Quiz — Cours-MI-Thierry-Fack.pdf

Question 1 sur 10 20:00

[{"id":98700,"question":"Quelle est la limite de la fonction f(x) = (3x² + 2x - 1) \/ (x² - 4) lorsque x tend vers +∞ ?","option_a":"A. 0","option_b":"B. 3","option_c":"C. +∞","option_d":"D. -∞","option_e":"","option_f":"","bonne_reponse":"b","explication":"La limite d'un polynôme de même degré au numérateur et au dénominateur est égale au rapport des coefficients dominants, soit 3\/1 = 3.","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. 0\", \"b\": \"B. 3\", \"c\": \"C. +∞\", \"d\": \"D. -∞\"}}","_debug_options_count":4},{"id":98701,"question":"Une suite géométrique de raison q = 2 et de premier terme u₀ = 3 est-elle croissante ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Une suite géométrique de raison q \u003E 1 et de premier terme positif est croissante.","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":98702,"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. x \/ (x² + 1)","option_d":"D. 2 \/ (x² + 1)","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de ln(u) est u'\/u, avec u = x² + 1, donc u' = 2x. Ainsi, f'(x) = 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. x \/ (x² + ","_debug_options_count":4},{"id":98703,"question":"La fonction f(x) = x³ - 3x² + 2 est-elle convexe 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 seconde f''(x) = 6x - 6 est négative sur [0, 1[ et positive sur ]1, 2], donc la fonction n'est pas convexe sur [0, 2].","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":98704,"question":"Quelle est la solution de l'équation différentielle y' + 2y = 0 ?","option_a":"A. y = Ce^(2x)","option_b":"B. y = Ce^(-2x)","option_c":"C. y = C + 2x","option_d":"D. y = Cx","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'équation différentielle linéaire du premier ordre y' + ay = 0 a pour solution y = Ce^(-ax), ici a = 2.","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. y = Ce^(2x)\", \"b\": \"B. y = Ce^(-2x)\", \"c\": \"C. y = C + 2x\", \"d","_debug_options_count":4},{"id":98705,"question":"Soit X une variable aléatoire suivant une loi normale N(0, 1). Quelle est la probabilité P(-1 ≤ X ≤ 1) ?","option_a":"A. 0,3413","option_b":"B. 0,6826","option_c":"C. 0,9544","option_d":"D. 0,9974","option_e":"","option_f":"","bonne_reponse":"b","explication":"Pour une loi normale centrée réduite, environ 68,26% des valeurs sont dans l'intervalle [-1, 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. 0,3413\", \"b\": \"B. 0,6826\", \"c\": \"C. 0,9544\", \"d\": \"D. 0,9974\"}","_debug_options_count":4},{"id":98706,"question":"La fonction f(x) = e^(1\/x) est-elle définie en x = 0 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction e^(1\/x) n'est pas définie en x = 0 car 1\/x tend vers +∞ ou -∞ selon la direction.","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":98707,"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, cela donne (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":98708,"question":"Une suite arithmétique de raison r = -3 et de premier terme u₀ = 5 est-elle décroissante ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Une suite arithmétique de raison négative est décroissante.","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":98709,"question":"Quelle est la valeur de l'intégrale impropre ∫(de 1 à +∞) 1\/x² dx ?","option_a":"A. 0","option_b":"B. 1","option_c":"C. +∞","option_d":"D. -1","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'intégrale ∫(1\/x²) dx = -1\/x, donc ∫(de 1 à +∞) 1\/x² dx = lim(x→+∞) (-1\/x + 1) = 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. 0\", \"b\": \"B. 1\", \"c\": \"C. +∞\", \"d\": \"D. -1\"}}","_debug_options_count":4}]