Quiz — 2. Practice Test Two.pdf

Question 1 sur 10 20:00

[{"id":15439,"question":"Quelle est la dérivée de la fonction f(x) = 3x² + 2x - 5 ?","option_a":"6x + 2","option_b":"3x² + 2","option_c":"6x - 5","option_d":"3x + 2","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'une fonction polynôme se calcule terme à terme : la dérivée de 3x² est 6x, celle de 2x est 2, et celle de -5 est 0. Donc f'(x) = 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\": \"6x - 5\", \"d\": \"3x + 2\"}}","_debug_options_count":4},{"id":15440,"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":15441,"question":"Quelle est la limite de la fonction f(x) = (2x² + 3x - 1)\/(x² - 4) lorsque x tend vers l'infini ?","option_a":"2","option_b":"0","option_c":"1","option_d":"∞","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour les fonctions rationnelles, la limite en l'infini est égale au rapport des coefficients des termes de plus haut degré. Ici, 2x²\/x² = 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\": \"0\", \"c\": \"1\", \"d\": \"∞\"}}","_debug_options_count":4},{"id":15442,"question":"Soit une suite géométrique de premier terme u₀ = 2 et de raison q = 3. Quelle est la valeur de u₃ ?","option_a":"18","option_b":"54","option_c":"162","option_d":"81","option_e":"","option_f":"","bonne_reponse":"b","explication":"La formule d'une suite géométrique est uₙ = u₀ × qⁿ. Donc u₃ = 2 × 3³ = 2 × 27 = 54.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"18\", \"b\": \"54\", \"c\": \"162\", \"d\": \"81\"}}","_debug_options_count":4},{"id":15443,"question":"L'équation x² - 5x + 6 = 0 admet deux solutions réelles distinctes.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le discriminant Δ = b² - 4ac = 25 - 24 = 1 \u003E 0. L'équation admet donc deux solutions réelles distinctes : 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":15444,"question":"Quelle est l'intégrale de la fonction f(x) = 4x³ entre 0 et 2 ?","option_a":"8","option_b":"16","option_c":"32","option_d":"64","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'intégrale de 4x³ est x⁴. Évaluée entre 0 et 2, on obtient 2⁴ - 0⁴ = 16.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"8\", \"b\": \"16\", \"c\": \"32\", \"d\": \"64\"}}","_debug_options_count":4},{"id":15445,"question":"Soit une variable aléatoire X suivant une loi normale centrée réduite. Quelle est la probabilité P(X ≤ 1) ?","option_a":"0.6826","option_b":"0.8413","option_c":"0.9772","option_d":"0.3413","option_e":"","option_f":"","bonne_reponse":"b","explication":"Pour une loi normale centrée réduite, P(X ≤ 1) ≈ 0.8413 (valeur lue dans la table de la loi normale).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"0.6826\", \"b\": \"0.8413\", \"c\": \"0.9772\", \"d\": \"0.3413\"}}","_debug_options_count":4},{"id":15446,"question":"La fonction f(x) = ln(x) est définie pour tout x ∈ ℝ.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction logarithme népérien est définie uniquement pour x \u003E 0. Elle n'est pas définie pour x ≤ 0.","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":15447,"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 multipliant par -1, on inverse le sens de l'inégalité).","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":15448,"question":"Soit une fonction f dérivable sur ℝ telle que f'(x) = 0 pour tout x ∈ ℝ. Que peut-on dire de f ?","option_a":"f est constante","option_b":"f est croissante","option_c":"f est décroissante","option_d":"f est périodique","option_e":"","option_f":"","bonne_reponse":"a","explication":"Si la dérivée d'une fonction est nulle sur tout son domaine, alors la fonction est constante (théorème fondamental de l'analyse).","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 est constante\", \"b\": \"f est croissante\", \"c\": \"f est décroissa","_debug_options_count":4}]