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

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

FIFTH SEMESTER – APRIL 2012

# MT 5405 – FLUID DYNAMICS

Date : 27-04-2012              Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

Section A

Answer ALL questions:                                                                                                           10 ´ 2 = 20

1. Define Stream tube.
2. Show that the velocity field the stream lines are circular.
1. Write down the boundary condition for the flow when it is in moving.
2. Prove that the fluid motion is possible if .
3. What is the complex potential of source with strength m situated at the origin?
4. Find the stream function y, if j = A(x2y2) represents a possible fluid motion
5. Define velocity potential.
6. Define vortex lines.
7. Find the vorticity components of a fluid motion, if the velocity components are

u = c(x + y), v = – c(x + y).

1. Define camber.

Section B

Answer any FIVE questions:                                                                               5 ´ 8 = 40

1. Find the equation of streamlines and path lines of a flow given by .
2. Explain pitot tube.
3. Derive the equation of continuity.
4. Derive the Bernoulli’s equation of motion for the fluid.
1. Prove that for the complex potential the streamlines and equipotentials are circles.
2. Obtain the complex potential due to the image of a doublet with respect to the circle.
3. Let , (A, B, C are constants) be the velocity vector of a fluid motion. Find the equation of vortex lines.
4. State and prove the theorem of Kutta-Joukowski.

Section C

Answer any TWO questions:                                                                         2 ´ 20 = 40

1. (a)The velocity components for a two dimensional fluid system can be given in the Eulerian system by . Find the displacement of a fluid particle in the Lagrangian system

(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. (a)Derive the Euler’s equation of motion.

(b)Draw and explain the working of a Venturi tube.                                               (12 + 8)

21.(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 April 2012 Complex Analysis Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

SIXTH SEMESTER – APRIL 2012

# MT 6603/MT 6600 – COMPLEX ANALYSIS

Date : 16-04-2012                   Dept. No.                                          Max. : 100 Marks

Time : 1:00 – 4:00

PART – A

1. Find the absolute value of
2. Define harmonic function.
3. Find the radius of convergence of the series .
4. Using Cauchy integral formulas evaluatewhere  is the unit circle .
5. Define zero and poles of a function.
6. Write Maclaurin series expansion of s.
7. Define residue of a function.
8. State Argument principle.
9. Define isogonal mapping.
10. Define critical point.

PART – B

1. Let f(z)= .  Show that f(z) satisfies CR equations at zero but not differential at .
2. Prove that is harmonic and find its Harmonic conjugate.
3. State and prove Liouvilles theorem and deduce Fundamental theorem of algebra.
4. Find the Taylors series to represent in .
5. Suppose is analytical in the region  and is not identically zero in .Show that the set of all zeros of  is isolated.
6. Use residue calculus to evaluate dz over  where  is the unit circle.
7. Show that any bilinear transformation can be expressed as a product of translation, rotation, magnification or contraction and inversion.
8. Find the bilinear transformation which maps the points  onto

PART – C

1. (a) Derive CR equations in polar coordinates.

(b)   Find the real part of the analytic function whose imaginary part isand construct the analytic function.

20 .       (a) State and prove Cauchy integral formula.

(b)   Let F be an analytic inside and on a simple closed curve C. Let z be a point inside C. Show that f’(z) = dt.

1. (a) Expand in a Laurants series in (i) , ( ii)  .

(b) Using method of contour integration evaluate   dx.

1. (a) Show that any bilinear transformation which maps the unit circle = 1 onto  = 1 can be written in the form  where is real.

(b)State and prove Rouche’s theorem.

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

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034B.Sc. DEGREE EXAMINATION – MATHEMATICSTHIRD SEMESTER – APRIL 2012MT 3502/MT 5503 – ASTRONOMY
Date : 26-04-2012 Dept. No.         Max. : 100 Marks                 Time : 9:00 – 12:00
PART – AAnswer ALL questions: (10×2=20)1. Define first point of Aries and first point of Libra.2. Write a short note on twilight.3. State the laws of refraction.4. What is a sidereal clock?5. State Kepler’s laws of planetary motion.6. The Nautical Almanac gives the Julian date of Jan.1. 1940 as 2429630.               Find the J.D of Jan 26, 1965 and the day of the week.7. Define the terms ‘Umbra’ and ‘Penumbra’.8. Prove that one eclipse must occur near a node.9. State Stefan’s and Wien’s laws of radiation.10. Define the term ‘comets’.PART – BAnswer any FIVE questions:         (5 x 8 = 40)
11. Define ‘Dip of Horizon’ and derive an expression for the same. List down the effects of dip.12. i) What are the variations in the durations of day and night during the year for a      place on the Tropic of Cancer?      (4 marks)ii) Find the condition that twilight may last throughout night.                (4 marks)13. Derive Cassini’s formula for refraction.14. Define geocentric parallax of a body and derive an expression for it.15. Explain the method of verifying the second law of Kepler in the case of the earth.16. Derive an analytic expression for the equation of time.17. Define sidereal and synodic months and find the relation between them.18. Write a brief note on ‘Neptune’.PART –CAnswer any TWO questions: (2×20=40)19. a) Explain the Horizontal system of celestial co – ordinates to fix the position of      a body in the celestial sphere.    (10 marks)    b) i) Prove that the latitude of a place is equal to the altitude of the celestial pole.      (5 marks)    ii) Find the condition that a star is circumpolar.                 (5 marks)
20. a) Derive an expression to find the effect of parallax on the longitude and latitude      of a star.    (12 marks) b) Define Aberration and prove that it varies as the sine of the Earth’s way. (8 marks)21. a) Derive Newton’s deductions from Kepler’s laws.    (12 marks) b) Write a short note on Asteroids.      (8 marks)22. a) Define phase of moon and prove that it is equal to  , where  is the     elongation of moon.    (10 marks)      b) Find the condition for the occurrence of a Lunar eclipse.                    (10 marks)

