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)

(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

- grad

(f) Prove that

3

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