B.Sc Mathematics Question Paper

Loyola College B.Sc. Mathematics Nov 2012 Modern Algebra Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

FOURTH SEMESTER – NOVEMBER 2012

MT 4502/4500 – MODERN ALGEBRA

Date: 03-11-2012                       Dept. No.                                                       Max. : 100 Marks

Time: 1.00 – 4.00

                                                                 

                                                                  SECTION – A                                       (10 ´ 2 = 20)

Answer ALL the questions.

  1. Define partially ordered set and give an example.
  2. Show that identity element of a group is unique.
  3. Let and . Prove that divides .
  4. Show that any group of order up to 5 is abelian.
  5. Define kernel of a homomorphism.
  6. State fundamental theorem of homomorphism.
  7. Define a ring and give an example.
  8. If is a ring with unit element , then for all , show that .
  9. State unique factorization theorem.
  10. If is a commutative ring with unity, prove that every maximum ideal of  is a prime ideal.

 

PART – B ( 5 ´ 8 = 40)

Answer any FIVE questions

  1. If is a group in which  for three consecutive integers  for all , show that  is abelian.
  2. Show that a subgroup of a group  is a normal subgroup of  if and only if every left coset of  in  is a right coset of in .
  3. `Prove that every group of prime order is cyclic.
  4. State and prove Cayley’s theorem.
  5. Show that any two finite cyclic groups of the same order are isomorphic.
  6. Define a subring of a ring. Show that the intersection of two subrings of a ring  is a subring of .
  7. Show that every finite integral domain is a field.
  8. Show that every Euclidean ring is a principal ideal domain.

 

PART – C (2 ´ 20 = 40)

Answer any TWO questions

  1. (a) State and prove the fundamental theorem of arithmetic.

(b)  Show that a nonempty subset  of a group  is a subgroup of  if and only if implies .

  1. (a) State and prove the Lagrange’s theorem.

(b)   Let  be a commutative ring with unit element whose only ideals are  and  itself.  Show that  is a field.

  1. (a) Determine which of the following are even permutations:
  • (ii)

(b)   If  is a group, then show that , the set of automorphisms of , is also a group.

  1. (a) Show that an ideal of the Euclidean ring  is a maximal ideal of  if and only if it is generated by a prime element of .

(b)   Show that , the set of all Gaussian integer, is a Euclidean ring.

 

 

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Loyola College B.Sc. Mathematics Nov 2012 Mechanics – II Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc., DEGREE EXAMINATION – MATHEMATICS

SIXTH SEMESTER – NOVEMBER 2012

MT 6604 / MT 5500 – MECHANICS – II

 

 

 

Date :5/11/2012                Dept. No.                                        Max. : 100 Marks

Time :1.00 – 4.00

 

PART – A

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

 

  1. State the conditions for non-existence of centre of gravity.
  2. Write down the co-ordinates of center of gravity for a solid cone.
  3. Define catenary.
  4. Define suspension bridge.
  5. A particle executing simple harmonic motion makes 100 complete oscillations per minute and its maximum speed is 15 feet per sec. What is the length of its path and maximum acceleration?
  1. Write down any two applications of simple harmonic motion.
  2. A point P describes with a constant angular velocity about O the equiangular

spiral r = a eθ. O being the pole of the spiral. Obtain the radial and transverse

acceleration of P.

  1. Define central orbit.
  2. Define Moment of Inertia of a particle about a straight line.
  3. Find the work that must be done on a uniform flywheel of mass 50 lbs and radius

6״  to increase its speed of rotation from 5 to 10 rotation per second.

 

PART –B

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

 

  1. Find the center of gravity of a hollow right circular cone of height h.
  2. Prove that if a dynamical system is in equilibrium, then the work done by the

applied forces in a virtual displacement is zero.

  1. Discuss the motion of a particle executing two simple harmonic motions in

perpendicular directions with same period.

  1. A square hole is punched out of a circular lamina of radius as its diagonal. Show

that the distance of Centre of gravity of the remainder from the centre of the circle

is  a/(4π-2).

  1. Derive the pedal p-r equation of a central orbit.
  2. If the law of acceleration is and the particle is projected from an

apse at a distance c with velocity , prove that the equation of the orbit is

.

  1. Find the moment of Inertia of a thin uniform parabolic lamina bounded by the

parabola  and y axis about the y-axis.

  1. Derive the equation of motion of a rigid body about a fixed axis.

 

 

PART –C

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

 

  1. (a) From a solid cylinder of height h, a cone whose base coincides with the base

of the cylinder is scooped out so that the mass centre of the remaining solid

coincides with the vertex of the cone. Find the height of the cone.

