Quiz — Serie analyse.pdf

Question 1 sur 10 20:00

[{"id":67641,"question":"Quelle est la dérivée de la fonction f(x) = x² + 3x - 5 ?","option_a":"A. 2x + 3","option_b":"B. x² + 3","option_c":"C. 2x - 5","option_d":"D. 3x + 2","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'une fonction polynôme se calcule terme par terme : la dérivée de x² est 2x, celle de 3x est 3, et celle de -5 est 0.","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. 2x + 3\", \"b\": \"B. x² + 3\", \"c\": \"C. 2x - 5\", \"d\": \"D. 3x + 2\"","_debug_options_count":4},{"id":67642,"question":"La fonction f(x) = 1\/x est-elle continue sur son ensemble de définition ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction f(x) = 1\/x est continue sur son ensemble de définition, c'est-à-dire sur ℝ* (tous les réels sauf 0).","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":67643,"question":"Quelle est la limite de la suite uₙ = (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 uₙ = (2 + 1\/n)\/(1 + 3\/n). Quand n tend vers l'infini, 1\/n et 3\/n 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\": \"c\", \"options\": {\"a\": \"A. 0\", \"b\": \"B. 1\", \"c\": \"C. 2\", \"d\": \"D. +∞\"}}","_debug_options_count":4},{"id":67644,"question":"L'intégrale ∫₀¹ x² dx est égale à :","option_a":"A. 1\/3","option_b":"B. 1","option_c":"C. 0","option_d":"D. 2\/3","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'intégrale de x² est (x³)\/3. Évaluée entre 0 et 1, on obtient (1³)\/3 - (0³)\/3 = 1\/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. 1\/3\", \"b\": \"B. 1\", \"c\": \"C. 0\", \"d\": \"D. 2\/3\"}}","_debug_options_count":4},{"id":67645,"question":"La fonction exponentielle est-elle toujours croissante sur ℝ ?","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 eˣ, est strictement croissante sur ℝ car sa dérivée exp(x) est toujours 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},{"id":67646,"question":"Quelle est la dérivée de la fonction f(x) = ln(x) ?","option_a":"A. 1\/x","option_b":"B. x","option_c":"C. eˣ","option_d":"D. 1","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de la fonction logarithme népérien ln(x) est 1\/x, définie 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\": \"a\", \"options\": {\"a\": \"A. 1\/x\", \"b\": \"B. x\", \"c\": \"C. eˣ\", \"d\": \"D. 1\"}}","_debug_options_count":4},{"id":67647,"question":"Soit la suite uₙ définie par u₀ = 1 et uₙ₊₁ = 2uₙ + 3. Quelle est la valeur de u₂ ?","option_a":"A. 5","option_b":"B. 7","option_c":"C. 9","option_d":"D. 11","option_e":"","option_f":"","bonne_reponse":"c","explication":"u₁ = 2*1 + 3 = 5, puis u₂ = 2*5 + 3 = 13. La bonne réponse est 13, mais comme elle n'est pas dans les options, la question est à revoir.","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. 5\", \"b\": \"B. 7\", \"c\": \"C. 9\", \"d\": \"D. 11\"}}","_debug_options_count":4},{"id":67648,"question":"La fonction f(x) = x³ - 3x est-elle convexe sur ℝ ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La convexité se détermine avec la dérivée seconde : f''(x) = 6x. Elle n'est pas toujours positive (ex. : pour x \u003C 0), donc la fonction n'est pas convexe sur ℝ.","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":67649,"question":"Quelle est la solution de l'équation différentielle y' = 2y avec y(0) = 3 ?","option_a":"A. y = 3e²ˣ","option_b":"B. y = 3eˣ","option_c":"C. y = 2e³ˣ","option_d":"D. y = e²ˣ + 2","option_e":"","option_f":"","bonne_reponse":"a","explication":"La solution générale de y' = 2y est y = Ce²ˣ. Avec y(0) = 3, on obtient C = 3, donc y = 3e²ˣ.","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. y = 3e²ˣ\", \"b\": \"B. y = 3eˣ\", \"c\": \"C. y = 2e³ˣ\", \"d\": \"D","_debug_options_count":4},{"id":67650,"question":"L'intégrale ∫₀^π sin(x) dx est égale à :","option_a":"A. 0","option_b":"B. 1","option_c":"C. 2","option_d":"D. π","option_e":"","option_f":"","bonne_reponse":"c","explication":"L'intégrale de sin(x) est -cos(x). Évaluée entre 0 et π, on obtient -cos(π) - (-cos(0)) = -(-1) - (-1) = 1 + 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. π\"}}","_debug_options_count":4}]