Quiz — TD1.docx

Question 1 sur 10 20:00

[{"id":119625,"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 s'obtient en appliquant la formule (ax^n)' = n*ax^(n-1). Ici, f'(x) = 3x² - 4x + 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\": \"3x² - 4x + 5\", \"b\": \"x² - 4x + 5\", \"c\": \"3x² - 4x + 1\", \"d\": \"","_debug_options_count":4},{"id":119626,"question":"La fonction f(x) = e^(-x²) admet un maximum en x = 0.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction est paire et sa dérivée f'(x) = -2x*e^(-x²) s'annule en x=0. Le tableau de variations confirme qu'il s'agit d'un maximum.","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":119627,"question":"Quelle est la limite de la suite u_n = (3n² + 2n - 1)\/(2n² - n + 4) quand n tend vers l'infini ?","option_a":"3\/2","option_b":"0","option_c":"1","option_d":"∞","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour les suites rationnelles, on compare les termes de plus haut degré : lim u_n = 3n²\/2n² = 3\/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\": \"3\/2\", \"b\": \"0\", \"c\": \"1\", \"d\": \"∞\"}}","_debug_options_count":4},{"id":119628,"question":"L'intégrale ∫(de 0 à 1) (2x + 1) dx vaut :","option_a":"2","option_b":"3","option_c":"1","option_d":"4","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'intégrale se calcule comme [x² + x] de 0 à 1 = (1 + 1) - (0 + 0) = 2. Attention à la primitive correcte.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"2\", \"b\": \"3\", \"c\": \"1\", \"d\": \"4\"}}","_debug_options_count":4},{"id":119629,"question":"Dans l'espace, l'équation x + 2y - z + 3 = 0 représente :","option_a":"Un plan","option_b":"Une droite","option_c":"Une sphère","option_d":"Un point","option_e":"","option_f":"","bonne_reponse":"a","explication":"Cette équation est de la forme ax + by + cz + d = 0, caractéristique d'un plan dans l'espace.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Un plan\", \"b\": \"Une droite\", \"c\": \"Une sphère\", \"d\": \"Un point\"}","_debug_options_count":4},{"id":119630,"question":"Quelle est la probabilité d'obtenir un double 6 en lançant deux dés équilibrés ?","option_a":"1\/6","option_b":"1\/12","option_c":"1\/36","option_d":"1\/18","option_e":"","option_f":"","bonne_reponse":"c","explication":"Il y a 6×6 = 36 issues possibles. Seul le couple (6,6) convient, 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\": \"1\/6\", \"b\": \"1\/12\", \"c\": \"1\/36\", \"d\": \"1\/18\"}}","_debug_options_count":4},{"id":119631,"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 ln(x) n'est définie que pour x \u003E 0. Elle n'est pas définie sur R tout entier.","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":119632,"question":"Quelle est la solution de l'équation différentielle y' + 2y = 0 avec y(0) = 3 ?","option_a":"y = 3e^(-2x)","option_b":"y = 3e^(2x)","option_c":"y = -3e^(-2x)","option_d":"y = 3e^(-x\/2)","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'équation est linéaire du premier ordre. La solution générale est y = Ce^(-2x). Avec y(0)=3, on trouve C=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\": \"y = 3e^(-2x)\", \"b\": \"y = 3e^(2x)\", \"c\": \"y = -3e^(-2x)\", \"d\": \"y ","_debug_options_count":4},{"id":119633,"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, deux vecteurs sont orthogonaux si et seulement si leur produit scalaire est nul.","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":119634,"question":"Quelle est la valeur de ∫(de 0 à π) sin(x) dx ?","option_a":"0","option_b":"2","option_c":"π","option_d":"1","option_e":"","option_f":"","bonne_reponse":"b","explication":"La primitive de sin(x) est -cos(x). L'intégrale vaut [-cos(x)] de 0 à π = -(-1) - (-1) = 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\": \"0\", \"b\": \"2\", \"c\": \"π\", \"d\": \"1\"}}","_debug_options_count":4}]