Quiz — 1572530552_Corrigé exercice 3.pdf

Question 1 sur 10 20:00

[{"id":65316,"question":"Soit la suite définie par u₀ = 2 et uₙ₊₁ = 3uₙ - 1. Quelle est la nature de cette suite ?","option_a":"Arithmétique de raison 3","option_b":"Géométrique de raison 3","option_c":"Arithmético-géométrique","option_d":"Constante","option_e":"","option_f":"","bonne_reponse":"c","explication":"La suite est arithmético-géométrique car elle s'écrit sous la forme uₙ₊₁ = auₙ + b avec a ≠ 1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"Arithmétique de raison 3\", \"b\": \"Géométrique de raison 3\", \"c\"","_debug_options_count":4},{"id":65317,"question":"La fonction f(x) = x³ - 3x² + 4 est dérivable sur ℝ.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Toute fonction polynôme est dérivable sur ℝ, donc la proposition est vraie.","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":65318,"question":"Quelle est la limite de la suite uₙ = (2n² + 3n - 1)\/(n² + 5) lorsque n tend vers l'infini ?","option_a":"2","option_b":"3","option_c":"0","option_d":"∞","option_e":"","option_f":"","bonne_reponse":"a","explication":"En divisant numérateur et dénominateur par n², on obtient lim uₙ = 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\": \"0\", \"d\": \"∞\"}}","_debug_options_count":4},{"id":65319,"question":"L'équation différentielle y' + 2y = 0 admet pour solution générale y(x) = Ce^(-2x).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est une équation différentielle linéaire du premier ordre à coefficients constants, la solution est bien de cette forme.","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":65320,"question":"Soit f une fonction continue sur [a, b] telle que f(a) = -2 et f(b) = 5. Le théorème des valeurs intermédiaires garantit l'existence d'un réel c dans [a, b] tel que :","option_a":"f(c) = 0","option_b":"f(c) = 3","option_c":"f(c) = -1","option_d":"f(c) = 6","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le théorème garantit l'existence d'une valeur intermédiaire entre f(a) et f(b), ici 3 est entre -2 et 5.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"f(c) = 0\", \"b\": \"f(c) = 3\", \"c\": \"f(c) = -1\", \"d\": \"f(c) = 6\"}}","_debug_options_count":4},{"id":65321,"question":"La dérivée de la fonction f(x) = ln(3x² + 1) est :","option_a":"f'(x) = 6x\/(3x² + 1)","option_b":"f'(x) = 3x\/(3x² + 1)","option_c":"f'(x) = 6x²\/(3x² + 1)","option_d":"f'(x) = 3\/(3x² + 1)","option_e":"","option_f":"","bonne_reponse":"a","explication":"En appliquant la formule de dérivation des fonctions composées, on obtient f'(x) = 6x\/(3x² + 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) = 6x\/(3x² + 1)\", \"b\": \"f'(x) = 3x\/(3x² + 1)\", \"c\": \"f'(x)","_debug_options_count":4},{"id":65322,"question":"Un dé à 6 faces est lancé deux fois. Quelle est la probabilité d'obtenir deux fois le même nombre ?","option_a":"1\/6","option_b":"1\/36","option_c":"1\/3","option_d":"1\/2","option_e":"","option_f":"","bonne_reponse":"a","explication":"Il y a 6 résultats favorables (1-1, 2-2, ..., 6-6) sur 36 résultats possibles, soit 6\/36 = 1\/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\": \"1\/6\", \"b\": \"1\/36\", \"c\": \"1\/3\", \"d\": \"1\/2\"}}","_debug_options_count":4},{"id":65323,"question":"La fonction f(x) = |x² - 4x + 3| est dérivable en x = 2.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction valeur absolue n'est pas dérivable aux points où l'expression à l'intérieur s'annule, ici x = 1 et x = 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":65324,"question":"Soit la matrice A = [[2, -1], [1, 3]]. Son déterminant est égal à :","option_a":"7","option_b":"5","option_c":"-5","option_d":"1","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le déterminant de A est (2×3) - (-1×1) = 6 + 1 = 7. (Erreur corrigée : réponse correcte est 7, pas 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\": \"7\", \"b\": \"5\", \"c\": \"-5\", \"d\": \"1\"}}","_debug_options_count":4},{"id":65325,"question":"La suite uₙ = (-1)ⁿ \/ n converge vers 0 lorsque n tend vers l'infini.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La suite est bornée par -1\/n et 1\/n, et ces deux suites convergent vers 0, donc uₙ converge vers 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4}]