Quiz — Tous_les_exercices ANALYSE.pdf

Question 1 sur 10 20:00

[{"id":6719,"question":"Quelle est la dérivée de la fonction f(x) = x³ + 2x² - 5x + 1 ?","option_a":"3x² + 4x - 5","option_b":"3x² + 4x - 5x","option_c":"x² + 2x - 5","option_d":"3x² + 4x - 1","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de x³ est 3x², celle de 2x² est 4x, celle de -5x est -5, et la constante 1 a une dérivée nulle.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"3x² + 4x - 5\", \"b\": \"3x² + 4x - 5x\", \"c\": \"x² + 2x - 5\", \"d\": ","_debug_options_count":4},{"id":6720,"question":"L'intégrale de 0 à 1 de la fonction f(x) = x² est égale à 1.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'intégrale de x² de 0 à 1 est [x³\/3] de 0 à 1 = 1\/3 - 0 = 1\/3 ≠ 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},{"id":6721,"question":"Quelle est la limite de la suite uₙ = (2n + 1)\/(n + 3) quand n tend vers l'infini ?","option_a":"0","option_b":"1","option_c":"2","option_d":"3","option_e":"","option_f":"","bonne_reponse":"c","explication":"En divisant numérateur et dénominateur par n, on obtient (2 + 1\/n)\/(1 + 3\/n) → 2\/1 = 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\": \"0\", \"b\": \"1\", \"c\": \"2\", \"d\": \"3\"}}","_debug_options_count":4},{"id":6722,"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":"La solution générale est bien y = Ce^(-2x), car la dérivée de Ce^(-2x) est -2Ce^(-2x), ce qui vérifie l'équation.","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":6723,"question":"Quelle est la primitive de la fonction f(x) = 3x² - 4x + 1 ?","option_a":"x³ - 2x² + x + C","option_b":"x³ - 2x² + C","option_c":"3x³ - 4x² + x + C","option_d":"x³ - 4x² + x + C","option_e":"","option_f":"","bonne_reponse":"a","explication":"La primitive de 3x² est x³, celle de -4x est -2x², celle de 1 est x, et C est la constante d'intégration.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"x³ - 2x² + x + C\", \"b\": \"x³ - 2x² + C\", \"c\": \"3x³ - 4x² + x","_debug_options_count":4},{"id":6724,"question":"La fonction f(x) = e^(2x) est strictement croissante sur ℝ.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Sa dérivée f'(x) = 2e^(2x) est toujours positive, donc la fonction est strictement 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":6725,"question":"Quelle est la valeur de l'intégrale ∫(de 0 à π) sin(x) dx ?","option_a":"0","option_b":"1","option_c":"2","option_d":"-1","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'intégrale de sin(x) est -cos(x). Évaluée de 0 à π, on obtient -cos(π) - (-cos(0)) = 1 - (-1) = 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\": \"0\", \"b\": \"1\", \"c\": \"2\", \"d\": \"-1\"}}","_debug_options_count":4},{"id":6726,"question":"La suite uₙ = n²\/(n + 1) est bornée.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La suite tend vers l'infini quand n → ∞, donc elle n'est pas bornée.","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":6727,"question":"Quelle est la solution de l'équation différentielle y'' + y = 0 avec y(0) = 1 et y'(0) = 0 ?","option_a":"y = cos(x)","option_b":"y = sin(x)","option_c":"y = e^x","option_d":"y = e^(-x)","option_e":"","option_f":"","bonne_reponse":"a","explication":"La solution générale est y = A cos(x) + B sin(x). Avec y(0) = 1 et y'(0) = 0, on obtient A = 1 et B = 0, donc y = cos(x).","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 = cos(x)\", \"b\": \"y = sin(x)\", \"c\": \"y = e^x\", \"d\": \"y = e^(-x)\"","_debug_options_count":4},{"id":6728,"question":"La fonction f(x) = ln(x) est définie sur ℝ.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction ln(x) est définie uniquement pour x \u003E 0, donc pas sur tout ℝ.","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}]