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# com00fA - DOCTORAL QUALIFYING EXAMINATION FALL 2000...

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Unformatted text preview: DOCTORAL QUALIFYING EXAMINATION FALL 2000 advanced Calculus & Linear Algebra NAME: ID#: Start your answer on each question sheet. Attach all extra sheets you use to the appropriate sheet. Hand in all question sheets. Date: September 5, 2000 Time o: Exam: l:30PMw3:30PM Place of Exam: Social qehavioral Science Bldg., Room 5228 ID #5 1. Let. b be rhy- SCI. I...Ifl‘L1EiUllLii numbers in [0:1] (:(ijiSLiilg of 0,11 and other I'alional numbers of lilEi form [:_;'_'J_..=q]‘_ where q 2”._ u a. }_)Linsit.i\'e inLegm; and p is odd. [a] \VVI'iaI are the} limit points of 5'.” lib} is *3. Lilt‘irstiﬂ'? \‘Ci‘iy 01‘ why not? '2. 131ml Lhc Vuhll'llt’ 01" the solid en<rh33€3d by intersecting right. circular cylinders 11:2 + y? x 4 2 . _,'.‘ .| 5-27.1(1'U r 1. -- u. L! ID if __ f-S. LE"! flip, w, 5': i], where is a continuous. functiozi 01' il'ldthIEI'lLiEI'lt- ma] variables {p: L'. 3'] and has (.‘(JI'JEiI'JUL‘JUS partial rierix-utiuas. Let. flip” no. injiﬁ. \Ioreomx the [iisL—order partial dcu‘ivatix’es fpij'i.._ﬁ do not. vanish at. (pg. co. [,0]. AS a consequence of linpiicit I5'1111ct,i011'l‘ln':('n'(-':1'11: iii a UGigi}i.')0[‘i'10{'){i ()i' {g.'),t.I(-,,1‘.[,} _, p 7- pm: I]: 'L-‘ : who, 1],: --. 1‘[\p,i_=j satisfy the equation fipti'b‘; t], vﬁp, 1‘}, tip: 5)) = {I}. Prove the iKOiifn-Villg idenliLieS: {it} 0}) Oil _ le 01* "'81.: p — (3-3.- :- {bi (9;) at at! ’i’hri .mbsr:ar“ipf. sin.r1.’r.(:arc.s the Daria-Me treated as a (:LJMMHE‘, in. 310,1"th diﬁ‘ere-niaiatio-n.. ID # 4. Uttm‘mme whezﬂwr the i‘o‘iluwingg‘ sysmm of linear ugljaxions has a umquc summon, no solution 01' :1. :‘mxliituﬁkt of solutions. Prove your assertion. (E;- ‘+ 51"} —-‘- 7:153 -- 134 J U :rl — (i .x:-_: - 81‘ 3 + 3 :54 = 1 3 1‘1 .-- .413 + :1“ 3 — 4 114 = .3 :1: g — 1:2 4 1:3 —:'— 3.11 : =1 -IJ. i La] .41 and 8 have. Lhe 5211119 eigenvahlcs. [b] If 41 is r10113ir1guku'. that, is] A“ exists, than H) = (1}) ‘1 ;; (CM—1:11" _ (viii dmmLes 1.11:: Ltorruspondiug transpose matrix. 3 :1 and B be LWO 5111'1ila11‘ n x :n: maniacs. P101563: 1.116. i'r:;ii(_1\\’111g: ID # U. A1: ('aliiprjer with mun-1' ((3.2:- 13 repu'eselncd by tbs: equation .- - 9 _. . . a . . '1 . \ 2231:“ + 201:3; 1- 40-53;" 3-10 — hat) '9 + 126:1 ‘— (J. By a. suitable (Jl'l'dligf: 01" variables, reduce the ahore equation to the (sammical fm'n'x .-\ a1" 1- by: '—- l. ...
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com00fA - DOCTORAL QUALIFYING EXAMINATION FALL 2000...

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