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Sunday, October 01, 2017

Dibrugarh University (BA - 1st Semester) Question Paper - Mathematics (Nov' 2012)

Course: 101
[(a) Classical Algebra, (b) Trigonometry, (c) Vector Calculus]
Full Marks: 80
Pass Marks: 32
Time: 3 hours
The figures in the margin indicate full marks for the questions
(Classical Algebra)

1. Answer the following questions: 1x4=4
  1. The range of a real sequence may contain a complex number. (State True or False)
  2. The elements of a real sequence can be put in a one-one correspondence with what set?
  3. Every equation of odd degree has at least one real root. (State True or False)
  4. Write the number of positive real roots of the equation
2. Answer the following questions: 2x4=8
  1. Write the limit point(s) of the sequence
  2. Write the interval of for which the sequence converges.
  3. Find the equation whose roots are the reciprocals of the roots of the equation
  4. Find the other root of the equation whose two roots being equal in magnitude but opposite in sing.
3. Find the value of 3
4. Prove that every convergent sequence is bounded. 4
Show that the sequence {1, 2, 3, 2, 5, 2, 7, 2, 3, 2, 11, 2, 13, …….}
oscillates infinitely.
5. Answer any two equations of the following: 5x2=10
  1. Show that the series does not converge.
  2. Test the convergence of the series
  3. Show that a series with positive term is convergent for
6. Solve the equation using Cardin’s method. 10
If be a root of the equation then show that is a root of the equation
7. Show that if a polynomial be divided by a binomial then remainder is 5


8. (a) Write the solution (s) of the equation 1
(b) Write the number of values of logarithm of a complex number. 1
9. (a) Determine the value of
(b) Write the sum of the series
. 2
10. If is a positive integer, then show that 5
Find the value of
11. Show that 4
Show that
12. Show that the coefficient of in the expansion of in powers of is. 4
13. Answer any two of the following: 3x2=6
  1. Find the sum to n terms the series
  2. Separate into real and imaginary parts.
  3. Prove that
(Vector Calculus)

14. (a) Find the value of 1
(b) Write the definition of an irrotational vector. 1
(c) Find, where 2
15. Prove that 3
16. Answer any two of the following: 4x2=8
  1. Find
  2. Evaluate:
  3. Show that is a vector perpendicular to the surface where is a constant.

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