Quiz — Corrigé type CTRTL.pdf

Question 1 sur 10 20:00

[{"id":19079,"question":"Quelle est la limite de la fonction f(x) = (2x² + 3x - 5)\/(x² - 1) lorsque x tend vers +∞ ?","option_a":"A. 2","option_b":"B. 1","option_c":"C. +∞","option_d":"D. 0","option_e":"","option_f":"","bonne_reponse":"a","explication":"La limite d'un polynôme en +∞ est donnée par le terme de plus haut degré. 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\": \"a\", \"options\": {\"a\": \"A. 2\", \"b\": \"B. 1\", \"c\": \"C. +∞\", \"d\": \"D. 0\"}}","_debug_options_count":4},{"id":19080,"question":"La fonction f(x) = x³ - 3x + 2 est-elle dérivable en x = 1 ?","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 f est dérivable en x = 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},{"id":19081,"question":"Soit la suite définie par u₀ = 2 et uₙ₊₁ = 3uₙ - 4. Quelle est la valeur de u₂ ?","option_a":"A. 2","option_b":"B. 6","option_c":"C. 14","option_d":"D. 20","option_e":"","option_f":"","bonne_reponse":"c","explication":"u₁ = 3*2 - 4 = 2, u₂ = 3*2 - 4 = 2. Attention à l'erreur courante : u₁ = 2, pas 6.","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. 2\", \"b\": \"B. 6\", \"c\": \"C. 14\", \"d\": \"D. 20\"}}","_debug_options_count":4},{"id":19082,"question":"La fonction f(x) = 1\/x est-elle continue en x = 0 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction 1\/x n'est pas définie en x = 0, donc elle n'est pas continue en ce point.","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":19083,"question":"Quelle est la dérivée de la fonction f(x) = e^(2x) ?","option_a":"A. 2e^(2x)","option_b":"B. e^(2x)","option_c":"C. 2xe^(2x)","option_d":"D. e^(x)","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.","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. 2e^(2x)\", \"b\": \"B. e^(2x)\", \"c\": \"C. 2xe^(2x)\", \"d\": \"D. e^(x)","_debug_options_count":4},{"id":19084,"question":"Soit la suite géométrique de premier terme u₀ = 5 et de raison q = -2. Quel est le terme u₃ ?","option_a":"A. -40","option_b":"B. 40","option_c":"C. -20","option_d":"D. 20","option_e":"","option_f":"","bonne_reponse":"a","explication":"uₙ = u₀ * qⁿ. Donc u₃ = 5 * (-2)³ = 5 * (-8) = -40.","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. -40\", \"b\": \"B. 40\", \"c\": \"C. -20\", \"d\": \"D. 20\"}}","_debug_options_count":4},{"id":19085,"question":"La fonction f(x) = x² - 4x + 3 admet-elle un minimum en x = 2 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée f'(x) = 2x - 4 s'annule en x = 2, et f''(x) = 2 \u003E 0, donc c'est un minimum.","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":19086,"question":"Quelle est la solution de l'équation ln(x) = 2 ?","option_a":"A. x = e²","option_b":"B. x = 2","option_c":"C. x = e","option_d":"D. x = 1\/e²","option_e":"","option_f":"","bonne_reponse":"a","explication":"ln(x) = 2 équivaut à x = e² par définition du logarithme népérien.","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. x = e²\", \"b\": \"B. x = 2\", \"c\": \"C. x = e\", \"d\": \"D. x = 1\/e²","_debug_options_count":4},{"id":19087,"question":"Soit la suite définie par uₙ = (n² + 1)\/n. Que vaut lim(n→+∞) uₙ ?","option_a":"A. 0","option_b":"B. 1","option_c":"C. +∞","option_d":"D. n","option_e":"","option_f":"","bonne_reponse":"b","explication":"uₙ = n + 1\/n. Lorsque n→+∞, 1\/n→0, donc uₙ→n→+∞, mais la limite est bien +∞.","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. n\"}}","_debug_options_count":4},{"id":19088,"question":"La fonction f(x) = |x| est-elle dérivable en x = 0 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction valeur absolue n'est pas dérivable en x = 0 car la dérivée à gauche (-1) ≠ dérivée à droite (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}]