Quiz — __Module 3 V1.ppt

Question 1 sur 10 20:00

[{"id":95109,"question":"Quelle est la dérivée de la fonction f(x) = x² + 3x - 5 ?","option_a":"A. 2x + 3","option_b":"B. x² + 3","option_c":"C. 2x - 5","option_d":"D. 3x + 2","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'une fonction polynôme se calcule terme par terme : (x²)' = 2x, (3x)' = 3, et (-5)' = 0. Donc f'(x) = 2x + 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. 2x + 3\", \"b\": \"B. x² + 3\", \"c\": \"C. 2x - 5\", \"d\": \"D. 3x + 2\"","_debug_options_count":4},{"id":95110,"question":"La limite de la fonction f(x) = (x² - 1)\/(x - 1) quand x tend vers 1 est égale à 2.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"En simplifiant l'expression, f(x) = (x - 1)(x + 1)\/(x - 1) = x + 1 pour x ≠ 1. Donc lim(x→1) f(x) = 2. 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":95111,"question":"Quelle propriété caractérise une suite géométrique de raison q ≠ 1 ?","option_a":"A. u_{n+1} = u_n + q","option_b":"B. u_{n+1} = q * u_n","option_c":"C. u_{n+1} = u_n - q","option_d":"D. u_{n+1} = u_n \/ q","option_e":"","option_f":"","bonne_reponse":"b","explication":"Par définition, une suite géométrique vérifie u_{n+1} = q * u_n, où q est la raison.","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. u_{n+1} = u_n + q\", \"b\": \"B. u_{n+1} = q * u_n\", \"c\": \"C. u_{n","_debug_options_count":4},{"id":95112,"question":"Si deux événements A et B sont indépendants, alors P(A ∩ B) = P(A) * P(B).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est la définition de l'indépendance de deux événements en probabilités.","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":95113,"question":"Quelle est la limite de la suite u_n = (3n + 1)\/(2n - 5) quand n tend vers l'infini ?","option_a":"A. 0","option_b":"B. 1","option_c":"C. 3\/2","option_d":"D. +∞","option_e":"","option_f":"","bonne_reponse":"c","explication":"En divisant numérateur et dénominateur par n, on obtient u_n = (3 + 1\/n)\/(2 - 5\/n). Quand n→∞, 1\/n et 5\/n tendent vers 0, donc u_n → 3\/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. 3\/2\", \"d\": \"D. +∞\"}}","_debug_options_count":4},{"id":95114,"question":"Le théorème des gendarmes s'applique-t-il pour déterminer la limite d'une suite ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Oui, le théorème des gendarmes permet de déterminer la limite d'une suite encadrée par deux suites convergentes de même limite.","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":95115,"question":"Quelle est la dérivée de la fonction f(x) = e^{2x} ?","option_a":"A. e^{2x}","option_b":"B. 2e^{2x}","option_c":"C. e^{x}","option_d":"D. 2x e^{2x}","option_e":"","option_f":"","bonne_reponse":"b","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\": \"b\", \"options\": {\"a\": \"A. e^{2x}\", \"b\": \"B. 2e^{2x}\", \"c\": \"C. e^{x}\", \"d\": \"D. 2x e^{2x","_debug_options_count":4},{"id":95116,"question":"Si une suite est croissante et majorée, alors elle converge.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est le théorème de la limite monotone : une suite croissante et majorée converge vers sa borne supérieure.","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":95117,"question":"Quelle est la probabilité d'obtenir exactement 3 fois pile en 5 lancers d'une pièce équilibrée ?","option_a":"A. 5\/16","option_b":"B. 10\/32","option_c":"C. 5\/32","option_d":"D. 1\/8","option_e":"","option_f":"","bonne_reponse":"b","explication":"C'est une loi binomiale de paramètres n=5 et p=0.5. P(X=3) = C(5,3) * (0.5)^3 * (0.5)^2 = 10\/32.","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. 5\/16\", \"b\": \"B. 10\/32\", \"c\": \"C. 5\/32\", \"d\": \"D. 1\/8\"}}","_debug_options_count":4},{"id":95118,"question":"La fonction f(x) = x^3 - 3x est strictement croissante sur ℝ.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée f'(x) = 3x² - 3 s'annule en x = ±1. La fonction n'est donc pas strictement croissante sur tout ℝ (elle décroît sur ]-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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4}]