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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 44 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'.
Mung5w
>+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, tanvs“. I’dh. 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 itbasic? . 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 swirlsh3 773333 3.3 5 it. Write down the equations with all suitable additional variables on Which shows that "it is not? . __ toputinto simplex tableau _ . 27» t 5' 20. 4' Y: ‘3 * Z t 53 = ‘1 391513536. 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 abanana 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)
5H  ' ‘l
o How many ways can you make a banana split which has
at least 2 scoops of chocolate?  99* if» 2. scoops of _ lOI 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+(Lln ' l 'I" ( .qtt sQ'I hasl 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'VﬂMllt in ThoHung.“
\o w.. All . marlat  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' {Hitl» 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 Putind 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‘“ CatId “5
n+5 5, aHo inducki‘mIhupﬂkosis: egsuﬂ tiar Jom in}! I M‘ a I  (9th) a (q Ho) 9+ bag, :(a'_‘__ +530» vb) = a”; d'io+ A
. \“uPI  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‘ “arch96h! QSSUML Gr 59..., “3'1
am “trees cm in whom km 05 tuft
2 MM Guam 1 ‘l’m on (n+1) varkm scion“ q hurl: (since it»; .‘hu is aback: this
We". is at insk en. Leaf). Cali H L. . 8%“
Mn it. This gives a fru tn n9 textcu; Cassi) The. n “me“hie. has hoe Remus «A A 4'. not “JJQM b‘Mwf‘ﬂdot. .%3)gg.¢£Mi:.tmﬁ um. .QuJ mun5mm.“th 7 (9) Find the least squares line which best describes the points
(1:1) {4.18)‘ 4——r__.__‘ ﬂ‘ .
.u~r_t_:‘._ '“ __30m+0E=5l+
[Oh1" 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: ‘15'
. (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 orr
00‘! oar (12) Give two different graphs on 6 vertices where breath first
i ‘ ..s,e_arch and depth first search give the same outputsequence. 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:
QJiMIIsia‘ I ' h I
 22's] (2')?“ = (314] _ if“  Esmei4; 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, butnow you have the choice of 4 different color
families, and there are 5 different shades of red? 350! BGR +oOR a 3012 ' (gm) + (ix 29(2) + (mm + elm.) " 19 (18) You have an 12x12 chess board, have many ways can you
place 8 nontaking rocks ‘? (In other_ words, no rocks can
share a row and column) 128’ .
(.25 ‘5 ‘. _
R claws cm rows 5 ColoInns on. but consi  (19)“Giveacombinatorlal‘proof'that js wakes for“ Hrs—l: to
(")4 ‘1); " ‘°' '0‘ We T . “idemanal M 3: (me o
_a wickloan item at rdnﬂdre:   g m" “M “J
b“§'a'0€e {‘M MMClrkin 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 1,4,1113 (using simplex method) wwuu' . 1.60; 1!? lnsa'cs. ...
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 Spring '09
 COSTELLO
 Math

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