Quiz — Correction de l' examen2 (juin 2020).pdf

Question 1 sur 10 20:00

[{"id":13844,"question":"Quelle est la dérivée de la fonction f(x) = 3x² - 5x + 2 ?","option_a":"6x - 5","option_b":"3x - 5","option_c":"6x² - 5","option_d":"3x² - 5","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'une fonction polynôme se calcule terme par terme : la dérivée de 3x² est 6x, celle de -5x est -5, et celle de 2 est 0. Ainsi, 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\": \"3x - 5\", \"c\": \"6x² - 5\", \"d\": \"3x² - 5\"}}","_debug_options_count":4},{"id":13845,"question":"L'équation x² - 4x + 4 = 0 admet deux solutions réelles distinctes.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le discriminant Δ = (-4)² - 4×1×4 = 16 - 16 = 0. Une équation du second degré avec Δ = 0 admet une solution réelle double (ici, x = 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":13846,"question":"Quelle est la limite de la fonction f(x) = (2x + 1)\/(x - 3) lorsque x tend vers l'infini ?","option_a":"2","option_b":"1\/2","option_c":"0","option_d":"∞","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour les fonctions rationnelles, la limite à l'infini est égale au rapport des coefficients des termes de plus haut degré : 2x\/x = 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\": \"1\/2\", \"c\": \"0\", \"d\": \"∞\"}}","_debug_options_count":4},{"id":13847,"question":"Dans un triangle ABC, si AB² + AC² = BC², alors le triangle est rectangle en A.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"D'après le théorème de Pythagore, si AB² + AC² = BC², alors le triangle ABC est rectangle en A.","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":13848,"question":"Quelle est la probabilité d'obtenir un nombre pair en lançant un dé équilibré ?","option_a":"1\/6","option_b":"1\/3","option_c":"1\/2","option_d":"2\/3","option_e":"","option_f":"","bonne_reponse":"c","explication":"Un dé équilibré a 6 faces numérotées de 1 à 6. Les nombres pairs sont 2, 4 et 6, soit 3 faces sur 6. La probabilité est donc 3\/6 = 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\": \"1\/6\", \"b\": \"1\/3\", \"c\": \"1\/2\", \"d\": \"2\/3\"}}","_debug_options_count":4},{"id":13849,"question":"La fonction f(x) = x³ est croissante sur ℝ.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de f(x) = x³ est f'(x) = 3x², qui est toujours positive ou nulle (sauf en x = 0 où elle est nulle). La fonction est donc croissante sur 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":13850,"question":"Quelle est la solution de l'inéquation 2x - 5 \u003E 3x + 1 ?","option_a":"x \u003C -6","option_b":"x \u003E -6","option_c":"x \u003C 6","option_d":"x \u003E 6","option_e":"","option_f":"","bonne_reponse":"a","explication":"En résolvant l'inéquation : 2x - 5 \u003E 3x + 1 → -x \u003E 6 → x \u003C -6 (en divisant par un nombre négatif, le sens de l'inégalité change).","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 \u003C -6\", \"b\": \"x \u003E -6\", \"c\": \"x \u003C 6\", \"d\": \"x \u003E 6\"}}","_debug_options_count":4},{"id":13851,"question":"Le produit scalaire de deux vecteurs orthogonaux est toujours égal à 0.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Par définition, le produit scalaire de deux vecteurs orthogonaux est égal à 0, car cos(90°) = 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},{"id":13852,"question":"Quelle est la somme des 5 premiers termes de la suite géométrique de premier terme 2 et de raison 3 ?","option_a":"242","option_b":"243","option_c":"486","option_d":"972","option_e":"","option_f":"","bonne_reponse":"a","explication":"La somme des n premiers termes d'une suite géométrique est S = u1 × (1 - q^n)\/(1 - q). Ici, S = 2 × (1 - 3^5)\/(1 - 3) = 2 × (1 - 243)\/(-2) = 242.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"242\", \"b\": \"243\", \"c\": \"486\", \"d\": \"972\"}}","_debug_options_count":4},{"id":13853,"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 logarithme népérien ln(x) est définie uniquement pour x \u003E 0. Elle n'est pas définie pour les nombres réels négatifs ou nuls.","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}]