• Épreuve de mathématiques

    Question : 1

    Soit L une liste finie d’entiers relatifs consécutifs dont le premier terme est -15. . L{ 15,14,... } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamitamaacm aabaGaeyOeI0IaaGymaiaaiwdacaGGSaGaeyOeI0IaaGymaiaaisda caGGSaGaaiOlaiaac6cacaGGUaaacaGL7bGaayzFaaaaaa@413C@ Si la somme de tous les éléments de L est égale à 51 alors le nombre totale des termes de la liste L est égale.

    Texte de la question

    Soit L une liste finie d’entiers relatifs consécutifs dont le premier terme est -15. . L{ 15,14,... } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamitamaacm aabaGaeyOeI0IaaGymaiaaiwdacaGGSaGaeyOeI0IaaGymaiaaisda caGGSaGaaiOlaiaac6cacaGGUaaacaGL7bGaayzFaaaaaa@413C@ Si la somme de tous les éléments de L est égale à 51 alors le nombre totale des termes de la liste L est égale.

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    Question : 2

    lim n+ (1) n 3 n+1 π n = MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci GGSbGaaiyAaiaac2gaaSqaaiaad6gacqGHsgIRcqGHRaWkcqGHEisP aeqaaOWaaSaaaeaacaGGOaGaeyOeI0IaaGymaiaacMcadaahaaWcbe qaaiaad6gaaaGccaaIZaWaaWbaaSqabeaacaWGUbGaey4kaSIaaGym aaaaaOqaaiabec8aWnaaCaaaleqabaGaamOBaaaaaaGccqGH9aqpaa a@49E9@

    Texte de la question

    lim n+ (1) n 3 n+1 π n = MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci GGSbGaaiyAaiaac2gaaSqaaiaad6gacqGHsgIRcqGHRaWkcqGHEisP aeqaaOWaaSaaaeaacaGGOaGaeyOeI0IaaGymaiaacMcadaahaaWcbe qaaiaad6gaaaGccaaIZaWaaWbaaSqabeaacaWGUbGaey4kaSIaaGym aaaaaOqaaiabec8aWnaaCaaaleqabaGaamOBaaaaaaGccqGH9aqpaa a@49E9@

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    Question : 3

    soit Z n = K=1 n e k1 π K+1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOwamaaBa aaleaacaWGUbaabeaakiabg2da9maaqahabaWaaSaaaeaacaWGLbWa aWbaaSqabeaacaWGRbGaeyOeI0IaaGymaaaaaOqaaiabec8aWnaaCa aaleqabaGaam4saiabgUcaRiaaigdaaaaaaaqaaiaadUeacqGH9aqp caaIXaaabaGaamOBaaqdcqGHris5aaaa@46E0@ : alors lim n+ Z n = MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci GGSbGaaiyAaiaac2gaaSqaaiaad6gacqGHsgIRcqGHRaWkcqGHEisP aeqaaOGaamOwamaaBaaaleaacaWGUbaabeaakiabg2da9aaa@414B@

    Texte de la question

    soit Z n = K=1 n e k1 π K+1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOwamaaBa aaleaacaWGUbaabeaakiabg2da9maaqahabaWaaSaaaeaacaWGLbWa aWbaaSqabeaacaWGRbGaeyOeI0IaaGymaaaaaOqaaiabec8aWnaaCa aaleqabaGaam4saiabgUcaRiaaigdaaaaaaaqaaiaadUeacqGH9aqp caaIXaaabaGaamOBaaqdcqGHris5aaaa@46E0@ : alors lim n+ Z n = MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci GGSbGaaiyAaiaac2gaaSqaaiaad6gacqGHsgIRcqGHRaWkcqGHEisP aeqaaOGaamOwamaaBaaaleaacaWGUbaabeaakiabg2da9aaa@414B@

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    Question : 4

    Une entreprise de fabrication de mixeurs a adopté pour l’année 2012 la stratégie de production suivante : la production connaitre une diminution mensuelle de 10% ; mais grâce à une commande destinée à l’export , l’entreprise produira chaque mois 150 mixeurs de plus. On note à présent par   t n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaGGGcGaamiDa8aadaWgaaWcbaWdbiaad6gaa8aabeaaaaa@3980@ la production de l’usine relative au mois N°n. L’expression relaint t n+1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWG0bWdamaaBaaaleaapeGaamOBaiabgUcaRiaaigdaa8aabeaa aaa@39F9@ et   t n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaGGGcGaamiDa8aadaWgaaWcbaWdbiaad6gaa8aabeaaaaa@3980@ est donnée par :

