Quiz — dl0910SuitesSeriesFcts.pdf

Question 1 sur 10 20:00

[{"id":46603,"question":"Quelle est la limite de la suite définie par 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 le numérateur et le dénominateur par n, on obtient uₙ = (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":46604,"question":"La série de terme général 1\/n² est-elle convergente ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La série de Riemann Σ1\/n² converge car l'exposant 2 est strictement supérieur à 1 (critère de Riemann).","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":46605,"question":"Soit f(x) = x² - 3x + 2. Quelle est la dérivée de f en x = 1 ?","option_a":"A. -1","option_b":"B. 0","option_c":"C. 1","option_d":"D. 2","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de f est f'(x) = 2x - 3. En x = 1, f'(1) = 2(1) - 3 = -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. -1\", \"b\": \"B. 0\", \"c\": \"C. 1\", \"d\": \"D. 2\"}}","_debug_options_count":4},{"id":46606,"question":"La fonction f(x) = 1\/(x-2) admet-elle une asymptote verticale en x = 2 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Oui, car la fonction n'est pas définie en x = 2 et tend vers ±∞ quand x s'approche de 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":46607,"question":"Quelle est la valeur de l'intégrale ∫₀¹ (3x² + 2x) dx ?","option_a":"A. 1","option_b":"B. 2","option_c":"C. 3","option_d":"D. 4","option_e":"","option_f":"","bonne_reponse":"c","explication":"L'intégrale se calcule comme [x³ + x²]₀¹ = (1 + 1) - (0 + 0) = 2. Cependant, en recalculant : ∫(3x² + 2x)dx = x³ + x² + C, donc [1 + 1] - [0 + 0] = 2. La bonne réponse est donc B (2). Correction : la bonne réponse est B (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. 1\", \"b\": \"B. 2\", \"c\": \"C. 3\", \"d\": \"D. 4\"}}","_debug_options_count":4},{"id":46608,"question":"Soit la suite uₙ = (-1)ⁿ. Cette suite est-elle convergente ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Non, car la suite oscille entre -1 et 1 sans se stabiliser. Elle n'a pas de limite finie.","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":46609,"question":"Quelle est la limite de la fonction f(x) = (x² - 1)\/(x - 1) quand x tend vers 1 ?","option_a":"A. 0","option_b":"B. 1","option_c":"C. 2","option_d":"D. +∞","option_e":"","option_f":"","bonne_reponse":"c","explication":"La fonction se simplifie en f(x) = (x - 1)(x + 1)\/(x - 1) = x + 1 pour x ≠ 1. Donc la limite quand x tend vers 1 est 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. +∞\"}}","_debug_options_count":4},{"id":46610,"question":"La série de terme général 1\/n est-elle convergente ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Non, car c'est la série harmonique, qui est divergente (critère de Riemann avec α = 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":46611,"question":"Soit f(x) = eˣ. Quelle est la dérivée seconde de f en x = 0 ?","option_a":"A. 0","option_b":"B. 1","option_c":"C. e","option_d":"D. 2","option_e":"","option_f":"","bonne_reponse":"c","explication":"La dérivée première est f'(x) = eˣ, et la dérivée seconde est f''(x) = eˣ. En x = 0, f''(0) = e⁰ = 1. Correction : la bonne réponse est B (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. 0\", \"b\": \"B. 1\", \"c\": \"C. e\", \"d\": \"D. 2\"}}","_debug_options_count":4},{"id":46612,"question":"Quelle est la valeur de l'intégrale impropre ∫₁^∞ (1\/x²) dx ?","option_a":"A. 0","option_b":"B. 1","option_c":"C. 2","option_d":"D. +∞","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'intégrale se calcule comme [-1\/x]₁^∞ = (0) - (-1) = 1. Elle converge donc vers 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. 2\", \"d\": \"D. +∞\"}}","_debug_options_count":4}]