Quiz — 63d8da7b220c4_Exercice 10.pdf

Question 1 sur 10 20:00

[{"id":16729,"question":"Quelle est la dérivée de la fonction f(x) = 3x² + 2x - 5 ?","option_a":"6x + 2","option_b":"3x² + 2","option_c":"6x - 5","option_d":"3x + 2","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'une fonction polynôme est obtenue en appliquant la règle de dérivation terme à terme : f'(x) = 6x + 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\": \"6x + 2\", \"b\": \"3x² + 2\", \"c\": \"6x - 5\", \"d\": \"3x + 2\"}}","_debug_options_count":4},{"id":16730,"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 de y' + 2y = 0 est bien y = Ce^(-2x), car en dérivant on retrouve l'équation initiale.","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":16731,"question":"Quelle est la limite de la fonction f(x) = (2x² + 3x - 1)\/(x² - 4) quand x tend vers l'infini ?","option_a":"2","option_b":"1","option_c":"0","option_d":"∞","option_e":"","option_f":"","bonne_reponse":"a","explication":"En divisant numérateur et dénominateur par x², on obtient lim (2 + 3\/x - 1\/x²)\/(1 - 4\/x²) = 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\": \"2\", \"b\": \"1\", \"c\": \"0\", \"d\": \"∞\"}}","_debug_options_count":4},{"id":16732,"question":"La fonction f(x) = x³ - 3x² + 4 est 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, 2], donc la fonction est concave sur cet intervalle.","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":16733,"question":"Quelle est l'intégrale de la fonction f(x) = 4x³ - 2x + 1 entre 0 et 2 ?","option_a":"16","option_b":"14","option_c":"12","option_d":"10","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'intégrale se calcule comme [x⁴ - x² + x] de 0 à 2 = (16 - 4 + 2) - 0 = 14.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"16\", \"b\": \"14\", \"c\": \"12\", \"d\": \"10\"}}","_debug_options_count":4},{"id":16734,"question":"Le nombre complexe z = 3 + 4i a pour module |z| = 5.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le module de z = a + bi est |z| = √(a² + b²) = √(9 + 16) = 5.","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":16735,"question":"Quelle est la solution de l'équation différentielle y'' + y = 0 ?","option_a":"y = Ce^x + De^(-x)","option_b":"y = Ccos(x) + Dsin(x)","option_c":"y = Cx + D","option_d":"y = Ce^(2x)","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'équation caractéristique r² + 1 = 0 a pour solutions r = ±i, donc la solution générale est y = Ccos(x) + Dsin(x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"y = Ce^x + De^(-x)\", \"b\": \"y = Ccos(x) + Dsin(x)\", \"c\": \"y = Cx +","_debug_options_count":4},{"id":16736,"question":"La fonction f(x) = ln(x) est définie pour x \u003E 0.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction logarithme népérien ln(x) est définie uniquement pour x \u003E 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":16737,"question":"Quelle est la valeur moyenne de la fonction f(x) = x² sur l'intervalle [0, 2] ?","option_a":"4\/3","option_b":"2\/3","option_c":"8\/3","option_d":"1","option_e":"","option_f":"","bonne_reponse":"a","explication":"La valeur moyenne est (1\/(2-0)) * ∫₀² x² dx = (1\/2) * [x³\/3]₀² = (1\/2)*(8\/3) = 4\/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\": \"4\/3\", \"b\": \"2\/3\", \"c\": \"8\/3\", \"d\": \"1\"}}","_debug_options_count":4},{"id":16738,"question":"L'intégrale ∫₀¹ e^(-x) dx est égale à 1 - 1\/e.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'intégrale se calcule comme [-e^(-x)]₀¹ = -1\/e + 1 = 1 - 1\/e.","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}]