    Texte de la question

    Une entreprise de fabrication de mixeurs a adopté pour l’année 2012 la stratégie de production suivante : la production connaitre une diminution mensuelle de 10% ; mais grâce à une commande destinée à l’export , l’entreprise produira chaque mois 150 mixeurs de plus. On note à présent par   t n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaGGGcGaamiDa8aadaWgaaWcbaWdbiaad6gaa8aabeaaaaa@3980@ la production de l’usine relative au mois N°n. L’expression relaint t n+1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWG0bWdamaaBaaaleaapeGaamOBaiabgUcaRiaaigdaa8aabeaa aaa@39F9@ et   t n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaGGGcGaamiDa8aadaWgaaWcbaWdbiaad6gaa8aabeaaaaa@3980@ est donnée par :

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    Question : 5

    Une entreprise de fabrication de mixeurs a adopté pour l’année 2012 la stratégie de production suivante : la production connaitre une diminution mensuelle de 10% ; mais grâce à une commande destinée à l’export , l’entreprise produira chaque mois 150 mixeurs de plus. Long terme La production mensuelle des mixeurs est estimée à P=

    Texte de la question

    Une entreprise de fabrication de mixeurs a adopté pour l’année 2012 la stratégie de production suivante : la production connaitre une diminution mensuelle de 10% ; mais grâce à une commande destinée à l’export , l’entreprise produira chaque mois 150 mixeurs de plus. Long terme La production mensuelle des mixeurs est estimée à P=

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    Question : 6

    soit ( u n ) n0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadw hadaWgaaWcbaGaamOBaaqabaGccaGGPaWaaSbaaSqaaiaad6gacqGH LjYScaaIWaaabeaaaaa@3D12@ une suite numérique à termes strictement positifs ( u n >0) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadw hadaWgaaWcbaGaamOBaaqabaGccqGH+aGpcaaIWaGaaiykaaaa@3B35@ vérifiant lim n+ u n+1 u n = 1 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci GGSbGaaiyAaiaac2gaaSqaaiaad6gacqGHsgIRcqGHRaWkcqGHEisP aeqaaOWaaSaaaeaacaWG1bWaaSbaaSqaaiaad6gacqGHRaWkcaaIXa aabeaaaOqaaiaadwhadaWgaaWcbaGaamOBaaqabaaaaOGaeyypa0Za aSaaaeaacaaIXaaabaGaaGOmaaaaaaa@46BD@ , Alors lim n+ u n =L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci GGSbGaaiyAaiaac2gaaSqaaiaad6gacqGHsgIRcqGHRaWkcqGHEisP aeqaaOGaamyDamaaBaaaleaacaWGUbaabeaakiabg2da9iaadYeaaa a@4237@ avec

    Texte de la question

    soit ( u n ) n0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadw hadaWgaaWcbaGaamOBaaqabaGccaGGPaWaaSbaaSqaaiaad6gacqGH LjYScaaIWaaabeaaaaa@3D12@ une suite numérique à termes strictement positifs ( u n >0) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadw hadaWgaaWcbaGaamOBaaqabaGccqGH+aGpcaaIWaGaaiykaaaa@3B35@ vérifiant lim n+ u n+1 u n = 1 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci GGSbGaaiyAaiaac2gaaSqaaiaad6gacqGHsgIRcqGHRaWkcqGHEisP aeqaaOWaaSaaaeaacaWG1bWaaSbaaSqaaiaad6gacqGHRaWkcaaIXa aabeaaaOqaaiaadwhadaWgaaWcbaGaamOBaaqabaaaaOGaeyypa0Za aSaaaeaacaaIXaaabaGaaGOmaaaaaaa@46BD@ , Alors lim n+ u n =L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci GGSbGaaiyAaiaac2gaaSqaaiaad6gacqGHsgIRcqGHRaWkcqGHEisP aeqaaOGaamyDamaaBaaaleaacaWGUbaabeaakiabg2da9iaadYeaaa a@4237@ avec

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    Question : 7

    Soit T n = p1 n 2 1 2p1 2 1 2p+1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaBa aaleaacaWGUbaabeaakiabg2da9maaqahabaGaaGOmamaaCaaaleqa baWaaSaaaeaacaaIXaaabaGaaGOmaiaadchacqGHsislcaaIXaaaaa aaaeaacaWGWbGaeyOeI0IaaGymaaqaaiaad6gaa0GaeyyeIuoakiab gkHiTiaaikdadaahaaWcbeqaamaalaaabaGaaGymaaqaaiaaikdaca WGWbGaey4kaSIaaGymaaaaaaaaaa@49CC@ ; alors lim n+ T n = MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci GGSbGaaiyAaiaac2gaaSqaaiaad6gacqGHsgIRcqGHRaWkcqGHEisP aeqaaOGaamivamaaBaaaleaacaWGUbaabeaakiabg2da9aaa@4145@

