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10 "2003-2004\n" }{TEXT 258 9 "Feuille 4" }{TEXT 259 19 " Al g\350bre lin\351aire 1" }}}{SECT 0 {PARA 256 "" 0 "" {TEXT -1 35 "Cons tructeurs et op\351rations de base" }}{SECT 0 {PARA 257 "" 0 "" {TEXT -1 26 "Constructeurs de matrices " }{TEXT 260 21 "( 2x2 pour le moment )" }}{EXCHG {PARA 258 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }} {PARA 351 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 350 "" 0 "" {TEXT -1 61 "# On \+ \"charge\" la biblioth\350que (de fonctions) LINear ALGebra " }}{PARA 259 "" 0 "" {TEXT -1 38 "Pour \"construire\" des matrices 2x2 :" }} {PARA 7 "" 1 "" {TEXT -1 32 "Warning, new definition for norm" }} {PARA 7 "" 1 "" {TEXT -1 33 "Warning, new definition for trace" }}} {EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 34 "A:= matrix(2,2,[a11,a12,a2 1,a22]);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'ma trixG6#7$7$%$a11G%$a12G7$%$a21G%$a22G" }}}{EXCHG {PARA 261 "" 0 "" {TEXT -1 48 "Notation ligne ( 1\260 indice) colonne (2\260 indice)." } }{PARA 262 "" 0 "" {TEXT -1 67 "On peut aussi avoir des matrices avec \+ des coefficients num\351riques :" }}}{EXCHG {PARA 263 "> " 0 "" {MPLTEXT 1 0 31 "L:=[5,6,7,8]:B:= matrix(2,2,L);" }}{PARA 264 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG-%'matrixG6#7$7$ \"\"&\"\"'7$\"\"(\"\")" }}}{EXCHG {PARA 265 "" 0 "" {TEXT -1 201 "Bien comprendre comment les \351l\351ments de L vont permettre de \"rempli r\" B : on \351crit les coefficients ( entries en anglais ) de B en l igne !\nOu encore certains coefficients num\351riques d'autres forme ls" }}}{EXCHG {PARA 266 "> " 0 "" {MPLTEXT 1 0 30 "L:=[2,1,3,m]:F:=mat rix(2,2,L);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG-%'matrixG6#7$7$ \"\"#\"\"\"7$\"\"$%\"mG" }}}{EXCHG {PARA 267 "" 0 "" {TEXT -1 90 "F es t la matrice ayant pour vecteurs-colonnes les vecteurs U1 et U2 de l' exo 1 feuille 3." }}{PARA 268 "" 0 "" {TEXT -1 32 "Pour \"construire\" des vecteurs :" }}}{EXCHG {PARA 269 "> " 0 "" {MPLTEXT 1 0 55 "U1:=ve ctor([2,3]);U2:=vector([1,m]);U3 :=vector([7,8]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#U1G-%'vectorG6#7$\"\"#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#U2G-%'vectorG6#7$\"\"\"%\"mG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#U3G-%'vectorG6#7$\"\"(\"\")" }}}{EXCHG {PARA 270 "" 0 "" {TEXT -1 85 "On dispose aussi (parce qu'on a \351crit des fonctio ns \"ad hoc\") et c'est en fran\347ais..." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 300 "> " 0 "" {MPLTEXT 1 0 234 "ra cine:=proc(n)\nlocal k;\nk:=1;\nwhile k*k <= n do k:=k+1; od;\nRETURN( k-1);\nend:\n\nmatrice:=proc(L)# nops(L) est un carr\351 \nlocal j; \nj:=racine(nops(L));\nRETURN(evalm(matrix(j,j,L)));\nend:\n\nvecteur: =proc(L)\nRETURN(evalm(vector(L)));\nend:" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 40 "G:=matrice([2,1,3,m]);V:=vecteur([1,m]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"GG-%'matrixG6#7$7$\"\"#\"\"\"7$\"\"$%\"m G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"VG-%'vectorG6#7$\"\"\"%\"mG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "racine(4);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "matrice(B);" }}{PARA 8 "" 1 "" {TEXT -1 64 "Error, (in matrix) 2nd index, 2, larger than upper array bound 1" }}}}{SECT 0 {PARA 272 "" 0 "" {TEXT -1 23 "Op\351rations \351l\351mentaires" }}{SECT 0 {PARA 273 "" 0 "" {TEXT -1 27 "Constructions pr\351paratoires" }}{EXCHG {PARA 274 "> " 0 "" {MPLTEXT 1 0 31 "L:=[2,1,-3,0]:A:=matrix(2,2,L):" }}}{EXCHG {PARA 275 "> " 0 "" {MPLTEXT 1 0 78 "racine:=proc(n)\nlocal \+ k;\nk:=1;\nwhile k*k <= n do k:=k+1; od;\nRETURN(k-1);\nend:" }}} {EXCHG {PARA 276 "> " 0 "" {MPLTEXT 1 0 16 "matrice:=proc(L)" }{TEXT -1 25 "# nops(L) est un carr\351 " }{MPLTEXT 1 0 65 "\nlocal j;\nj:= racine(nops(L));\nRETURN(evalm(matrix(j,j,L)));\nend:\n" }}}{EXCHG {PARA 277 "> " 0 "" {MPLTEXT 1 0 11 "matrice(L);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7$\"\"#\"\"\"7$!\"$\"\"!" }}}{EXCHG {PARA 278 "> " 0 "" {MPLTEXT 1 0 47 "vecteur:=proc(L)\nRETURN(evalm(ve ctor(L)));\nend:" }}{PARA 279 "> " 0 "" {MPLTEXT 1 0 25 "L:=[2,1,-3,0] :vecteur(L);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7&\"\"#\" \"\"!\"$\"\"!" }}}{EXCHG {PARA 280 "> " 0 "" {MPLTEXT 1 0 23 "B:=matri ce([-1,1,2,3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG-%'matrixG6#7 $7$!\"\"\"\"\"7$\"\"#\"\"$" }}}{EXCHG {PARA 281 "> " 0 "" {MPLTEXT 1 0 24 "C:=matrice([5,-1,-2,4]):" }}}{EXCHG {PARA 282 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 283 "" 0 "" {TEXT -1 32 "Somme et \+ produit par un scalaire" }}{EXCHG {PARA 284 "> " 0 "" {MPLTEXT 1 0 27 "evalm(A);evalm(B);evalm(C);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'mat rixG6#7$7$\"\"#\"\"\"7$!\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-% 'matrixG6#7$7$!\"\"\"\"\"7$\"\"#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7$\"\"&!\"\"7$!\"#\"\"%" }}}{EXCHG {PARA 285 "> " 0 " " {MPLTEXT 1 0 11 "evalm(3*A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'m atrixG6#7$7$\"\"'\"\"$7$!\"*\"\"!" }}}{EXCHG {PARA 286 "" 0 "" {TEXT -1 55 "Donc Maple sait multiplier une matrice par un scalaire." }}} {EXCHG {PARA 287 "> " 0 "" {MPLTEXT 1 0 20 "F:=evalm(3*A+2*B-C);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG-%'matrixG6#7$7$!\"\"\"\"'7$!