Quiz — 6654865a5681a_Enoncé-Révision.pdf

Question 1 sur 10 20:00

[{"id":82727,"question":"Quelle est la limite de la fonction f(x) = (2x² + 3x - 5)\/(x² - 4) lorsque x tend vers l'infini ?","option_a":"A. 2","option_b":"B. 0","option_c":"C. +∞","option_d":"D. -∞","option_e":"","option_f":"","bonne_reponse":"a","explication":"La limite d'un polynôme de même degré au numérateur et au dénominateur 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\": \"a\", \"options\": {\"a\": \"A. 2\", \"b\": \"B. 0\", \"c\": \"C. +∞\", \"d\": \"D. -∞\"}}","_debug_options_count":4},{"id":82728,"question":"Une suite géométrique de raison q = -2 et de premier terme u₀ = 3 est-elle convergente ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Une suite géométrique converge uniquement si la raison q vérifie |q| \u003C 1. Ici, |q| = 2 \u003E 1, donc la suite diverge.","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":82729,"question":"Soit f une fonction dérivable sur ℝ telle que f'(x) = 3x² - 6x + 2. Quel est le sens de variation de f sur l'intervalle [0, 2] ?","option_a":"A. Croissante","option_b":"B. Décroissante","option_c":"C. Non monotone","option_d":"D. Constante","option_e":"","option_f":"","bonne_reponse":"c","explication":"Le signe de f'(x) = 3x² - 6x + 2 change sur [0, 2] (racines en x ≈ 0.42 et x ≈ 1.58). f est donc non monotone sur cet intervalle.","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. Croissante\", \"b\": \"B. Décroissante\", \"c\": \"C. Non monotone\", ","_debug_options_count":4},{"id":82730,"question":"Dans une urne contenant 5 boules rouges et 3 boules bleues, quelle est la probabilité de tirer deux boules rouges sans remise ?","option_a":"A. 5\/8","option_b":"B. 10\/28","option_c":"C. 25\/64","option_d":"D. 15\/56","option_e":"","option_f":"","bonne_reponse":"b","explication":"La probabilité est (5\/8) * (4\/7) = 20\/56 = 10\/28. On multiplie les probabilités successives sans remise.","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. 5\/8\", \"b\": \"B. 10\/28\", \"c\": \"C. 25\/64\", \"d\": \"D. 15\/56\"}}","_debug_options_count":4},{"id":82731,"question":"L'équation différentielle y' + 2y = 0 admet-elle une solution de la forme y = Ce^(-2x) ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"En substituant y = Ce^(-2x) dans l'équation, on obtient y' = -2Ce^(-2x), donc y' + 2y = -2Ce^(-2x) + 2Ce^(-2x) = 0. C'est bien une solution.","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":82732,"question":"Quelle est la dérivée de la fonction f(x) = ln(3x² + 1) ?","option_a":"A. 6x\/(3x² + 1)","option_b":"B. 3x\/(3x² + 1)","option_c":"C. 6x²\/(3x² + 1)","option_d":"D. 1\/(3x² + 1)","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de ln(u) est u'\/u. Ici, u = 3x² + 1, donc u' = 6x. Ainsi, f'(x) = 6x\/(3x² + 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. 6x\/(3x² + 1)\", \"b\": \"B. 3x\/(3x² + 1)\", \"c\": \"C. 6x²\/(3x² +","_debug_options_count":4},{"id":82733,"question":"Un dé équilibré à 6 faces est lancé deux fois. Quelle est la probabilité d'obtenir un double 4 ?","option_a":"A. 1\/6","option_b":"B. 1\/36","option_c":"C. 1\/18","option_d":"D. 1\/12","option_e":"","option_f":"","bonne_reponse":"b","explication":"Il y a 6 * 6 = 36 issues possibles. Seul un cas (4,4) est favorable, donc la probabilité est 1\/36.","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\/6\", \"b\": \"B. 1\/36\", \"c\": \"C. 1\/18\", \"d\": \"D. 1\/12\"}}","_debug_options_count":4},{"id":82734,"question":"La fonction f(x) = x³ - 3x + 2 est-elle strictement croissante sur ℝ ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée f'(x) = 3x² - 3 s'annule en x = -1 et x = 1. f n'est donc pas strictement croissante 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":82735,"question":"Quelle est la solution générale de l'équation différentielle y'' - 5y' + 6y = 0 ?","option_a":"A. y = Ce^x + De^(-6x)","option_b":"B. y = Ce^(2x) + De^(3x)","option_c":"C. y = Ce^(3x) + De^(2x)","option_d":"D. y = Ce^(-2x) + De^(-3x)","option_e":"","option_f":"","bonne_reponse":"c","explication":"L'équation caractéristique est r² - 5r + 6 = 0, de racines r = 2 et r = 3. La solution générale est donc y = Ce^(2x) + De^(3x).","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. y = Ce^x + De^(-6x)\", \"b\": \"B. y = Ce^(2x) + De^(3x)\", \"c\": \"C","_debug_options_count":4},{"id":82736,"question":"Dans un triangle rectangle, si l'hypoténuse mesure 10 cm et un angle aigu mesure 30°, quelle est la longueur du côté opposé à cet angle ?","option_a":"A. 5 cm","option_b":"B. 5√3 cm","option_c":"C. 10 cm","option_d":"D. 10\/√3 cm","option_e":"","option_f":"","bonne_reponse":"a","explication":"Dans un triangle rectangle, sin(30°) = côté opposé \/ hypoténuse. Donc côté opposé = 10 * sin(30°) = 10 * 0.5 = 5 cm.","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. 5 cm\", \"b\": \"B. 5√3 cm\", \"c\": \"C. 10 cm\", \"d\": \"D. 10\/√3 c","_debug_options_count":4}]