Quiz — Exercice2- corrigé - vidéo.pdf

Question 1 sur 10 20:00

[{"id":97499,"question":"Quelle est la dérivée de la fonction f(x) = x² + 3x - 5 ?","option_a":"A) 2x + 3","option_b":"B) x² + 3","option_c":"C) 2x + 5","option_d":"D) 2x - 5","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'une fonction polynôme se calcule terme par terme. La dérivée de x² est 2x, celle de 3x est 3, et celle de -5 est 0. Donc f'(x) = 2x + 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\": \"A) 2x + 3\", \"b\": \"B) x² + 3\", \"c\": \"C) 2x + 5\", \"d\": \"D) 2x - 5\"","_debug_options_count":4},{"id":97500,"question":"Une suite géométrique de raison q = -2 et de premier terme u₀ = 3 est-elle convergente ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Une suite géométrique converge si et seulement si |q| \u003C 1. Ici, |q| = 2 \u003E 1, donc la suite diverge.","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":97501,"question":"Quelle est la solution générale de l'équation différentielle y' + 2y = 0 ?","option_a":"A) y = Ce^(-2x)","option_b":"B) y = Ce^(2x)","option_c":"C) y = Cx + C","option_d":"D) y = C\/x","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\": \"A) y = Ce^(-2x)\", \"b\": \"B) y = Ce^(2x)\", \"c\": \"C) y = Cx + C\", \"d","_debug_options_count":4},{"id":97502,"question":"Soit une variable aléatoire X suivant une loi normale N(μ, σ²). Que vaut P(μ - σ ≤ X ≤ μ + σ) ?","option_a":"A) 0,68","option_b":"B) 0,95","option_c":"C) 0,997","option_d":"D) 0,5","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour une loi normale, environ 68% des valeurs se situent dans l'intervalle [μ - σ, μ + σ].","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"A) 0,68\", \"b\": \"B) 0,95\", \"c\": \"C) 0,997\", \"d\": \"D) 0,5\"}}","_debug_options_count":4},{"id":97503,"question":"La matrice A = [[1, 2], [3, 4]] est-elle inversible ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Une matrice est inversible si son déterminant est non nul. Ici, det(A) = (1*4) - (2*3) = -2 ≠ 0, donc A est inversible.","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":97504,"question":"Quelle est la limite de la fonction f(x) = (x² - 1)\/(x - 1) quand x tend vers 1 ?","option_a":"A) 0","option_b":"B) 1","option_c":"C) 2","option_d":"D) +∞","option_e":"","option_f":"","bonne_reponse":"c","explication":"On peut simplifier f(x) = (x-1)(x+1)\/(x-1) = x+1 pour x ≠ 1. Donc lim(x→1) f(x) = 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\": \"A) 0\", \"b\": \"B) 1\", \"c\": \"C) 2\", \"d\": \"D) +∞\"}}","_debug_options_count":4},{"id":97505,"question":"Soit une suite définie par uₙ₊₁ = 0,5uₙ + 1 avec u₀ = 2. Cette suite est-elle monotone ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La suite est décroissante car uₙ₊₁ - uₙ = -0,5uₙ + 1 ≤ 0 pour u₀ = 2 (et reste vrai pour tout 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":97506,"question":"Quelle est la probabilité d'obtenir un double 6 en lançant deux dés équilibrés ?","option_a":"A) 1\/6","option_b":"B) 1\/12","option_c":"C) 1\/36","option_d":"D) 1\/18","option_e":"","option_f":"","bonne_reponse":"c","explication":"Il y a 6 faces sur chaque dé, donc 6*6 = 36 issues possibles. Seul un cas correspond à un double 6, donc P = 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\": \"A) 1\/6\", \"b\": \"B) 1\/12\", \"c\": \"C) 1\/36\", \"d\": \"D) 1\/18\"}}","_debug_options_count":4},{"id":97507,"question":"Soit f une fonction dérivable sur ℝ telle que f'(x) = 3x² - 2x + 1. Quelle est la forme de f(x) ?","option_a":"A) f(x) = x³ - x² + x + C","option_b":"B) f(x) = 3x³ - 2x² + x + C","option_c":"C) f(x) = x³ - x + C","option_d":"D) f(x) = 3x² - 2x + 1","option_e":"","option_f":"","bonne_reponse":"a","explication":"La primitive de f'(x) = 3x² - 2x + 1 est f(x) = x³ - x² + x + C, où C est une constante.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"A) f(x) = x³ - x² + x + C\", \"b\": \"B) f(x) = 3x³ - 2x² + x + C","_debug_options_count":4},{"id":97508,"question":"La fonction f(x) = x³ - 3x + 2 admet-elle un extremum local en x = 1 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"f'(x) = 3x² - 3. En x=1, f'(1)=0, mais f''(x)=6x, donc f''(1)=6\u003E0 : c'est un minimum local. L'affirmation est donc fausse.","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}]