    Texte de la question

    Soit T n = p1 n 2 1 2p1 2 1 2p+1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaBa aaleaacaWGUbaabeaakiabg2da9maaqahabaGaaGOmamaaCaaaleqa baWaaSaaaeaacaaIXaaabaGaaGOmaiaadchacqGHsislcaaIXaaaaa aaaeaacaWGWbGaeyOeI0IaaGymaaqaaiaad6gaa0GaeyyeIuoakiab gkHiTiaaikdadaahaaWcbeqaamaalaaabaGaaGymaaqaaiaaikdaca WGWbGaey4kaSIaaGymaaaaaaaaaa@49CC@ ; alors lim n+ T n = MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci GGSbGaaiyAaiaac2gaaSqaaiaad6gacqGHsgIRcqGHRaWkcqGHEisP aeqaaOGaamivamaaBaaaleaacaWGUbaabeaakiabg2da9aaa@4145@

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    Question : 8

    On considère la courbe représentative de la fonction f(x)= e x 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacI cacaWG4bGaaiykaiabg2da9iaadwgadaahaaWcbeqaaiabgkHiTiaa dIhadaahaaadbeqaaiaaikdaaaaaaaaa@3E29@ . On désigne par R(x),x>0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuaiaacI cacaWG4bGaaiykaiaacYcacaGG4bGaeyOpa4JaaGimaaaa@3C92@ le rectangle symétrique inscrit à l’intérieur de la courbe et dont l’un des cotés est le segment d’extrémités (x,0) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiabgk HiTiaadIhacaGGSaGaaGimaiaacMcaaaa@3AA4@ et (x,0) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadI hacaGGSaGaaGimaiaacMcaaaa@39B7@ . La surface maximale de ce rectangle est égale à :

    Texte de la question

    On considère la courbe représentative de la fonction f(x)= e x 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacI cacaWG4bGaaiykaiabg2da9iaadwgadaahaaWcbeqaaiabgkHiTiaa dIhadaahaaadbeqaaiaaikdaaaaaaaaa@3E29@ . On désigne par R(x),x>0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuaiaacI cacaWG4bGaaiykaiaacYcacaGG4bGaeyOpa4JaaGimaaaa@3C92@ le rectangle symétrique inscrit à l’intérieur de la courbe et dont l’un des cotés est le segment d’extrémités (x,0) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiabgk HiTiaadIhacaGGSaGaaGimaiaacMcaaaa@3AA4@ et (x,0) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadI hacaGGSaGaaGimaiaacMcaaaa@39B7@ . La surface maximale de ce rectangle est égale à :

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    Question : 9

    lim x 0 + sinπx 1cos πx = MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci GGSbGaaiyAaiaac2gaaSqaaiaadIhacqGHsgIRcaaIWaWaaWbaaWqa beaacqGHRaWkaaaaleqaaOWaaSaaaeaaciGGZbGaaiyAaiaac6gacq aHapaCcaWG4baabaGaaGymaiabgkHiTiGacogacaGGVbGaai4Camaa kaaabaGaeqiWdaNaamiEaaWcbeaaaaGccqGH9aqpaaa@4BCB@

    Texte de la question

    lim x 0 + sinπx 1cos πx = MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci GGSbGaaiyAaiaac2gaaSqaaiaadIhacqGHsgIRcaaIWaWaaWbaaWqa beaacqGHRaWkaaaaleqaaOWaaSaaaeaaciGGZbGaaiyAaiaac6gacq aHapaCcaWG4baabaGaaGymaiabgkHiTiGacogacaGGVbGaai4Camaa kaaabaGaeqiWdaNaamiEaaWcbeaaaaGccqGH9aqpaaa@4BCB@

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    Question : 10

    lim h0 1 h e e+h 1 (lnx) 2 dx= MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci GGSbGaaiyAaiaac2gaaSqaaiaadIgacqGHsgIRcaaIWaaabeaakmaa laaabaGaaGymaaqaaiaadIgaaaWaa8qmaeaadaWcaaqaaiaaigdaae aacaGGOaGaciiBaiaac6gacaWG4bGaaiykamaaCaaaleqabaGaaGOm aaaaaaGccaWGKbGaamiEaiabg2da9aWcbaGaamyzaaqaaiaadwgacq GHRaWkcaWGObaaniabgUIiYdaaaa@4D06@

    Texte de la question

    lim h0 1 h e e+h 1 (lnx) 2 dx= MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci GGSbGaaiyAaiaac2gaaSqaaiaadIgacqGHsgIRcaaIWaaabeaakmaa laaabaGaaGymaaqaaiaadIgaaaWaa8qmaeaadaWcaaqaaiaaigdaae aacaGGOaGaciiBaiaac6gacaWG4bGaaiykamaaCaaaleqabaGaaGOm aaaaaaGccaWGKbGaamiEaiabg2da9aWcbaGaamyzaaqaaiaadwgacq GHRaWkcaWGObaaniabgUIiYdaaaa@4D06@

