## Saturday, January 05, 2019

2014
(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 question
GROUP – A
(Classical Algebra)

1. (a) Write the range set of the sequence 1
(b) Prove that every convergent sequence is bounded. 4
(c) State and prove Cauchy’s general principle of convergence. 1+4=5
Or
Define monotonic sequence. Prove that is convergent, when
2. (a) State the Cauchy’s root test for convergence.
(b) Prove that a necessary condition for a infinite series to be convergent that
(c) Test for convergence: 3
(d) Discuss the convergency or divergency of the following (any one): 5

3. (a) Write the fundamental theorem of algebra. 1
(b) Show that has at least two imaginary roots. 2
(c) Solve the equation
Where the sum of two roots is zero 3
(d) Solve by Cardan’s method (any one):
(e) If is a factor of then show that

GROUP – B
(Trigonometry)

4. (a) Expressin De Moivre’s form.
(b) Find the values of.
(c) If n is a positive integer, prove that
Or
If then prove that
.
5. If then prove that 5
6. Prove that 4

7. (a) Find the sum of the series:
(b) Find the sum of the series (any one):

GROUP – C
(Vector Calculus)
8. (a) Define space curve.
(b) A particle moves along a curve where time is. Find the velocity and acceleration at
(c) If , then find the unit normal vector to the surface of at .     3
(d) If then prove that 2
(e) Prove any one of the following: 4
(f) Prove that
3

***