Quiz — 1565780365_Exercice 4.pdf

Question 1 sur 10 20:00

[{"id":30150,"question":"Soit f une fonction dérivable sur ℝ telle que f'(x) = 2x + 1. Quelle est la primitive F de f vérifiant F(0) = 3 ?","option_a":"F(x) = x² + x + 3","option_b":"F(x) = x² + x + 2","option_c":"F(x) = x² + x + 1","option_d":"F(x) = x² + x","option_e":"","option_f":"","bonne_reponse":"a","explication":"La primitive de f'(x) = 2x + 1 est F(x) = x² + x + C. Avec F(0) = 3, on trouve C = 3. Ainsi, F(x) = x² + x + 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\": \"F(x) = x² + x + 3\", \"b\": \"F(x) = x² + x + 2\", \"c\": \"F(x) = x² ","_debug_options_count":4},{"id":30151,"question":"La fonction exponentielle est toujours positive sur ℝ.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction exponentielle e^x est définie pour tout x réel et vérifie e^x \u003E 0 pour tout x ∈ ℝ.","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":30152,"question":"Quelle est la limite de la suite (u_n) définie par u_n = (3n² + 2n - 1)\/(n² + 5) lorsque n tend vers +∞ ?","option_a":"3","option_b":"1","option_c":"0","option_d":"∞","option_e":"","option_f":"","bonne_reponse":"a","explication":"En divisant le numérateur et le dénominateur par n², on obtient u_n = (3 + 2\/n - 1\/n²)\/(1 + 5\/n²). Lorsque n → +∞, u_n → 3\/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\": \"3\", \"b\": \"1\", \"c\": \"0\", \"d\": \"∞\"}}","_debug_options_count":4},{"id":30153,"question":"Dans un triangle ABC, si AB² + AC² = BC², alors le triangle est rectangle en A.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"D'après le théorème de Pythagore, si AB² + AC² = BC², alors l'angle en A est droit, donc le triangle ABC est rectangle en 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":30154,"question":"Quelle est la dérivée de la fonction f(x) = ln(2x + 1) ?","option_a":"f'(x) = 2\/(2x + 1)","option_b":"f'(x) = 1\/(2x + 1)","option_c":"f'(x) = 2x\/(2x + 1)","option_d":"f'(x) = 1\/x","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de ln(u) est u'\/u. Ici, u = 2x + 1, donc u' = 2. Ainsi, f'(x) = 2\/(2x + 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\": \"f'(x) = 2\/(2x + 1)\", \"b\": \"f'(x) = 1\/(2x + 1)\", \"c\": \"f'(x) = 2x\/","_debug_options_count":4},{"id":30155,"question":"Un dé équilibré à 6 faces est lancé deux fois. Quelle est la probabilité d'obtenir deux fois le même nombre ?","option_a":"1\/6","option_b":"1\/3","option_c":"1\/36","option_d":"1\/2","option_e":"","option_f":"","bonne_reponse":"a","explication":"Il y a 6 cas favorables (1-1, 2-2, ..., 6-6) sur 36 résultats possibles, soit 6\/36 = 1\/6.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"1\/6\", \"b\": \"1\/3\", \"c\": \"1\/36\", \"d\": \"1\/2\"}}","_debug_options_count":4},{"id":30156,"question":"La fonction f(x) = x³ - 3x + 2 admet un minimum local en x = 1.","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 et x = -1. La dérivée seconde f''(x) = 6x vaut 6 en x = 1 (f''(1) \u003E 0), donc x = 1 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":30157,"question":"Quelle est la solution de l'équation différentielle y' + 2y = 0 avec y(0) = 5 ?","option_a":"y = 5e^(-2x)","option_b":"y = 5e^(2x)","option_c":"y = -2e^(-2x)","option_d":"y = 2e^(-x)","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'équation différentielle y' + 2y = 0 a pour solution générale y = Ce^(-2x). Avec y(0) = 5, on trouve C = 5. Ainsi, y = 5e^(-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\": \"y = 5e^(-2x)\", \"b\": \"y = 5e^(2x)\", \"c\": \"y = -2e^(-2x)\", \"d\": \"y ","_debug_options_count":4},{"id":30158,"question":"Dans une classe de 30 élèves, 12 sont des filles. Quelle est la probabilité qu'un élève choisi au hasard soit une fille ?","option_a":"0,4","option_b":"0,3","option_c":"0,6","option_d":"0,25","option_e":"","option_f":"","bonne_reponse":"a","explication":"La probabilité est le rapport du nombre de filles au nombre total d'élèves, soit 12\/30 = 0,4.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"0,4\", \"b\": \"0,3\", \"c\": \"0,6\", \"d\": \"0,25\"}}","_debug_options_count":4},{"id":30159,"question":"La fonction f(x) = x² - 4x + 3 est croissante sur l'intervalle ]2 ; +∞[.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée f'(x) = 2x - 4 est positive pour x \u003E 2, donc la fonction est croissante sur ]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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4}]