## Saturday, January 05, 2019

2013
(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. Answer the following questions: 1x3=3
1. Define real sequence.
2. State the condition for convergence of infinite geometric series
1. If a polynomial is divided by where is a constant, then state the remainder.
2. Answer the following questions: 2x5=10
1. Prove that a convergent sequence cannot tend to two distinct limits.
2. State the conditions of d’ Alembert’s ratio test for convergence of a series.
3. Test for convergence of the series:
1. If are the roots of the equation then show that
1. Show that the equation has a root between – 3 and – 2.
3. (a) Show that the sequence is convergent, if
3
(b) Use general principle of Cauchy’s criterion to show that is not convergent, where
4
4. Test the convergence of any two of the following: 5x2=10
5. (a) If be the roots of the equation form the equation whose roots are
.
Or
Prove that every algebraic equation of degree n has exactly n real or imaginary roots.
(b) Solve the equation using Cardan’s method.

GROUP – B
(Trigonometry)

6. (a) What is the 2nd term in the expansion of in terms of ?
(b) State Gregory’s series. 1
(c) Prove that 2
7. State De Moivre’s theorem and prove it when n is any integer. 5
Or
If Prove that
8. (a) Express in the form 2
(b) Prove that
where a and b are two real quantities. 3

(c) Prove that
3
Or
If prove that
9. (a) If , prove that and are the roots of the equation 3
(b) Find the sum: 5
Or
Find the sum of the sines of a series of n angles which are in arithmetical progression.
GROUP – C
(Vector Calculus)
10. (a) Write the following expression correctly:
1
(b) Give the definition of vector differential operator. 1
(c) If where prove that 2
11. What do you mean by directional derivative of at the point Find its values in the direction of coordinate axes.
Or
Find the directional derivative of at the point in the direction of the vector
12. Answer any two of the following: 4x2=8
1. Prove that
2. If determine and at.
3. Prove that: