Nalanda Open University 2009 B.Sc Mathematics Bachelor of Science ( Hors), Part-III Term End , -VI - Question Paper
Bachelor of Science (Mathematics Honours), Part-III Term End Examination, 2009
Paper-VI (Mathematics)
Bachelor of Science (Mathematics Honours), Part-HI Term End Examination* 2009 Paper-VI (Mathematics) lime: 3.00 Hi's. Full Marks: 75
Answer any Six Questions, selecting at least one ques-tion frome.acla.group.
Group-A
1. (a) Define an antomorphism of a group G. Let x e G. Then prove that the function-f defined by
f(g) = x_1gx for g in G is an automorphism of (S':
(b) Let C(G) denote the centre of a group G and 1(G) be the set of all inner automorphisms on G. Then prove that Vc(G) -1(G).
2. (a) Let Gbe a finite group and | G | = pn, n > 0, p a prime. Show that G has a nori-trivai centre
of order at.least p-
(b) Define a ring homomorphism. Let f: R > T be a homomorphism of a ring R onto a ring T. Then show that.fis ail isomorphism if and only if kerf={0}.
3.. Let I be an ideal in a ring R. Show that the quotient ring R/I is a ring. Further if R is commutative, then so is R/I
j,
4. Prove that eve.rintegjal domain can be embedded into afield...
Group-B
5. State and prove Cantor's theorem.
5.. (a) Define sum and product of cardinal numbers. Give an example of each.
(b) Prove that n+a=Q!, where a is an infinite cardinal number and n a natural number.
7. Prove that 2* = C, symbols, haying usyal.me.njngsi;
Group-C
3. (a) Define a partition of a set and equivalence classes. Show that the equivalent classes are
either disjoint or same.
(b) Find'the number of solutions of the Equation x+y+z=l 5, x, y,.z being n on-negaUy-'e integers;
3. (a) Find all solutions of the recurrence relation an = 3an l+2n.
What is the solution when a=3 ?.
(b) Prove that
v ' n
J, C(n, E)2 = C(2n, n):
using generating function (neN).
Group-D
10. (a) Show .thatifa function f(z) = u(x,y)4iv(x,y) is differentiable at any point z=x+iy, theirijke partial derivatives uK, vK, uu, exist and satisfy the Cauchy-Riemann equations.
(b) Iff(z) is an analytic function ofz, show that
+ ) |#X).|3 =4| f!(zj \2,. fo2 fr2 /_
1.1.. (a) If f(z) is regular within,andr.priia .plo s. e d ;p ont-our G a.nd if | be. any .jsoint within C, then f>roye that , ,
Z-|'.
(b) Obtain'tli'eTslpir-k -an'lureht1 s series whichrepresents the funi:ticdi.
, (s+2) (z+3)
m the region-
(1) | z | < 2 {ji) .2 < | z | < 3.
12. (a) Show that a fanctioii wliich 'has no singularity lii the t'init* part of the plane or at infinity is constant.
(b) Prove that z = 00 is a non-isolated essential.smgularity for the function 1 -ez
|
Attachment: |
| Earning: Approval pending. |
