CS(M) EXAM2012 SI. No. F-DTN-M-NUIB MATHEMATICS Paper H Time Allowed : Three Hours Maximum Marks : 300 INSTRUCTIONS Each question is printed both in Hindi and in English. Answers must be written in the medium specified in the Admission Certificate issued to you, which must be stated clearly on the cover of the answer-book in the space provided for the purpose. No marks will be given for the answers written in a medium other than that specified in the Admission Certificate. Candidates should attempt Question Nos. 1 and 5 which are compulsory and any three of the remaining questions selecting at least one question from each Section. Assume suitable data if considered necessary and indicate the same clearly. Symbols and notations carry usual meaning, unless otherwise indicated. All questions carry equal marks. A graph sheet is attached to this question paper for use by the candidate. The graph sheet is to be carefully detached from the question paper and securely fastened to the answer-book. Important : Whenever a Question is being attempted, all its parts/sub-parts must be attempted contiguously. This means that before moving on to the next Question to be attempted, candidates must finish attempting all parts/sub-pans of the previous Question atteMpted. This is to be strictly followed. Pages left blank in the answer-book are to be clearly struck out in ink. Any answers that follow pages left blank may not be given credit. erc1 timn Tig 9-47-3v *PIS ga- iszn9- : gflsvl t1 Section 'A' 1. (a) How many elements of order 2 are there in the group of order 16 generated by a and b such that the order of a is 8, the order of b is 2 and -i 1 bab = a . 12 (b) Let 0, if x < 1 n +1 f,,(x). 1 sin—, if — n +1 0, if x > x 1 1 n Show that f„ (x) converges to a continuous function but not uniformly. 12 (c) Show that the function defined by 3 y5 (x + iy) f (z) = {x x6 + y10 Z *° 0, z— 0 is not analytic at the origin though it satisfies Cauchy-Riemann equations at the origin. 12 F-DTN-M-NUIB 2 (Contd.) 1. CO a aft b WRTwftie a1r 16 k fig A' ctre 2%ffm1-4)A tat, ke)- ft aft4P. 2*311Tba/;1 =iii erl 0, fn (x) = rift x < 12 1 n +1 1 n +1 LE X tS. 1 . sin, x iffr., — 0, 1 za. x >—, T46-47 k fn(x) tioo TERff I tft9 t;chttilici 31firiTiCa" rdT t, 12 (Ti) Tfri-47 x3 y5 (x + iy x6 4' y 10 f(z) = Z #0 0, z= 0 fa'A 4ch 9* trirgrrEff EFF94%14 Wf .1ttet 1:R40-414111 4-141chtui11 wli7 Th-K-dr t I 12 k F-DTN-M-NUIB 3 (Contd.) (d) For each hour per day that Ashok studies mathematics, it yields him 10 marks and for each hour that he studies physics, it yields him 5 marks. He can study at most 14 hours a day and he must get at least 40 marks in each. Determine graphically how many hours a day he should study mathematics and physics each, in order to maximize his marks ? 12 (e) Show that the series i g — n n6 is conver„=1( g +1 gent. 12 2. (a) How many conjugacy classes does the permutation group S5 of permutations 5 numbers have ? Write down one element in each class (preferably in terms of cycles). 15 + )1)2 , Y) # (0, 0) (b) Let f(x, y) = x 2 + y- ' If 1 , if (x, y) = (0, 0). af af ax ay Show that — and — exist at (0, 0) though f(x, y) is not continuous at (0, 0). 15 (c) Use Cauchy integral formula to evaluate ,3z fc (z +1)4 where c is the circle I z I = 2. 15 F-DTN-M-NUIB 4 (Contd.) 3Thz179. clvk (r) Raft* 31 -5N9. t t1UU t 110 349: tt 9-6r .R7 ‘Itich vir-dft k alum- * raw trit5 are? it 9-ftr t1a1 t i a-6 9-F-dfqff 367 4 atrEw 14 E* 3TETRF T.( 9cht0 311T 3041 k7 -TE Wil" 40 31-W 711:1 cht•1I 311-4"4/17 t I VFW 1,141,11: Wm- eF-47 f* 31.74 31-4: *-r Tgfa-R9. ka4 3TiWItTTat (=IA *M-7, Ert:r .11,31d afr 474 ErE4 qtrafr 31}-AER .113 (iii) 03 (iii) 0