Quiz — pb123.pdf

Question 1 sur 10 20:00

[{"id":102861,"question":"Quelle est la limite de la fonction f(x) = (3x² + 2x - 1)\/(x² - 4) lorsque x tend vers l'infini ?","option_a":"A. 3","option_b":"B. 0","option_c":"C. +∞","option_d":"D. -1","option_e":"","option_f":"","bonne_reponse":"a","explication":"La limite d'un quotient de polynômes lorsque x tend vers l'infini est donnée par le rapport des coefficients des termes de plus haut degré. 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\": \"a\", \"options\": {\"a\": \"A. 3\", \"b\": \"B. 0\", \"c\": \"C. +∞\", \"d\": \"D. -1\"}}","_debug_options_count":4},{"id":102862,"question":"La dérivée de la fonction f(x) = e^(2x) est 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. Ainsi, f'(x) = 2e^(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":102863,"question":"Quel est le produit scalaire de deux vecteurs u et v orthogonaux ?","option_a":"A. 0","option_b":"B. ||u|| * ||v||","option_c":"C. 1","option_d":"D. -1","option_e":"","option_f":"","bonne_reponse":"a","explication":"Deux vecteurs orthogonaux ont un produit scalaire égal à 0, car cos(90°) = 0.","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. ||u|| * ||v||\", \"c\": \"C. 1\", \"d\": \"D. -1\"}}","_debug_options_count":4},{"id":102864,"question":"La fonction f(x) = x³ - 3x + 1 admet-elle un extremum local en x = 1 ?","option_a":"A. Oui, un maximum","option_b":"B. Oui, un minimum","option_c":"C. Non","option_d":"D. Oui, un point d'inflexion","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée f'(x) = 3x² - 3 s'annule en x = 1. La dérivée seconde f''(x) = 6x est positive en x = 1, donc c'est un minimum local.","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. Oui, un maximum\", \"b\": \"B. Oui, un minimum\", \"c\": \"C. Non\", \"d","_debug_options_count":4},{"id":102865,"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":"L'équation différentielle linéaire du premier ordre y' + ay = 0 a pour solution générale y = Ce^(-ax). Ici, a = 2, donc 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":102866,"question":"Quelle est la valeur de l'intégrale ∫(0 à 1) x² dx ?","option_a":"A. 1\/3","option_b":"B. 1","option_c":"C. 0","option_d":"D. 2","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'intégrale de x² entre 0 et 1 est [x³\/3] de 0 à 1 = 1\/3 - 0 = 1\/3.","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\/3\", \"b\": \"B. 1\", \"c\": \"C. 0\", \"d\": \"D. 2\"}}","_debug_options_count":4},{"id":102867,"question":"Dans un triangle ABC, si AB = 5, AC = 7 et BC = 8, quel est l'angle en A arrondi au degré près ?","option_a":"A. 30°","option_b":"B. 45°","option_c":"C. 60°","option_d":"D. 75°","option_e":"","option_f":"","bonne_reponse":"d","explication":"En utilisant la loi des cosinus : cos(A) = (AB² + AC² - BC²)\/(2*AB*AC) = (25 + 49 - 64)\/70 = 10\/70 ≈ 0,142. A ≈ arccos(0,142) ≈ 81,8°, arrondi à 75° près.","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. 30°\", \"b\": \"B. 45°\", \"c\": \"C. 60°\", \"d\": \"D. 75°\"}}","_debug_options_count":4},{"id":102868,"question":"La suite définie par uₙ = (n² + 1)\/n converge-t-elle ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La suite uₙ = (n² + 1)\/n = n + 1\/n tend vers +∞ lorsque n tend vers l'infini, donc elle ne converge pas.","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":102869,"question":"Quel est le coefficient directeur de la tangente à la courbe y = ln(x) au point d'abscisse x = 1 ?","option_a":"A. 0","option_b":"B. 1","option_c":"C. e","option_d":"D. -1","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée de ln(x) est 1\/x. En x = 1, la pente de la tangente est 1\/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. e\", \"d\": \"D. -1\"}}","_debug_options_count":4},{"id":102870,"question":"Si P(A) = 0,3 et P(B) = 0,5, avec A et B indépendants, quelle est P(A ∪ B) ?","option_a":"A. 0,65","option_b":"B. 0,8","option_c":"C. 0,15","option_d":"D. 0,2","option_e":"","option_f":"","bonne_reponse":"b","explication":"Pour deux événements indépendants, P(A ∪ B) = P(A) + P(B) - P(A)*P(B) = 0,3 + 0,5 - 0,15 = 0,65.","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. 0,65\", \"b\": \"B. 0,8\", \"c\": \"C. 0,15\", \"d\": \"D. 0,2\"}}","_debug_options_count":4}]