3= 0 given that is a root of it.

1. Solve the equation x4-2x3-21x2+22x+40= 0 given the roots are in A.P.

1. Prove that cos 6ө in terms of sin ө.
2. Prove that 32 sin6ө = 10 -15 cos2ө + 6 cos4ө -cos6ө.
3. If P and D are extremities of conjugate diameters of an ellipse, prove that the locus of middle point of PD is .

SECTION –C

## Answer any two:                                                                                  2x 20 = 40

1. State and prove Leibnitz theorem and , prove that

.

20 a) Find the evolute of the parabola y2= 4ax.

1. b) Prove that p-r equation of r= a(1+ cos ө) is p2 =  .

(10+10)

21 a) Find the real root of the equation  x3+6x-2 = 0.

1. b) If a+b+c+d = 0 , prove that

.

(10+10)

22 a) Separate tan-1(x + iy) into real and imaginary parts.

1. b) Derive the polar equation of a conic.

(10+10)

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## Loyola College B.Sc. Mathematics April 2007 Real Analysis Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc.

 CV 13

DEGREE EXAMINATION – MATHEMATICS

FIFTH SEMESTER – APRIL 2007

MT 5501REAL ANALYSIS

Date & Time: 28/04/2007 / 1:00 – 4:00          Dept. No.                                                     Max. : 100 Marks

Answer all the questions:                                                                           10 x 2 = 10

1. Define an inductive set with an example.

1. Prove that every positive integer n (except 1) is either a prime or a product of primes.

1. State and prove Euler’s theorem for real numbers.

1. Define a Metric space.

1. State Cantor’s intersection theorem for closed sets.

1. Define an interior point and an open set.

7.Give an example of a uniformly continuous function.

1. Define a Cauchy sequence.

1. Suppose f and g are defined on (a, b) and are both differentiable at c Î (a, b), then prove

that the function fg is also differentiable at c.

1. Define total variation of a function f on .

Answer any five questions:                                                                                         5 x 8=40

1. Prove that the set R of all real numbers is uncountable.

1. State and prove Bolzano-Weirstass theorem for R.

1. Prove that every compact subset of a metric space is complete.

1. Let (X, d1) and (Y, d2) be metric spaces and f: X Y be continuous on X. If X is compact, then prove that f (X) is a compact subset of Y.

1. Let (X, d1) and (Y, d2) be metric spaces and f: X Y be continuous on X. Then show that a map f: X Y is continuous on X if and only if f -1 (G) is open in X for every open set G in Y.

16    Prove that in a metric space (X, d)

( i ) Arbitrary union of open sets in X is open in X

( ii) Arbitrary intersection of closed sets in X  is closed in X.

1. Let f: R and f have a local maximum or a local minimum at a point c.

Then prove that f ’(c) = 0.

1. Let f be of bounded variation onand xÎ (a, b) Define V:  R as   follows:

V (a) = 0

V (x) =Vf , a <  x ≤ b.

Then show that the functions V and V – f are both increasing functions on.

Answer any two questions:                                                                                                      2 x 20 = 40

19   State and prove Intermediate value theorem for continuous functions.

1.   State and prove Lagrange’s theorem for a function f :  R

21.(a) Suppose c Î (a ,b) and two of the three integrals f da ,f da , and f da

exists. Then show that the third also exists andf da =f da +f da.

(b) When do we say f is Riemann-Stieltjes integrable?

1. (a) State and prove Unique factorization theorem for real numbers.

(b) If F is a countable family of countable sets then show that  is also countable.

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## Loyola College B.Sc. Mathematics April 2007 Physics For Mathematics Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

 SC 06

THIRD SEMESTER – APRIL 2007

# PH 3100 – PHYSICS FOR MATHEMATICS

Date & Time: 28/04/2007 / 9:00 – 12:00 Dept. No.                                            Max. : 100 Marks

PART A

Answer ALL questions:                                                                 10 x 2 = 20 marks

1. Define Simple Harmonic motion.
2. A heavy fly wheel of moment of inertia 0.3 kg m2 is mounted on a horizontal axle of radius 0.01 m and negligible mass compared with the flywheel. Neglecting friction, find the angular acceleration if a force of 40 N is applied tangentially to the axle.
3. State any two Kepler’s laws.
4. Define Gravitational Potential.
5. Explain the terms stress and strain.
6. Define Surface tension of a liquid.
7. State Einstein mass-energy relation.
8. Name the transformation under which Maxwell’s equations are invariant.
9. Explain Doppler Effect.
10. A tuning fork A of frequency 384 Hz gives 6 beats per second when sounded with another tuning fork B. On loading B with a little wax, the number of beats per second becomes 4.  What is the frequency of B?

