Quiz — 674f23dcbe031_corrigé_série N° 15.pdf

Question 1 sur 10 20:00

[{"id":5616,"question":"Quelle est la dérivée de la fonction f(x) = e^(2x) ?","option_a":"A. e^(2x)","option_b":"B. 2e^(2x)","option_c":"C. e^(x)","option_d":"D. 2e^(x)","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée de e^(u(x)) est u'(x)*e^(u(x)). Ici, u(x)=2x, donc u'(x)=2. Ainsi, f'(x)=2e^(2x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"A. e^(2x)\", \"b\": \"B. 2e^(2x)\", \"c\": \"C. e^(x)\", \"d\": \"D. 2e^(x)\"}","_debug_options_count":4},{"id":5617,"question":"Soit une suite (u_n) définie par u_0 = 3 et u_{n+1} = 2u_n - 1. Cette suite est-elle arithmétique ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Une suite arithmétique a une différence constante entre deux termes consécutifs. Ici, u_{n+1} - u_n = u_n - 1, qui n'est pas constant. La suite n'est donc pas arithmétique.","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":5618,"question":"Quelle est la limite de la suite (v_n) définie par v_n = (3n + 1)\/(2n - 5) quand n tend vers l'infini ?","option_a":"A. 0","option_b":"B. 3\/2","option_c":"C. 1","option_d":"D. +∞","option_e":"","option_f":"","bonne_reponse":"b","explication":"Pour les suites rationnelles, la limite est le rapport des coefficients dominants. Ici, 3n\/2n = 3\/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\": \"A. 0\", \"b\": \"B. 3\/2\", \"c\": \"C. 1\", \"d\": \"D. +∞\"}}","_debug_options_count":4},{"id":5619,"question":"Dans une classe de 30 élèves, 18 étudient l'anglais et 12 étudient l'espagnol. 5 élèves étudient les deux langues. Combien d'élèves n'étudient ni l'anglais ni l'espagnol ?","option_a":"A. 5","option_b":"B. 7","option_c":"C. 9","option_d":"D. 11","option_e":"","option_f":"","bonne_reponse":"a","explication":"Utilisons la formule des probabilités totales : 30 - (18 + 12 - 5) = 30 - 25 = 5 élèves n'étudient ni l'anglais ni l'espagnol.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"A. 5\", \"b\": \"B. 7\", \"c\": \"C. 9\", \"d\": \"D. 11\"}}","_debug_options_count":4},{"id":5620,"question":"La fonction f(x) = x^3 - 3x^2 + 4 admet-elle un point d'inflexion en x=1 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Un point d'inflexion est un point où la dérivée seconde s'annule et change de signe. f''(x)=6x-6, donc f''(1)=0. De plus, f''(x) change de signe en x=1 (négatif avant, positif après). Le point (1,2) est bien un point d'inflexion.","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":5621,"question":"Quelle est la solution de l'équation ln(x) + ln(2) = 3 ?","option_a":"A. e^3","option_b":"B. 2e^3","option_c":"C. e^3\/2","option_d":"D. e^(3\/2)","option_e":"","option_f":"","bonne_reponse":"d","explication":"ln(x) + ln(2) = ln(2x) = 3 ⇒ 2x = e^3 ⇒ x = e^3\/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\": \"A. e^3\", \"b\": \"B. 2e^3\", \"c\": \"C. e^3\/2\", \"d\": \"D. e^(3\/2)\"}}","_debug_options_count":4},{"id":5622,"question":"Soit une variable aléatoire X suivant une loi binomiale B(n=10, p=0.3). Quelle est la probabilité P(X=3) ?","option_a":"A. C(10,3)*(0.3)^3*(0.7)^7","option_b":"B. C(10,3)*(0.3)^7*(0.7)^3","option_c":"C. (0.3)^3","option_d":"D. (0.7)^7","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour une loi binomiale, P(X=k) = C(n,k)*p^k*(1-p)^(n-k). Ici, P(X=3) = C(10,3)*(0.3)^3*(0.7)^7.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"A. C(10,3)*(0.3)^3*(0.7)^7\", \"b\": \"B. C(10,3)*(0.3)^7*(0.7)^3\", \"","_debug_options_count":4},{"id":5623,"question":"La fonction f(x) = x^2 - 4x + 5 admet-elle un minimum en x=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 s'annule en x=2. La dérivée seconde f''(x)=2 est positive, donc la fonction admet un minimum en x=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":5624,"question":"Quelle est la valeur de la somme S = 1 + 2 + 4 + ... + 2^n ?","option_a":"A. 2^(n+1) - 1","option_b":"B. 2^n - 1","option_c":"C. n*2^(n+1)","option_d":"D. 2^(n)","option_e":"","option_f":"","bonne_reponse":"a","explication":"Il s'agit d'une suite géométrique de raison 2. La somme des n+1 premiers termes est S = 2^(n+1) - 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\": \"A. 2^(n+1) - 1\", \"b\": \"B. 2^n - 1\", \"c\": \"C. n*2^(n+1)\", \"d\": \"D.","_debug_options_count":4},{"id":5625,"question":"Soit A et B deux événements indépendants avec P(A)=0.4 et P(B)=0.5. Quelle est P(A ∩ B) ?","option_a":"A. 0.2","option_b":"B. 0.9","option_c":"C. 0.1","option_d":"D. 0.45","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour des événements indépendants, P(A ∩ B) = P(A)*P(B) = 0.4*0.5 = 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\": \"A. 0.2\", \"b\": \"B. 0.9\", \"c\": \"C. 0.1\", \"d\": \"D. 0.45\"}}","_debug_options_count":4}]