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## Loyola College B.Sc. Mathematics April 2012 Algebra,Analytical Geometry And Calculus Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc., DEGREE EXAMINATION – MATHEMATICS

SECOND SEMESTER – APRIL 2012

MT 2501/MT2500- ALGEBRA,ANALYTICAL GEOMETRY AND CALCULUS

Date: 16-04-2012           Dept. No.                                                         Max. : 100 marks

Time: 9.00 – 12.00

PART-A

1. Evaluate .
2. Find x dx.
3. Define exact differential equations.
4. Solve (D22D + 1) y = 0.
5. Show that the series is convergent.
6. State Cauchy root test for convergence of a series.
7. Find the coefficient of in the expansion of 1 +  +  +  + …
8. Prove that
9. Find the direction cosines of the line joining the points (3,-5,4) and (1,-8,-2).
10. Find the angle between the planes and .

PART –B

1. Evaluate
2. Find the surface area of the solid formed by revolving the cardiodabout the initial line.
3. Solve
4. Solve .
5. Examine the convergence of
6. Assuming that the square and the higher powers of x may be neglected

show that

1. Sum to infinity the series
2. Find the shortest distance between the linesand and the equation of the line.

PART – C

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

1. (a) Evaluate.

(b)  Find the length of the curve  between the points given by and.

1. (a) Solve

(b) Solve the following equation by the method of variation of parameter:

.

1. (a) Test the convergence of the series

(b)  Find the equation of the sphere passing through the points (2,3,1),(5,-1, 2),
(4,3,-1) and (2,5,3).

1. (a) Show that

(b)  Sum the series

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

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

FIRST SEMESTER – APRIL 2012

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

Date : 28-04-2012              Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

PART – A

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

1. Find the nth derivative of .
2. Find the slope of the straight line .
3. Write the formula for the radius of curvature in Cartesian form.
4. Define Cartesian equation of the circle of the curvature.
5. If ,are the roots of the equation x3+px2+qx+r=0. Find the value of .
6. Diminish the roots x4+x3-3x2+2x-4 =0 by 2.
7. Evaluate
8. Prove that
9. Define Pole and Polar of a ellipse.
10. In the hyperbola 16x2-9y2 = 144, find the equation of the diameter conjugate to the diameter x =2y.

PART – B

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

1. Find the nth derivative of .
2. Find the angle between the radius vector and tangent for the curve at

.

1. Solve the equation x3-4x2-3x+18=0 given that two of its roots are equal.
2. Solve the equation x4-5x3+4x2+8x-8=0 given that 1-is a root.
3. Expand in terms .
4. Separate real and imaginary parts .
5. P and Q are extremities of two conjugate diameters of the ellipse and S is a focus. Prove that
6. The asymptotes of a hyperbola are parallel to 2x+3y=0 and 3x-2y =0 . Its centre is at (1,2) and it passes through the point (5,3). Find its equation and its conjugate.

PART – C

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

1. (a) If , show that

(b) Prove that the sub-tangent at any point on                is constant ant the subnormal is

(10 +10)

1. (a) Find the radius of curvature at any point on the curve

(b) Show that the evolute  of the cycloid   is another

cycloid .                                                                                                                     (10+10)

1. (a) Solve 6x5+11x4-33x3-33x2+11x+6=0.

(b)  Find by Horner’s method, the roots of the equation  which lies between 1 and 2

correct to two decimal places.                                                                                                                             (10+10)

1. (a) Prove that

(b) Prove that the Product of the perpendicular drawn from any point on a hyperbola to its

asymptotes is constant.                                                                                                   (10+10)

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