 

(b) Find the centre of gravity of the arc of the cardiod r=a(1+cosθ) lying above

the initial line.                                                                                                             (10 + 10)

 

  1. (a) Derive the equation to the common catenary in the form y = C cosh x/c.

 

(b) A chain of length 2l is to be suspended from two points  A and B in the same

horizontal level so that either terminal tension is n times that at the lowest

point. Show that the span AB must be

  • + 10)
  1. (a) A particle executing simple harmonic motion in a straight line has velocities

8,7,4 at three points distant one foot from each other. Find the period.

 

(b) Find the resultant of two simple harmonic motions of the same period in the

same straight line.                                                                                                  (10 + 10)

 

  1. (a) State and prove Perpendicular axis theorem

 

(b) Show that the moment of inertia of the part of the parabola  cut off

by the double ordinate  is about the tangent at the vertex and

about its axis, M being the mass.                                                               (6 + 14)

 

 

 

 

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Loyola College B.Sc. Mathematics Nov 2012 Mechanics – I Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

FIFTH SEMESTER – NOVEMBER 2012

MT 5506/MT 4501 – MECHANICS – I

 

 

 

Date : 03/11/2012             Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

PART – A

 

 

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

 

  1. State the conditions for equilibrium of a system of concurrent forces.
  2. State the law of parallelogram of forces.
  3. Define torque of a force.
  4. State any two laws of friction.
  5. Find the resultant of two velocities 6 mt/sec and 8 mt/sec inclined to each other at an angle

of 30.

  1. Define angular velocity.
  2. Define momentum.
  3. State the principle of conservation of linear momentum.
  4. Define range of flight for a projectile.
  5. Define the coefficient of elasticity.

 

PART – B

 

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

 

  1. State and prove Lami’s theorem.

 

  1. A uniform plane lamina in the form of a rhombus one of whose angles is 120° is supported by two forces of magnititudes P and Q applied at the centre in the directions of the diagonals so that one side is horizontal. Show that if P > Q, then P2 = 3Q2.

 

  1. State and prove Varignon’s theorem on moments.

 

  1. Two particles weighing 2 kg and 1 kg are placed on the equally rough slopes of a double inclined plane whose inclinations with the horizontal are 60° and 30° The particles are connected by a light string passing over a smooth pulley at the common vertex of planes.  If the heavier particle is on the point of slipping downwards, show that the coefficient of friction is

 

  1. A particle is dropped from an aeroplane which is rising with acceleration f and t secs after this; another stone is dropped. Prove that the distance of between the stones at time t’ after the second stone is dropped is .
  2. 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.

 

 

  1. Show that when masses P and Q are connected by a string over the edge of a table, the tension is the same wheter P hangs and Q is on the table or Q hangs and P is on the table.

 

  1. Two balls impinge directly and the interchange their velocities after impact. Show that they are perfectly elastic and of equal mass.

 

 

PART – C

 

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

 

  1. a) Two strings AB and AC are knotted at A, where a weight W is attached. If the weight

hangs freely and in the position of equilibrium, with BC horizontal,

AB : BC : CA = 2 : 4 : 3, show that the tensions in the strings are

 

  1. b) A system of forces in the plane of D ABC is equivalent to a single force at A, 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 (b+c) G1 = bG2 + cG3.

(10 + 10)

 

  1. a) Two unlike parallel forces P and Q (P>Q) act at A and B respectively. Show that if the

direction of P be reversed, the resultant is displaced through the distance .

 

  1. b) A particle moving in a st. line is subject to a resistance KV3 producing retardation

where v is the velocity.  Show that if v is the velocity at any time t when the distance is

s,  and  where u is the initial velocity.                              (10 + 10)

 

  1. Derive the equation to the path of the projectile in the form

 

  1. A particle falls under gravity in a medium where the resistance varies as the square of the velocity. Discuss the motion.

 

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Loyola College B.Sc. Mathematics Nov 2012 Linear Algebra Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

FIFTH SEMESTER – NOVEMBER 2012

MT 5508/MT 5502 – LINEAR ALGEBRA

 

 

Date : 08/11/2012             Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

PART – A

 

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

 

  1. Define a vector space over a field F.
  2. Show that the vectors (1,1) and (-3, 2) in R2 are linearly independent over R, the field of real numbers.
  3. Define homomorphism of a vector space into itself.
  4. Define rank and nullity of a vector space homomorphism T: u®
  5. Define an orthonormal set.
  6. Normalise in R3 relative to the standard inner product.
  7. Define a skew symmetric matrix and give an example.
  8. Show that is orthogonal.
  9. Show that is unitary.
  10. Define unitary linear transformation.

