Quiz — 67509d142497d_série de révision n°15_mat_tri_enrg_corrigé.pdf

Question 1 sur 10 20:00

[{"id":28840,"question":"Quelle est la dérivée de la fonction f(x) = sin(2x) ?","option_a":"2cos(2x)","option_b":"cos(2x)","option_c":"2sin(2x)","option_d":"cos(x)","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de sin(u) est u'cos(u). Ici, u = 2x donc u' = 2. Ainsi, f'(x) = 2cos(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\": \"2cos(2x)\", \"b\": \"cos(2x)\", \"c\": \"2sin(2x)\", \"d\": \"cos(x)\"}}","_debug_options_count":4},{"id":28841,"question":"L'équation différentielle y' + 3y = 0 a pour solution générale y = Ce^{-3x}.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est une équation différentielle linéaire du premier ordre à coefficients constants. La solution générale est bien y = Ce^{-3x}, où C est une constante réelle.","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":28842,"question":"Quelle est l'intégrale de f(x) = 3x^2 - 2x + 1 ?","option_a":"x^3 - x^2 + x + C","option_b":"x^3 - x^2 + C","option_c":"3x^3 - x^2 + x + C","option_d":"x^3 - x^2 + x","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'intégrale de x^n est x^{n+1}\/(n+1). Ainsi, ∫(3x^2 - 2x + 1)dx = x^3 - x^2 + x + C.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"x^3 - x^2 + x + C\", \"b\": \"x^3 - x^2 + C\", \"c\": \"3x^3 - x^2 + x + ","_debug_options_count":4},{"id":28843,"question":"La fonction f(x) = tan(x) est périodique de période π.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction tangente a une période de π, car tan(x + π) = tan(x) pour tout x où la fonction est définie.","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":28844,"question":"Quelle est la solution de l'équation sin(x) = √2\/2 dans l'intervalle [0, 2π] ?","option_a":"π\/4 et 3π\/4","option_b":"π\/6 et 5π\/6","option_c":"π\/3 et 2π\/3","option_d":"π\/4 et 7π\/4","option_e":"","option_f":"","bonne_reponse":"a","explication":"Les solutions de sin(x) = √2\/2 dans [0, 2π] sont x = π\/4 et x = 3π\/4, car sin(π\/4) = sin(3π\/4) = √2\/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\": \"π\/4 et 3π\/4\", \"b\": \"π\/6 et 5π\/6\", \"c\": \"π\/3 et 2π\/3\", \"d\": ","_debug_options_count":4},{"id":28845,"question":"L'intégrale définie de f(x) = x^2 entre 0 et 2 est égale à 8\/3.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"∫(0 à 2) x^2 dx = [x^3\/3](0 à 2) = 8\/3 - 0 = 8\/3. 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\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":28846,"question":"Quelle est la dérivée de la fonction f(x) = ln(x^2 + 1) ?","option_a":"2x\/(x^2 + 1)","option_b":"1\/(x^2 + 1)","option_c":"2x","option_d":"x\/(x^2 + 1)","option_e":"","option_f":"","bonne_reponse":"a","explication":"En utilisant la règle de la chaîne, f'(x) = (2x)\/(x^2 + 1), car la dérivée de ln(u) est u'\/u avec u = x^2 + 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\": \"2x\/(x^2 + 1)\", \"b\": \"1\/(x^2 + 1)\", \"c\": \"2x\", \"d\": \"x\/(x^2 + 1)\"}","_debug_options_count":4},{"id":28847,"question":"L'équation différentielle y'' + 4y = 0 a pour solution générale y = Acos(2x) + Bsin(2x).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est une équation différentielle linéaire du second ordre à coefficients constants. La solution générale est bien y = Acos(2x) + Bsin(2x), où A et B sont des constantes réelles.","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":28848,"question":"Quelle est la valeur 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":"c","explication":"∫(0 à π) sin(x) dx = [-cos(x)](0 à π) = -cos(π) - (-cos(0)) = -(-1) - (-1) = 1 + 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\": \"-1\"}}","_debug_options_count":4},{"id":28849,"question":"La fonction f(x) = cos(2x) peut s'écrire sous la forme 2cos^2(x) - 1.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"En utilisant la formule de duplication cos(2x) = 2cos^2(x) - 1, 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}]