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Final Exam Solutions

Final Exam Solutions - Math 2602 M1/M2 Final Exam May 2...

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Unformatted text preview: Math 2602 M1/M2 Final Exam May 2, 2007 1 j MATH 2602 M1 M2 fZNAfi EXAM Name TURN” V0 LW Section fl This is a closed book exam. A Letter size (2—sided) formula sheet is allowed. Any calculators (or any electronic devices) are not allowed TO RECEIVE CREDIT, YOU MUST SHOW YOUR WORK. CHOOSE 10 OF THE FOLLOWING 12 PROBLEMS. fCROSS OUT THE SCORE BOX BELOW FOR THE ONE YOU ELIMINATED} START : 11:30 am END: 2:20 pm. GOOD LUCK ‘ Problems Scores P1(10pts) P2(10pts) P3(10pts) ‘l P4(10pts) P5(10pts) P6(10pts) ’ P7(lOpts) P8(10pts) P9(10pts) P10(10pts) P11(10pts) P12(10pts) Total wkmfivnwumm’nflrmumtmmuwv'14;vawnzuflvxmwawkwummflflm“mmnmymrrmwtu4mnzmmwhawx«m»'Mugwrxwxalémm‘mmwmeul,wmm‘xmzr .W »Mflvawwrw§.mnww&g Math 2602 M1/M2 Final Exam May 2, 2007 2 Problem#1 (10 points) An electricity company wants to interconnect five town in Graph county: town A, town B, town C, town D and town E. It is not necessary to build a direct link between all the pairs of the towns, but it will be enough to be possible to transport the electric power between every pair of towns using links leading through other towns in the county. The lengths of direct electricity links between different pairs of towns are given in the following table: Town A B C D E A — 20 miles 25 miles 50 miles 40 miles B 20 miles k 15 miles 45 miles 35 miles C ‘5 25 miles 15 miles - 55 miles 30 miles D 50 miles 45 miles 55 miles — 5 miles E 40 miles 35 miles 30 miles 5 miles —— Because of building costs, the company wants the total length of the links built in the county to be the least possible. Formulate this problem as one of the graph theory problems introduced during the lecture and explain why the problem models the situation. Giyeuanfloptimum solution of the problem you have just described. siSaW / __________ Woblém. r to A A. gem-sis a} A ‘Tlm‘g ig “B . Wv1r»wm‘sfllwzznmu1&thWMquonKwMWm/Xmuwmw'kwmxamnwmuwmmmvmw (.mgmflmmmmmwm wmmmwzmzmmmwmmam ”a mmmnmswwmumi, Math 2602 M1 /M2 Final Exam May 2, 2007 V 3 Problem#2 (3+5+2 points) Let on be the number of ways to write an integer n as the sum of four positive integers in any order. For example, 05 = 10 because 6 = 2 + 2 + 1 + 1 = 2+1+2+1=2+1+1+2:1+2+1+2—1:2:2:1:1:1+2:2:3:1:1:1: 1+3+1+1=1+1+3+1=i+i+i+a § (a) Derive a closed formula for the generating function g(:c) : 042:4 + C5335 + - . ’ as a ratio of two polynomials. " 4 éf t ’L 3 , M—B L ,. 551,07; (QC +7C+9L+" +X 0V WCK): <X+X+~W*> ; VMMM mmmmmmmmmmm g :> ‘l "L i ’X ' 300: xii (HHX tr)“: XL“ = Lt. O‘KY‘ (PK?) _ (b) Obtain an explicit formula for cn for n 2 4 and write it in the form of a binomial E coeflicient. : 90 "W M w k, r X“ : xkl. Q“) -_—_~ x‘f 24(360 gfifl"ny>% bro fimwlwmmwwm‘fiuvwwmm«« wwmwwmmmmmmmm (c) What is 02007? 34,906 ‘ 91):) 5 I Q90 L‘ Cflm’t: 3’: é) “2.1mm l'mmeij-fiwma/M Math 2602 M1/M2 Final Exam May 2, 2007 4 Problem#3 (4+3+3 pointsfla) Calculate gcd( 1387 24) by using Euler Algorithm. (b) Use parta (a ) to find integers k and m such that 138k + 24m— gcd(138 24). (9-3 a 6;». M42 : an (88 aqrfl— 4-1) 3544354; W> 6:: gcACl33,3l—$7: (”4“) W33 -+ 6, 941.! I“: :2?) LL: “\ ) “:6 (C) Use part (b) to find the general solution of 2355 + 4y 2 1. LLJ Winn my?