 

PART – B

 

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

 

  1. Prove that the intersection of two subspaces of a vector space v is a subspace of V.
  2. If S and T are subsets of a vector space V over F, then prove that
  3. S T implies that L(S) ≤ L(T)
  4. L(L(S)) = L(S)
  5. L(S U T) = L(S) + L(T).
  6. Determine whether the vectors (1,3,2), (1, -7, -8) and (2, 1, -1) in R3 are linearly dependent on independent over R.
  7. If V is a vector space of finite dimension and W is a subspace of V, then prove that

dim V/W = dim V – dim W.

  1. For any two vectors u, v in V, Prove that .
  2. If and l Î F, then prove that l is an eigen value of T it and only if [l I – T] is singular.
  3. Show that any square matrix can be expressed as the sum of a symmetric matrix and a skew symmetric matrix.
  4. For what values of T, the system of equations over the rational field is consistent?

PART – C

 

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

 

  1. a) Prove that the vector space V over F is a direct sum of two of its subspaces W1 and W2

if and only if V = W1 + W2 and W1  W2 = {0}.

 

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

W, than prove that dim V = dim U + dim W.                                                                   (10 + 10)

 

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

vector space Hom (U, V) is of dimension mn.

 

  1. Apply the 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) and (1, 0, -1, 2).

 

  1. a) Prove that TÎA(V) is singular if and only it there exists an element v ≠ 0 in V such that

T(v) = 0.

  1. b) Prove that the linear transformation T on V is unitary of and only if it takes an

orthonormal basis of V onto an orthonormal basis of V.                                                       (10 +10)

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Loyola College B.Sc. Mathematics Nov 2012 Fluid Dynamics Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

FIFTH SEMESTER – NOVEMBER 2012

MT 5405 – FLUID DYNAMICS

 

 

Date : 10/11/2012             Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

Section A

Answer ALL questions:                                                                                                                         10 ´ 2 = 20

 

  1. Define stream tube.
  2. Show that is a possible motion.
  3. The velocity vector q is given by determine the equation of stream line.
  4. Write down the boundary condition for the flow when it is moving.
  5. What is the complex potential of a source with strength m situated at the points z=z1
  6. Find the stream function y, if j = A(x2 – y2) represents a possible fluid motion
  7. Find the vorticity vector for the velocity
  8. Define vortex tube and vortex filament.
  9. What is lift of an aerofoil?
  10. Define camber.

Section B

Answer any FIVE questions:                                                                                        5 ´ 8 = 40

  1. Explain Material, Local and Convective derivative fluid motion.
  1. Find the equation of streamlines and path lines of a flow given by
  1. Explain the construction of a Venturi tube.
  2. Prove that for the complex potential the streamlines and equipotentials are circles.
  3. For an incompressible fluid. Find the vorticity vector and equations of stream line.
  4. Derive the equation of continuity.
  5. Find the stream function y(x, y, t) for a given velocity field u = 2Axy, v = A(a2 + x2 – y2).
  6. State and prove the theorem of Kutta-Joukowski.

 

Section C

Answer any TWO questions:                                                                                                      2 ´ 20 = 40

  1. (a) For a two-dimensional flow the velocities at a point in a fluid may be expressed in the Eulerian coordinates by u = x + y + t and v = 2x+2y + t. Determine the Lagrange coordinates as functions of the initial positions, and the time t.

(b) If the velocity of an incompressible fluid at the point (x, y, z) is given by  where . Prove that the fluid motion is possible and the velocity potential is .                                                                       (10 + 10)

 

  1. Derive the Euler’s equation of motion and deduce the Bernoulli’s equation of motion.

 

  1. (a)What arrangement of sources and sinks will give rise to the function?

(b)Obtain the complex potential due to the image of a source with respect to a circle.    (12+8)

 

  1. (a)Discuss the structure of an aerofoil.

(b)Derive Joukowski transformation.                                                                   (8+12 )

 

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Loyola College B.Sc. Mathematics Nov 2012 Complex Analysis Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

SIXTH SEMESTER – NOVEMBER 2012

MT 6603/6600 – COMPLEX ANALYSIS

 

 

 

Date : 05/11/2012               Dept. No.                                          Max. : 100 Marks

Time : 1:00 – 4:00

PART-A

Answer ALL questions                                                                                                                 (10×2=20 )

  1. Show that the function is nowhere differentiable.
  2. When do we say that a function is harmonic.
  3. Find the radius of convergence of the series .
  4. State Cauchy Goursat theorem.
  5. Expand as a Taylor’s series about the point .
  6. Define meromorphic function with an example.
  7. Define residue of a function at a point.
  8. State argument principle.
  9. Define the cross ratio of a bilinear transformation.
  10. Define an isogonal mapping.

