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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 268 14 "Calcul Formel " }{TEXT 
-1 17 "MASS 1 2003-2004\n" }{TEXT 269 9 "Feuille 6" }{TEXT -1 20 "   I
nterrogation 1 (" }{TEXT 270 27 "~/calcul_mass1_2003_f6.mws)" }{TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT 295 67 "Dans ce qui suit , il vous fau
dra  remplacer les \":\"  par des \" ;\")" }}}{SECT 0 {PARA 257 "" 0 "
" {TEXT -1 58 "Calculs pr\351paratoires :  circulez! il n'y a rien \+
\340 voir..." }}{EXCHG {PARA 258 "> " 0 "" {MPLTEXT 1 0 42 "tr:=proc(M
)\nRETURN(linalg[trace](M));\nend:" }}}{EXCHG {PARA 292 "> " 0 "" 
{MPLTEXT 1 0 13 "with(linalg):" }{TEXT -1 30 "# ici ne pas modifier le
 \":\" !" }}{PARA 7 "" 1 "" {TEXT -1 32 "Warning, new definition for n
orm" }}{PARA 7 "" 1 "" {TEXT -1 33 "Warning, new definition for trace
" }}}}{SECT 0 {PARA 259 "" 0 "" {TEXT -1 10 "Exercice 1" }{TEXT 287 6 
"\n  Soi" }{TEXT 256 2 "t " }{TEXT 263 1 "A" }{TEXT 264 12 " la matric
e " }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 39 "A:= matrix(2,2,[a11,
 a12,  a21, a22 ]):" }}}{EXCHG {PARA 299 "" 0 "" {TEXT -1 3 "et " }}}
{EXCHG {PARA 300 "> " 0 "" {MPLTEXT 1 0 27 "Id2:=matrix(2,2,[1,0,0,1])
:" }}}{EXCHG {PARA 261 "" 0 "" {TEXT -1 23 "On note  trA le nombre " }
}}{EXCHG {PARA 262 "> " 0 "" {MPLTEXT 1 0 11 "trA:=tr(A):" }}}{EXCHG 
{PARA 263 "" 0 "" {TEXT 260 2 "Q1" }{TEXT -1 48 " Calculer la matrice \+
C:=A^2- trA*A +det(A)*Id2;\n" }{TEXT 261 2 "Q2" }{TEXT -1 54 "  Pour q
uelles valeurs de a11,a12,a21,a22 a-t-on C=0 ?" }}}{SECT 0 {PARA 264 "
" 0 "" {TEXT -1 7 "R\351ponse" }}{EXCHG {PARA 265 "> " 0 "" {MPLTEXT 
1 0 36 "C:= evalm(A^2- trA*A + det(A)*Id2):\n" }}}{EXCHG {PARA 266 "" 
0 "" {TEXT -1 61 "Si on d\351veloppe \"\340 la main\" les coefficients
 de C , on trouve" }}}{EXCHG {PARA 267 "> " 0 "" {MPLTEXT 1 0 61 "expa
nd(C[1,1]), expand(C[2,1]),expand(C[1,2]),expand(C[2,2]):" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT 257 7 "Moralit" }{TEXT -1 3 "\351 :" }{TEXT 258 
91 " tous les coefficients de C sont nuls, quels que soient  a11, a12,
 a21, a22 !  donc  C= 0 ." }}}}}{SECT 0 {PARA 268 "" 0 "" {TEXT 262 
10 "Exercice 2" }{TEXT -1 8 "\n  Soit " }{TEXT 259 1 "A" }{TEXT -1 11 
" la matrice" }}{SECT 0 {PARA 269 "" 0 "" {TEXT -1 14 "Calculs cach
\351s" }}{EXCHG {PARA 270 "> " 0 "" {MPLTEXT 1 0 17 "tr(A)^2-4*det(A):
" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 23 "sol:=evalf(solve(g=x)
):" }}}{EXCHG {PARA 272 "> " 0 "" {MPLTEXT 1 0 25 "M:=max(sol): m:=min
(sol):" }}{PARA 8 "" 1 "" {TEXT -1 57 "Error, (in simpl/max) arguments
 must be of type algebraic" }}{PARA 8 "" 1 "" {TEXT -1 57 "Error, (in \+
simpl/min) arguments must be of type algebraic" }}}}{EXCHG {PARA 273 "
