Page 1 :
Total No. of Questions - 12], , (Total No. of Printed Pages-3, B.A./B.Sc. 41] Semester Examination, CBS-INIS/11, 237557, , MATHEMATICS APPLIED MATHEMATI CS, , Time Allowed- 2% Hours, , Course No. : UMTTC- 301, , Maximum Marks-80, , Note: Attempt all questions from Sections (A) and (B), each, question of Section A is of 4 marks, each question of, Section B is of 8 marks and each question of, Section Cisof ., 18 marks,, : Section-A °, , 1., , 3300, , Letx and y be two real numbers such that x > 0. Show that there, exist n €N such that nx> ve, , - Define Cauchy sequence. Show that Cauchy sequence is bounded,, , Discuss the convergence of the series > "where > 0,, n=0, , Define uniformly continuous function. Give an example of, uniformly continuous function., , : nx, Show that the sequence LPs where J, (x)= ees,, nx, pointwise convergent but not unifomly convergent,, , 1S, , [Turn Over

Page 2 :
1., , 2., , (2) CBS-IIIS/11-237557, , Section-B, , Show that every Cauchy sequence need not be convergent in, , Q but every Cauchy sequence is convergent in R ., , Show that there exist an irrational number between two real, numbers and hence there exists infinitely many irrational numbers, , between two real numbers., , x, +1, , , , Let >s be a positive term series wd that lim ‘ al., n=), , Show that the series is convergent if/< 1 and divergent if/> 1., Section-C (Do any two), , a) | Showthat countable union of countable sets is countable., Then show that N x Nis countable., , b) Find the supremum and infimum of the following sets, , i) S = {sin x—2cosx:x €R} and, , ii)” r= {m+2smn ex}, a) Let {x,} be a'sequence of real numbers such that, , a 1 ae Px, lim x, =/, Then show that lim at l, , b). Let {x,} be a sequence of‘real numbers such ‘that, , lim Xns =], where |/| < 1. Then show that lim x, =0,

Page 3 :
MONE ef pig PS EN TOE t410, , (3) (BS-INS/11-237557, , ef a" ye, ¢) Prove the lim —| =e, , n>o\ yf, , Test the convergence of the following series :, , . a 2 3, Debt, 468 4.6.80, 5 8). 579, , , , , , , , i) (/2-1)+(V5-2)+(Vi0-3)+.., a) Defineabsoluteand conditional convergent series. Illustrate, , with examples., , b) - Show that every absolutely convergent series is convergent, , but converse need not be true., , 6) Showthat every-continuous function attains its bounds on, closed and bounded interval., , wet, , Mh, , 4