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

Wednesday, 01 May 2013 11:45Web

[BS-284/B-282]

**-MATHEMATICS**

**PART I**I(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)

Answer ALL the 4 ques..

Each ques. carries 15 marks.

**1.**

**(**

**a)****(**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.

**i)****(**Prove that a subgroup H of a group G is normal iff xHx-1 = H, for all x ? G.

**i****i)**Or

**(**

**b)****(**If G is a group and H is a subgroup of index two in G. Then prove that H is a normal

**i)**subgroup G.

**(**State and prove Cayley's theorem.

**i****i)****2.**

**(**

**a)****(**Show that a field has no zero divisors.

**i)****(**If M is a maximal ideal of the ring of integers z then prove that M is generated by prime integer.

**i****i)**Or

**(**

**b)****(**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.

**i)****(**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}.

**i****i)****3.**

**(**

**a)****(**Prove that every infinite bounded set of real numbers has a limit point.

**i)****(**Prove that the sequence {sn} described by sn= c > 0, sn+1= c+sn , for all

**i****i)**n ? z+ converges to the positive root of x2- x-c=0.

Or

**(**

**b)****(**State and prove D'Alembert's Ratio test.

**i)****(**Examine the convergence of n=18-1n+1n+1- n.

**i****i)****4.**

**(**

**a)****(**obtain the points of discontinuity of fx= 12n for 12n+1

**i)****(**Show that fx= x+x-1 is not derivable at x=0 and x=1.

**i****i)**Or

**(**

**b)****(**State and prove Cauchy's mean value theorem.

**i)****(**Prove that 38=p4p3sinxx dx =26.

**i****i)**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+…

**1**Prove that limx?81+1xx=e.

**1.****1**Prove that x-x22

**2.**part C-(10X2=20 marks0

ans ALL the TEN ques..

every ques. carries two marks.

**1**Show that the identity of a subgroup H of a group G is identical as the identity of G.

**3.****1**Show that if a is a generator of a cyclic group G, then a-1 is also a generator of G.

**4.****1**If f=1 two three four five eight seven 6then obtain f-1.

**5.**16.Give an example of a division ring which is not a field.

**1**Write the quotient ring z6U under +6 when U=0,3.

**7.****1**Prove that fx=e1x2+1e1x2-1 if x?0 and f0=2 is discontinous at the origin.

**8.****1**Approximate 1.2 if for x>0, 1+12x-18x2=1+x=1+12x.

**9.****20.**Write Taylor's series of f at 'a'.

**2**Evaluate limx?1+1xt-1dtsinx-1.

**1.**22.Evaluate:-1111+x2dx by substituting x=1t.

Earning: Approval pending. |