Quiz — cntrl 2 math.pdf

Question 1 sur 10 20:00

[{"id":27090,"question":"Quelle est la dérivée de la fonction f(x) = 3x² + 2x - 5 ?","option_a":"6x + 2","option_b":"3x² + 2","option_c":"6x + 2x","option_d":"6x - 5","option_e":"","option_f":"","bonne_reponse":"a","explication":"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 + 2x\", \"d\": \"6x - 5\"}}","_debug_options_count":4},{"id":27091,"question":"Une suite convergente est toujours bornée.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Une suite convergente est toujours bornée, car elle se rapproche d'une limite finie et ne peut donc pas tendre vers l'infini.","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":27092,"question":"Soit une fonction f définie par f(x) = (2x + 1)\/(x - 3). Quelle est sa limite en +∞ ?","option_a":"2","option_b":"0","option_c":"1","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). En +∞, 1\/x et 3\/x tendent vers 0, donc 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\": \"0\", \"c\": \"1\", \"d\": \"∞\"}}","_debug_options_count":4},{"id":27093,"question":"Dans une loi binomiale de paramètres n et p, l'espérance est égale à :","option_a":"n + p","option_b":"n * p","option_c":"p","option_d":"n \/ p","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'espérance d'une loi binomiale B(n, p) est n * p, où n est le nombre d'épreuves et p la probabilité de succès.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"n + p\", \"b\": \"n * p\", \"c\": \"p\", \"d\": \"n \/ p\"}}","_debug_options_count":4},{"id":27094,"question":"La fonction exponentielle est toujours positive.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction exponentielle, notée exp(x) ou e^x, est toujours 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},{"id":27095,"question":"Résoudre l'équation différentielle y' = 2y avec y(0) = 3.","option_a":"y = 3e^(2x)","option_b":"y = 2e^(3x)","option_c":"y = 3e^(-2x)","option_d":"y = 2e^(-3x)","option_e":"","option_f":"","bonne_reponse":"a","explication":"La solution générale de y' = 2y est y = Ce^(2x). Avec y(0) = 3, on trouve C = 3, donc y = 3e^(2x).","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 = 2e^(3x)\", \"c\": \"y = 3e^(-2x)\", \"d\": \"y = ","_debug_options_count":4},{"id":27096,"question":"Soit une variable aléatoire X suivant une loi normale N(μ, σ²). Que vaut P(μ - σ ≤ X ≤ μ + σ) ?","option_a":"0,68","option_b":"0,95","option_c":"0,99","option_d":"0,5","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour une loi normale, environ 68% des valeurs se situent dans l'intervalle [μ - σ, μ + σ].","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"0,68\", \"b\": \"0,95\", \"c\": \"0,99\", \"d\": \"0,5\"}}","_debug_options_count":4},{"id":27097,"question":"La fonction f(x) = x^3 est convexe sur ℝ.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction f(x) = x^3 est concave sur ℝ car sa dérivée seconde f''(x) = 6x est négative pour x \u003C 0 et positive pour x \u003E 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":27098,"question":"Soit une suite définie par u₀ = 1 et u_{n+1} = 2u_n + 3. Quelle est la limite de cette suite ?","option_a":"3","option_b":"∞","option_c":"0","option_d":"2","option_e":"","option_f":"","bonne_reponse":"a","explication":"La suite converge vers une limite L telle que L = 2L + 3, donc L = -3. Cependant, comme u₀ = 1 \u003E -3, la suite diverge vers +∞. La bonne réponse est donc ∞.","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\", \"b\": \"∞\", \"c\": \"0\", \"d\": \"2\"}}","_debug_options_count":4},{"id":27099,"question":"Soit f une fonction dérivable sur ℝ. Si f'(x) \u003E 0 pour tout x, alors f est strictement croissante.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Si f'(x) \u003E 0 pour tout x, alors f est strictement croissante sur ℝ, car sa dérivée ne s'annule jamais et reste positive.","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}]