 

PART-B

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

  1. Show that the function is discontinuous at  given that when and .
  2. Find the analytic function of which the real part is .
  3. Evaluate along the closed curve containing paths and .
  4. State and prove Morera’s theorem.
  5. State and prove Maxmimum modulus principle.
  6. Find out the zeros and discuss the nature of the singularity of .

 

  1. State and prove Rouche’s theorem.
  2. Find the bilinear transformation which maps the points into the points

PART C

Answer any TWO questions                                                                                                       (2×20=40)

  1. (a) Let be a function defined in a region such that  and their first order partial derivatives are continuous in . If the first order partial derivatives of  satisfy the Cauchy-Riemann equations at a point  in D then show that f is differentiable at .

(b) Prove that every power series represents an analytic function inside its circle of convergence.

  1. (a) State and prove Cauchy’s integral formula.

(b)          Expand in a Laurent’s series for (i) (ii)
(iii) .

  1. (a) State and prove Residue theorem.

(b) Using contour integration evaluate  .

  1. (a) Let be analytic in a region  and  for .Prove that f is conformal at .

(b) Find the bilinear transformation which maps the unit circle onto the unit circle .

 

 

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Loyola College B.Sc. Mathematics Nov 2012 Graphs, Diff. Equ., Matrices & Fourier Series Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

FIRST SEMESTER – NOVEMBER 2012

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

 

Date : 10/11/2012               Dept. No.                                          Max. : 100 Marks

Time : 1:00 – 4:00

 

PART – A

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

  1. Let f: be defined by.  Find the  range of the function .
  2. Find the equation of the line passing through (-3,4) and (1,6).
  3. Write the normal equation of y = a+bx.
  4. Reduce y = aebx to normal form .
  5. Define Difference equation with an example.
  6. Solve yx+2 – 4yx=0.
  7. State Cayley Hamilton theorem.
  8. Find the eigen value of the matrix .
  9. Find the Fourier coefficient a0 for the function f(x) = ex in (0,2π).
  10. Define odd and even function.

PART – B

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

  1. A company has a total cost function represented by the equation y = 2x3-3x2+12x, where y

represents cost and x represent quantity.  (i) what is the equation for the Marginal cost function?

(ii) What is the equation for average cost function? What point average cost is at its minimum?

  1. The total cost in Rs.of output x is given by . Find  Cost when output is 4 units.

(i)   Average cost of output of 10 units.

(ii)  Marginal cost when output is 3 units.

  1. Fit a straight line to the following data. Estimate the sale for 1977.
Year: 1969 1970 1971 1972 1973 1974 1975 1976
Sales(lakhs) 38 40 65 72 69 60 87 95
  1. Solve yx+2 – 4yx = 9x2.
  2. Find the eigen vectors of the matrix .
  3. Verify Cayley Hamilton theorem for the matrix A =
  4. Expand f(x) = x (-π <x< π) as a Fourier series with period 2 π.
  5. Obtain a Fourier series expansion for f(x) = – x  in the range (-π,0)

=      x   in the range [0, π).

PART – C

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

  1. (a) Fit a curve of the form y = a + bx +c x2 for the following table:
                X: 0 1 2 3 4
Y: 1 1.8 1.3 2.5 6.3

 

(b)  The total profit y in rupees of a drug company from the manufacture and sale of x drug

bottles is given by  .

(i) How many drug bottles must the company sell to achieve the maximum  profit?

(ii) What is the profit per drug bottle when this maximum is achieved ?                                                     (10+10)

  1. (a) Solve yx+2 – 7yx+1 – 8yx = x(x-1) 2x.

(b) So lve u(x+1) – au(x) = cosnx.                                                                                                                                 (10+10)

  1. (a) Find the Fourier series of period 2 π for  f(x) = x2 in (o,2 π) . Deduce

(b)  Find the Fourier (i) Cosine series (ii) Sine series for the function f(x) =  π-x  in (0, π).                        (10+10)

 

 

  1. (a) Determine the Characteristic roots and corresponding vectors for the matrix

.Hence diagonalise A.