“ an} 6 l 6\ 7?” E,» m 37 y ww’w‘wfimwnxwquwwflwkrmow < MM wwmw“): mantel)»: ~wam mmwwm Math 2602 M1/M2 Final Exam May 2, 2007 5 Problem#4 (4+6 points) (a) Conjecture an explicit formula for (b) Prove that your conjecture is true in part (a) by using mathematical induction. a A i 05") wfifil—efivl w W‘sa’k Y‘s ‘ gem : l§+\(¢i*\} 1'4 w l l {M 3’71 “fa “ #7 v9 «fig I : KHS‘ wmhnwiw'mwnmmemwwwi'mwmnmn>7 mew‘wcwxwmmnaw wtmmmmnwm Math 2602 M1/M2 Final Exam A4ay 2, 2007 6 Problem#5 (2+4+4 points) For the following, let C(V, E) be the digraph given by V 2 {11171127713} and E _= {(111702), (112,711), (027113), (037713)} (a) Draw the digraph C(V, E). (1)) Find the adjacency matrix A for G(V, E)7 as well as, 242,143. A: .93? ~'>A1*lo©w~—>A3=O‘ OOl OO’lj 0 (C) List all paths of lengths 2 and 3 between the vertices v1 and '03. AvwWWm'flwfl1mflwwwmvmtfimwnan/auénw WWQM‘MW mmlmwfiwm. mm» "mm: swam:mmawmmmwmvm:,wwmymmlwimmxwrm mw‘mmmwuwm Vufaxm Math 2602 Ml/M2 Final Exam May 2, 2007 7 Problem#6 {6+4 points) HOW many positive integer k from 1 to 6000 (a) share a, common prime divisor with 6000 7 : : , Li [:3 9. 6000 3 ‘6600 ,9 5‘ gam; 3W.Q i rig? “WEN“ Lfiégggjfléwm “35 (flooj 6000 § (b) satisfy gcd(k, 6000) = 1 ? _ .. épQO—~(LQOO «wwmmmmmm wmmmwmmmvw: v mummmwmamwmmmrme wrwwmmm. .mmw nmai W .2: ’i £600 www—m,mmmmm 1; Math 2602 IVIl/MZ Final Exam May 2, 2007 96 lMlCny “9le 8 M Pflblem#7 (10 pomts) In how many ways can three dozen identical robots be assigned to {MEX/e assembly li®with at least two, but no more than ten, robots assigned to each line ? Hint: 1+33+$ +$3 + 112(1- 11+11/(1-1.) 5 11111111 11111 W (11111”) a“) t. MW: 3 e, ‘5 g ‘0 .45.}..— fflw(1+x+x+m+X\g “ 7C " 1~x @ Gee/W %(fl 1 36 L1 1851 (11“)? 0-13%., x“: (1-6-1 xq—r ox 234m +51. ——X l l ‘53 xwmm\mmmm\t>ommmwxmn\w rmwzmwmvwmw flown. mm' w 111 “11151111111111.1111 “1111,1111,“ 3 é E 1, r: g 5 1 Math 2602 M1/M2 Final Exam May 2, 2007 9 Problem#8 ( 5+5 p02‘nts)(a C ons1der a set with m + n elements. Give a combinatorial 0y). } argument by interpreting both sides separately to Show that: M 0? 0F“ c ‘2“ fiwp‘e’ MVM we 9er + (m? ”l = (’3‘) (11‘) 5G) t 1‘ MW AZ K} CW Ira/twink b m we reWS ’9‘” V‘ W’ 0W2 M {M mime amok am we, WQ J 5 M4 mam «we 2m mm W "3 ”1MB 19% n 4.» g CW 1 den“? 49‘?" «mete/m 6AA Malawi 4% xx mam QHWW mils/47M aflhw urging Sella/t rm owl—4 WWW ML 9 (b) There are 10 cars of the same kind that should be painted with the red, green and blue colors. What is the number of ways in which the 16 cars can be colored using the 3 colors? Tm Tatum Mn; ,M in MW“ 4; KO animal 194,015 Mutual m0 3 beve- » 0000000003 M W N g .6, ’2 3 E E 5; a i 3 E E a E g i Q 5 3 a 5 s, ‘1 mm,,mmmwmwmwm WWW m mwuwmmwuw'awwuvmrxm W imzmmmwmw IVIath 2602 M1/M2 Final Exam May 2, 2007 10 Problem#9 (5+5 points) Consider the following graph G: (a) Determine whether the graph G contains a Closed Eulerian cycle (tour). If yes7 write down an Eulerian cycle starting at 121. (£2044): H: '3‘“ uah'WiG- 7> Wit is W Eu/QJ/w'éw Cacti” \3“ {09‘ iwé (DB [07" 31)?) ENQL‘ i wilymfly 1V6 9 fl)“ {Wow ”rm/Um) “0‘3; 205 {VI} 33 0m Eagle/«é 9/ youth flax/We? 03‘» ”0% \M“ HIM/”W ngcotmfiai 4; {flxxwaNeNfi NU; 313,; W? ,fliai’vie; wraiwn Sq)?! 