Quiz — 1565781790_Corrigé Exercice 3.pdf

Question 1 sur 10 20:00

[{"id":14599,"question":"Soit f une fonction dérivable sur ℝ telle que f'(x) = 2x + 1. Quelle est la fonction f si f(0) = 3 ?","option_a":"f(x) = x² + x + 3","option_b":"f(x) = x² + x","option_c":"f(x) = 2x² + x + 3","option_d":"f(x) = x² + 2x + 3","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de f(x) = x² + x + 3 est f'(x) = 2x + 1, et f(0) = 3. C'est la seule fonction qui vérifie ces conditions.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"f(x) = x² + x + 3\", \"b\": \"f(x) = x² + x\", \"c\": \"f(x) = 2x² + x","_debug_options_count":4},{"id":14600,"question":"La suite (uₙ) définie par uₙ = (-1)ⁿ \/ n est convergente.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La suite (uₙ) n'est pas convergente car elle oscille entre des valeurs positives et négatives sans tendre vers une limite finie.","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":14601,"question":"Quelle est la primitive de la fonction f(x) = 3x² + 2x + 1 ?","option_a":"F(x) = x³ + x² + x + C","option_b":"F(x) = x³ + x² + C","option_c":"F(x) = 6x + 2 + C","option_d":"F(x) = x³ + 2x² + x + C","option_e":"","option_f":"","bonne_reponse":"a","explication":"La primitive de f(x) = 3x² + 2x + 1 est F(x) = x³ + x² + x + C, car sa dérivée est bien 3x² + 2x + 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\": \"F(x) = x³ + x² + x + C\", \"b\": \"F(x) = x³ + x² + C\", \"c\": \"F(x","_debug_options_count":4},{"id":14602,"question":"Soit X une variable aléatoire suivant une loi binomiale de paramètres n = 10 et p = 0.3. Quelle est l'espérance de X ?","option_a":"3","option_b":"7","option_c":"0.3","option_d":"10","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'espérance d'une loi binomiale B(n, p) est E(X) = n × p = 10 × 0.3 = 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\": \"3\", \"b\": \"7\", \"c\": \"0.3\", \"d\": \"10\"}}","_debug_options_count":4},{"id":14603,"question":"La fonction f(x) = e^(2x) est solution de l'équation différentielle y' = 2y.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de f(x) = e^(2x) est f'(x) = 2e^(2x) = 2f(x), donc elle vérifie bien l'équation différentielle y' = 2y.","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":14604,"question":"Quelle est la limite de la suite (uₙ) définie par uₙ = (2n + 1)\/(n + 3) lorsque n tend vers l'infini ?","option_a":"0","option_b":"2","option_c":"1\/3","option_d":"2\/3","option_e":"","option_f":"","bonne_reponse":"b","explication":"En divisant le numérateur et le dénominateur par n, on obtient uₙ = (2 + 1\/n)\/(1 + 3\/n). Lorsque n tend vers l'infini, uₙ tend vers 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\/3\", \"d\": \"2\/3\"}}","_debug_options_count":4},{"id":14605,"question":"Soit f une fonction continue sur [a, b]. Quelle est la valeur moyenne de f sur cet intervalle ?","option_a":"(f(a) + f(b))\/2","option_b":"(b - a) × ∫[a à b] f(x) dx","option_c":"(1\/(b - a)) × ∫[a à b] f(x) dx","option_d":"∫[a à b] f(x) dx","option_e":"","option_f":"","bonne_reponse":"c","explication":"La valeur moyenne d'une fonction continue sur [a, b] est donnée par (1\/(b - a)) × ∫[a à b] f(x) dx.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"(f(a) + f(b))\/2\", \"b\": \"(b - a) × ∫[a à b] f(x) dx\", \"c\": \"(1","_debug_options_count":4},{"id":14606,"question":"La fonction f(x) = ln(x) est définie pour tout x réel.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction ln(x) n'est définie que pour x \u003E 0. Elle n'est pas définie pour les valeurs négatives ou nulles de 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":14607,"question":"Quelle est la dérivée de la fonction f(x) = sin(2x) ?","option_a":"cos(2x)","option_b":"2cos(2x)","option_c":"cos(x)","option_d":"2sin(x)","option_e":"","option_f":"","bonne_reponse":"b","explication":"En utilisant la formule de dérivation des fonctions composées, f'(x) = 2 × cos(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\": \"cos(2x)\", \"b\": \"2cos(2x)\", \"c\": \"cos(x)\", \"d\": \"2sin(x)\"}}","_debug_options_count":4},{"id":14608,"question":"Soit (uₙ) une suite géométrique de raison q = -2 et de premier terme u₀ = 1. Quelle est la valeur de u₃ ?","option_a":"-8","option_b":"8","option_c":"-4","option_d":"4","option_e":"","option_f":"","bonne_reponse":"a","explication":"La suite géométrique est définie par uₙ = u₀ × qⁿ. Donc u₃ = 1 × (-2)³ = -8.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"-8\", \"b\": \"8\", \"c\": \"-4\", \"d\": \"4\"}}","_debug_options_count":4}]