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[{"id":931,"question":"Quelle est la dérivée de la fonction f(x) = x³ - 2x² + 5x - 7 ?","option_a":"3x² - 4x + 5","option_b":"x² - 2x + 5","option_c":"3x² - 4x + 5x","option_d":"x³ - 2x² + 5","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'une fonction polynôme s'obtient en appliquant la règle de dérivation terme à terme : (x^n)' = n*x^(n-1). Ici, f'(x) = 3x² - 4x + 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\": \"3x² - 4x + 5\", \"b\": \"x² - 2x + 5\", \"c\": \"3x² - 4x + 5x\", \"d\": ","_debug_options_count":4},{"id":932,"question":"Quelle est la limite de la suite u_n = (2n + 1)\/(n + 3) quand n tend vers l'infini ?","option_a":"0","option_b":"2","option_c":"1","option_d":"∞","option_e":"","option_f":"","bonne_reponse":"b","explication":"Pour trouver la limite d'une suite rationnelle, on divise le numérateur et le dénominateur par la plus haute puissance de n (ici n). On obtient : lim(u_n) = lim(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\": \"b\", \"options\": {\"a\": \"0\", \"b\": \"2\", \"c\": \"1\", \"d\": \"∞\"}}","_debug_options_count":4},{"id":933,"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 + 2x","option_d":"y = Cx²","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est une équation différentielle linéaire du premier ordre. La solution générale est y = Ce^(-2x), 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\": \"y = Ce^(-2x)\", \"b\": \"y = Ce^(2x)\", \"c\": \"y = C + 2x\", \"d\": \"y = C","_debug_options_count":4},{"id":934,"question":"Quelle est la probabilité d'obtenir un double 6 en lançant deux dés équilibrés ?","option_a":"1\/6","option_b":"1\/12","option_c":"1\/36","option_d":"1\/18","option_e":"","option_f":"","bonne_reponse":"c","explication":"Il y a 36 issues possibles (6 faces x 6 faces). Une seule issue correspond à un double 6 : (6,6). Donc P(double 6) = 1\/36.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"1\/6\", \"b\": \"1\/12\", \"c\": \"1\/36\", \"d\": \"1\/18\"}}","_debug_options_count":4},{"id":935,"question":"Quelle est la formule de la somme des n premiers termes d'une suite arithmétique de raison r et de premier terme u_1 ?","option_a":"S_n = n\/2 * (2u_1 + (n-1)r)","option_b":"S_n = n * u_1","option_c":"S_n = n\/2 * (u_1 + u_n)","option_d":"S_n = u_1 + (n-1)r","option_e":"","option_f":"","bonne_reponse":"a","explication":"La somme des n premiers termes d'une suite arithmétique est donnée par S_n = n\/2 * (2u_1 + (n-1)r) ou S_n = n\/2 * (u_1 + u_n), où u_n est le n-ième terme.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"S_n = n\/2 * (2u_1 + (n-1)r)\", \"b\": \"S_n = n * u_1\", \"c\": \"S_n = n","_debug_options_count":4}]