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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.
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‘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
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gﬁﬂ"ny>% bro ﬁmwlwmmwwm‘ﬁuvwwmm«« wwmwwmmmmmmmm (c) What is 02007? 34,906 ‘ 91):) 5 I Q90 L‘ Cﬂm’t: 3’: é) “2.1mm l'mmeijﬁwma/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). (93 a 6;». M42 : an (88 aqrﬂ— 41) 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
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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
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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/(11.) 5 11111111 11111 W (11111”) a“) t. MW: 3 e, ‘5
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argument by interpreting both sides separately to Show that: M 0? 0F“ c ‘2“ ﬁwp‘e’ MVM
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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
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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
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< 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 ?
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‘ 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. Dig4L? 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
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{trig}: BKktGAtméy: ”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\kmWynnmm: >2 :mzrwmkw Math 2602 M1 /M2 Final Exam May 2, 2007 13 Problem#12 {8+2 points) As a part of a weightreduction 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
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