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    Question : 11

    0 π 2 ln 1+sinx 1+cosx dx= MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qmaeaaci GGSbGaaiOBamaalaaabaGaaGymaiabgUcaRiGacohacaGGPbGaaiOB aiaadIhaaeaacaaIXaGaey4kaSIaci4yaiaac+gacaGGZbGaamiEaa aacaWGKbGaamiEaiabg2da9aWcbaGaaGimaaqaamaalaaabaGaeqiW dahabaGaaGOmaaaaa0Gaey4kIipaaaa@4B22@

    Texte de la question

    0 π 2 ln 1+sinx 1+cosx dx= MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qmaeaaci GGSbGaaiOBamaalaaabaGaaGymaiabgUcaRiGacohacaGGPbGaaiOB aiaadIhaaeaacaaIXaGaey4kaSIaci4yaiaac+gacaGGZbGaamiEaa aacaWGKbGaamiEaiabg2da9aWcbaGaaGimaaqaamaalaaabaGaeqiW dahabaGaaGOmaaaaa0Gaey4kIipaaaa@4B22@

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    Question : 12

    1 2 1 2 dx 4 x 2 +4x+5 = MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qmaeaada WcaaqaaiaadsgacaWG4baabaGaaGinaiaadIhadaahaaWcbeqaaiaa ikdaaaGccqGHRaWkcaaI0aGaamiEaiabgUcaRiaaiwdaaaaaleaacq GHsisldaWcaaqaaiaaigdaaeaacaaIYaaaaaqaamaalaaabaGaaGym aaqaaiaaikdaaaaaniabgUIiYdGccqGH9aqpaaa@460D@

    Texte de la question

    1 2 1 2 dx 4 x 2 +4x+5 = MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qmaeaada WcaaqaaiaadsgacaWG4baabaGaaGinaiaadIhadaahaaWcbeqaaiaa ikdaaaGccqGHRaWkcaaI0aGaamiEaiabgUcaRiaaiwdaaaaaleaacq GHsisldaWcaaqaaiaaigdaaeaacaaIYaaaaaqaamaalaaabaGaaGym aaqaaiaaikdaaaaaniabgUIiYdGccqGH9aqpaaa@460D@

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    Question : 13

    La surface formée par la courbe de f(x)= (lnx) 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacI cacaWG4bGaaiykaiabg2da9iaacIcaciGGSbGaaiOBaiaadIhacaGG PaWaaWbaaSqabeaacaaIYaaaaaaa@3F61@ et par les droits x=1 et x=e est égale

    Texte de la question

    La surface formée par la courbe de f(x)= (lnx) 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacI cacaWG4bGaaiykaiabg2da9iaacIcaciGGSbGaaiOBaiaadIhacaGG PaWaaWbaaSqabeaacaaIYaaaaaaa@3F61@ et par les droits x=1 et x=e est égale

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    Question : 14

    soit ( V n ) n3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadA fadaWgaaWcbaGaamOBaaqabaGccaGGPaWaaSbaaSqaaiaad6gacqGH LjYScaaIZaaabeaaaaa@3CF6@ la suite définie par V n = e n 1 x (lnx) 3 dx MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBa aaleaacaWGUbaabeaakiabg2da9maapedabaWaaSaaaeaacaaIXaaa baGaamiEamaakaaabaGaaiikaiGacYgacaGGUbGaamiEaiaacMcada ahaaWcbeqaaiaaiodaaaaabeaaaaaabaGaamyzaaqaaiaad6gaa0Ga ey4kIipakiaadsgacaWG4baaaa@45E8@ alors lim n+ V n = MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci GGSbGaaiyAaiaac2gaaSqaaiaad6gacqGHsgIRcqGHRaWkcqGHEisP aeqaaOGaamOvamaaBaaaleaacaWGUbaabeaakiabg2da9aaa@4147@

    Texte de la question

    soit ( V n ) n3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadA fadaWgaaWcbaGaamOBaaqabaGccaGGPaWaaSbaaSqaaiaad6gacqGH LjYScaaIZaaabeaaaaa@3CF6@ la suite définie par V n = e n 1 x (lnx) 3 dx MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBa aaleaacaWGUbaabeaakiabg2da9maapedabaWaaSaaaeaacaaIXaaa baGaamiEamaakaaabaGaaiikaiGacYgacaGGUbGaamiEaiaacMcada ahaaWcbeqaaiaaiodaaaaabeaaaaaabaGaamyzaaqaaiaad6gaa0Ga ey4kIipakiaadsgacaWG4baaaa@45E8@ alors lim n+ V n = MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci GGSbGaaiyAaiaac2gaaSqaaiaad6gacqGHsgIRcqGHRaWkcqGHEisP aeqaaOGaamOvamaaBaaaleaacaWGUbaabeaakiabg2da9aaa@4147@

