Quiz — 6379~1.DOC

Question 1 sur 10 20:00

[{"id":11903,"question":"Quelle est la limite de la suite définie par uₙ = (3n² + 2n - 1)\/(2n² + 5) lorsque n tend vers l'infini ?","option_a":"A. 3\/2","option_b":"B. 0","option_c":"C. +∞","option_d":"D. 1","option_e":"","option_f":"","bonne_reponse":"a","explication":"La limite d'une suite rationnelle est égale au rapport des coefficients des termes de plus haut degré. Ici, 3n²\/2n² = 3\/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. 3\/2\", \"b\": \"B. 0\", \"c\": \"C. +∞\", \"d\": \"D. 1\"}}","_debug_options_count":4},{"id":11904,"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, donc la fonction n'est pas strictement croissante sur tout 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":11905,"question":"Quelle est la solution générale de l'équation différentielle y'' + 4y = 0 ?","option_a":"A. y = Ae^(2x) + Be^(-2x)","option_b":"B. y = A cos(2x) + B sin(2x)","option_c":"C. y = Ax + B","option_d":"D. y = Ae^(4x) + Be^(-4x)","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'équation caractéristique est r² + 4 = 0, soit r = ±2i. La solution générale est donc y = A cos(2x) + B sin(2x).","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. y = Ae^(2x) + Be^(-2x)\", \"b\": \"B. y = A cos(2x) + B sin(2x)\", ","_debug_options_count":4},{"id":11906,"question":"Si X suit une loi normale N(0, 1), quelle est la probabilité P(-1 ≤ X ≤ 1) ?","option_a":"A. 0,3413","option_b":"B. 0,6826","option_c":"C. 0,9544","option_d":"D. 0,9974","option_e":"","option_f":"","bonne_reponse":"b","explication":"Pour une loi normale centrée réduite, environ 68,26% des valeurs sont dans l'intervalle [-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. 0,3413\", \"b\": \"B. 0,6826\", \"c\": \"C. 0,9544\", \"d\": \"D. 0,9974\"}","_debug_options_count":4},{"id":11907,"question":"Le nombre complexe z = 3 + 4i a pour module |z| = 5. Vrai ou faux ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le module de z = a + bi est √(a² + b²). Ici, |z| = √(3² + 4²) = 5, donc l'affirmation 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":11908,"question":"Quelle est la dérivée de la fonction f(x) = ln(2x + 1) ?","option_a":"A. 1\/(2x + 1)","option_b":"B. 2\/(2x + 1)","option_c":"C. 1\/(x + 1)","option_d":"D. 2x\/(2x + 1)","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée de ln(u) est u'\/u. Ici, u = 2x + 1, donc u' = 2. Ainsi, f'(x) = 2\/(2x + 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. 1\/(2x + 1)\", \"b\": \"B. 2\/(2x + 1)\", \"c\": \"C. 1\/(x + 1)\", \"d\": \"","_debug_options_count":4},{"id":11909,"question":"Soit une variable aléatoire X suivant une loi binomiale B(n=10, p=0,3). Quelle est l'espérance E(X) ?","option_a":"A. 3","option_b":"B. 7","option_c":"C. 10","option_d":"D. 0,3","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour une loi binomiale B(n, p), l'espérance est E(X) = n × p = 10 × 0,3 = 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. 3\", \"b\": \"B. 7\", \"c\": \"C. 10\", \"d\": \"D. 0,3\"}}","_debug_options_count":4},{"id":11910,"question":"La fonction f(x) = e^(-x²) est-elle paire ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Une fonction f est paire si f(-x) = f(x). Ici, f(-x) = e^(-(-x)²) = e^(-x²) = f(x), donc la fonction est paire.","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":11911,"question":"Quelle est la primitive de la fonction f(x) = 5x⁴ - 3x² + 2 ?","option_a":"A. x⁵ - x³ + 2x + C","option_b":"B. 5x⁵ - x³ + 2x + C","option_c":"C. x⁵ - 3x³ + 2x + C","option_d":"D. 20x³ - 6x + C","option_e":"","option_f":"","bonne_reponse":"a","explication":"La primitive de xⁿ est x^(n+1)\/(n+1). Ainsi, ∫(5x⁴ - 3x² + 2) dx = x⁵ - x³ + 2x + C.","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. x⁵ - x³ + 2x + C\", \"b\": \"B. 5x⁵ - x³ + 2x + C\", \"c\": \"C.","_debug_options_count":4},{"id":11912,"question":"Le produit scalaire de deux vecteurs u et v est-il commutatif ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le produit scalaire est commutatif : u·v = v·u. Cette propriété est fondamentale en géométrie vectorielle.","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}]