practice_final_solns

# practice_final_solns - Practice Questions for the Final...

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Unformatted text preview: Practice Questions for the Final Name: Sol/v40}: nS - Math 2602 Spring 2006 A. '(t) The f‘oliowing matrices are in the form [AIE]. For each of them, say how many soiutions there are to the equation An? = I? there are and give a brief explanation showing how you know. (a) :4: 1 3 8 1 4 ' 0 4 4-4 4 - 0 0.4 _5 1 . I Semi—ran I (b) ‘ I '" 5 3 6 1 4 F _ _ . 0 4 4 4 4 mo OF terns Q'l'emd m4», Posovqim; 0 0 8 10 —4 in 105+ (01 ‘ 0 0 8 10 1 " ' ' ' ' 5-; 6i Li ' _ ' o H‘Lt U; _ I (c) _\ o 0%I0 -L| ‘ ' o ()0 00 5‘ Ogobhbnj'. Mung-5w >+ku¢vouus “Nagy/lb We»; 6 a 21 Li (2)“,Take the. following limits: 3 Shut. +9f "‘ barks-«vs “at Polyphomfcxb of" +LQ . m)n'o or Leadtrj (Mm-(Mrs ; M. - —n 3 nl—Eoe ” m”) =- O c" = J— v‘krda has i C” . (if) _______ﬁ_ﬂ q, tan-vs“. I’d-h. of May . _ lim (1+—)” _ 7" ‘ - _: u -‘ n—>oo n. "" 6 r- . 'HVI.\$ [-5 q- ﬂ“ . n r (1 +7; =8“ ‘ (3) Given the following triples of functionsLtwo of them have the same order (Le. f(n) = 0(g(n)) and g(n) = O(f(n))). Circle :the.function which has a different order. h. (4) Given the followingset of linear equations, - 23f+3y+22 520 2w+y+3z \$12 m+§y+z 59 3;,va is (191.1) in the feasible solutions space of this problem? EIf yes, is it-basic? . if he, give an equatl Yes . I :bq‘sig'. (“on of h ..c%ns n .h’3k4) 'ls (1,4,4) in the'feasibie solutiOns space of this problem? lf__yes, is it basic? . . - If no, give an equation which shows that itis not? "No. Doesn‘t swirls-h3- 773333 3.3 5 it. Write down the equations with all suitable additional variables on Which shows that "it is not? . __ toput-into simplex tableau _ . 27» t 5' 2-0. 4' Y: ‘3 * Z t 53 = ‘1 39151-3536. S, )0 . q; ' .‘ ':U_sing.the'ob1ective function maxla: _+ 23; + 32) show an initial '- : simplex tableau: ' ' X Circle the entry that you would pivot on to make a: basic (5) At your favorite ice cream parlor there is a-banana split which calls for 12 scoops of ice cream. You get to decide how many scoops of each ﬂavor are included. There are 5 different ﬂa- vors. Huts is an indfﬁinauiskably His _ /J.ﬂana vim urns . How many different banana splits can you make? - P you I 17. + 5 III! 3 C in) 5-H - ' ‘l o How many ways can you make a banana split which has at least 2 scoops of chocolate? - 99* if» 2. scoops of _ lO-I- A) .cq) s—tm have to a. di'HfibU'ie 5" ti -' 0' How many ways are'there so there are at most 2 scoops ct pistachio ice cream? _ . Wis _ ‘ i -- ' i 12+(Ll-n ' l 'I" ( .qtt s-Q'I has-l- 2)_+ 1) . i L}; > h o How many ways arethere with at mosta scoops of choco- a ‘- ii ‘ 5. I 3 __ _ late. and at least25coopsof vanilla? . ‘ + i W mm 2. i'VﬂMll-t in Tho-Hung.“ \o w..- All -. marl-at - canons to 0 Write down the generating function if all ﬂavors must re- 14) _ ceive an even number of scoops. (Do not multiply out the (It ( " t polynomials!) j (in x *+ x"+ X‘ 1' All"' VIE“? K12) ., o . Give the closed form for the polynomial above. ' H ' 5' H \" X . f I t _ Xe. \ﬂ'““‘vF—W- “c—r—mw tmdlﬁ? 1' {Hit-l» lb 1- {Mid-'3: (6) You have an alphabet of 3 letters, {A,B,C}. - o How many words of length 8 can you form“? 