Quiz — CONT 03 4M 2010-2011 LPA.pdf

Question 1 sur 10 20:00

[{"id":32140,"question":"Quelle est la dérivée de la fonction f(x) = x³ + 2x² - 5x + 1 ?","option_a":"3x² + 4x - 5","option_b":"x² + 2x - 5","option_c":"3x² + 4x + 1","option_d":"x³ + 2x - 5","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 dérivée de 1 est 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\": \"3x² + 4x - 5\", \"b\": \"x² + 2x - 5\", \"c\": \"3x² + 4x + 1\", \"d\": \"","_debug_options_count":4},{"id":32141,"question":"L'intégrale de f(x) = 4x³ entre 0 et 2 est égale à 16.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'intégrale de 4x³ est x⁴. Entre 0 et 2, cela donne 2⁴ - 0⁴ = 16.","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":32142,"question":"Quelle est la solution générale de l'équation différentielle y' + 2y = 0 ?","option_a":"y = Ce^(-2x)","option_b":"y = Ce^(2x)","option_c":"y = C + e^(-2x)","option_d":"y = C - 2x","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'équation différentielle y' + 2y = 0 a pour solution y = Ce^(-2x), où C est une constante.","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 = Ce^(-2x)\", \"b\": \"y = Ce^(2x)\", \"c\": \"y = C + e^(-2x)\", \"d\": \"","_debug_options_count":4},{"id":32143,"question":"L'aire sous la courbe de f(x) = x² entre x = 0 et x = 3 est égale à 9.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'intégrale de x² entre 0 et 3 est [x³\/3]₀³ = 27\/3 = 9. L'affirmation est donc vraie.","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":32144,"question":"Quelle est la dérivée seconde de la fonction f(x) = e^(3x) ?","option_a":"9e^(3x)","option_b":"3e^(3x)","option_c":"e^(3x)","option_d":"27e^(3x)","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée première de e^(3x) est 3e^(3x), et sa dérivée seconde est 9e^(3x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"9e^(3x)\", \"b\": \"3e^(3x)\", \"c\": \"e^(3x)\", \"d\": \"27e^(3x)\"}}","_debug_options_count":4},{"id":32145,"question":"L'intégrale indéfinie de 1\/x est ln|x| + C.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'intégrale indéfinie de 1\/x est bien ln|x| + C, où C est une 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":32146,"question":"Quelle est la solution de l'équation différentielle y'' - 4y = 0 ?","option_a":"y = Ce^(2x)","option_b":"y = C₁e^(2x) + C₂e^(-2x)","option_c":"y = C₁cos(2x) + C₂sin(2x)","option_d":"y = Ce^(-2x)","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'équation caractéristique est r² - 4 = 0, dont les solutions sont r = 2 et r = -2. La solution générale est donc y = C₁e^(2x) + C₂e^(-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\": \"y = Ce^(2x)\", \"b\": \"y = C₁e^(2x) + C₂e^(-2x)\", \"c\": \"y = C₁","_debug_options_count":4},{"id":32147,"question":"La dérivée de la fonction f(x) = ln(x) est 1\/x.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de ln(x) est bien 1\/x 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":32148,"question":"Quelle est l'intégrale de f(x) = cos(x) entre 0 et π\/2 ?","option_a":"0","option_b":"1","option_c":"-1","option_d":"π\/2","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'intégrale de cos(x) est sin(x). Entre 0 et π\/2, cela donne sin(π\/2) - sin(0) = 1 - 0 = 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\": \"0\", \"b\": \"1\", \"c\": \"-1\", \"d\": \"π\/2\"}}","_debug_options_count":4},{"id":32149,"question":"La fonction f(x) = x² + 3x + 2 est toujours positive.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le discriminant de x² + 3x + 2 est 9 - 8 = 1 \u003E 0, donc la fonction s'annule en deux points et n'est pas toujours positive.","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}]