 

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Loyola College B.Sc. Mathematics Nov 2012 Business Mathematics Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

FOURTH SEMESTER – NOVEMBER 2012

MT 4205 – BUSINESS MATHEMATICS

 

 

Date : 05/11/2012             Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

 

PART A

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

 

  1. Find the equilibrium price and quantity for the functions  and
  2. If the demand law is find the elasticity of demand in terms of x.
  3. Find if .
  4. Find the first order partial derivatives of .
  5. Evaluate
  6. Prove that .
  7. If and , find .
  8. If , find .
  9. If  then find A and B
  10. Define Linear Programming Problem.

PART B

Answer any FIVE of the following:                                                                               (5x 8=40)

  1. The total cost C for output x is given by . Find (i) Cost when output is 4 units (ii) Average cost of output of 10 units (iii) Marginal cost when output is 3 units.
  2. If then prove that .
  3. Find the first and second order partial derivatives of .
  4. Integrate with respect to x.
  5. Prove that (i) , if f(x) is an even function.

(ii) , if f(x) is an odd function.

  1. If  then show that .
  2. Compute the inverse of the matrix .
  3. Integrate with respect to x.

PART C

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

  1. (a) If AR and MR denote the average and marginal revenue at any output, show that elasticity of demand is equal to . Verify this for the linear demand law .

(b) If the marginal revenue function for output x is given by , find the total revenue by integration. Also deduce the demand function.

  1. (a) Let the cost function of a firm be given by the following equation: where C stands for cost and x for output. Calculate (i) output, at which marginal cost is minimum (ii) output, at which average cost is minimum (iii) output, at which average cost is equal to marginal cost .

(b) Evaluate .

  1. (a) Find the maximum and minimum values of the function .

(b) Solve the equations  by Crammer’s rule.

  1. (a) The demand and supply function under perfect competition are and Find the market price, consumer’s surplus and producer’s surplus.

(b) Food X contains 6 units of vitamin A per gram and 7 units of vitamin B per gram and costs 12  paise  per gram. Food Y contains 8 units of vitamin A per gram and 12 units of vitamin B per gram and costs 20 paise per gram. The daily minimum requirements of vitamin A and vitamin B are 100 units and 120 units respectively. Find the minimum cost of the product mix using graphical method.

 

 

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Loyola College B.Sc. Mathematics Nov 2012 Bioinformatics – I Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS & PHYSICS

THIRD SEMESTER – NOVEMBER 2012

PB 3208 – BIOINFORMATICS – I

 

 

Date : 09/11/2012             Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

                Part –A                                                               (20 marks)

 

Answer the following, each answer within 50 words:                                                    (10×2=20 marks)

 

  1. Mention the central dogma of molecular biology.
  2. Give the types of Mrna.
  3. What is DDBJ?
  4. Expand MIPS and mention its function.
  5. What is a domain?
  6. Give any two objectives of learning bioinformatics.
  7. What is BLAST?
  8. Name any two secondary structures of proteins.
  9. Which database can be accessed to retrieve protein sequences?
  10. Name any two protein structure models.

 

Part B                                                            

 

Answer the following each within 500 words. Draw diagrams wherever necessary:

(5×7=35 marks)

  1. a) Explain the structure of chromosome.

 

(OR)

 

  1. b) Explain the secondary structure of Proteins.

 

  1. a) Define PIR database and explain its types.

 

(OR)

 

  1. b) Define Genomic database and explain the subfields in Genomics.

 

  1. a) Mention the uses of FASTA and BLAST in sequence alignment.

 

(OR)

 

  1. b) Write about Local alignment and Multiple Sequence alignment.

 

 

  1. a) How to study the physical properties of proteins using internet?

 

(OR)

 

  1. b) What are repetitive sequences and how are they masked?

 

  1. a) Compare the usage of WEBTHERMODYN and DNAlive in predicting the physical properties of

DNA.

 

(OR)

 

  1. b) Write about any one protein visualization tool.

 

 

 

                                         Part C                                                                  

 

Answer any three of the following each within 1200 words. Draw diagrams wherever necessary:                                                                                                                                (3×15=45 marks)

 

  1. Explain the structure and function of DNA.
  2. Describe the projects carried out by HGP and mention their applications.
  3. What is OMIM? Mention its significance and the procedure.
  4. Explain Needleman- Wunsch and Smith-Waterman Algorithms.
  5. Describe the steps involved in Gene Finding.

