LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

           B.Sc. DEGREE EXAMINATION – MATHEMATICS

XZ 8

THIRD SEMESTER – APRIL 2008

MT 3501 – ALGEBRA, CALCULUS AND VECTOR ANALYSIS

 

 

 

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

Time : 1:00 – 4:00

SECTION – A

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

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

.

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

 

SECTION – B

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

1. By the changing the order of integration evaluate

 

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

 

 

 

SECTION – C

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

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

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

1. a) Solve

1. b) Solve the following by Charpit’s method

 

1. c) Solve

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

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

1. a) State and prove Wilson’s theorem.

1. b) Solve given using Laplace

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

           B.Sc. DEGREE EXAMINATION – MATHEMATICS

XZ 7

THIRD SEMESTER – APRIL 2008

MT 3500 – ALGEBRA, CALCULUS & VECTOR ANALYSIS

 

 

 

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

Time : 1:00 – 4:00

PART – A

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

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

 

PART – B

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

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

of

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

for

1. With how many zeros does         end.

                                                            

PART – C

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

1. a) Evaluate

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

1. a) Find the complete integral of using charpits method.

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

1. a) Verify Stoke’s theorem for

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

1. b) Find (i) ë^ùand (ii) ë^ù

1. a) Solve using Laplace transforms

given that

and at t = 0.

1. b) Find the highest power of 11 in     .

 

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034        LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034B.Sc. DEGREE EXAMINATION – MATHEMATICSTHIRD SEMESTER – November 2008MT 3501 – ALGEBRA, CALCULUS AND VECTOR ANALYSIS
Date : 06-11-08 Dept. No.         Max. : 100 Marks                 Time : 9:00 – 12:00                                               PART – A        (10 × 2 = 20 marks)
Answer ALL the questions
1. Evaluate   .2. What is  ?3. Find the complete integral of  q = 2yp2
4. Write down the complete integral of  z = px + qy + pq.
5. Find the constant k, so that the divergence of the vector   is zero.
6. State Gauss Divergence theorem.
7. Find L(cos23t).8. Find  .9. Find Φ(360).
10. Find the highest power of 5 in 79!

PART – B         (5 × 8 = 40 marks)
Answer any FIVE questions
11. Change the order of integration and evaluate  .12. Prove that β(m,n+1 )+ β(m+1,n) = β(m,n).
13. Solve p tanx + q tany = tanz.
14. If   are irrotational, prove that
(a)   is  solenoidal.
(b)Find the unit vector normal to the surface   z = x2 + y2 – 3 at (2,-1,2).                              (4+4)15. Evaluate   by Stokes Theorem where   & C is the boundary of the        triangle with vertices (0,0,0), (1,0,0) and (1,1,0).
16. Find  (a) L(te-t sint).
(b)L(sin3t cosh2t).
17. Find  .18. (a) If N is an integer, prove that N5-N is divisible by 30.                                                            (6+2)
(b)State Fermat’s Theorem.
PART – C        (2 × 10 = 20 marks)
Answer any TWO questions
19. (a) Evaluate   over the positive octant of the sphere x2+y2+z2 = a2
(b)Establish β(m,n) =  .                                                                                 (10+10)
20. (a) Solve  .
(b) Solve by Charpit’s Method, pxy + pq + qy = yz.                                                           (10+10)
21. (a) Verify Green’s theorem for   where C is the boundary of the              region  x=0, y=0, x+y=1.
(b) Evaluate   .                                                                                         (10+10)22. (a) Using Laplace Transform, solve   given that y(0)=1, y`(0)=0..
(b) Using Wilson’s Theorem, prove that 10!+111   0 mod 143.                                               (12+8)

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

   B.Sc. DEGREE EXAMINATION – MATHEMATICS

AB 07

THIRD SEMESTER – November 2008

MT 3500 – ALGEBRA, CALCULUS & VECTOR ANALYSIS

 

 

 

Date : 06-11-08                     Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

SECTION-A

Answer all questions:                                                                        (10 x 2=20)

 

1. Evaluate .
2. If .
3. Obtain the partial differential equation by eliminating
4. Solve
5. If j =j  at  (1, -2, -1).
6. State Stoke’s theorem.
7. Find .
8. Find .
9. Find the number and sum of all the divisors of 360.
10. Find the number of integers less that 720 and prime to it.

 

 

SECTION-B

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

 

1. Evaluate  where R is the region bounded by the curves and .
2. Express interms of Gamma function and evaluate .
3. Solve .
4. Solve .
5. Show that .
6. (a) Find .

(b) Find .

1. Find .
2. Show that (18) is divisible by 437.

 

 

SECTION-C

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

 

1. (a) Change the order of integration  and evaluate the integral.

(b) Evaluate  taken over the volume bounded by the plane .

(c) Evaluate .

1. (a) Solve

(b) Find a complete integral of .

1. Verify Gauss divergence theorem for  for the cylinderical region S given by .
2. Solve .

 

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE Examination – Mathematics

Third Semester – OCT/NOV 2010

MT 3501/MT 3500 – Algebra, Calculus and Vector Analysis

 

 

Date & Time:                                Dept. No.                                                    Max. : 100 Marks

 

 

PART – A

Answer ALL questions.                                                                                                  (10 ´ 2 = 20)

1. Evaluate
2. Find the Jacobian of the transformation x = u (1 + v) ; y = v (1 + u).
3. Find the complete solution of z = xp + yq + p² – q².
4. Solve
5. For , find div at (1, -1, 1)
6. State Green’s theorem.
7. What is L(f¢¢ (t))?
8. Compute
9. Find the sum and number of all the divisors of 360.
10. Define Euler’s function f(n) for a positive integer n.

 

PART – B

Answer any FIVE questions                                                                                          (5 ´ 8 = 40)

11. Evaluate by changing the order of the integration.
12. Express in terms Gamma functions.
13. Solve z²( p²+q² + 1 ) = b²
14. Solve p² + q² = z²(x + y).
15. Find
16. Find
17. Prove that
18. Show that 18! + 1 is divisible by 437

 

PART – C

Answer any THREE questions.                                                                               (2 ´ 20 =40)

19. (a) Evaluate  taken through the positive octant of the sphere x² + y² + z² = a².

(b)  Show that

20. (a) Solve (p² + q²) y = qz.

(b)  Solve (x² – y²)p + (y² – zx)q = z² – xy

21. (a) Verify Gauss divergence theorem for taken over the region bounded by the planes x = 0, x = a, y = 0 y = a, z = 0 and z = a.

(b)  State and prove Fermat’s theorem.

22. (a) Using Laplace transform solve  given that .

(b)  Show that if n is a prime and r < n, then (n – r)!  (r – 1)! + (-1)^{r – 1} º 0 mod n.

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