2012

(November)

MATHEMATICS

(Major)

Course: 101

(Classical Algebra, Trigonometry and 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)

- Define a null sequence.
- Give an example of an oscillatory series.
- Write the necessary condition for convergence of an infinite series.
- Write the equation whose roots are the roots of the following equation with opposite signs:

2. Answer the following questions:

- Write the limit point(s) and the range set of the following sequence: 2

Or

Show that the series

does not converge.

- Prove that every convergent sequence is bounded. Is the converse true? 2+1=3
- Using comparison test, find whether the series is convergent. 3

Or

Using Leibnitz test for the convergence of alternating series, show that the series

Converges for

3. (a) State and prove the factor theorem for polynomial equations. 4

(b) Show that the sequence, where is convergent. 4

(c) Test for convergence of the series 5

(d) Test the convergence of the series

5

4. (a) From the equation whose roots are the squares of the differences of the roots of the cubic equation

5

Or

If are the roots of the equation then find the equation whose roots are

(b) Discuss the Cardan’s method of solving a general cubic equation. 5

Or

Solve the equation by Cardan’s method.

GROUP – B

(Trigonometry)

5. (a) How many different values can be obtained for the following expression? 1

(b) Distinguish between and where is a complex quantity. 1

6. Answer (any two): 2x2=4

- Express in the form.
- Find the real part of.
- Find all the values of.

7. (a) Using De Moivre’s theorem, prove that are the roots of the cubic equation 5

(b) Sum to n terms the series

4

(c) Expand in ascending powers of

(d) If

then show that

5

GROUP – C

(Vector Calculus)

8. (a) What is the physical interpretation of directional derivative of in the direction

(b) What do you mean by a solenoidal vector?

9. If has constant magnitude, then show that and are perpendicular to each other provided

Or

Show that

Where is the magnitude of. 2

10. (a) Show that is a vector perpendicular to the surface , where K is a constant. 3

(b) Prove that 4

(c) If and then find curlat the point . 4

Or

Prove that

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