Quiz — M211-Correction-Devoir-2.pdf

Question 1 sur 10 20:00

[{"id":38870,"question":"Quelle est la limite de la suite définie par uₙ = (3n² + 2n - 1)\/(2n² + 5) lorsque n tend vers l’infini ?","option_a":"A) 3\/2","option_b":"B) 0","option_c":"C) +∞","option_d":"D) 1","option_e":"","option_f":"","bonne_reponse":"a","explication":"La limite d’une suite rationnelle est égale au rapport des coefficients des termes de plus haut degré. 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\": \"a\", \"options\": {\"a\": \"A) 3\/2\", \"b\": \"B) 0\", \"c\": \"C) +∞\", \"d\": \"D) 1\"}}","_debug_options_count":4},{"id":38871,"question":"Soit f(x) = x³ - 3x² + 4. La fonction f est croissante sur l’intervalle [-1, 3].","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée f’(x) = 3x² - 6x s’annule en x=0 et x=2. Le signe de f’(x) montre que f est décroissante sur [0, 2], donc l’affirmation est fausse.","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":38872,"question":"Dans une classe de 30 élèves, 18 sont des filles. On choisit un élève au hasard. Quelle est la probabilité que ce soit une fille sachant qu’il porte des lunettes (sachant que 10 élèves portent des lunettes dont 6 filles) ?","option_a":"A) 6\/10","option_b":"B) 6\/30","option_c":"C) 6\/18","option_d":"D) 18\/30","option_e":"","option_f":"","bonne_reponse":"c","explication":"C’est une probabilité conditionnelle : P(fille | lunettes) = P(fille ∩ lunettes) \/ P(lunettes) = 6\/10.","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) 6\/10\", \"b\": \"B) 6\/30\", \"c\": \"C) 6\/18\", \"d\": \"D) 18\/30\"}}","_debug_options_count":4},{"id":38873,"question":"L’équation différentielle y’ + 2y = 0 admet pour solution générale y(x) = Ce^(-2x), où C est une constante réelle.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C’est une équation différentielle linéaire du premier ordre. La solution générale est bien y(x) = Ce^(-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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":38874,"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) 25","option_e":"","option_f":"","bonne_reponse":"a","explication":"La formule d’une suite arithmétique est uₙ = u₀ + n*r. Donc u₁₀ = 5 + 10*(-2) = -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) 25\"}}","_debug_options_count":4},{"id":38875,"question":"La fonction f(x) = e^(2x) a pour dérivée f’(x) = 2e^(2x).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de e^(u(x)) est u’(x)*e^(u(x)). Ici, u(x) = 2x, donc u’(x) = 2. La dérivée est bien 2e^(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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":38876,"question":"Un sac contient 5 boules rouges et 3 boules bleues. On tire successivement et sans remise 3 boules. Quelle est la probabilité d’obtenir exactement 2 boules rouges ?","option_a":"A) 15\/56","option_b":"B) 30\/56","option_c":"C) 10\/56","option_d":"D) 5\/8","option_e":"","option_f":"","bonne_reponse":"a","explication":"C’est une loi hypergéométrique. Le nombre de cas favorables est C(5,2)*C(3,1) = 10*3 = 30. Le nombre total de tirages est C(8,3) = 56. Donc P = 30\/56.","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\/56\", \"b\": \"B) 30\/56\", \"c\": \"C) 10\/56\", \"d\": \"D) 5\/8\"}}","_debug_options_count":4},{"id":38877,"question":"L’équation différentielle y’ = y² admet pour solution y(x) = 1\/(C - x), où C est une constante réelle.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C’est une équation différentielle à variables séparables. La solution générale est bien y(x) = 1\/(C - x).","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":38878,"question":"Soit f(x) = ln(x² + 1). Quelle est la dérivée de f en x = 1 ?","option_a":"A) 1","option_b":"B) 0","option_c":"C) 2","option_d":"D) 1\/2","option_e":"","option_f":"","bonne_reponse":"c","explication":"La dérivée de ln(u(x)) est u’(x)\/u(x). Ici, u(x) = x² + 1, donc u’(x) = 2x. En x=1, f’(1) = 2*1\/(1² + 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\": \"c\", \"options\": {\"a\": \"A) 1\", \"b\": \"B) 0\", \"c\": \"C) 2\", \"d\": \"D) 1\/2\"}}","_debug_options_count":4},{"id":38879,"question":"La suite (uₙ) définie par u₀ = 1 et uₙ₊₁ = (uₙ + 2)\/3 est convergente.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Cette suite est contractante (|uₙ₊₁ - L| ≤ k|uₙ - L| avec k \u003C 1), donc elle converge vers la solution de L = (L + 2)\/3, soit L = 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4}]