Quiz — Lycée Pilote Sfax - DC 3 2015 (corrigé).pdf

Question 1 sur 10 20:00

[{"id":23049,"question":"Quelle est la solution de l'équation x² - 5x + 6 = 0 ?","option_a":"x = 2 ou x = 3","option_b":"x = -2 ou x = -3","option_c":"x = 1 ou x = 6","option_d":"x = 0 ou x = 5","option_e":"","option_f":"","bonne_reponse":"a","explication":"Factorisons l'équation : (x-2)(x-3) = 0. Les solutions sont donc x = 2 et x = 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 ou x = 3\", \"b\": \"x = -2 ou x = -3\", \"c\": \"x = 1 ou x = 6\", ","_debug_options_count":4},{"id":23050,"question":"La fonction f(x) = x³ - 3x² + 2x est-elle paire ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Une fonction paire vérifie f(-x) = f(x). Ici, f(-x) = -x³ - 3x² - 2x ≠ f(x). Donc elle n'est pas paire.","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":23051,"question":"Quelle est la limite de (2x² + 3x - 5)\/(x² - 1) quand x tend vers +∞ ?","option_a":"2","option_b":"3","option_c":"1","option_d":"0","option_e":"","option_f":"","bonne_reponse":"a","explication":"En divisant numérateur et dénominateur par x², on obtient (2 + 3\/x - 5\/x²)\/(1 - 1\/x²). Quand x→+∞, la limite est 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\": \"2\", \"b\": \"3\", \"c\": \"1\", \"d\": \"0\"}}","_debug_options_count":4},{"id":23052,"question":"Si f'(x) = 3x² - 6x + 2, alors f(x) est croissante sur l'intervalle :","option_a":"]-∞; 0[","option_b":"[0; 2]","option_c":"[1; +∞[","option_d":"[2; +∞[","option_e":"","option_f":"","bonne_reponse":"d","explication":"Étudions le signe de f'(x) = 3(x-1)(x-2). f'(x) \u003E 0 pour x \u003C 1 ou x \u003E 2. Donc f 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\": \"d\", \"options\": {\"a\": \"]-∞; 0[\", \"b\": \"[0; 2]\", \"c\": \"[1; +∞[\", \"d\": \"[2; +∞[\"}}","_debug_options_count":4},{"id":23053,"question":"La fonction f(x) = 1\/x est-elle continue en x = 0 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction 1\/x n'est pas définie en x = 0, donc elle ne peut pas être continue 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":23054,"question":"Quelle est la dérivée de f(x) = (3x² + 2x)(x - 1) ?","option_a":"f'(x) = 6x + 2","option_b":"f'(x) = 9x² - 4x - 2","option_c":"f'(x) = 3x² + 8x - 2","option_d":"f'(x) = 6x² + 2x - 2","option_e":"","option_f":"","bonne_reponse":"b","explication":"En utilisant la formule (uv)' = u'v + uv', on obtient f'(x) = (6x+2)(x-1) + (3x²+2x)(1) = 9x² - 4x - 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\": \"f'(x) = 6x + 2\", \"b\": \"f'(x) = 9x² - 4x - 2\", \"c\": \"f'(x) = 3x²","_debug_options_count":4},{"id":23055,"question":"Si lim(x→2) f(x) = 5 et f(2) = 3, alors la fonction f est :","option_a":"continue en x = 2","option_b":"discontinue en x = 2","option_c":"dérivable en x = 2","option_d":"paire en x = 2","option_e":"","option_f":"","bonne_reponse":"b","explication":"Pour qu'une fonction soit continue en un point, il faut que lim(x→a) f(x) = f(a). Ici, 5 ≠ 3, donc f est discontinue en 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\": \"continue en x = 2\", \"b\": \"discontinue en x = 2\", \"c\": \"dérivable","_debug_options_count":4},{"id":23056,"question":"L'équation x³ - 4x + 2 = 0 admet-elle une solution dans l'intervalle [0; 2] ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction f(x) = x³ - 4x + 2 est continue sur [0; 2]. f(0) = 2 \u003E 0 et f(2) = 2 \u003E 0. Cependant, f(1) = -1 \u003C 0, donc par le théorème des valeurs intermédiaires, il existe une solution dans [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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":23057,"question":"Quelle est la dérivée de f(x) = √(x² + 1) ?","option_a":"f'(x) = x\/√(x² + 1)","option_b":"f'(x) = 1\/√(x² + 1)","option_c":"f'(x) = x\/2√(x² + 1)","option_d":"f'(x) = (x² + 1)\/2√(x² + 1)","option_e":"","option_f":"","bonne_reponse":"a","explication":"En utilisant la formule de dérivation des fonctions composées, f'(x) = (1\/2)(x² + 1)^(-1\/2) * 2x = x\/√(x² + 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) = x\/√(x² + 1)\", \"b\": \"f'(x) = 1\/√(x² + 1)\", \"c\": \"f'(","_debug_options_count":4},{"id":23058,"question":"Si f est une fonction dérivable sur ℝ et f'(x) \u003E 0 pour tout x ∈ ℝ, alors f est :","option_a":"constante","option_b":"croissante","option_c":"décroissante","option_d":"périodique","option_e":"","option_f":"","bonne_reponse":"b","explication":"Si la dérivée est strictement positive sur tout son domaine, la fonction est strictement croissante.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"constante\", \"b\": \"croissante\", \"c\": \"décroissante\", \"d\": \"pério","_debug_options_count":4}]