B.Sc Corporate Mathematics Question Paper 2006

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

by

             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)

 

 

Go To Main page

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

by

            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.

 

Go To Main page

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

by

LOYOLA COLLEGE  ( AUTONOMOUS ) , CHENNAI – 600 034

BSc DEGREE EXAMINATION  -MATHEMATICS

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.

 

 

Go To Main Page

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

by

             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.

 

Go To Main Page

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

by

             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

adjacent vertices.

  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)

 

Go To Main Page

 

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

by

   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)

 

Go To Main Page

 

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

by

             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)

 

 

Go To Main Page

© Copyright Entrance India - Engineering and Medical Entrance Exams in India | Website Maintained by Firewall Firm - IT Monteur