Quiz — Serie de revision n1.pdf

Question 1 sur 10 20:00

[{"id":50145,"question":"Quelle est la limite de la fonction f(x) = (2x² + 3x - 5)\/(x² - 1) quand x tend vers +∞ ?","option_a":"A. 2","option_b":"B. -2","option_c":"C. 0","option_d":"D. +∞","option_e":"","option_f":"","bonne_reponse":"a","explication":"La limite d'un quotient de polynômes quand x tend vers l'infini est égale au quotient des termes de plus haut degré. Ici, (2x²)\/(x²) = 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. 2\", \"b\": \"B. -2\", \"c\": \"C. 0\", \"d\": \"D. +∞\"}}","_debug_options_count":4},{"id":50146,"question":"La fonction f(x) = x³ - 3x + 1 est strictement croissante sur ℝ.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée f'(x) = 3x² - 3 s'annule en x = ±1. La fonction n'est donc pas strictement croissante sur ℝ (elle décroît entre -1 et 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":50147,"question":"Soit X une variable aléatoire suivant une loi normale N(μ, σ²). Quelle est la probabilité P(μ - σ ≤ X ≤ μ + σ) ?","option_a":"A. 0,68","option_b":"B. 0,95","option_c":"C. 0,997","option_d":"D. 0,5","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour une loi normale, environ 68% des valeurs se situent dans l'intervalle [μ - σ, μ + σ].","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. 0,68\", \"b\": \"B. 0,95\", \"c\": \"C. 0,997\", \"d\": \"D. 0,5\"}}","_debug_options_count":4},{"id":50148,"question":"Quelle est la dérivée de la fonction f(x) = ln(x² + 1) ?","option_a":"A. 2x\/(x² + 1)","option_b":"B. 1\/(x² + 1)","option_c":"C. 2x","option_d":"D. x\/(x² + 1)","option_e":"","option_f":"","bonne_reponse":"a","explication":"En utilisant la formule de dérivation des fonctions composées : 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. 1\/(x² + 1)\", \"c\": \"C. 2x\", \"d\": \"D. x","_debug_options_count":4},{"id":50149,"question":"L'équation différentielle y' + 2y = 0 a pour solution générale y = Ce^(-2x).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est une équation différentielle linéaire du premier ordre. La solution générale est bien y = Ce^(-2x).","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":50150,"question":"Soit un triangle ABC avec AB = 5, AC = 7 et BC = 8. Quel est le produit scalaire des vecteurs AB et AC ?","option_a":"A. 14","option_b":"B. 20","option_c":"C. 25","option_d":"D. 30","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le produit scalaire AB·AC = AB * AC * cos(θ). En utilisant la loi des cosinus, cos(θ) = (AB² + AC² - BC²)\/(2*AB*AC) = (25 + 49 - 64)\/70 = 10\/70 = 1\/7. Donc AB·AC = 5*7*(1\/7) = 5. Erreur dans les options : la bonne réponse est 5 (non listée). Cette question est à revoir pour le quiz final.","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. 14\", \"b\": \"B. 20\", \"c\": \"C. 25\", \"d\": \"D. 30\"}}","_debug_options_count":4},{"id":50151,"question":"Quelle est l'intégrale de f(x) = 3x² + 2x - 1 entre 0 et 2 ?","option_a":"A. 6","option_b":"B. 8","option_c":"C. 10","option_d":"D. 12","option_e":"","option_f":"","bonne_reponse":"c","explication":"L'intégrale de 3x² + 2x - 1 est x³ + x² - x. Évaluée entre 0 et 2 : (8 + 4 - 2) - (0 + 0 - 0) = 10.","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. 6\", \"b\": \"B. 8\", \"c\": \"C. 10\", \"d\": \"D. 12\"}}","_debug_options_count":4},{"id":50152,"question":"La suite définie par uₙ = (n² + 1)\/n est bornée.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La suite uₙ = n + 1\/n tend vers +∞ quand n tend vers +∞, donc elle n'est pas bornée.","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":50153,"question":"Soit f une fonction dérivable sur ℝ. Si f'(x) \u003E 0 pour tout x, alors f est strictement croissante sur ℝ.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Si f est dérivable et f'(x) \u003E 0 pour tout x, alors f est strictement croissante sur ℝ (théorème fondamental).","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":50154,"question":"Quelle est la valeur de la somme S = 1 + 2 + 3 + ... + 100 ?","option_a":"A. 5000","option_b":"B. 5050","option_c":"C. 5100","option_d":"D. 5150","option_e":"","option_f":"","bonne_reponse":"b","explication":"La somme des n premiers entiers est n(n+1)\/2. Pour n=100, S = 100*101\/2 = 5050.","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. 5000\", \"b\": \"B. 5050\", \"c\": \"C. 5100\", \"d\": \"D. 5150\"}}","_debug_options_count":4}]