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    Question : 15

    soit g(x)= 1 tgx 1 arctgu du MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zaiaacI cacaWG4bGaaiykaiabg2da9maapedabaWaaSaaaeaacaaIXaaabaGa amyyaiaadkhacaWGJbGaamiDaiaadEgacaWG1baaaaWcbaGaaGymaa qaaiaadshacaWGNbGaamiEaaqdcqGHRiI8aOGaamizaiaadwhaaaa@4861@ , alors la tangente à la courbe de g en x= π 4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabg2 da9maalaaabaGaeqiWdahabaGaaGinaaaaaaa@3A85@ admet pour équation

    Texte de la question

    soit g(x)= 1 tgx 1 arctgu du MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zaiaacI cacaWG4bGaaiykaiabg2da9maapedabaWaaSaaaeaacaaIXaaabaGa amyyaiaadkhacaWGJbGaamiDaiaadEgacaWG1baaaaWcbaGaaGymaa qaaiaadshacaWGNbGaamiEaaqdcqGHRiI8aOGaamizaiaadwhaaaa@4861@ , alors la tangente à la courbe de g en x= π 4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabg2 da9maalaaabaGaeqiWdahabaGaaGinaaaaaaa@3A85@ admet pour équation

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    Question : 16

    0 π 4 dx cos 2 x+4 sin 2 x = MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qmaeaada WcaaqaaiaadsgacaWG4baabaGaci4yaiaac+gacaGGZbWaaWbaaSqa beaacaaIYaaaaOGaamiEaiabgUcaRiaaisdaciGGZbGaaiyAaiaac6 gadaahaaWcbeqaaiaaikdaaaGccaWG4baaaaWcbaGaaGimaaqaamaa laaabaGaeqiWdahabaGaaGinaaaaa0Gaey4kIipakiabg2da9aaa@4996@

    Texte de la question

    0 π 4 dx cos 2 x+4 sin 2 x = MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qmaeaada WcaaqaaiaadsgacaWG4baabaGaci4yaiaac+gacaGGZbWaaWbaaSqa beaacaaIYaaaaOGaamiEaiabgUcaRiaaisdaciGGZbGaaiyAaiaac6 gadaahaaWcbeqaaiaaikdaaaGccaWG4baaaaWcbaGaaGimaaqaamaa laaabaGaeqiWdahabaGaaGinaaaaa0Gaey4kIipakiabg2da9aaa@4996@

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    Question : 17

    lim n+ (n!) 2 (2n)! = MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci GGSbGaaiyAaiaac2gaaSqaaiaad6gacqGHsgIRcqGHRaWkcqGHEisP aeqaaOWaaSaaaeaacaGGOaGaamOBaiaacgcacaGGPaWaaWbaaSqabe aacaaIYaaaaaGcbaGaaiikaiaaikdacaWGUbGaaiykaiaacgcaaaGa eyypa0daaa@46E4@

    Texte de la question

    lim n+ (n!) 2 (2n)! = MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci GGSbGaaiyAaiaac2gaaSqaaiaad6gacqGHsgIRcqGHRaWkcqGHEisP aeqaaOWaaSaaaeaacaGGOaGaamOBaiaacgcacaGGPaWaaWbaaSqabe aacaaIYaaaaaGcbaGaaiikaiaaikdacaWGUbGaaiykaiaacgcaaaGa eyypa0daaa@46E4@

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    Question : 18

    soit B={ u,v,w } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOqaiabg2 da9maacmaabaGaamyDaiaacYcacaWG2bGaaiilaiaadEhaaiaawUha caGL9baaaaa@3E46@ une base de ( 3 ,+,) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiabl2 riHoaaCaaaleqabaGaaG4maaaakiaacYcacqGHRaWkcaGGSaGaeyOe I0Iaaiykaaaa@3CE3@ , on considère les familles suivantes E={ u+v,v+w,u+w } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyraiabg2 da9maacmaabaGaamyDaiabgUcaRiaadAhacaGGSaGaamODaiabgUca RiaadEhacaGGSaGaamyDaiabgUcaRiaadEhaaiaawUhacaGL9baaaa a@43E0@ N={ u,v,u+w } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtaiabg2 da9maacmaabaGaamyDaiaacYcacaWG2bGaaiilaiaadwhacqGHRaWk caWG3baacaGL7bGaayzFaaaaaa@402E@ S={ u,v+w,vu+w } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uaiabg2 da9maacmaabaGaeyOeI0IaamyDaiaacYcacaWG2bGaey4kaSIaam4D aiaacYcacaWG2bGaeyOeI0IaamyDaiabgUcaRiaadEhaaiaawUhaca GL9baaaaa@44E6@ A={ uvw,u+v+w,u } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaiabg2 da9maacmaabaGaamyDaiabgkHiTiaadAhacqGHsislcaWG3bGaaiil aiaadwhacqGHRaWkcaWG2bGaey4kaSIaam4DaiaacYcacaWG1baaca GL7bGaayzFaaaaaa@45CE@ Alors laquelle (ou lesquelles) des familles formé une base ?