\"$ \"\"#" }}}{EXCHG {PARA 288 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 289 "" 0 "" {TEXT -1 21 "Produits de matrices " }}{EXCHG {PARA 290 "> " 0 "" {MPLTEXT 1 0 44 "A:=matrice([1,2,3,4]):B:=matrice([5,6,7 ,8]):" }}}{EXCHG {PARA 291 "" 0 "" {TEXT -1 72 "Pour effectuer le prod uit de A par B, on dispose de l'\"op\351ration\" &* : " }}}{EXCHG {PARA 292 "> " 0 "" {MPLTEXT 1 0 31 "F:=evalm(A&*B);G:=evalm(B &*A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG-%'matrixG6#7$7$\"#>\"#A7$\"#V \"#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"GG-%'matrixG6#7$7$\"#B\"#M 7$\"#J\"#Y" }}}{EXCHG {PARA 293 "" 0 "" {TEXT -1 53 "Eh voil\340, le t our est jou\351 !... G est diff\351rent de F!" }}{PARA 294 "" 0 "" {TEXT -1 36 "Et pour le calcul d'un d\351terminant ?" }}}{EXCHG {PARA 295 "> " 0 "" {MPLTEXT 1 0 4 "?det" }}}{EXCHG {PARA 296 "> " 0 "" {MPLTEXT 1 0 49 "f:=det(F);g:=det(G);a:=det(A);b:=det(B);c := a*b;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG! \"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cG\"\"%" }}}{EXCHG {PARA 297 "" 0 "" {TEXT -1 24 "F in de l'exo 5 feuille 3" }}}{EXCHG {PARA 298 "> " 0 "" {MPLTEXT 1 0 53 "A:=matrice([1,-1,-1,1]):B:=evalm(A&*A);C:=evalm(2*A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG-%'matrixG6#7$7$\"\"#!\"#7$F+F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"CG-%'matrixG6#7$7$\"\"#!\"#7$F+F*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "M:=matrice([-1,-1,-1,-1]);N: =matrice([-2,2,2,-2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG-%'mat rixG6#7$7$!\"\"F*F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG-%'matrix G6#7$7$!\"#\"\"#7$F+F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "e valm(M&*N);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7$\"\"!F( F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "M:=matrice([1,-1,-1,1 ]);N:=matrice([2,-2,-2,2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG- %'matrixG6#7$7$\"\"\"!\"\"7$F+F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"NG-%'matrixG6#7$7$\"\"#!\"#7$F+F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "evalm(M&*N);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'ma trixG6#7$7$\"\"%!\"%7$F)F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "evalm(M&*M);evalm(N&*N);evalm(2*N);evalm(4*N);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#-%'matrixG6#7$7$\"\"#!\"#7$F)F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7$\"\")!\")7$F)F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7$\"\"%!\"%7$F)F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7$\"\")!\")7$F)F(" }}}}}}{SECT 0 {PARA 299 "" 0 "" {TEXT -1 9 "Exercices" }}{SECT 0 {PARA 301 "" 0 "" {TEXT -1 3 "E1 " }{TEXT 266 17 "(exo 8 feuille 3)" }{TEXT -1 3 " : " }{TEXT 261 4 "soit" }}{EXCHG {PARA 302 "> " 0 "" {MPLTEXT 1 0 22 "T:=matrice( [2,1,0,1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"TG-%'matrixG6#7$7$ \"\"#\"\"\"7$\"\"!F+" }}}{EXCHG {PARA 303 "" 0 "" {TEXT -1 75 "Calcule r T^n pour n = 2..7 (\340 la main exclusivement !)