Quiz — 63b2de6cc8672_Corrigé_Serie n°2.pdf

Question 1 sur 10 20:00

[{"id":4946,"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 s'obtient en appliquant la formule : (ax^n)' = n*ax^(n-1). Ici, 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":4947,"question":"Une suite géométrique de raison q = 2 et de premier terme u0 = 3 a pour terme général :","option_a":"u_n = 3 * 2^n","option_b":"u_n = 3 + 2n","option_c":"u_n = 2 * 3^n","option_d":"u_n = 3 * n^2","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour une suite géométrique, u_n = u0 * q^n. Ici, u_n = 3 * 2^n.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"u_n = 3 * 2^n\", \"b\": \"u_n = 3 + 2n\", \"c\": \"u_n = 2 * 3^n\", \"d\": \"","_debug_options_count":4},{"id":4948,"question":"Vrai ou Faux ? La fonction exponentielle est toujours positive.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction exponentielle, notée exp(x) ou e^x, prend toujours des valeurs strictement positives pour tout x réel.","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":4949,"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":"1","option_c":"2","option_d":"∞","option_e":"","option_f":"","bonne_reponse":"b","explication":"En divisant numérateur et dénominateur par n, on obtient u_n = (2 + 1\/n)\/(1 + 3\/n). Quand n → ∞, u_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\": \"1\", \"c\": \"2\", \"d\": \"∞\"}}","_debug_options_count":4},{"id":4950,"question":"Vrai ou Faux ? Si f est une fonction dérivable sur R, alors f' est continue sur R.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Un contre-exemple est la fonction f(x) = x² * sin(1\/x) pour x ≠ 0 et f(0) = 0. f est dérivable en 0 mais f' n'est pas continue en 0.","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":4951,"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 = Cx + C","option_d":"y = C\/x","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'équation y' + 2y = 0 est une équation différentielle linéaire du premier ordre. Sa solution générale est y = Ce^(-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\": \"y = Ce^(-2x)\", \"b\": \"y = Ce^(2x)\", \"c\": \"y = Cx + C\", \"d\": \"y = C","_debug_options_count":4},{"id":4952,"question":"Vrai ou Faux ? La probabilité conditionnelle P(A|B) est toujours inférieure à P(A).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"P(A|B) = P(A ∩ B)\/P(B). Si A et B sont indépendants, P(A|B) = P(A). Sinon, P(A|B) peut être supérieure ou inférieure à P(A).","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":4953,"question":"Quelle est l'équation cartésienne d'une sphère de centre O(0,0,0) et de rayon R ?","option_a":"x² + y² + z² = R","option_b":"x² + y² + z² = R²","option_c":"x + y + z = R","option_d":"x² + y² = R² + z²","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'équation cartésienne d'une sphère de centre O et de rayon R est x² + y² + z² = R².","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"x² + y² + z² = R\", \"b\": \"x² + y² + z² = R²\", \"c\": \"x + y +","_debug_options_count":4},{"id":4954,"question":"Vrai ou Faux ? Toute fonction continue sur un intervalle fermé est dérivable sur cet intervalle.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Un contre-exemple est la fonction f(x) = |x| sur l'intervalle [-1, 1]. Elle est continue mais non dérivable en x = 0.","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":4955,"question":"Quelle est la valeur de l'intégrale ∫(0 à 1) x² dx ?","option_a":"0","option_b":"1\/3","option_c":"1\/2","option_d":"1","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'intégrale de x² entre 0 et 1 est [x³\/3] de 0 à 1 = 1\/3 - 0 = 1\/3.","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\/3\", \"c\": \"1\/2\", \"d\": \"1\"}}","_debug_options_count":4}]