> " 0 "" {MPLTEXT 1 0 19 "A:=randmatrix(2,2):" }}}{EXCHG {PARA 274 "" 
0 "" {TEXT 303 12 "ATTENTION : " }{TEXT -1 20 "Si det(A)  est nul  " }
{TEXT 274 2 "OU" }{TEXT -1 26 "  trA^2 - 4*det(A) < 0 ,  " }{TEXT 272 
13 "se \"redonner\"" }{TEXT -1 14 " la matrice A " }}{PARA 301 "" 0 "
" {TEXT 271 10 "jusqu'\340 ce" }{TEXT -1 26 "  que det(A) soit non nul
 " }{TEXT 273 2 "ET" }{TEXT -1 71 "  que trA^2-4*det(A) soit positif o
u nul.\nOn d\351finit alors la fonction " }{TEXT 275 2 "fA" }{TEXT -1 
6 "  par:" }}}{EXCHG {PARA 275 "> " 0 "" {MPLTEXT 1 0 46 "fA:= x-> (A[
1,1]*x+A[1,2])/(A[2,1]*x+ A[2,2]):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
276 16 "et l'expression " }{TEXT 277 1 "g" }{TEXT 278 6 " par :" }}}
{EXCHG {PARA 276 "> " 0 "" {MPLTEXT 1 0 9 "g:=fA(x):" }}}{EXCHG {PARA 
277 "" 0 "" {TEXT 265 2 "Q3" }{TEXT -1 24 "   R\351soudre l'\351quatio
n  " }{TEXT 279 3 "g=x" }{TEXT -1 63 "  (on notera M la plus grande de
s solutions, m laplus petite) ." }}{PARA 278 "" 0 "" {TEXT -1 5 "Soit \+
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "sol:=evalf(solve(g=x));
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$solG6$$\"+s'*fp?!\"*$!+.K[#=(!#
5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "M:=sol[1];" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"MG$\"+s'*fp?!\"*" }}}{EXCHG {PARA 279 ">
 " 0 "" {MPLTEXT 1 0 11 "u0:= 100*M:" }}}{EXCHG {PARA 280 "" 0 "" 
{TEXT -1 11 "La suite ( " }{TEXT 266 1 "u" }{TEXT -1 3 "(n)" }{TEXT 
299 1 " " }{TEXT -1 41 ") d\351finie par la relation de r\351currence \+
 " }}{PARA 281 "" 0 "" {TEXT -1 36 "                                  \+
  " }{TEXT 288 1 "u" }{TEXT -1 30 "(0) := u0  et,  pour n >= 0,  " }
{TEXT 289 1 "u" }{TEXT -1 8 "(n+1) :=" }{TEXT 267 3 " fA" }{TEXT -1 2 
"( " }{TEXT 290 1 "u" }{TEXT -1 7 "(n) )  " }}{PARA 282 "" 0 "" {TEXT 
-1 64 "vous semble-t-elle convergente ? Si oui, quelle est  sa limite \+
?" }}}{SECT 0 {PARA 283 "" 0 "" {TEXT -1 7 "R\351ponse" }}{EXCHG 
{PARA 284 "> " 0 "" {MPLTEXT 1 0 111 "suite:= proc(val,nb)\nlocal S,u,
j;\nu:=val:S:=u:\nfor j from 1 to nb do\n   u:=fA(u);\n   S:=S,u\nod:
\nRETURN(S);\nend;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&suiteGR6$%$va
lG%#nbG6%%\"SG%\"uG%\"jG6\"F-C&>8%9$>8$F0?(8&\"\"\"F69%%%trueGC$>F0-%#
fAG6#F0>F36$F3F0-%'RETURNG6#F3F-F-F-" }}}{EXCHG {PARA 285 "> " 0 "" 
{MPLTEXT 1 0 13 "suite(u0,20);" }}{PARA 12 "" 1 "" {XPPMATH 20 "67$\"+
s'*fp?!\"($\"+&o%*RH#!\"*$\"+x(p`3#F($\"+VUyq?F($\"+\"3*op?F($\"+Akgp?
F($\"+\"=+'p?F($\"+5(*fp?F($\"+u'*fp?F($F$F($\"+q'*fp?F($\"+r'*fp?F(F:
F:F:F:F:F:F:F:F:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(pl
ots):" }}}{EXCHG {PARA 286 "> " 0 "" {MPLTEXT 1 0 25 "listplot([suite(
u0,20)]);" }}{PARA 13 "" 1 "" {INLPLOT "6#-%'CURVESG6#777$$\"\"\"\"\"!
