Quiz — Série Ensemble et Application Cor 4.pdf

Question 1 sur 10 20:00

[{"id":84910,"question":"Quelle est la dérivée de la fonction f(x) = x³ - 2x² + 5x - 1 ?","option_a":"3x² - 4x + 5","option_b":"x² - 4x + 5","option_c":"3x² - 4x + 1","option_d":"x³ - 2x + 5","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'un polynôme se calcule terme par terme : (x³)' = 3x², (-2x²)' = -4x, (5x)' = 5, et la dérivée de -1 est 0.","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 - 4x + 5\", \"c\": \"3x² - 4x + 1\", \"d\": \"","_debug_options_count":4},{"id":84911,"question":"L'intégrale ∫(2x + 3) dx est égale à :","option_a":"x² + 3x + C","option_b":"x² + 3x","option_c":"2x² + 3x + C","option_d":"x² + 2x + C","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'intégrale d'une somme est la somme des intégrales. ∫2x dx = x² + C1 et ∫3 dx = 3x + C2, donc la somme est x² + 3x + C (où C = C1 + C2).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"x² + 3x + C\", \"b\": \"x² + 3x\", \"c\": \"2x² + 3x + C\", \"d\": \"x² +","_debug_options_count":4},{"id":84912,"question":"Vrai ou Faux ? La fonction f(x) = 1\/x est dérivable en x = 0.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction f(x) = 1\/x n'est pas définie en x = 0, donc elle ne peut pas y être dérivable.","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":84913,"question":"Quelle est la solution de l'équation ln(x) = 2 ?","option_a":"x = e²","option_b":"x = 2","option_c":"x = 100","option_d":"x = √2","option_e":"","option_f":"","bonne_reponse":"a","explication":"Par définition du logarithme, si ln(x) = a, alors x = e^a. Ici, a = 2, donc 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\": \"x = e²\", \"b\": \"x = 2\", \"c\": \"x = 100\", \"d\": \"x = √2\"}}","_debug_options_count":4},{"id":84914,"question":"Vrai ou Faux ? La fonction f(x) = x² est convexe sur ℝ.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Une fonction est convexe si sa dérivée seconde est positive. Ici, f''(x) = 2 \u003E 0 pour tout x ∈ ℝ.","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":84915,"question":"Quelle est la valeur de l'intégrale ∫₀¹ (3x² + 2x) dx ?","option_a":"2","option_b":"3","option_c":"1","option_d":"4","option_e":"","option_f":"","bonne_reponse":"a","explication":"Calculons l'intégrale : ∫(3x² + 2x) dx = x³ + x² + C. Évaluons entre 0 et 1 : (1 + 1) - (0 + 0) = 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\": \"2\", \"b\": \"3\", \"c\": \"1\", \"d\": \"4\"}}","_debug_options_count":4},{"id":84916,"question":"Vrai ou Faux ? La suite uₙ = n² est croissante pour tout n ∈ ℕ.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour tout n ∈ ℕ, u_{n+1} = (n+1)² = n² + 2n + 1 \u003E n² = uₙ, donc la suite est strictement croissante.","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":84917,"question":"Quelle est la limite de (3x² - 2x + 1)\/(x² + 1) quand x tend vers +∞ ?","option_a":"3","option_b":"1","option_c":"0","option_d":"∞","option_e":"","option_f":"","bonne_reponse":"a","explication":"Divisons le numérateur et le dénominateur par x² : (3 - 2\/x + 1\/x²)\/(1 + 1\/x²). Quand x → +∞, les termes en 1\/x tendent vers 0, donc la limite est 3\/1 = 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\": \"1\", \"c\": \"0\", \"d\": \"∞\"}}","_debug_options_count":4},{"id":84918,"question":"Vrai ou Faux ? La fonction f(x) = e^x est sa propre dérivée.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de e^x est e^x, donc la fonction est bien sa propre dérivée.","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":84919,"question":"Quelle est la solution de l'inéquation x² - 5x + 6 ≤ 0 ?","option_a":"x ∈ [2, 3]","option_b":"x ∈ ]-∞, 2] ∪ [3, +∞[","option_c":"x ∈ [1, 6]","option_d":"x ∈ ℝ","option_e":"","option_f":"","bonne_reponse":"a","explication":"Résolvons l'équation x² - 5x + 6 = 0 : (x-2)(x-3) = 0 ⇒ x = 2 ou x = 3. Le polynôme est négatif entre ses racines, donc x ∈ [2, 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\": \"x ∈ [2, 3]\", \"b\": \"x ∈ ]-∞, 2] ∪ [3, +∞[\", \"c\": \"x ∈ ","_debug_options_count":4}]