    Texte de la question

    soit B={ u,v,w } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOqaiabg2 da9maacmaabaGaamyDaiaacYcacaWG2bGaaiilaiaadEhaaiaawUha caGL9baaaaa@3E46@ une base de ( 3 ,+,) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiabl2 riHoaaCaaaleqabaGaaG4maaaakiaacYcacqGHRaWkcaGGSaGaeyOe I0Iaaiykaaaa@3CE3@ , on considère les familles suivantes E={ u+v,v+w,u+w } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyraiabg2 da9maacmaabaGaamyDaiabgUcaRiaadAhacaGGSaGaamODaiabgUca RiaadEhacaGGSaGaamyDaiabgUcaRiaadEhaaiaawUhacaGL9baaaa a@43E0@ N={ u,v,u+w } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtaiabg2 da9maacmaabaGaamyDaiaacYcacaWG2bGaaiilaiaadwhacqGHRaWk caWG3baacaGL7bGaayzFaaaaaa@402E@ S={ u,v+w,vu+w } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uaiabg2 da9maacmaabaGaeyOeI0IaamyDaiaacYcacaWG2bGaey4kaSIaam4D aiaacYcacaWG2bGaeyOeI0IaamyDaiabgUcaRiaadEhaaiaawUhaca GL9baaaaa@44E6@ A={ uvw,u+v+w,u } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaiabg2 da9maacmaabaGaamyDaiabgkHiTiaadAhacqGHsislcaWG3bGaaiil aiaadwhacqGHRaWkcaWG2bGaey4kaSIaam4DaiaacYcacaWG1baaca GL7bGaayzFaaaaaa@45CE@ Alors laquelle (ou lesquelles) des familles formé une base ?

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    Question : 19

    Soit S={ ( x,y,z ) 3 /x+2y=0 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uaiabg2 da9maacmaabaWaaeWaaeaacaWG4bGaaiilaiaadMhacaGGSaGaamOE aaGaayjkaiaawMcaaiabgIGiolabl2riHoaaCaaaleqabaGaaG4maa aakiaac+cacaWG4bGaey4kaSIaaGOmaiaadMhacqGH9aqpcaaIWaaa caGL7bGaayzFaaaaaa@49DD@ Lequel des système suivante forme une base pour S ?

    Texte de la question

    Soit S={ ( x,y,z ) 3 /x+2y=0 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uaiabg2 da9maacmaabaWaaeWaaeaacaWG4bGaaiilaiaadMhacaGGSaGaamOE aaGaayjkaiaawMcaaiabgIGiolabl2riHoaaCaaaleqabaGaaG4maa aakiaac+cacaWG4bGaey4kaSIaaGOmaiaadMhacqGH9aqpcaaIWaaa caGL7bGaayzFaaaaaa@49DD@ Lequel des système suivante forme une base pour S ?

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    Question : 20

    On considère les ensembles suivants E={ ( x,y,z ) 3 /x+yz=0 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyraiabg2 da9maacmaabaWaaeWaaeaacaWG4bGaaiilaiaadMhacaGGSaGaamOE aaGaayjkaiaawMcaaiabgIGiolabl2riHoaaCaaaleqabaGaaG4maa aakiaac+cacaWG4bGaey4kaSIaamyEaiaadQhacqGH9aqpcaaIWaaa caGL7bGaayzFaaaaaa@4A12@ N={ ( x,y,z ) 3 /xyz=0 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtaiabg2 da9maacmaabaWaaeWaaeaacaWG4bGaaiilaiaadMhacaGGSaGaamOE aaGaayjkaiaawMcaaiabgIGiolabl2riHoaaCaaaleqabaGaaG4maa aakiaac+cacaWG4bGaamyEaiaadQhacqGH9aqpcaaIWaaacaGL7bGa ayzFaaaaaa@4939@ S={ ( x,y,z ) 3 /z=2 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uaiabg2 da9maacmaabaWaaeWaaeaacaWG4bGaaiilaiaadMhacaGGSaGaamOE aaGaayjkaiaawMcaaiabgIGiolabl2riHoaaCaaaleqabaGaaG4maa aakiaac+cacaWG6bGaeyypa0JaaGOmaaGaay5Eaiaaw2haaaaa@4745@ A={ ( x,y,z ) 3 /x+y=z } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaiabg2 da9maacmaabaWaaeWaaeaacaWG4bGaaiilaiaadMhacaGGSaGaamOE aaGaayjkaiaawMcaaiabgIGiolabl2riHoaaCaaaleqabaGaaG4maa aakiaac+cacaWG4bGaey4kaSIaamyEaiabg2da9iaadQhaaiaawUha caGL9baaaaa@4954@ Lesquels parmi ces ensembles sont des sous espaces vectoriels de ?

