Quiz — corrct2 rattrapg2010-2011.pdf

Question 1 sur 10 20:00

[{"id":13243,"question":"Quelle est la dérivée de la fonction f(x) = 3x² + 2x - 5 ?","option_a":"6x + 2","option_b":"3x² + 2","option_c":"6x² + 2x","option_d":"6x + 2x - 5","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'une fonction polynôme s'obtient en appliquant la formule : (ax^n)' = n*ax^(n-1). Ici, f'(x) = (3x²)' + (2x)' - (5)' = 6x + 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\": \"6x + 2\", \"b\": \"3x² + 2\", \"c\": \"6x² + 2x\", \"d\": \"6x + 2x - 5\"}}","_debug_options_count":4},{"id":13244,"question":"La limite de la fonction f(x) = (2x + 1)\/(x - 3) quand x tend vers +∞ est égale à 2.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"En divisant numérateur et dénominateur par x, on obtient f(x) = (2 + 1\/x)\/(1 - 3\/x). Quand x → +∞, 1\/x → 0, donc f(x) → 2\/1 = 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":13245,"question":"Quelle est l'intégrale indéfinie de f(x) = 4x³ ?","option_a":"x⁴ + C","option_b":"12x² + C","option_c":"x⁴\/4 + C","option_d":"16x⁴ + C","option_e":"","option_f":"","bonne_reponse":"c","explication":"L'intégrale de x^n est (x^(n+1))\/(n+1) + C. Ici, ∫4x³ dx = 4*(x⁴\/4) + C = x⁴ + C.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"x⁴ + C\", \"b\": \"12x² + C\", \"c\": \"x⁴\/4 + C\", \"d\": \"16x⁴ + C\"","_debug_options_count":4},{"id":13246,"question":"La fonction f(x) = x² - 4x + 3 est croissante sur l'intervalle [2, +∞[.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée f'(x) = 2x - 4 est positive pour x ≥ 2, donc la fonction est croissante sur [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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":13247,"question":"Quelle est la solution de l'équation ln(x) = 2 ?","option_a":"x = e²","option_b":"x = 2","option_c":"x = ln(2)","option_d":"x = 1\/2","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'équation ln(x) = a a pour solution x = e^a. Ici, 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 = ln(2)\", \"d\": \"x = 1\/2\"}}","_debug_options_count":4},{"id":13248,"question":"La fonction f(x) = 1\/x est définie et continue sur ℝ.","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 et présente une discontinuité en ce point.","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":13249,"question":"Quelle est la dérivée seconde de f(x) = x³ - 3x² + 2x ?","option_a":"6x - 6","option_b":"3x² - 6x + 2","option_c":"6x² - 6x","option_d":"x³ - 3x²","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée première est f'(x) = 3x² - 6x + 2. La dérivée seconde est f''(x) = 6x - 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\": \"6x - 6\", \"b\": \"3x² - 6x + 2\", \"c\": \"6x² - 6x\", \"d\": \"x³ - 3x²","_debug_options_count":4},{"id":13250,"question":"L'intégrale de f(x) = cos(x) entre 0 et π\/2 est égale à 1.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"∫cos(x) dx = sin(x) + C. Entre 0 et π\/2, sin(π\/2) - sin(0) = 1 - 0 = 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":13251,"question":"Quelle est la limite de f(x) = (x² - 1)\/(x - 1) quand x tend vers 1 ?","option_a":"0","option_b":"1","option_c":"2","option_d":"∞","option_e":"","option_f":"","bonne_reponse":"c","explication":"En factorisant, f(x) = (x - 1)(x + 1)\/(x - 1) = x + 1 pour x ≠ 1. Donc lim(x→1) f(x) = 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\": \"0\", \"b\": \"1\", \"c\": \"2\", \"d\": \"∞\"}}","_debug_options_count":4},{"id":13252,"question":"La fonction f(x) = e^x est toujours positive sur ℝ.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction exponentielle e^x est strictement positive pour tout x réel.","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}]