Quiz — série 10.pdf

Question 1 sur 10 20:00

[{"id":74025,"question":"Quelle est la limite de la suite définie par uₙ = (3n² + 2n - 1)\/(2n² - n + 4) quand n tend vers +∞ ?","option_a":"A. 0","option_b":"B. 3\/2","option_c":"C. +∞","option_d":"D. 1","option_e":"","option_f":"","bonne_reponse":"b","explication":"La limite d'une suite rationnelle est égale au rapport des coefficients dominants. Ici, 3n²\/2n² = 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. +∞\", \"d\": \"D. 1\"}}","_debug_options_count":4},{"id":74026,"question":"Soit f une fonction dérivable sur ℝ telle que f'(x) = 2x - 3 pour tout x. Si f(0) = 1, alors f(2) est égal à :","option_a":"A. 1","option_b":"B. 3","option_c":"C. -1","option_d":"D. 0","option_e":"","option_f":"","bonne_reponse":"d","explication":"f(x) = x² - 3x + C. Avec f(0)=1, on trouve C=1. Donc f(2)=4-6+1=-1.","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. 1\", \"b\": \"B. 3\", \"c\": \"C. -1\", \"d\": \"D. 0\"}}","_debug_options_count":4},{"id":74027,"question":"Vrai ou Faux ? La fonction exponentielle est strictement croissante sur ℝ.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction exponentielle a une dérivée toujours positive (eˣ \u003E 0), donc elle est strictement croissante.","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":74028,"question":"Quelle est la solution générale de l'équation différentielle y' + 2y = 0 ?","option_a":"A. y = Ceˣ","option_b":"B. y = Ce⁻²ˣ","option_c":"C. y = Ce²ˣ","option_d":"D. y = Cx","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'équation est de la forme y' + ay = 0. La solution est y = Ce⁻ᵃˣ, ici a=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. y = Ceˣ\", \"b\": \"B. y = Ce⁻²ˣ\", \"c\": \"C. y = Ce²ˣ\", \"d\":","_debug_options_count":4},{"id":74029,"question":"Vrai ou Faux ? Si deux événements A et B sont incompatibles, alors P(A ∪ B) = P(A) + P(B).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est la définition de l'additivité des probabilités pour des événements incompatibles.","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":74030,"question":"Soit f(x) = ln(x² + 1). Quelle est la dérivée de f en x = 1 ?","option_a":"A. 0","option_b":"B. 1\/2","option_c":"C. 1","option_d":"D. 2","option_e":"","option_f":"","bonne_reponse":"b","explication":"f'(x) = (2x)\/(x²+1). En x=1, f'(1)=2\/2=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. 0\", \"b\": \"B. 1\/2\", \"c\": \"C. 1\", \"d\": \"D. 2\"}}","_debug_options_count":4},{"id":74031,"question":"Vrai ou Faux ? Toute fonction continue sur [a,b] est dérivable sur ]a,b[.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Contre-exemple : la fonction valeur absolue est continue mais non dérivable en 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":74032,"question":"Quelle est la probabilité d'obtenir exactement 2 fois 'Pile' en 4 lancers d'une pièce équilibrée ?","option_a":"A. 1\/4","option_b":"B. 3\/8","option_c":"C. 1\/2","option_d":"D. 1\/16","option_e":"","option_f":"","bonne_reponse":"b","explication":"C'est une loi binomiale B(4,1\/2). P(X=2) = C(4,2)*(1\/2)^4 = 6\/16 = 3\/8.","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\/4\", \"b\": \"B. 3\/8\", \"c\": \"C. 1\/2\", \"d\": \"D. 1\/16\"}}","_debug_options_count":4},{"id":74033,"question":"Soit (uₙ) une suite arithmétique de raison r = -2 et de premier terme u₀ = 5. Quelle est la valeur de u₁₀ ?","option_a":"A. -15","option_b":"B. 5","option_c":"C. -5","option_d":"D. 15","option_e":"","option_f":"","bonne_reponse":"a","explication":"uₙ = u₀ + nr. Donc u₁₀ = 5 + 10*(-2) = 5 - 20 = -15.","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. -15\", \"b\": \"B. 5\", \"c\": \"C. -5\", \"d\": \"D. 15\"}}","_debug_options_count":4},{"id":74034,"question":"Vrai ou Faux ? Si une fonction f est dérivable en a, alors elle est continue en a.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivabilité implique la continuité (théorème fondamental).","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}]