Nalanda Open University 2009 B.Sc Mathematics Bachelor of Science ( Hors ), Part-II Final , - Question Paper
Bachelor of Science (Mathematics Honours ), Part-II
Final Examination, 2009
Nrilranin Open University Bachelor of Science (Mathematics Honours), Part U Final E\ajninatio n, 2009 Paper-Ill (Mathematics (Hons.))
Tiine: 3.00 Hrs. FullMarks: 75
Answer any Six Questions, selectingat least one questionfromeachgiuup.
Group-A
I. (a) Define sum of two Dedekind cuts. Show that sum of two cuts is also a cut.
(b) Prove that siny" no n- empt y se t o f re al numbers whic h is b ounde d ab ove has a least upp er bound.
J:::j(a) .Show that the inteisectionof a finite number of open sets is open. What can be said about ' inters ection o f an infinit e number of open set s?
(b) State and prove Bo lzano - We ierstrass theorem.
(a) Show that a funct ion f: R R is c ont inuo us on R if and only if for every close d se t B in R, f _1(B) is closed inR.
(b) If a function f is cont inuo us o n a c lose d andbo unde d interval [a,b] the n prove that it attains its bounds on [a,b].
I. ta5 A funct ion f is defined as folio ws:.
f(x) = xP Cos 1/x, x =* .&
= 0 , x = 0-
What condition should be imposedonpsqJt.f.be differentiable
(b) State and prove Rolle's theorem.
(b) UsingMaclaurin's theorem, find the expansion of fx) = log (1+x) with Lagrange's form of ...... . h. . . . .....
5. (a) Define a sequence and its convergence. Prove that the limit of a convergent sequence is unique. (b) Show that {x} is convergent and find its. .limit wiue
.n'.n" 'o ri ! .... r'-;
7. (a) St ate and prove Cauchy's conde nsation test.
.... . .......... % . --------...... . .. ........
|b) Hence show that 2 is convergent if p > 1 and diveigeht if p jMa). St at e and prove Gaus s s ratio test. (b) Test for convergence the series whose nth termis \iognx). ?. (a) Show that t he sum of an ab solute ly convergent series is indep endent of the o ide r o f t enns fb), Give an example to show that the Cauchy product of two divergent series may be absolutely convergent.
10. (a) Show that a necessary and sufficient condition for anon-empty subset Wo fa vector space :V dverF to be a subspace of V is that y e_W and a,b t'E +by e W._
IJb) If Wj and be the subspaces if V(F), then show that Wj + is a subspace of V(F).
11. (a) Show that any two bases of a finite-dimensional vector space have the same number of elements?
(b) If a function T : U > V be a linear transformation from U(F) to V(F),-.then prove that:.
(1) T(a-|0) = T(o:)-T(i5) for all ot, j0 in U.
n. M: ;.
(ii) T (2 ai oij) = 2 ai T(ai) for all 24's'iiiT 'and for all s in IT. i=l i=l
Find the characteristic e quation of the matrix
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A = |
2 |
1 |
r |
|
I |
& | ||
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I |
and verify Cayley - Hamilton theorem. Hence calculate theinverse of A.
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Attachment: |
| Earning: Approval pending. |
