Quiz — Série_TD_2 (1).pdf

Question 1 sur 10 20:00

[{"id":4676,"question":"Quelle est la limite de la fonction f(x) = (3x² - 2x + 1)\/(x² + 1) lorsque x tend vers +∞ ?","option_a":"A. 0","option_b":"B. 1","option_c":"C. 3","option_d":"D. +∞","option_e":"","option_f":"","bonne_reponse":"c","explication":"La limite d’un quotient de polynômes de même degré est égale au rapport des coefficients dominants. Ici, 3x²\/x² = 3.","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. 3\", \"d\": \"D. +∞\"}}","_debug_options_count":4},{"id":4677,"question":"La fonction f(x) = x³ - 3x² + 2 est décroissante sur l’intervalle [0, 2].","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée f’(x) = 3x² - 6x s’annule en x=0 et x=2. Sur ]0,2[, f’(x) \u003C 0, donc la fonction est décroissante sur [0,2]. L’affirmation est donc VRAIE.","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":4678,"question":"Quel est le produit scalaire de deux vecteurs u(2, -1) et v(3, 4) ?","option_a":"A. 2","option_b":"B. 5","option_c":"C. 10","option_d":"D. -1","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le produit scalaire se calcule par u·v = (2×3) + (-1×4) = 6 - 4 = 2. La bonne réponse est donc 2 (option A).","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. 2\", \"b\": \"B. 5\", \"c\": \"C. 10\", \"d\": \"D. -1\"}}","_debug_options_count":4},{"id":4679,"question":"L’intégrale ∫(de 0 à 1) (2x + 1) dx est égale à :","option_a":"A. 1","option_b":"B. 2","option_c":"C. 3","option_d":"D. 4","option_e":"","option_f":"","bonne_reponse":"c","explication":"L’intégrale se calcule comme [x² + x] de 0 à 1 = (1 + 1) - (0 + 0) = 2. La bonne réponse est donc 2 (option B).","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\", \"b\": \"B. 2\", \"c\": \"C. 3\", \"d\": \"D. 4\"}}","_debug_options_count":4},{"id":4680,"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":"a","explication":"La suite uₙ = n + 1\/n tend vers +∞ lorsque n tend vers +∞, donc elle n’est pas bornée. L’affirmation est donc FAUSSE.","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":4681,"question":"Quelle est la dérivée de la fonction f(x) = ln(2x + 1) ?","option_a":"A. 1\/(2x + 1)","option_b":"B. 2\/(2x + 1)","option_c":"C. 1\/x","option_d":"D. 2x\/(2x + 1)","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée de ln(u) est u’\/u. Ici, u = 2x + 1, donc u’ = 2. La dérivée est donc 2\/(2x + 1).","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. 1\/(2x + 1)\", \"b\": \"B. 2\/(2x + 1)\", \"c\": \"C. 1\/x\", \"d\": \"D. 2x\/","_debug_options_count":4},{"id":4682,"question":"L’équation x² - 4x + 3 = 0 admet deux solutions réelles distinctes.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le discriminant Δ = 16 - 12 = 4 \u003E 0, donc l’équation admet deux solutions réelles distinctes. L’affirmation est donc VRAIE.","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":4683,"question":"Quelle est la valeur de l’intégrale ∫(de -1 à 1) x³ dx ?","option_a":"A. 0","option_b":"B. 1","option_c":"C. 2","option_d":"D. -1","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction x³ est impaire, donc son intégrale sur un intervalle symétrique autour de 0 est nulle. La bonne réponse est 0 (option A).","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\", \"b\": \"B. 1\", \"c\": \"C. 2\", \"d\": \"D. -1\"}}","_debug_options_count":4},{"id":4684,"question":"La fonction f(x) = e^(2x) a pour dérivée f’(x) = 2e^(2x).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de e^(u(x)) est u’(x)e^(u(x)). Ici, u(x) = 2x, donc u’(x) = 2. La dérivée est donc 2e^(2x). L’affirmation est VRAIE.","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":4685,"question":"Quel est le résultat de l’équation 3^(x+1) = 27 ?","option_a":"A. x = 1","option_b":"B. x = 2","option_c":"C. x = 3","option_d":"D. x = 0","option_e":"","option_f":"","bonne_reponse":"b","explication":"27 = 3³, donc 3^(x+1) = 3³ implique x + 1 = 3, soit x = 2. La bonne réponse est x = 2 (option B).","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. x = 1\", \"b\": \"B. x = 2\", \"c\": \"C. x = 3\", \"d\": \"D. x = 0\"}}","_debug_options_count":4}]