PART B

Answer any FOUR questions:                                                      4 x 7.5 = 30 marks

1. Derive expressions for the periods of oscillation of the mass suspended when two springs are connected (i) in series (ii) in parallel.
2. Obtain an expression for the escape velocity of a body from the surface of the earth. Calculate its value.
3. Derive an expression for the work done in stretching a wire.
4. Obtain the time dilation formula of special theory of relativity.
5. Two trains traveling in opposite directions at 100 km/hr each, cross each other while one of them is whistling. If the frequency of the note is 800 Hz find the apparent pitch as heard by an observer in the other train: (i) before the trains cross each other (ii) after the train have crossed each other.

PART C

Answer any FOUR questions:                                                    4 x 12.5 = 50 marks

1. An object rolls without slipping down a smooth inclined plane find its acceleration.  Compare the acceleration of a solid cylinder with that of a hollow cylinder of the same mass.
2. State Newton’s law of gravitation.  With a neat diagram describe Boy’s experiment to determine G.  (2.5 + 10)
3. Define coefficient of viscosity and derive Poiseuille’s formula for the flow of a liquid through a capillary tube.
4. State the postulates of the Special Theory of Relativity.   On the basis of Lorentz transformation, derive an expression for length contraction.
5. State the laws of vibration of strings and describe experiments to verify the law concerning (i) length (ii) tension and (iii) linear density.

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## Loyola College B.Sc. Mathematics April 2007 Operations Research Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

 CV 16

B.Sc.  DEGREE EXAMINATION –MATHEMATICS

FIFTH SEMESTER – APRIL 2007

MT 5504 – OPERATIONS RESEARCH

Date & Time: 03/05/2007 / 1:00 – 4:00          Dept. No.                                                     Max. : 100 Marks

SECTION –A

## Answer All:                                                                                     2 x 10 = 20

1. Define Operations Research.
2. What are the three methods to find Initial Basic Feasible solution in Transportation problem ?
3. Solve the Transportation problem by Least Cost Method.
 A1 A2 A3 Supply B1 4 6 2 5 B2 3 1 5 15 B3 4 5 3 15 Demand 10 10 10

1. Solve the game:
 2 1 4 1 4 3 2 2 6

1. Define Unbalanced situation in Transportation problem.
2. Define Feasible Solution.
3. Solve the Assignment problem
 3 7 5 4 7 2 5 4 6

1. What is Dummy activity in Network problem ?
2. Define Optimistic Time Estimate.
3. Define Economic Order Quantity.

SECTION –B

## Answer any five:                                                                              5x 8 = 40

1. Find the Initial Basic Feasible solution in Transportation problem using i) North West Corner Rule ii) Least Cost Method.
 A1 A2 A3 A4 Supply B1 4 2 1 3 20 B2 8 4 2 4 20 B3 1 2 3 4 30 B4 5 2 4 6 20 Demand 10 30 10 40

1. Using Graphical method solve the Linear Programming Problem

Max z = 2x1+4x2    subject to the constraints

2x1+4x≤ 5 ,       2x1+4x2 ≤ 4,         x1, x2 ≥ 0.

1. Solve the Assignment problem

 M1 M2 M3 M4 M5 J1 9 22 58 11 19 J2 43 78 72 50 63 J3 41 28 91 37 45 J4 74 42 27 49 39 J5 36 11 57 22 25

1. Solve using Matrix Oddment method

 -1 2 1 1 -2 2 3 4 -3

1. Define critical path and draw the Network diagram for

Activity:   A    B   C    D    E   F    G    H     I      J       K

Immediate predecessor:    –     –    –     A    B   B    C    D    E    H,I    F,G

1. Solve using Dominance property
 1 7 3 4 5 6 4 5 7 2 0 3

1. Solve the Transportation problem
 A1 A2 A3 A4 Supply B1 1 2 1 4 30 B2 3 3 2 1 50 B3 4 2 5 9 20 Demand 20 40 30 10

1. The probability distribution of monthly sales of certain item is as follows:

Number of items:    0         1          2        3         4          5          6

P(d)    :   0.02    0.05    0.30    0.27    0.20     0.10     0.06

The cost of carrying inventory  is Rs 10 per unit per month . Find the

shortage cost for one item for one unit of time.