    Texte de la question

    On considère les ensembles suivants E={ ( x,y,z ) 3 /x+yz=0 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyraiabg2 da9maacmaabaWaaeWaaeaacaWG4bGaaiilaiaadMhacaGGSaGaamOE aaGaayjkaiaawMcaaiabgIGiolabl2riHoaaCaaaleqabaGaaG4maa aakiaac+cacaWG4bGaey4kaSIaamyEaiaadQhacqGH9aqpcaaIWaaa caGL7bGaayzFaaaaaa@4A12@ N={ ( x,y,z ) 3 /xyz=0 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtaiabg2 da9maacmaabaWaaeWaaeaacaWG4bGaaiilaiaadMhacaGGSaGaamOE aaGaayjkaiaawMcaaiabgIGiolabl2riHoaaCaaaleqabaGaaG4maa aakiaac+cacaWG4bGaamyEaiaadQhacqGH9aqpcaaIWaaacaGL7bGa ayzFaaaaaa@4939@ S={ ( x,y,z ) 3 /z=2 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uaiabg2 da9maacmaabaWaaeWaaeaacaWG4bGaaiilaiaadMhacaGGSaGaamOE aaGaayjkaiaawMcaaiabgIGiolabl2riHoaaCaaaleqabaGaaG4maa aakiaac+cacaWG6bGaeyypa0JaaGOmaaGaay5Eaiaaw2haaaaa@4745@ A={ ( x,y,z ) 3 /x+y=z } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaiabg2 da9maacmaabaWaaeWaaeaacaWG4bGaaiilaiaadMhacaGGSaGaamOE aaGaayjkaiaawMcaaiabgIGiolabl2riHoaaCaaaleqabaGaaG4maa aakiaac+cacaWG4bGaey4kaSIaamyEaiabg2da9iaadQhaaiaawUha caGL9baaaaa@4954@ Lesquels parmi ces ensembles sont des sous espaces vectoriels de ?

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    Question : 21

    soit A une matrice carrée d’orde n vérifiant A 2 =2 I n A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqamaaCa aaleqabaGaaGOmaaaakiabg2da9iaaikdacaWGjbWaaSbaaSqaaiaa d6gaaeqaaOGaeyOeI0Iaamyqaaaa@3D1C@ ( I n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaBa aaleaacaWGUbaabeaaaaa@37E4@ est la matrice identité) On considère les égalités suivantes (i) detA=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciizaiaacw gacaGG0bGaaiyqaiabg2da9iaaicdaaaa@3B47@ (ii) A 1 = 1 2 (A+ I n ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqamaaCa aaleqabaGaeyOeI0IaaGymaaaakiabg2da9maalaaabaGaaGymaaqa aiaaikdaaaGaaiikaiaacgeacqGHRaWkcaWGjbWaaSbaaSqaaiaad6 gaaeqaaOGaaiykaaaa@4020@ (iii) detA0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciizaiaacw gacaGG0bGaamyqaiabgcMi5kaaicdaaaa@3C09@ (iv) A 1 =2 I n +A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqamaaCa aaleqabaGaeyOeI0IaaGymaaaakiabg2da9iaaikdacaWGjbWaaSba aSqaaiaad6gaaeqaaOGaey4kaSIaamyqaaaa@3DFD@ (v) det(A+ I n )= 2 detA MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciizaiaacw gacaGG0bGaaiikaiaadgeacqGHRaWkcaWGjbWaaSbaaSqaaiaad6ga aeqaaOGaaiykaiabg2da9maalaaabaGaaGOmaaqaaiGacsgacaGGLb GaaiiDaiaadgeaaaaaaa@431D@ Alors

