Quiz — Exercice 7.pdf

Question 1 sur 10 20:00

[{"id":32320,"question":"Quelle est la dérivée de la fonction f(x) = 3x² - 5x + 2 ?","option_a":"6x - 5","option_b":"3x² - 5","option_c":"6x - 5x","option_d":"6x² - 5","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'une fonction polynôme se calcule terme à terme : (3x²)' = 6x, (-5x)' = -5, et 2' = 0. Donc f'(x) = 6x - 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\": \"6x - 5\", \"b\": \"3x² - 5\", \"c\": \"6x - 5x\", \"d\": \"6x² - 5\"}}","_debug_options_count":4},{"id":32321,"question":"La limite de (sin x)\/x quand x tend vers 0 est égale à 1.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est un résultat fondamental en analyse : lim(x→0) (sin x)\/x = 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":32322,"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 = Cx^(-2)","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'équation est de la forme y' + ay = 0, dont la solution générale est y = Ce^(-ax). Ici a = 2, donc 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 = C + e^(-2x)\", \"d\": \"","_debug_options_count":4},{"id":32323,"question":"Soit une suite définie par u₀ = 1 et uₙ₊₁ = 2uₙ + 3. Quelle est la limite de cette suite si elle converge ?","option_a":"-3","option_b":"3","option_c":"1","option_d":"La suite diverge","option_e":"","option_f":"","bonne_reponse":"b","explication":"Si la suite converge vers L, alors L = 2L + 3 ⇒ L = -3. Mais comme u₀ = 1 \u003E -3 et la suite est croissante, elle ne peut converger vers -3. La suite diverge donc vers +∞.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"-3\", \"b\": \"3\", \"c\": \"1\", \"d\": \"La suite diverge\"}}","_debug_options_count":4},{"id":32324,"question":"Dans une expérience aléatoire, la probabilité d'un événement A est 0,4. Quelle est la probabilité de l'événement contraire ?","option_a":"0,6","option_b":"0,4","option_c":"0,2","option_d":"1","option_e":"","option_f":"","bonne_reponse":"a","explication":"La probabilité de l'événement contraire est P(A̅) = 1 - P(A) = 1 - 0,4 = 0,6.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"0,6\", \"b\": \"0,4\", \"c\": \"0,2\", \"d\": \"1\"}}","_debug_options_count":4},{"id":32325,"question":"La fonction f(x) = x³ - 3x + 1 est strictement croissante sur ℝ.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée f'(x) = 3x² - 3 s'annule en x = ±1. La fonction n'est donc pas strictement croissante sur ℝ (elle décroît sur ]-1,1[).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":32326,"question":"Quelle est l'intégrale de f(x) = 2x + 1 entre 0 et 2 ?","option_a":"4","option_b":"6","option_c":"8","option_d":"2","option_e":"","option_f":"","bonne_reponse":"b","explication":"∫(2x + 1)dx = x² + x + C. Entre 0 et 2 : (2² + 2) - (0² + 0) = 6.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"4\", \"b\": \"6\", \"c\": \"8\", \"d\": \"2\"}}","_debug_options_count":4},{"id":32327,"question":"Soit un vecteur u(2, -3, 1) dans l'espace. Quelle est la norme de ce vecteur ?","option_a":"√14","option_b":"√13","option_c":"√11","option_d":"√10","option_e":"","option_f":"","bonne_reponse":"a","explication":"La norme est √(2² + (-3)² + 1²) = √(4 + 9 + 1) = √14.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"√14\", \"b\": \"√13\", \"c\": \"√11\", \"d\": \"√10\"}}","_debug_options_count":4},{"id":32328,"question":"Si deux événements A et B sont indépendants, alors P(A ∩ B) = P(A) × P(B).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est la définition de l'indépendance de deux événements.","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":32329,"question":"Quelle est la solution de l'équation ln(x) = 2 ?","option_a":"e²","option_b":"2","option_c":"ln(2)","option_d":"√e","option_e":"","option_f":"","bonne_reponse":"a","explication":"ln(x) = 2 ⇒ x = e².","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"e²\", \"b\": \"2\", \"c\": \"ln(2)\", \"d\": \"√e\"}}","_debug_options_count":4}]