Quiz — ana2.pdf

Question 1 sur 10 20:00

[{"id":53387,"question":"Quelle est la limite de la fonction f(x) = (2x² + 3x - 5)\/(x² - 4) lorsque x tend vers +∞ ?","option_a":"A. 0","option_b":"B. 2","option_c":"C. +∞","option_d":"D. -∞","option_e":"","option_f":"","bonne_reponse":"b","explication":"La limite d'un quotient de polynômes de même degré est égale au rapport des coefficients dominants. Ici, 2x²\/x² = 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. 2\", \"c\": \"C. +∞\", \"d\": \"D. -∞\"}}","_debug_options_count":4},{"id":53388,"question":"La fonction f(x) = x³ - 3x² + 2 est dérivable sur ℝ.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Toute fonction polynôme est dérivable sur ℝ, donc la proposition est vraie.","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":53389,"question":"Quelle est la dérivée de la fonction f(x) = ln(3x + 1) ?","option_a":"A. 3\/(3x + 1)","option_b":"B. 1\/(3x + 1)","option_c":"C. 3x\/(3x + 1)","option_d":"D. 1\/(x + 1)","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de ln(u) est u'\/u. Ici, u = 3x + 1, donc u' = 3. Ainsi, f'(x) = 3\/(3x + 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. 3\/(3x + 1)\", \"b\": \"B. 1\/(3x + 1)\", \"c\": \"C. 3x\/(3x + 1)\", \"d\":","_debug_options_count":4},{"id":53390,"question":"Soit la suite définie par uₙ = (2n + 1)\/(n + 1). Que vaut sa limite lorsque n tend vers +∞ ?","option_a":"A. 1","option_b":"B. 2","option_c":"C. 0","option_d":"D. +∞","option_e":"","option_f":"","bonne_reponse":"b","explication":"En divisant numérateur et dénominateur par n, on obtient uₙ = (2 + 1\/n)\/(1 + 1\/n). La limite est donc 2\/1 = 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. 1\", \"b\": \"B. 2\", \"c\": \"C. 0\", \"d\": \"D. +∞\"}}","_debug_options_count":4},{"id":53391,"question":"L'intégrale ∫₀¹ (3x² + 2x) dx vaut :","option_a":"A. 1","option_b":"B. 2","option_c":"C. 3","option_d":"D. 4","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'intégrale d'un polynôme se calcule en trouvant une primitive. Ici, ∫(3x² + 2x) dx = x³ + x². Évaluée entre 0 et 1, cela donne (1 + 1) - (0 + 0) = 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. 1\", \"b\": \"B. 2\", \"c\": \"C. 3\", \"d\": \"D. 4\"}}","_debug_options_count":4},{"id":53392,"question":"La fonction f(x) = e^(2x) est solution de l'équation différentielle y' = 2y.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de e^(2x) est 2e^(2x), qui est bien égale à 2 fois la fonction elle-même. La proposition est donc vraie.","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":53393,"question":"Quelle est la limite de la suite uₙ = n\/(n + 1) lorsque n tend vers +∞ ?","option_a":"A. 0","option_b":"B. 1","option_c":"C. +∞","option_d":"D. -1","option_e":"","option_f":"","bonne_reponse":"b","explication":"En divisant numérateur et dénominateur par n, on obtient uₙ = 1\/(1 + 1\/n). La limite est donc 1\/1 = 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\", \"c\": \"C. +∞\", \"d\": \"D. -1\"}}","_debug_options_count":4},{"id":53394,"question":"L'intégrale ∫₀^π sin(x) dx vaut :","option_a":"A. 0","option_b":"B. 1","option_c":"C. 2","option_d":"D. π","option_e":"","option_f":"","bonne_reponse":"c","explication":"La primitive de sin(x) est -cos(x). Évaluée entre 0 et π, cela donne -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},{"id":53395,"question":"Soit f une fonction dérivable sur ℝ telle que f'(x) = 3x² + 2x. Que vaut f(x) ?","option_a":"A. x³ + x² + C","option_b":"B. 6x + 2","option_c":"C. 3x³ + 2x² + C","option_d":"D. x³ + x + C","option_e":"","option_f":"","bonne_reponse":"a","explication":"La primitive de 3x² + 2x est x³ + x² + C, où C est une constante réelle.","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. x³ + x² + C\", \"b\": \"B. 6x + 2\", \"c\": \"C. 3x³ + 2x² + C\", \"","_debug_options_count":4},{"id":53396,"question":"La suite uₙ = (-1)^n est convergente.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La suite (-1)^n oscille entre -1 et 1 et ne converge pas vers une limite unique. Elle est donc divergente.","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}]