Quiz — ds4mpsi1213pro1.pdf

Question 1 sur 10 20:00

[{"id":94093,"question":"Quelle est la limite de la suite définie par \u003Cstrong\u003Eu\u003Csub\u003En\u003C\/sub\u003E = (n² + 3n - 1)\/(2n² - 5)\u003C\/strong\u003E quand \u003Cem\u003En\u003C\/em\u003E tend vers l'infini ?","option_a":"A. 1\/2","option_b":"B. 1","option_c":"C. +∞","option_d":"D. 0","option_e":"","option_f":"","bonne_reponse":"a","explication":"La limite d'une suite rationnelle est égale au rapport des coefficients dominants des termes de plus haut degré. Ici, (n²)\/(2n²) = 1\/2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"A. 1\/2\", \"b\": \"B. 1\", \"c\": \"C. +∞\", \"d\": \"D. 0\"}}","_debug_options_count":4},{"id":94094,"question":"Soit \u003Cstrong\u003Ef(x) = ln(x² + 1)\u003C\/strong\u003E. La dérivée \u003Cstrong\u003Ef'(x)\u003C\/strong\u003E est égale à :","option_a":"A. 2x\/(x² + 1)","option_b":"B. 2x","option_c":"C. 1\/(x² + 1)","option_d":"D. 2\/(x² + 1)","option_e":"","option_f":"","bonne_reponse":"a","explication":"En utilisant la règle de la chaîne, f'(x) = (1\/(x² + 1)) * (2x) = 2x\/(x² + 1).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"A. 2x\/(x² + 1)\", \"b\": \"B. 2x\", \"c\": \"C. 1\/(x² + 1)\", \"d\": \"D. 2","_debug_options_count":4},{"id":94095,"question":"Vrai ou Faux ? \u003Cstrong\u003EToute fonction continue sur un intervalle fermé est dérivable sur cet intervalle.\u003C\/strong\u003E","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Une fonction continue n'est pas nécessairement dérivable. Exemple : f(x) = |x| est continue sur [-1,1] mais non dérivable en x=0.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":94096,"question":"Quelle est la solution générale de l'équation différentielle \u003Cstrong\u003Ey'' + 4y' + 4y = 0\u003C\/strong\u003E ?","option_a":"A. y = e\u003Csup\u003E-2x\u003C\/sup\u003E(C\u003Csub\u003E1\u003C\/sub\u003E + C\u003Csub\u003E2\u003C\/sub\u003Ex)","option_b":"B. y = C\u003Csub\u003E1\u003C\/sub\u003Ee\u003Csup\u003E-2x\u003C\/sup\u003E + C\u003Csub\u003E2\u003C\/sub\u003Ee\u003Csup\u003E2x\u003C\/sup\u003E","option_c":"C. y = C\u003Csub\u003E1\u003C\/sub\u003Ecos(2x) + C\u003Csub\u003E2\u003C\/sub\u003Esin(2x)","option_d":"D. y = C\u003Csub\u003E1\u003C\/sub\u003Ee\u003Csup\u003E-x\u003C\/sup\u003E + C\u003Csub\u003E2\u003C\/sub\u003Ee\u003Csup\u003E-3x\u003C\/sup\u003E","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'équation caractéristique est r² + 4r + 4 = 0, qui a une racine double r = -2. La solution générale est donc y = e\u003Csup\u003E-2x\u003C\/sup\u003E(C\u003Csub\u003E1\u003C\/sub\u003E + C\u003Csub\u003E2\u003C\/sub\u003Ex).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"A. y = e\u003Csup\u003E-2x\u003C\/sup\u003E(C\u003Csub\u003E1\u003C\/sub\u003E + C\u003Csub\u003E2\u003C\/sub\u003Ex)\", \"b\": \"B.","_debug_options_count":4},{"id":94097,"question":"Soit \u003Cstrong\u003EA\u003C\/strong\u003E une matrice carrée de taille 3x3. Si \u003Cstrong\u003Edet(A) = 0\u003C\/strong\u003E, alors :","option_a":"A. A est inversible","option_b":"B. A a au moins une valeur propre nulle","option_c":"C. A est symétrique","option_d":"D. A est diagonalisable","option_e":"","option_f":"","bonne_reponse":"b","explication":"Si le déterminant est nul, la matrice n'est pas inversible et admet au moins une valeur propre nulle (car le produit des valeurs propres est égal au déterminant).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"A. A est inversible\", \"b\": \"B. A a au moins une valeur propre nul","_debug_options_count":4},{"id":94098,"question":"Vrai ou Faux ? \u003Cstrong\u003EUne matrice symétrique réelle est toujours diagonalisable.\u003C\/strong\u003E","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est un théorème fondamental : toute matrice symétrique réelle est diagonalisable dans une base orthonormée.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":94099,"question":"Quelle est la valeur de l'intégrale \u003Cstrong\u003E∫\u003Csub\u003E0\u003C\/sub\u003E\u003Csup\u003Eπ\u003C\/sup\u003E sin(x) dx\u003C\/strong\u003E ?","option_a":"A. 0","option_b":"B. 1","option_c":"C. 2","option_d":"D. π","option_e":"","option_f":"","bonne_reponse":"c","explication":"L'intégrale de sin(x) entre 0 et π est [-cos(x)]\u003Csub\u003E0\u003C\/sub\u003E\u003Csup\u003Eπ\u003C\/sup\u003E = -cos(π) + cos(0) = -(-1) + 1 = 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"A. 0\", \"b\": \"B. 1\", \"c\": \"C. 2\", \"d\": \"D. π\"}}","_debug_options_count":4},{"id":94100,"question":"Soit \u003Cstrong\u003EE\u003C\/strong\u003E un espace vectoriel de dimension 3. Quelle est la dimension maximale d'un sous-espace vectoriel de \u003Cstrong\u003EE\u003C\/strong\u003E ?","option_a":"A. 0","option_b":"B. 1","option_c":"C. 2","option_d":"D. 3","option_e":"","option_f":"","bonne_reponse":"d","explication":"La dimension maximale d'un sous-espace vectoriel d'un espace de dimension n est n. Ici, E est de dimension 3, donc la dimension maximale est 3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"d\", \"options\": {\"a\": \"A. 0\", \"b\": \"B. 1\", \"c\": \"C. 2\", \"d\": \"D. 3\"}}","_debug_options_count":4},{"id":94101,"question":"Vrai ou Faux ? \u003Cstrong\u003ELa série ∑\u003Csub\u003En=1\u003C\/sub\u003E\u003Csup\u003E∞\u003C\/sup\u003E 1\/n² converge.\u003C\/strong\u003E","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est la série de Riemann avec α = 2 \u003E 1, donc elle converge (vers π²\/6).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":94102,"question":"Quelle est la probabilité d'obtenir un double 6 en lançant deux dés équilibrés ?","option_a":"A. 1\/6","option_b":"B. 1\/12","option_c":"C. 1\/36","option_d":"D. 1\/18","option_e":"","option_f":"","bonne_reponse":"c","explication":"Il y a 36 issues possibles (6 faces × 6 faces), et une seule issue favorable (6,6). La probabilité est donc 1\/36.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"A. 1\/6\", \"b\": \"B. 1\/12\", \"c\": \"C. 1\/36\", \"d\": \"D. 1\/18\"}}","_debug_options_count":4}]