2012

(November)

MATHEMATICS

(General)

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

GROUP – A

(Classical Algebra)

- The range of a real sequence may contain a complex number. (State True or False)
- The elements of a real sequence can be put in a one-one correspondence with what set?
- Every equation of odd degree has at least one real root. (State True or False)
- Write the number of positive real roots of the equation

2. Answer the following questions: 2x4=8

- Write the limit point(s) of the sequence
- Write the interval of for which the sequence converges.
- Find the equation whose roots are the reciprocals of the roots of the equation
- 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

Or

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

- Show that the series does not converge.
- Test the convergence of the series
- Show that a series with positive term is convergent for

6. Solve the equation using Cardin’s method. 10

Or

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

GROUP – B

(Trigonometry)

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

Or

Find the value of

11. Show that 4

Or

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

- Find the sum to n terms the series
- Separate into real and imaginary parts.
- Prove that

GROUP – C

(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

- Find
- Evaluate:
- Show that is a vector perpendicular to the surface where is a constant.

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