Quiz — 664a2e490991a_Epreuve N°4.pdf

Question 1 sur 10 20:00

[{"id":92528,"question":"Quelle est la limite de la suite définie par u_n = (2n + 1)\/(n + 3) quand n tend vers l'infini ?","option_a":"A. 0","option_b":"B. 1","option_c":"C. 2","option_d":"D. +∞","option_e":"","option_f":"","bonne_reponse":"c","explication":"En divisant numérateur et dénominateur par n, on obtient u_n = (2 + 1\/n)\/(1 + 3\/n). Quand n tend vers l'infini, 1\/n tend 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\": \"c\", \"options\": {\"a\": \"A. 0\", \"b\": \"B. 1\", \"c\": \"C. 2\", \"d\": \"D. +∞\"}}","_debug_options_count":4},{"id":92529,"question":"La fonction f(x) = x^3 - 3x^2 + 2 est-elle croissante sur l'intervalle [0, 2] ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée f'(x) = 3x^2 - 6x s'annule en x=0 et x=2. Sur [0,2], f'(x) ≤ 0, donc la fonction est décroissante sur cet intervalle.","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":92530,"question":"Soit X une variable aléatoire suivant une loi binomiale B(10, 0.3). Quelle est l'espérance de X ?","option_a":"A. 3","option_b":"B. 7","option_c":"C. 10","option_d":"D. 0.3","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour une loi binomiale B(n,p), l'espérance est E(X) = n*p = 10*0.3 = 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\": \"A. 3\", \"b\": \"B. 7\", \"c\": \"C. 10\", \"d\": \"D. 0.3\"}}","_debug_options_count":4},{"id":92531,"question":"L'équation x^2 - 5x + 6 = 0 admet-elle 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 = 25 - 24 = 1 \u003E 0, donc l'équation admet 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":92532,"question":"Quelle est la dérivée de la fonction f(x) = ln(2x + 1) ?","option_a":"A. 1\/(2x + 1)","option_b":"B. 2\/(2x + 1)","option_c":"C. 1\/(x + 1)","option_d":"D. 2x\/(2x + 1)","option_e":"","option_f":"","bonne_reponse":"b","explication":"En utilisant la formule de dérivation de ln(u), on obtient f'(x) = u'(x)\/u(x) = 2\/(2x + 1).","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. 1\/(2x + 1)\", \"b\": \"B. 2\/(2x + 1)\", \"c\": \"C. 1\/(x + 1)\", \"d\": \"","_debug_options_count":4},{"id":92533,"question":"Soit une suite géométrique de premier terme u_0 = 5 et de raison q = -2. Quel est le terme u_3 ?","option_a":"A. -40","option_b":"B. 40","option_c":"C. 20","option_d":"D. -20","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le terme général d'une suite géométrique est u_n = u_0 * q^n. Donc u_3 = 5 * (-2)^3 = 5 * (-8) = -40.","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. -40\", \"b\": \"B. 40\", \"c\": \"C. 20\", \"d\": \"D. -20\"}}","_debug_options_count":4},{"id":92534,"question":"La fonction f(x) = e^x est-elle convexe sur ℝ ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée seconde f''(x) = e^x est toujours positive sur ℝ, donc la fonction est convexe.","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":92535,"question":"Quel est le coefficient directeur de la tangente à la courbe de f(x) = x^2 + 3x - 1 au point d'abscisse x=1 ?","option_a":"A. 1","option_b":"B. 5","option_c":"C. 3","option_d":"D. 2","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée f'(x) = 2x + 3 donne le coefficient directeur. En x=1, f'(1) = 2*1 + 3 = 5.","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. 1\", \"b\": \"B. 5\", \"c\": \"C. 3\", \"d\": \"D. 2\"}}","_debug_options_count":4},{"id":92536,"question":"Soit X une variable aléatoire suivant une loi normale N(0,1). Quelle est la probabilité P(X ≤ 1) ?","option_a":"A. 0.6826","option_b":"B. 0.8413","option_c":"C. 0.9772","option_d":"D. 0.5","option_e":"","option_f":"","bonne_reponse":"b","explication":"D'après la table de la loi normale centrée réduite, P(X ≤ 1) ≈ 0.8413.","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. 0.6826\", \"b\": \"B. 0.8413\", \"c\": \"C. 0.9772\", \"d\": \"D. 0.5\"}}","_debug_options_count":4},{"id":92537,"question":"L'inéquation x^2 - 4x + 3 \u003E 0 est-elle vérifiée pour x ∈ ]1, 3[ ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Les racines de l'équation x^2 - 4x + 3 = 0 sont x=1 et x=3. Le trinôme est positif à l'extérieur des racines, donc l'inéquation est fausse sur ]1,3[.","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}]