'fiwioiyiLfl’UF'} 94535} (b) Determine the Clique and Chromatic numbers of the graph G. 7 www.mflW—LA @342» know that 3 g ’)(((1\ g 5 Oymg F?“ if? 271 Chgpvj ‘ {4: vm’iw our M 91" “HM gagggkwg So )ch: 9:} Math 2602 M1 /M2 Final Exam May 2, 2007 11 Problem#10 (2+2+3+3 points) How many of the words of length 16, which can be formed by using the alphabet {A, C, G, T} have (a) exactly two A’s 7 46 t (lace N5 - < l\ we? 0 V‘s \ 3x0 a‘l'l'fiil we“)? QCQ M (QM/WY 3 MR’YS (b) at least two A’s 7 TvM l Lt l4) (c) exactly one A and exactly one C ? 46 r‘ , A w’ G all 4 5 ”lav W GHQ/(l New «We/re owe M 390% ”W9“? Wee. «Me, Com wlwgg A 2<C WW) Mom; £71K? wag}? “toggle“: « 46 V; . l 38 Wg‘fig‘) 0%) oi“ > a 9.x”: 2 \[3 \ScQ‘Llrls'c; (d) exactly one A or exactly one C (or both) 7 _ Bap” lég<‘%\ _ <\6‘(933» _._ W W“ W exadM {Mg A anal ‘ a ww/rm‘WKWfliamémln; mm nmmxm/vm zzwmwu wmwmuma‘yAmww‘W rvmemmammm Math 2602 Ml/M2 Final Exam May 2, 2007 12 Problem#11 ( 5+3+1 +1 points) The integer programming problem below has the solution ,2 : 39.1 at (00,31): (8.7,13). maximize z = 327 + y 10:10 + y g 100 10:0 + 7y 3 178 x, y nonnegative integers When the corresponding integer programming problem is to be solved by using the branch and bound approach, two new subproblems are created, one with the added constraint :r S 8, and one with x 2 9. (a) Write the itial tableaux for the problem with x S 8 added, and use it to find the maximum value and its location in this case. Dig-4L? kw: (WW; £~3s 2:38 (a: 635%} (b) The problem with x Z 9 can be solved by letting x 2 t + 9, and rewriting z and the original constraints in terms of t and y. Give the initial tableaux for this new formulation. “(HQWE too <2?) WW8 5 to i9 i t 0 \Q \0 {Wm + 35 fig 4:» mews” lo ”t o \ g3 {trig}: BKktGA-tméy: ”aria—var? ”g” p DWWSW“ 4—8? trajizawnéé 44%; crew (c) The solution to the problem in (b) occurs at t : 0, and y z 10. What is the maximum value ? tram: B'Oriowi .2: a; r (d) W' hat is the solution to the integer programming problem 7 38 at (an) , mm‘arworw. 2h ,Wms».¢mrw.a a am. a ,, «my, «.«W. . wmwwmwwmmwamwmww w» m ,rmmwwm- :mmww/mn 2i:iwvmWWz;vmWVle<r\kmWynn-mm: >2 :mzrwmkw Math 2602 M1 /M2 Final Exam May 2, 2007 13 Problem#12 {8+2 points) As a part of a weight-reduction program, a man designs a monthly exercise program consisting of bicycling, jogging, and swimming. He would like to exercise at most 30 hours, devote at most 4 hours to swimming, and jog for no more than the total number of hours bicycling and swimming. The calories burned per hour by bicycling, jogging, and swimming are 200, 475, and 275, respectively. (a) How many hours should be allotted to each activity to maximize the number of calories burned? (b) If he loses 1 pound of weight for each 3500 calories burned, how many pounds will he g lose each month exercising 7 i‘ ”HMWi\+2%s./ A J (of) M07 %; 30/59? WW 0% .>< 96m LJSWWTW> (May t 272;? gal} 3 fomés Gerbil/«Li; floox l\ \+ lfirEx l6 4— 9&ng 353:) 0 l i WW3, W ,_ .. A. , M a :3. 9’43 «e 6‘ 2% O o D 39f fir’S “22’ lgggo Hofqakl‘W—lg 1 E g , E _g g 0 W??? {1317* “3%" as %~%'F a’.xfi tam a i , WW> 3““ ’ 33%. > \ @336“? l Ugh g bar\5:9\/€> =2 09H Mi} fl“) é ...
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