$\"+s'*fp?!\"(7$$\"\"#F*$\"+&o%*RH#!\"*7$$\"\"$F*$\"+x(p`3#F37$$\"\"%F
*$\"+VUyq?F37$$\"\"&F*$\"+\"3*op?F37$$\"\"'F*$\"+Akgp?F37$$\"\"(F*$\"+
\"=+'p?F37$$\"\")F*$\"+5(*fp?F37$$\"\"*F*$\"+u'*fp?F37$$\"#5F*$F,F37$$
\"#6F*$\"+q'*fp?F37$$\"#7F*$\"+r'*fp?F37$$\"#8F*F]o7$$\"#9F*F]o7$$\"#:
F*F]o7$$\"#;F*F]o7$$\"#<F*F]o7$$\"#=F*F]o7$$\"#>F*F]o7$$\"#?F*F]o7$$\"
#@F*F]o" 2 649 649 649 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 12338 0 0 0 0 0 0 }}}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{SECT 0 {PARA 287 "" 0 "" 
{TEXT -1 10 "Exercice 3" }}{EXCHG {PARA 293 "" 0 "" {TEXT 286 11 "D
\351finition " }{TEXT -1 6 ": si  " }{TEXT 284 1 "B" }{TEXT -1 16 " es
t la matrice " }}}{EXCHG {PARA 294 "> " 0 "" {MPLTEXT 1 0 31 "B := mat
rix(2,3,[1,2,6,3,4,7]):" }}}{EXCHG {PARA 295 "" 0 "" {TEXT -1 7 "et si
  " }{TEXT 291 9 "equation1" }{TEXT -1 4 " et " }{TEXT 292 9 "equation
2" }{TEXT -1 25 " d\351signent les \351quations " }}}{EXCHG {PARA 296 
"> " 0 "" {MPLTEXT 1 0 71 "equation1:= B[1,1]*x+B[1,2]*y - 6 :\nequati
on2:= B[2,1]*x+B[2,2]*y - 7 :" }}}{EXCHG {PARA 297 "" 0 "" {TEXT -1 
65 "On dira que  le syst\350me lin\351aire associ\351 \340 la matrice \+
 B est   \{ " }{TEXT 293 9 "equation1" }{TEXT -1 7 " = 0 , " }{TEXT 
294 9 "equation2" }{TEXT -1 7 " = 0 \}." }}{PARA 0 "" 0 "" {TEXT 281 
5 "Soit " }{TEXT 282 1 "L" }{TEXT 283 21 " la liste de matrices" }}}
{EXCHG {PARA 288 "> " 0 "" {MPLTEXT 1 0 35 "L:= [seq(randmatrix(2,3),k
=1..20)]:" }}}{EXCHG {PARA 289 "" 0 "" {TEXT -1 10 "Noter que " }
{TEXT 298 1 "L" }{TEXT -1 36 "[j] est la j\350me matrice de la liste \+
" }{TEXT 297 1 "L" }{TEXT -1 29 " ; par exemple, pour  j:=3,  " }
{TEXT 300 1 "L" }{TEXT -1 29 "[3] est la troisi\350me matrice." }}}
{EXCHG {PARA 290 "> " 0 "" {MPLTEXT 1 0 20 "verification:= L[3]:" }}}
{EXCHG {PARA 291 "" 0 "" {TEXT 280 3 "Q1 " }{TEXT -1 32 "Pour chaque m
atrice de la liste " }{TEXT 301 1 "L" }{TEXT -1 73 ", r\351soudre (si \+
c'est possible) le syst\350me  lin\351aire qui lui est associ\351 ." }
}{PARA 298 "" 0 "" {TEXT -1 49 "Chaque solution [x,y] d\351finit un po
int du plan; \n" }{TEXT 285 2 "Q2" }{TEXT -1 150 " Faire afficher  tou
s les points du plan ainsi obtenus (indication : consulter l'aide en l
igne pour la fonction pointplot et utiliser la biblioth\350que " }
{TEXT 302 5 "plots" }{TEXT -1 2 " )" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 
296 20 "Une r\351ponse possible" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 166 "S:=[]:\nfor j from 1 to nops(L) do\n    A:=L[j];\n    M:= mat
rix(2,2,[ A[1,1],A[1,2],A[2,1],A[2,2]]):\n    b:= vector([A[1,3],A[2,3
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