 

 

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Loyola College B.Sc. Mathematics Nov 2012 Astronomy Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

THIRD SEMESTER – NOVEMBER 2012

MT 3502/5503 – ASTRONOMY

 

 

 

Date : 05/11/2012             Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

PART – A

Answer all questions:                                                                                           (10 x 2 = 20)

  1. What are cardinal points?
  2. Define Zenith and Nadir.
  3. Define Aberration of a celestial body.
  4. What is the use of a Gnomon?
  5. Define sidereal year.
  6. Define dynamical mean sun.
  7. Define an umbra.
  8. What is Harvest Moon?
  9. What are the chief elements present in sun?
  10. Define chromosphere.

PART – B

 

Answer any five questions:                                                                                 (5 X 8 = 40)

 

  1. Write a note on the equatorial system of coordinates.
  2. Trace the variations in the duration of day and night in a place of latitude 5N.
  3. Derive the tangent formula for refraction.
  4. Find an analytical expression for the equation of time.
  5. What are Astronomical seasons? Find their duration.
  6. Give a brief description of the surface structure of moon.
  7. Explain how solar and lunar eclipses are caused?
  8. Write a note on comets.

PART – C

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

  1. (i) What is Twilight? Find the number of days that twilight may last throughout night

in a given place.

(ii) Write a note on Morning stars, Evening stars and Circumpolar stars.

  1. (i) Explain any one astronomical instrument with a neat diagram.

(ii) Write a note on the different types of Calendar.

  1. (i) Describe the successive phases of moon with a neat diagram.

(ii) Find the maximum number of eclipses possible in a year.

  1. (i) Describe any two planets of the solar system.

(ii) Write a note on any four constellations visible over Chennai.

 

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Loyola College B.Sc. Mathematics Nov 2012 Analytical Geometry Of 2D,Trig. & Matrices Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

FIRST SEMESTER – NOVEMBER 2012

MT 1503 – ANALYTICAL GEOMETRY OF 2D,TRIG. & MATRICES

 

 

Date : 10/11/2012             Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

 

PART – A

Answer all questions:                                                                                           (10 x 2 = 20)

  1. Write down the expression of cos 4θ in terms of cosθ and sinθ.
  2. Give the expansion of sinθin ascending powers of θ.
  3. Express sin ix and cosix in terms of sin hx and coshx.
  4. Find the value of log(1 + i).
  5. Find the characteristic equation of A = .
  6. If the characteristic equation of a matrix is , what are its eigen values?
  7. Find pole of lx + my + n = 0 with respect to the ellipse
  8. Give the focus, vertex and axis of the parabola
  9. Find the equation of the hyperbola with centre (6, 2), focus (4, 2) and e = 2.
  10. What is the polar equation of a straight line?

PART – B

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

  1. Expandcos6θ in terms of sinθ .
  2. If sinθ = 0.5033 show thatθ is approximately .
  3. Prove that .
  4. If tany = tanα tanhβ ,tanz = cotα tanhβ, prove that tan (y+z) = sinh2βcosec2α.
  5. Verify Cayley Hamilton theorem for A =
  6. Prove that the eccentric angles of the extremities of a pair of semi-conjugate diameters of an ellipse differ by a right angle.
  7. Find the locus of poles of all tangents to the parabola with respect to

 

  1. Prove that any two conjugate diameters of a rectangular hyperbola are equally inclined to the asymptotes.

 

PART – C

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

  1. (i) Prove that .

 

(ii) Prove that .

  1. (i) Prove that if

(ii) Separate into real and imaginary parts tanh(x + iy).

  1. Diagonalise A =
  2. (i) Show that the locus of the point of intersection of the tangent at the extremities of a pair of

conjugate diameters of the ellipse is the ellipse

(ii) Show that the locus of the perpendicular drawn from the pole to the tangent to the circle r = 2a

     cosθ  isr = a(1+cosθ).

 

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Loyola College B.Sc. Mathematics Nov 2012 Algebra, Calculus And Vector Analysis Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

FIRST SEMESTER – NOVEMBER 2012

MT 1503 – ANALYTICAL GEOMETRY OF 2D,TRIG. & MATRICES

 

 

Date : 10/11/2012             Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

 

PART – A

Answer all questions:                                                                                           (10 x 2 = 20)

  1. Write down the expression of cos 4θ in terms of cosθ and sinθ.
  2. Give the expansion of sinθin ascending powers of θ.
  3. Express sin ix and cosix in terms of sin hx and coshx.
  4. Find the value of log(1 + i).
  5. Find the characteristic equation of A = .
  6. If the characteristic equation of a matrix is , what are its eigen values?
  7. Find pole of lx + my + n = 0 with respect to the ellipse
  8. Give the focus, vertex and axis of the parabola
  9. Find the equation of the hyperbola with centre (6, 2), focus (4, 2) and e = 2.
  10. What is the polar equation of a straight line?

