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Andhra University 2010 B.Sc Computer Science II Maths - Question Paper

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[BS-284/B-282]
PART II -MATHEMATICS
(Common core scheme)
Paper II- ABSTRACT ALGEBRA AND REAL ANALYSIS
(Revised syllabus w.e.f. 2009-10)
(Common for B.sc./B.A.)
Time: 3 hours Maximum: 100 marks
Note: Follow the instructions carefully provided in every part.
SECTION A-(4X15 = 60 marks)
Each ques. carries 15 marks.

1. (a) (i) Prove that the set G of rational numbers other than one with operation T such that a T b = a + b – ab for a,b ? G is an abelian group.
(ii) Prove that a subgroup H of a group G is normal iff xHx-1 = H, for all x ? G.
Or
(b) (i) If G is a group and H is a subgroup of index two in G. Then prove that H is a normal
subgroup G.
(ii) State and prove Cayley's theorem.
2. (a) (i) Show that a field has no zero divisors.
(ii) If M is a maximal ideal of the ring of integers z then prove that M is generated by prime integer.
Or
(b) (i) Prove that an ideal U of a commutative ring R with unity is maximal if and only if the quotient ring R/U is a field.
(ii) Prove that if f is a homomorphism of a ring R into the ring R' then f is an onto isomorphism if and only if ker f = {0}.
3. (a) (i) Prove that every infinite bounded set of real numbers has a limit point.
(ii) Prove that the sequence {sn} described by sn= c > 0, sn+1= c+sn , for all
n ? z+ converges to the positive root of x2- x-c=0.
Or
(b) (i) State and prove D'Alembert's Ratio test.
(ii) Examine the convergence of n=18-1n+1n+1- n.
4. (a) (i) obtain the points of discontinuity of fx= 12n for 12n+1 0,1,2,……and f0= 0.
(ii) Show that fx= x+x-1 is not derivable at x=0 and x=1.
Or
(b) (i) State and prove Cauchy's mean value theorem.
(ii) Prove that 38=p4p3sinxx dx =26.
part B-(5X4=20 marks)
ans any 5 out of 8 ques..
every questioncarries four marks.

5. If G is an abelian group, then show that the center of G is G.
6. describe a simple group. Show that every group of prime order is simple.
7. Show that every field is an integral domain.
8. If f is a homomorphism of a ring R into a ring R', then show that ker f is an ideal of R.
9. Show that if sn is a Cauchy sequence then sn is convergent.
10. Test for the convergence of 2-133-1+3-143-1+4-153-1+…
11. Prove that limx?81+1xx=e.
12. Prove that x-x220.

part C-(10X2=20 marks0
ans ALL the TEN ques..
every ques. carries two marks.

13. Show that the identity of a subgroup H of a group G is identical as the identity of G.
14. Show that if a is a generator of a cyclic group G, then a-1 is also a generator of G.
15. If f=1 two three four five eight seven 6then obtain f-1.
16.Give an example of a division ring which is not a field.
17. Write the quotient ring z6U under +6 when U=0,3.
18. Prove that fx=e1x2+1e1x2-1 if x?0 and f0=2 is discontinous at the origin.
19. Approximate 1.2 if for x>0, 1+12x-18x2=1+x=1+12x.
20. Write Taylor's series of f at 'a'.
21. Evaluate limx?1+1xt-1dtsinx-1.
22.Evaluate:-1111+x2dx by substituting x=1t.