(P.T.O)

SECTION –C

## Answer any two:                                                                              2x 20 = 40

1. Solve the following Linear Programming Problem using Simplex method

Max z = 3x1+2x2    subject to the constraints

x1+2x≤ 6 ,       2x1+x2 ≤ 8,       -x1+x2 ≤ 1,         x2 ≤ 2,        x1, x2 ≥ 0.         (20)

20 a) Solve the Transportation problem to maximize the profit

 A1 A2 A3 A4 Supply B1 40 25 22 33 100 B2 44 35 30 30 30 B3 38 38 38 30 70 Demand 40 20 60 30

1. b) Solve the following traveling sales man problem

 A B C D E A – 3 6 2 3 B 3 – 5 2 3 C 6 5 – 6 4 D 2 2 6 – 6 E 3 3 4 6 –

(10+10)

21 a) Solve the game graphically

 1 0 4 -1 -1 1 2 5

1. b) The annual demand for an item is 3200 units, the unit cost is Rs 6 and inventory

Carrying charges 25% per annum. If the cost of one procurement is Rs 150. Find

1. i) Economic Order Quantity ii) Time between two consecutive orders

iii) Number of orders per year             iv) The optimal total cost                      (10+10)

22 a) Draw the Network diagram ,the Critical path ,the project duration and the total float

for the following activities

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

Duration:     3          4          4           4           2          2           3           2

1. b) What is the probability that the project will be completed in 27 days? Draw the

network diagram also .

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

T0          :      3          2          6           2          5            3           3           1           2

Tm         :      6          5         12          5         11           6           9           4           5

Tp          :      15       14        30          8         17          15         27          7           8

(10+10)

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## Loyola College B.Sc. Mathematics April 2007 Graph Theory Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

 CV 17

B.Sc.  DEGREE EXAMINATION –MATHEMATICS

FIFTH SEMESTER – APRIL 2007

MT 5400GRAPH THEORY

Date & Time: 03/05/2007 / 9:00 – 12:00        Dept. No.                                                     Max. : 100 Marks

Part A

Answer all the questions. Each question carries 2 marks.

1. Show that Kpv = Kp – 1.
2. Give an example of a self-complementary graph.
3. Write down the incidence and adjacency matrices of the following graph.

1. Give an example of a disconnected graph with three components each of which is

3-regular.

1. Give an example of a non-Eulerian graph which is Hamiltonian.
2. For what values of m and n is Km,n Eulerian?
3. Draw all non-isomorphic trees on 5 vertices.
4. Give an example of a closed walk of even length which does not contain a cycle.
5. Define a planar graph and give an example of a non-planar graph.
6. Define the chromatic number of a graph.

Part B

Answer any 5 questions. Each question carries 8 marks.

1. (a). Prove that in any graph G, the sum of degrees of the vertices is twice the number

of edges. Deduce that the number of vertices of odd degree in any graph is even.

(b). Draw the eleven non-isomorphic graphs on 4 vertices.                           (4+4)

1. (a). Let G be a (p, q)-graph all of whose vertices have degree k or k + 1. If G has t

vertices of degree k then show that t = p(k+1)-2q.

(b). Define isomorphism of graphs. If two graphs have the same number of

vertices and same number of edges are they isomorphic? Justify your answer.                                                                                                                           (4+4)

1. (a). Define product and composition of two graphs. Illustrate with examples.

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

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

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

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

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

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

are of even length.

1. Let v be a vertex of a connected graph. Then prove that the following statements are equivalent:
1. v is a cut-vertex of G.
2. There exists a partition of V – {v} into subsets U and W such that for each

uU and  wW, the point v is on every (u, w) – path.

1. There exist two vertices u and w distinct from v such that v is on every (u, w)-

path.

1. Let G be graph with p ≥ 3 and. Then prove that G is Hamiltonian.

Part C

Answer any 2 questions. Each question carries 20 marks.

1. Prove that the maximum number of edges among all graphs with p vertices with no triangles is [p2 / 4], where [x] denotes the greatest integer not exceeding the real number x.                                     (20)
2. (a).Prove that every connected graph has a spanning tree.