PART – B

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

  1. Expandcos6θ in terms of sinθ .
  2. If sinθ = 0.5033 show thatθ is approximately .
  3. Prove that .
  4. If tany = tanα tanhβ ,tanz = cotα tanhβ, prove that tan (y+z) = sinh2βcosec2α.
  5. Verify Cayley Hamilton theorem for A =
  6. Prove that the eccentric angles of the extremities of a pair of semi-conjugate diameters of an ellipse differ by a right angle.
  7. Find the locus of poles of all tangents to the parabola with respect to

 

  1. Prove that any two conjugate diameters of a rectangular hyperbola are equally inclined to the asymptotes.

 

PART – C

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

  1. (i) Prove that .

 

(ii) Prove that .

  1. (i) Prove that if

(ii) Separate into real and imaginary parts tanh(x + iy).

  1. Diagonalise A =
  2. (i) Show that the locus of the point of intersection of the tangent at the extremities of a pair of

conjugate diameters of the ellipse is the ellipse

(ii) Show that the locus of the perpendicular drawn from the pole to the tangent to the circle r = 2a

     cosθ  isr = a(1+cosθ).

 

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Loyola College B.Sc. Mathematics Nov 2012 Algebra, Analy. Geo., Calculus & Trigonometry Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc., DEGREE EXAMINATION – MATHEMATICS

THIRD SEMESTER – NOVEMBER 2012

MT 3501/3500 – ALGEBRA, CALCULUS AND VECTOR ANALYSIS

 

 

 

Date : 02-11-2012              Dept. No.                                        Max. : 100 Marks

Time : 9.00 – 12.00

 

PART – A

 

ANSWER ALL THE QUESTIONS:                                                                                                (10 x 2 =  20)

 

  1. Evaluate .
  2. Evaluate .
  3. Eliminate the arbitrary constants from .
  4. Find the complete solution for
  5. Find , if .
  6. Prove that div , where is the position vector.
  7. Find L(Sin2t).
  8. Find .
  9. Find the number and sum of all the divisors of 360.
  10. State Fermat’s theorem.

 

PART – B

ANSWER ANY FIVE QUESTIONS:                                                                                  (5 x 8 = 40)

  1. Change the order of integration and evaluate the integral  .
  2. Express  in terms of Gamma functions and evaluate the integral .
  3. Solve
  4. Solve
  5. Find .
  6. Find .
  7. Show that
  8. Show that is divisible by 22.

 

PART – C

 

ANSWER ANY TWO QUESTIONS                                                                                               (2x 20 = 40)

 

  1. (a) Evaluate taken over the positive quadrant of the circle .

(b)  Prove that

  1. (a) Solve

 

(b) Solve (y+z)p + (z+x)q = x+y.

 

21.(a)  Verify Stoke’s theorem for  taken over the upper half surface of

the  sphere  x2+y2 +z2 = 1, z 0 and the boundary curve C, the x2+y2  = 1, z=0.

 

(b)  State and prove Wilson’s Theorem.

  1. Using Laplace transform solve the equation given y(0) = 0 , y1(0)= -1.

 

 

 

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Loyola College B.Sc. Mathematics Nov 2012 Algebra, Anal.Geo & Calculus – II Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

FIRST SEMESTER – NOVEMBER 2012

MT 1500 – ALGEBRA, ANALY. GEO., CALCULUS & TRIGONOMETRY

 

 

 

Date : 08/11/2012             Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

 

PART – A

 

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

 

  1. Write the nth derivative of
  2. If y = a show that
  3. Define the evolute of a curve.
  4. Find the p-r equation of the curve r = a sin q.
  5. Determine the quadratic equation having 3 – 2 i as a root.
  6. Diminish the roots of by 2.
  7. Show that
  8. Express in locus of logarithmic function.
  9. Define a rectangular hyporbola.
  10. Write down the angle between the asymptotes of the hyperbola

PART – B

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

 

  1. Show that in the parabola the subtangent at any point is double the abscissa and the subnormal is a constant.
  2. Find the radius of curvature at the point ‘O’ on
  3. Show that if the roots of
  4. Find the p-r equation of the curve with respect to the focus as the pole.
  5. Separate into real and uniaguinary parts.
  6. Find the sum of the series
  7. Find the locus of poles of ale Laugets to with respect to
  8. Derive the polar equation of a comic.