    Texte de la question

    soit A une matrice carrée d’orde n vérifiant A 2 =2 I n A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqamaaCa aaleqabaGaaGOmaaaakiabg2da9iaaikdacaWGjbWaaSbaaSqaaiaa d6gaaeqaaOGaeyOeI0Iaamyqaaaa@3D1C@ ( I n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaBa aaleaacaWGUbaabeaaaaa@37E4@ est la matrice identité) On considère les égalités suivantes (i) detA=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciizaiaacw gacaGG0bGaaiyqaiabg2da9iaaicdaaaa@3B47@ (ii) A 1 = 1 2 (A+ I n ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqamaaCa aaleqabaGaeyOeI0IaaGymaaaakiabg2da9maalaaabaGaaGymaaqa aiaaikdaaaGaaiikaiaacgeacqGHRaWkcaWGjbWaaSbaaSqaaiaad6 gaaeqaaOGaaiykaaaa@4020@ (iii) detA0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciizaiaacw gacaGG0bGaamyqaiabgcMi5kaaicdaaaa@3C09@ (iv) A 1 =2 I n +A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqamaaCa aaleqabaGaeyOeI0IaaGymaaaakiabg2da9iaaikdacaWGjbWaaSba aSqaaiaad6gaaeqaaOGaey4kaSIaamyqaaaa@3DFD@ (v) det(A+ I n )= 2 detA MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciizaiaacw gacaGG0bGaaiikaiaadgeacqGHRaWkcaWGjbWaaSbaaSqaaiaad6ga aeqaaOGaaiykaiabg2da9maalaaabaGaaGOmaaqaaiGacsgacaGGLb GaaiiDaiaadgeaaaaaaa@431D@ Alors

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    Question : 22

    12345 2 12343×12347 = MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaaca aIXaGaaGOmaiaaiodacaaI0aGaaGynamaaCaaaleqabaGaaGOmaaaa kiabgkHiTiaaigdacaaIYaGaaG4maiaaisdacaaIZaGaey41aqRaaG ymaiaaikdacaaIZaGaaGinaiaaiEdaaSqabaGccqGH9aqpaaa@462C@

    Texte de la question

    12345 2 12343×12347 = MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaaca aIXaGaaGOmaiaaiodacaaI0aGaaGynamaaCaaaleqabaGaaGOmaaaa kiabgkHiTiaaigdacaaIYaGaaG4maiaaisdacaaIZaGaey41aqRaaG ymaiaaikdacaaIZaGaaGinaiaaiEdaaSqabaGccqGH9aqpaaa@462C@

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    Question : 23

    lim n+ ( 2 )( 2 4 )( 2 8 )...( 2 2 n )= MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci GGSbGaaiyAaiaac2gaaSqaaiaad6gacqGHsgIRcqGHRaWkcqGHEisP aeqaaOGaaiikamaakaaabaGaaGOmaaWcbeaakiaacMcacaGGOaWaaO qaaeaacaaIYaaaleaacaaI0aaaaOGaaiykaiaacIcadaGcbaqaaiaa ikdaaSqaaiaaiIdaaaGccaGGPaGaaiOlaiaac6cacaGGUaGaaiikam aakeaabaGaaGOmaaWcbaGaaGOmamaaCaaameqabaGaamOBaaaaaaGc caGGPaGaeyypa0daaa@4D9E@

    Texte de la question

    lim n+ ( 2 )( 2 4 )( 2 8 )...( 2 2 n )= MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci GGSbGaaiyAaiaac2gaaSqaaiaad6gacqGHsgIRcqGHRaWkcqGHEisP aeqaaOGaaiikamaakaaabaGaaGOmaaWcbeaakiaacMcacaGGOaWaaO qaaeaacaaIYaaaleaacaaI0aaaaOGaaiykaiaacIcadaGcbaqaaiaa ikdaaSqaaiaaiIdaaaGccaGGPaGaaiOlaiaac6cacaGGUaGaaiikam aakeaabaGaaGOmaaWcbaGaaGOmamaaCaaameqabaGaamOBaaaaaaGc caGGPaGaeyypa0daaa@4D9E@

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    Question : 24

    Si 0 x h(t)dt=xarctgx MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qmaeaaca WGObGaaiikaiaadshacaGGPaGaamizaiaadshacqGH9aqpcaWG4bGa amyyaiaadkhacaWGJbGaamiDaiaadEgacaWG4baaleaacaaIWaaaba GaamiEaaqdcqGHRiI8aaaa@46A2@ alors h(1)= MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiAaiaacI cacaaIXaGaaiykaiabg2da9aaa@39FE@

    Texte de la question

    Si 0 x h(t)dt=xarctgx MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qmaeaaca WGObGaaiikaiaadshacaGGPaGaamizaiaadshacqGH9aqpcaWG4bGa amyyaiaadkhacaWGJbGaamiDaiaadEgacaWG4baaleaacaaIWaaaba GaamiEaaqdcqGHRiI8aaaa@46A2@ alors h(1)= MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiAaiaacI cacaaIXaGaaiykaiabg2da9aaa@39FE@

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    Question : 25

    dx t g 3 x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qaaeaada WcaaqaaiaadsgacaWG4baabaGaamiDaiaadEgadaahaaWcbeqaaiaa iodaaaGccaWG4baaaaWcbeqab0Gaey4kIipaaaa@3DBE@

    Texte de la question

    dx t g 3 x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qaaeaada WcaaqaaiaadsgacaWG4baabaGaamiDaiaadEgadaahaaWcbeqaaiaa iodaaaGccaWG4baaaaWcbeqab0Gaey4kIipaaaa@3DBE@

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