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

1. G is Eulerian.
2. Every vertex of G has even degree.
3. The set of edges of G can be partitioned into cycles. (5+15)
4. Let G be a (p, q)-graph. Prove that the following statements are equivalent.
5. G is a tree.
6. Any two vertices of G are joined by a unique path.
7. G is connected and p = q + 1.
8. G is acyclic and p = q + 1. (20)

1. (a).Let G be a connected plane graph with V, E and F as the sets of vertices, edges

and faces respectively. Then prove that | V | – | E | + | F | = 2.

(b). State and prove the five-colour theorem.                                                 (10+10)

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## Loyola College B.Sc. Mathematics April 2007 Formal Languages And Automata Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

 CV 19

B.Sc.  DEGREE EXAMINATION –MATHEMATICS

FIFTH SEMESTER – APRIL 2007

MT 5404FORMAL LANGUAGES AND AUTOMATA

Date & Time: 04/05/2007 / 1:00 – 4:00          Dept. No.                                                     Max. : 100 Marks

PART A

Answer all questions. Each question carries two marks.                                                                         10×2=20

1. Define context – free grammar and give an example.
2. Write a grammar to generate all palindromes over {a,b}
3. Show that every regular language is a context free language.
4. Define the concatenation of two languages and give an example.
5. Show that the PSL is closed under reflection.
6. Define an unambiguous grammar.
7. Show that SSS, Sa, Sb is ambiguous.
8. Construct a finite automation that accepts exactly those input strings of 0’s and 1’s that end in 11.
9. Construct a DFA to test whether a given positive integer is divisible by 2.
10. Define a non – deterministic finite automation.

PART B

Answer any five questions. Each question carries 8 marks.                                                          5×8=40

1. Prove that PSL is closed under union.
2. Let G be a grammar with SAaS|SS|a, ASbA|SS|ba. For the string aabbaaa find
1. a left most derivation
2. a right most derivation
3. Construct a grammar G for the language L(G) = {anbam / n, m 1}
4. Discuss about the Chomskian hierarchy
5. Prove that L={ai / i is a prime} is not a CFL.
6. Give an ambiguous and an unambiguous grammar to generate L={anbn / n1}.
7. Construct a DFA to test whether a given positive integer is divisible by 3.
8. Give a deterministic finite automation accepting the set of all strings over {0,1} with three consecutive 0’s.

PART C

Answer any two questions. Each question carries 20 marks.                                                            2×20=40

1. a) Write a note on the construction of CNF
2. b) Find a grammar in CNF equivalent to a grammar whose productions are

SaAbB,  AaA|a,  BbB|b                                                           (5+15)

1. State and prove uvwxy theorem.
 b
 a
1. Construct a DFA for the NDFA given below

a                           b

1. a) Construct a DFA accepting all strings over {0,1} having even number of 0’s.
2. b) Let G = ( N, T, P, S), N = {S, A},   T = {a,b},   P = { SaA,  A bS, Ab}.

Find L(G) and also construct an NDFA accepting L(G)         (10+10)

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## Loyola College B.Sc. Mathematics April 2007 Fluid Dynamics Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

 CV 18

B.Sc.  DEGREE EXAMINATION –MATHEMATICS

FIFTH SEMESTER – APRIL 2007

MT 5401FLUID DYNAMICS

Date & Time: 03/05/2007 / 9:00 – 12:00        Dept. No.                                                     Max. : 100 Marks

SECTION A

Answer ALL Questions.                     (10 x 2 = 20)

1. Define Lagrangian method of fluid motion.
2. State the components of acceleration in Cartesian coordinates?
3. What is the equation of continuity for (i) a homogeneous steady flow of fluid, (ii) a non-homogeneous incompressible flow of fluid.
4. Show that u = a+ by – cz, v = d – bx + ez, w = f + cx – ey are the velocity components of a possible liquid motion.
5. Write down the boundary condition when a liquid is in contact with a rigid surface.
6. Write down the stream function in terms of fluid velocity.
7. If = A(x2 – y2) represents a possible flow phenomena, determine the stream function.
8. State the Bernoulli’s equation for a steady irrotational flow?
9. What is the complex potential of sources at a1, a2, ….,an with strengths m1, m2,…,mn respectively?
10. Describe the shape of an aerofoil.

