## Friday, January 04, 2019 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)

1. Answer the following questions: 1x4=4
1. Define a null sequence.
2. Give an example of an oscillatory series.
3. Write the necessary condition for convergence of an infinite series.
4. Write the equation whose roots are the roots of the following equation with opposite signs: 1. Write the limit point(s) and the range set of the following sequence: 2 Or
Show that the series does not converge.
1. Prove that every convergent sequence is bounded. Is the converse true? 2+1=3
2. 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
1. Express in the form .
2. Find the real part of .
3. 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 curl at the point . 4
Or
Prove that 