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Unformatted text preview: Math 2602 M1/M2 Test3 April 19, 2007 1 Math 2602 TEST3 Name: VOLC“ Problems P1/12 P2/9 P3/19 P46 Total/40
H l l I / l H Scores 1 l ] J H TIME ALLOWED: 50 minutes JSHOW YOUR REASONING TO RECEIVE CREDITS j No Calculator may be used Problem#1 (12 points) Determine the clique and chromatic numbers of the following graphs. In all cases, you are required to construct a coloring with the least number of
colors‘ Math 2602 M1/M2 Test3 April 19, 2007 2 Problem#2 (9 points ) Find the largest value of the function f (as, y, z) = x + 23/ + 3z on the
3dimensional region determined by O < a: < 2 , 0 < y < 3, O < z < 1, 3:1; + 2y + 42 S 12. "Z Math 2602 M1 /M2 Test3 April 19, 2007 a 3 Problem#3 (19 points) Assume that 70% of people that voted republican in the last election will vote republican and that the remaining 30% Will vote democrat in the next election. Also assume that 60% of those who voted democrat in the last election will vote democrat and that the remaining 40% will switch to republican in the next election. Suppose that in the last election 80 million voted republican and 60 million voted democrat. Assume that this happens in every election. (a) (3 points) Turn this data into a matrix A and vector x(0), the distribution of votes in the 326% last election and determine x(1), the distribution of votes in t m t election. /\r\,
Rm : 0? Ho) + 04mm 4r> H0 : 0?— m, 2(a)
b(\\ : ()8 mm + 0:6'D(,o\ 13‘“ 0.3 0.6 9(a) ‘
g M :
> {ROD V 0.} on] 20 my] ~ swam mi\. CA :: 3Q » . ‘ 63 4  «36 ‘a : OM v; o 3 0 6 9; (in + )m l (0M (b) { 6 points) Find the eigenvalues of A and the corresponding eigenvectors. . J 0‘ in: 0H 2 =0 <=—> az~h’5D+9‘f:0
.« 0. IA a 3 {213.com (an) {20.3}:0 :> 34 ,2‘0'3 1:1: ~06 .m ~=> ti? Fl 06 «ow; . 4' \
9:064 0‘ o'ﬁvhma Cd’é?‘ g V" W 3 0:; o Math 2602 Mi/M2 Test3 April 19, 2007 4 (c) (4 points) Compute A" by using diagonalization. “ 1/71— V4"
\ \ o _, “L _}_ A l:
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, 3 '\ 71km) JAG”) * %‘%(03)n %_%®3)  (d) (5’ points) Express x(10), the distribution of votes in the 10th election, as a product. Do
not compute! m
—  — X 0}
><(\o\= Mm : Azx(8\ ,  A (
l A m;
: —%r%(03)\0 %,%(0.3)° 20 \
‘ b 3 i 03 ‘° 60 «*L
%»§@3) 3}(_> (e) (3 points) What should be the number (in million) of people who voted republican and
the number (in million) of people who voted democrat in the last election so that the numbers
can not Change in every successive election ‘? (Hint: ﬁnd the steady state vector p for A, then
express your answer in million suitably) V s ~ «A k ,
if: 6W] tort/3 our J ’3» Li
team it is He “twine Mai/EAJUJTV v» [3]
Cd(%‘»n&\(( \‘0 ﬂip/tqu i, nms~wi 34e:?%wmﬁ“
Etc“: 60. «(Mien A/[ath 2602 Nil/M2 Test3 April 19, 2007 Problem#4 ( 6 points, bonus) Consider the following system of equations 3:5 — y + a2 : 1 —6x ~ 2y + (2a+1)z : 3 92: — 33/ + (4~a)z : 5—!) For which values of a and I) does this system have
(a) (2 points) no solution ?
(b) ( points) inﬁnitely many solutions 7
(C) (2 points) a unique solution 7
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 Spring '09
 COSTELLO
 Math

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