\nMeme question po ur" }}}{EXCHG {PARA 304 "> " 0 "" {MPLTEXT 1 0 47 "A:=matrice([cos(u), -2*sin(u),sin(u)/2,cos(u)]):" }}}{SECT 0 {PARA 305 "" 0 "" {TEXT -1 7 "r\351ponse" }}{EXCHG {PARA 306 "> " 0 "" {MPLTEXT 1 0 16 "T2:=evalm(T &*T);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#T2G-%'matrixG6#7$7$\"\"%\" \"$7$\"\"!\"\"\"" }}}{EXCHG {PARA 307 "> " 0 "" {MPLTEXT 1 0 17 "T3:=e valm(T2&*T);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#T3G-%'matrixG6#7$7$ \"\")\"\"(7$\"\"!\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 " T4:=evalm(T3&*T);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#T4G-%'matrixG6 #7$7$\"#;\"#:7$\"\"!\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "T5:=evalm(T4&*T);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#T5G-%'matr ixG6#7$7$\"#K\"#J7$\"\"!\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "T6:=evalm(T5&*T);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#T6G-%' matrixG6#7$7$\"#k\"#j7$\"\"!\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "T7:=evalm(T6&*T);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%#T7G-%'matrixG6#7$7$\"$G\"\"$F\"7$\"\"!\"\"\"" }}}{EXCHG {PARA 308 " > " 0 "" {MPLTEXT 1 0 29 "T7:=matrice([2^7,2^7-1,0,1]);" }{TEXT -1 14 "# Vrai ou faux" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#T7G-%'matrixG6#7 $7$\"$G\"\"$F\"7$\"\"!\"\"\"" }}}{EXCHG {PARA 309 "> " 0 "" {MPLTEXT 1 0 17 "A2:= evalm(A&*A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A2G-%' matrixG6#7$7$,&*$)-%$cosG6#%\"uG\"\"#\"\"\"\"\"\"*$)-%$sinGF/F1F2!\"\" ,$*&F-F3F6F3!\"%7$F:F*" }}}{EXCHG {PARA 310 "> " 0 "" {MPLTEXT 1 0 18 "A2s:=simplify(A2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$A2sG-%'matri xG6#7$7$,&*$)-%$cosG6#%\"uG\"\"#\"\"\"F1!\"\"\"\"\",$*&F-F4-%$sinGF/F4 !\"%7$F6F*" }}}{EXCHG {PARA 311 "" 0 "" {TEXT -1 79 "Ce qui prouve que Maple ne connait pas ses formules de trigonom\351trie, et vous ?" }}} }}{SECT 0 {PARA 312 "" 0 "" {TEXT -1 35 "E2 r\351solution de syst\350m es lin\351aires" }}{SECT 0 {PARA 313 "" 0 "" {TEXT 262 22 "syst\350mes num\351riques : " }{TEXT 263 5 "soit " }}{EXCHG {PARA 314 "> " 0 "" {MPLTEXT 1 0 41 "A:=matrice([7,8,9,10]):b:=vecteur([2,4]):" }}}{EXCHG {PARA 315 "" 0 "" {TEXT -1 64 "Pour r\351soudre le syst\350me Ax = b, \+ on peut calculer l'inverse de A" }}}{EXCHG {PARA 316 "> " 0 "" {MPLTEXT 1 0 7 "det(A);" }}{PARA 317 "" 1 "" {XPPMATH 20 "6#!\"#" }}} {EXCHG {PARA 318 "" 0 "" {TEXT -1 40 "det(A) non nul donc A est invers ible et " }}}{EXCHG {PARA 319 "> " 0 "" {MPLTEXT 1 0 14 "B:=inverse(A) ;" }}{PARA 320 "" 1 "" {XPPMATH 20 "6#>%\"BG-%'matrixG6#7$7$!\"&\"\"%7 $#\"\"*\"\"##!