31 o How many of these'wOrds begin with the letter A? o How many of the words of lengh 8 have m and; z A‘s -_ M M each other? ‘3 _ Andix z x. ml of W“ o How many words oilengths do not use the letter ? a l 2. . o How many o_f the words at length 8 begin With A or end with A? Md Put-ind inalieinn/udmton 3’+ +3 - 5 o How many of the words of length 8 use exactly 2 A’syand 2 3’9? ' - ~ Gill 3.) . How menyfworcls of length: 8 etartfandjerid with C and use exactly 2 A's and 2 3'5? ' ' (7) Prove the following statement using induction. If a, b > 0, then (a + b)“ 2 a" + b” for any positive integer n. ‘3‘“ Cat-Id “5| n+5 5, a-Ho inducki‘m-Ihupﬂkosis: egsuﬂ tiar- Jom in}! I M‘ a I - (9th) a (q Ho) 9+ bag, -:(a'_‘__ +530» vb) = a”; d'io+ A . \“uP-I - bﬁa .* _ 7/ Omelet“ (8) Prove using induction that all trees have atleast two leaves. Do induction on n=number of verticee and note that if you have a tree on n + 1 vertices, and remove a leaf then you get . a tree on n verticee. beﬁecase: n=’2 I I has, 27‘ “arch-96h! QSSUML Gr 59..., “3'1 am “trees cm in whom km 05- tuft 2 MM Guam 1 ‘l’m on (n+1) var-km scion“ q hurl: (since it»; .‘hu is aback: this We". is at ins-k- en. Leaf). Cali H- L. . 8%“ Mn it. This gives a fru- tn n9 text-cu; Cassi) The. n “me-“hie. has hoe Remus «A A 4'. not “JJQM b‘M-wf-‘ﬂdot. .%3)gg.¢£M-i:.tmﬁ um. .Qu-J mun-5mm.“th 7 (9) Find the least squares line which best describes the points (1:1) {4.18)‘ 4——r__.-__‘ ﬂ‘ . .u~r_t_:|‘._ '“ __30m+|0-E=5l+ [Oh-1" b: I? (10) 152: = 0(2.1)w? _ _ _ __ _ vs ' ' ' _ ' 3am res»; =9 _ -_ IS \$2+17w_=o(2i.1)”’? I ‘ __ ‘I -_ ‘Zb 3 5.3 b: ‘1-5' . (PS , I s. mdé“? a an: m: a IS loggtc) = .. 3 : -_ . n0 is __= 08.1)”? no (11) Let A be the matrix below. 100140 can or-r 00‘! oar (12) Give two different graphs on 6 vertices where breath first i ‘ -..s,e_arch and depth first search give the same output-sequence. p... (13) Give an example of a complete bipartite graph which is also planar. Give a planar drawing. - - E' 1th.“ r5 Impl- {H K... Ate munch, (14) Give an example of a graph which has x(G) aé w(G). (15) Simplify the following sums: QJiMII-sia‘ I ' h I - 22's] (2')?“ = (314] -_ if“ - Esme-i4; z (u Ln": 5» (16) Write down the polynomials which correspond tothe following ' generating functions: 1 - ' . I 1,235 '- H «2&5 fr ‘ix'°‘r 3x5+.... (17) There are 5 different shades of blue, 4 different shades of green, 3'shades of orange. How many different ways can you choose .thegpaint samples? You_want to pick six paint ' Samples, 2 of each of three color families. (WW?) What if you still wanted six paint samples, 2'of each from 3 color families, but-now you have the choice of 4 different color families, and there are 5 different shades of red? 350-!- BGR +o-OR a 3012 ' (gm) -+ (ix 29(2) + (mm + elm.) " 19 (18) You have an 12x12 chess board, have many ways can you place 8 non-taking rocks ‘? (In other_ words, no rocks can share a row and column) 128’- . (.25 ‘5 ‘. _ R claws cm rows 5 Colo-Inns on. but consi - (19)“Giveacombinatorlal‘proof'that js wakes for“ Hrs—l: to (")4 ‘1); " ‘°' '0‘ We T . “idem-anal M 3: (me o _a wick-loan item at r-dnﬂdre: - - g- m" “M “J b“§'a'0€e {‘M MMCl-rkin 0M (20) There are bothquestionsand solutions in the files on the web: RH 5 : [notes 2] has lots of workedout optimization problems. (The solutions for all sections are at the end of the document) ” " ¢h°3¢ Some which might be useful to look at are ' ‘- st . section 20 1 2,3 (drawing feasible regions and visually finding . maxima) . e m H'- mm" 21 1,4,6 (finding maxima) - 'l china... 22 1,4,11-13 (using simplex method) wwuu' -. 1.60; 1!? -lnsa'cs. ...
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practice_final_solns - Practice Questions for the Final...

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