SECTION B

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

1. (a) Define a streamline. Derive the differential equation of streamline.

(b) Determine the equation of streamline for the flow given by .         (4 + 4)

1. Explain local, convective and material derivatives.
2. The velocity field at a point is . Obtain pathlines and streaklines.
3. Show that the velocity potential satisfies the Laplace equation. Also find the streamlines.
4. Derive Euler’s equation of motion for one-dimensional flow.
5. Explain how to measure the flow rate of a fluid using a Venture tube.
6. Derive the complex potential of a doublet.
7. Explain the image system of a source with regard to a plane.

SECTION C

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

1. The velocity components of a two-dimensional flow system can be given in Eulerian system by . Find the displacement of the fluid particle in the Lagrangian system.
2. (a) Show that is a possible form of a bounding surface of a liquid.

(8 + 12 marks)

1. (a) Derive Bernoulli’s equation.

(b) Explain the functions of a pitot tube with a neat diagram.                             (10 + 10 marks)

1. State and prove the theorem of Kutta and Joukowski.

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## Loyola College B.Sc. Mathematics April 2007 Astronomy Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc.

 CV 15

DEGREE EXAMINATION –MATHEMATICS

FIFTH SEMESTER – APRIL 2007

MT 5503ASTRONOMY

Date & Time: 02/05/2007 / 1:00 – 4:00          Dept. No.                                                     Max. : 100 Marks

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## Loyola College B.Sc. Mathematics April 2007 Alg.,Anal.Geomet. Cal. & Trign. – I Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034B.Sc. DEGREE EXAMINATION – MATHEMATICSFIRST SEMESTER – APRIL 2007MT 1500 – ALG.,ANAL.GEOMET. CAL. & TRIGN. – I
Date & Time: 24/04/2007 / 1:00 – 4:00 Dept. No. Max. : 100 Marks
SECTION –AAnswer all:                                                                              2 x 10 = 20
1. If y = a cos5x + b sin5x, show that  . 2. Write down the nth derivative of eax. 3. What is the formula for radius of curvature in parametric form.                  4. Find the sub tangent and the sub normal for y = 3×3. 5. If x = sin 2ө and y = cos 2ө, find  . 6. Determine the quadratic equation having 3-2i  as one of its roots. 7. Derive the relation sin ix = i sinh x. 8. Separate into real and imaginary parts for cos (x+iy). 9. Define conjugate diameters.10. Write the polar form of the conic.
SECTION –BAnswer any five:                                                                              5x 8 = 40
11. Find the nth derivative of sin2x sin4x sin6x.12. Find the angle of intersection of the cardioids r = a(1+cosө) and r = b(1-cosө).13. Find the lengths of the sub tangent and the sub normal at the point (a, a)      for the curve y = x3+ 3x+4.  14. Show that the roots of the equation x3+px2+qx+r =0 are in A.P if    2p3-9pq+27r =0.15. Solve the equation 6×5+11×4-33×3-33×2+11x+6= 0.
16. Prove that  = 7 – 56 sin2ө + 11 2 sin4ө – 64 sin6ө.                        17. Prove that 32 cos6ө = cos6ө + 6 cos4ө +15 cos2ө + 10.                    18. Prove that the product of the focal distances of a point on an ellipse is equal to the           square of the semi-diameter which is conjugate to the diameter through the point.

SECTION –CAnswer any two:                                                                              2x 20 = 40
19. State and prove Leibnitz theorem and prove that (1-x2)y2 –xy1+m2y = 0 and       (1-x2) yn+2 –(2n+1)xyn+1+(m2-n2)yn = 0  for  y = sin( msin-1x).                                                                                                                                               (P.T.O)20 a) Find the evolute of the ellipse  .     b) Find the p-r equation of rm = am sinm ө.                                                (10+10)
21 a) Find the equation whose roots are the roots of the equation          x4-x3-10×2+4x+24 = 0 increased by 2 and hence solve the equation.      b) Find the sum of the fourth power of the roots of the equation             x3-2×2+x+1 = 0.                                                                                   (10+10)                                                                                          22 a) Prove that  .
b) Prove that the tangent to a rectangular hyperbola terminated by its asymptotes          is bisected at the point of contact and encloses a triangle of constant area.                                                                                                                                                                                                                                                                                       (10+10)

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