\"(F/" }}}{EXCHG {PARA 321 "" 0 "" {TEXT -1 42 "Impressi onnant , non ? et on conclut par :" }}}{EXCHG {PARA 322 "> " 0 "" {MPLTEXT 1 0 15 "x:=evalm(B&*b);" }}{PARA 323 "" 1 "" {XPPMATH 20 "6#> %\"xG-%'vectorG6#7$\"\"'!\"&" }}}{EXCHG {PARA 324 "" 0 "" {TEXT 268 8 "Question" }{TEXT -1 58 " : traiter les deux syst\350mes propos\351s \+ \340 la feuille 3 exo 9" }}}}{SECT 0 {PARA 325 "" 0 "" {TEXT 264 26 "S yst\350mes \"\340 param\350tres\" : " }{TEXT 265 4 "soit" }}{EXCHG {PARA 326 "> " 0 "" {MPLTEXT 1 0 43 "A:=matrice([1,3,2*m,-1]):u:=vecte ur([a,b]):" }}}{EXCHG {PARA 327 "> " 0 "" {MPLTEXT 1 0 7 "det(A);" }} {PARA 328 "" 1 "" {XPPMATH 20 "6#,&!\"\"\"\"\"%\"mG!\"'" }}}{EXCHG {PARA 329 "" 0 "" {TEXT -1 78 "Dans la mesure o\371 m est \"formel\", \+ det(A) est \"formellement\" non nul et donc :" }}}{EXCHG {PARA 330 "> " 0 "" {MPLTEXT 1 0 14 "B:=inverse(A);" }}{PARA 331 "" 1 "" {XPPMATH 20 "6#>%\"BG-%'matrixG6#7$7$*&\"\"\"F+,&\"\"\"F-%\"mG\"\"'!\"\",$F*\" \"$7$,$*&F.F+F,F0\"\"#,$F*!\"\"" }}}{EXCHG {PARA 332 "" 0 "" {TEXT -1 2 "et" }}}{EXCHG {PARA 333 "> " 0 "" {MPLTEXT 1 0 25 "x:=simplify(eval m(B&*u));" }}{PARA 334 "" 1 "" {XPPMATH 20 "6#>%\"xG-%'vectorG6#7$*&,& %\"aG\"\"\"%\"bG\"\"$\"\"\",&F,F,%\"mG\"\"'!\"\"*&,&*&F1F,F+F,\"\"#F-! \"\"F/F0F3" }}}{EXCHG {PARA 335 "" 0 "" {TEXT -1 195 "Les ennuis comme ncent si on veut on veut donner des \"valeurs\" au param\350tre m.\nSi cette valeur est diff\351rente de -1/6 (par exemple 2), on peut \"r emplacer m par sa valeur\" dans l'expression de x:" }}}{EXCHG {PARA 336 "> " 0 "" {MPLTEXT 1 0 23 "x_num1:=subs(m=2,x[1]);" }}{PARA 337 " " 1 "" {XPPMATH 20 "6#>%'x_num1G,&%\"aG#\"\"\"\"#8%\"bG#\"\"$F)" }}} {EXCHG {PARA 338 "> " 0 "" {MPLTEXT 1 0 23 "x_num2:=subs(m=2,x[2]);" } }{PARA 339 "" 1 "" {XPPMATH 20 "6#>%'x_num2G,&%\"aG#\"\"%\"#8%\"bG#!\" \"F)" }}}{EXCHG {PARA 340 "" 0 "" {TEXT -1 71 "Dans le cas contraire, \+ on revient \"\340 la case d\351part\" et on red\351finit A:" }}} {EXCHG {PARA 341 "> " 0 "" {MPLTEXT 1 0 30 "A:=matrice([1,3,2*(-1/6),- 1]);" }}{PARA 342 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7$7$\"\"\" \"\"$7$#!\"\"F+F." }}}{EXCHG {PARA 343 "" 0 "" {TEXT -1 181 "On consta te que les deux vecteurs colonnes de A sont proportionnels; u est donc dans l'EV engendr\351 s'il est lui aussi proportionnel \340 l'un de s deux ce qui implique que a = - 3 b." }}}{EXCHG {PARA 344 "> " 0 "" {MPLTEXT 1 0 24 "x_f1:=subs(a=-3*b,x[1]);" }}{PARA 345 "" 1 "" {XPPMATH 20 "6#>%%x_f1G\"\"!" }}}{EXCHG {PARA 346 "> " 0 "" {MPLTEXT 1 0 34 "x_f2:=simplify(subs(a=-3*b,x[2]));" }}{PARA 347 "" 1 "" {XPPMATH 20 "6#>%%x_f2G,$%\"bG!\"\"" }}}{EXCHG {PARA 348 "" 0 "" {TEXT -1 5 "Hop !" }}{PARA 349 "" 0 "" {TEXT 267 9 "Question " }{TEXT -1 38 ": traiter la fin de l'exo 10 feuille 3" }}}{SECT 0 {PARA 20 "" 0 "" {TEXT -1 8 "solution" }}}}}}}{MARK "2 2 2 16" 0 }{VIEWOPTS 1 1 0 1 1 1803 }