 

PART – C

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

 

  1. a) If prove that
  2. b) Show that r = a sec2 and r = b cosec2 intersect at right angles.
  3. a) Find the minimum value of
  4.        b) Find the radius of  curvature of .
  5. a) Solve: given that the roots are in geometric progression.

 

  1. b) Solve: .

 

  1. a) Express cos8q in locus of power of sinq.

 

  1. b) If e1 and e2 are the eccentricities of a hyperbola and its conjugate show that .

 

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Loyola College B.Sc. Mathematics Nov 2012 Algebra And Calculus – I Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

SECOND SEMESTER – NOVEMBER 2012

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

 

 

 

Date : 07/11/2012             Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

 

PART-A

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

  1. Evaluate
  2. Evaluate
  3. Verify whether is exact?
  4. Solve
  5. Prove that the series is convergent.
  6. Find the values of k for which the series
  7. Prove that
  8. Using binomial expansion, find the value of correct to two decimal places.
  9. Find the direction cosines of the line
  10. Find the centre and radius of the sphere

PART-B

 

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

  1. Evaluate
  2. Prove that the area and the perimeter of the cardioids are and
  3. Solve
  4. Solve
  5. Using D’Alembert’s ratio test, examine the convergence of the series
  6. Prove that
  7. When is small , prove that
  8. Find the equation of the plane passing through and and perpendicular to the plane

PART-C

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

 

  1. a) Evaluate
  2. b) Define the length of the curve and find the length of one loop of the curve
  3. a) Solve when
  4. b) Apply the method of variation of parameters to solve
  5. a) Using Rabee’s test, Examine the convergence and divergence for the series
  6. b) Prove that
  7. a) Prove that the lines and are coplanar. Find the point of intersection .Also find the equation of the plane determined by the lines.
  8. b) Find the equation of the sphere passing through the points,and having the centre of the sphere on the line

 

 

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Loyola College B.Sc. Mathematics April 2013 Modern Algebra April 2013 Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

FIRST SEMESTER – NOVEMBER 2012

MT 1502 – ALGEBRA AND CALCULUS – I

 

 

 

Date : 08/11/2012             Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

 

PART – A

ANSWER ALL QUESTIONS:                                                                (10×2=20)

 

  1. Find the nth derivative of
  2. Show that, in the curve, the polar sub tangent varies as the square of the

radius vector and the polar subnormal is a constant.

  1. Write the conditions for maxima and minima of two variables.
  2. What is the radius of curvature of the curve at the point (1, 1)?
  3. Find the co-ordinates of the centre of curvature of the curve at the point

(2, 1).

  1. Form a rational cubic equation which shall have the roots 1,
  2. If are the roots of the biquadratic equation

find

  1. State Newton’s theorem on the sum of the powers of the roots.
  2. State Descartes’ rule of signs for negative roots.
  3. Determine if Cardon’s method can be applied to solve the equation

 

PART – B                               

 

ANSWER ANY FIVE QUESTIONS:                                                      (5×8=40)

  1. a) Find the angle at which the radius vector cuts the curve
  2. b) Find the slope of the tangent with the initial line for the cardioid

at                                                                                            (4 + 4)

  1. Discuss the maxima and minima of the function

 

  1. Prove that the (p-r) equation of the cardioid is and hence

prove that its radius of curvature is

  1. Show that the evolute of the cycloid ; is another

cycloid.

  1. Solve the equation
  2. Show that the sum of the eleventh powers of the roots of is zero.
  3. a) If are the roots of the equation find the value of

 

  1. b) Determine completely the nature of the roots of the equation

(5 + 3)

  1. If be a real root of the cubic equation of which the coefficients

are real, show that the other two roots of the equation are real, if

 

PART – C                               

 

ANSWER ANY TWO QUESTIONS:                                                      (2 x 20 = 40)

 

  1. a) Find the nth differential coefficient of .
  2. b) If, prove that               (10 +10)

 

  1. A tent having the form of a cylinder surmounted by a cone is to contain a given

volume. If the canvass required is minimum, show that the altitude of the cone is

twice that of the cylinder.

 

  1. a) Find the asymptotes of

 

  1. b) Show that the roots of the equation are in Arithmetical

progression if  Show that the above condition is satisfied by the

equation and hence solve it.                                              (10 + 10)

 

  1. Determine the root of the equation which lies between 1 and 2

correct to three places of decimals by Horner’s method.

 

 

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Loyola College B.Sc. Mathematics Nov 2013 Physics For Mathematics-I Nov 2013 Question Paper PDF Download

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Loyola College B.Sc. Mathematics April 2014 Physics For Mathematics-I Question Paper PDF Download

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