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Posted Date: 11 Nov 2010      Posted By:: CH.Basaveswara Rao    Member Level: Silver  Points: 5 (Rs. 1)

# 2009 Acharya Nagarjuna University B.Sc Electronics Maths-II Question paper

 Course: B.Sc Electronics University/board: Acharya Nagarjuna University

ACHARYA NAGARJUNA UNIVERSITY
B.A. / B.Sc. DEGREE EXAMINATION, MODEL PAPER
(Examination at the end of second year, for 2009 - 2010 onwards)
MATHEMATICS - II
PAPER II - ABSTRACT ALGEBRA AND REAL ANALYSIS
Time : 3 Hours Max. Marks : 100
SECTION - A (6 X 6 = 36 Marks)
Answer any SIX questions.
1. Prove that a subgroup H of a group G is a normal subgroup of G iff every left coset
of H in G is a right coset of H in G .
2. Prove that every subgroup of a cyclic group is cyclic.
3. Prove that an ideal M of a commutative ring R with unity is maximal iff R / M is a
field.
4. State and prove division algorithm for polynomials.
5. Prove that a monotonic sequence is convergent iff it is bounded.
6. Examine the continuity of the function f defined by f (x) =| x|+| x -1| at x = 0 and 1.
7. State and prove Cauchy's mean value theorem.
8. If f : [a, b]?R is continuous on [a, b] then prove that f is R -integrable on [a, b] .
SECTION - B (4 X 16 = 64 Marks)
Answer any ALL questions.
9.(a) Let H and K be two subgroups of a group G . Prove that H ? K is a subgroup of
G iff either H ? K or K ? H .
(b) If G = Q- {1} and * is defined on G as a *b = a + b - ab then show that (G, *) is
an abelian group.
OR
10.(a) State and prove Lagrange's theorem on finite groups.
(b) Let G be a group. Show that the mapping f :G ?G defined by f (a) = a-1 ?a?G
is one one onto. Also show that f is an automorphism iff G is abelian.
11.(a) Define the terms integral domain and field.
Prove that every finite field is an integral domain.
(b) If Q( 2) = {a + b 2 :a,b ?Q} then show that Q( 2) is a field.
OR
12.(a) State and prove fundamental theorem of homomorphism of rings.
(b) Find the sum and product of the polynomials f (x) = 2x3 + 4x2 + 3x + 2 and
g(x) = 3x4 + 2x + 4 over Z5 .
13(a) State and prove D'Alembert's ratio test.
(b) Test for convergence of
( )
.....
-
? + + + +
? ?
?
? ?
=
8S
1
1
1
2
1
3
1
1
n
n n n
OR
14(a) If f in continuous on [a,b] then prove that f is bounded on [a,b] .
(b) If s
n n = 1+ + + +
1
1
1
2
1
! !
......
!
then show that {s } n converges.
15(a) State and prove Rolle's theorem.
(b) Derive the expansion of sin x by using Maclaurin's theorem.
OR
16(a) State and prove fundamental theorem of integral calculus.
(b) Prove that
p p p
4 2 2 0
4
= ? sec x dx = using the first mean value theorem.
2

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