Quiz — M-PT-SRF-JMF-2.pdf

Question 1 sur 10 20:00

[{"id":101202,"question":"Quelle est la limite de la suite définie par uₙ = (3n² + 2n - 1)\/(n² + 5) quand n tend vers +∞ ?","option_a":"A. 0","option_b":"B. 3","option_c":"C. +∞","option_d":"D. -∞","option_e":"","option_f":"","bonne_reponse":"b","explication":"La limite d'une suite rationnelle est égale au rapport des coefficients dominants des polynômes du numérateur et du dénominateur. Ici, 3n²\/n² = 3.","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. 3\", \"c\": \"C. +∞\", \"d\": \"D. -∞\"}}","_debug_options_count":4},{"id":101203,"question":"La fonction f(x) = e^(2x) - 3x est convexe sur ℝ.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée seconde f''(x) = 4e^(2x) \u003E 0 pour tout x, donc la fonction est convexe sur ℝ.","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":101204,"question":"Quel est le module d'un nombre complexe z = 3 - 4i ?","option_a":"A. 1","option_b":"B. 5","option_c":"C. 7","option_d":"D. 12","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le module de z = a + bi est √(a² + b²). Ici, √(3² + (-4)²) = √(9 + 16) = √25 = 5.","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. 5\", \"c\": \"C. 7\", \"d\": \"D. 12\"}}","_debug_options_count":4},{"id":101205,"question":"La probabilité d'obtenir un double 6 en lançant deux dés équilibrés est :","option_a":"A. 1\/6","option_b":"B. 1\/12","option_c":"C. 1\/36","option_d":"D. 1\/18","option_e":"","option_f":"","bonne_reponse":"c","explication":"Il y a 36 issues possibles équiprobables, et une seule issue favorable (6,6), donc P = 1\/36.","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\/6\", \"b\": \"B. 1\/12\", \"c\": \"C. 1\/36\", \"d\": \"D. 1\/18\"}}","_debug_options_count":4},{"id":101206,"question":"L'équation différentielle y' + 2y = 0 a pour solution générale y = Ce^(-2x).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'équation est linéaire du 1er ordre. La solution générale est y = Ce^(-2x), 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":101207,"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","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de ln(u) est u'\/u. Ici, u = 3x + 1, donc u' = 3, et 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":101208,"question":"Dans une classe de 30 élèves, 18 sont des filles. On choisit un élève au hasard. Quelle est la probabilité que ce soit une fille ?","option_a":"A. 0,4","option_b":"B. 0,6","option_c":"C. 0,5","option_d":"D. 0,3","option_e":"","option_f":"","bonne_reponse":"b","explication":"La probabilité est le rapport du nombre de cas favorables sur le nombre total de cas. Ici, 18\/30 = 0,6.","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. 0,4\", \"b\": \"B. 0,6\", \"c\": \"C. 0,5\", \"d\": \"D. 0,3\"}}","_debug_options_count":4},{"id":101209,"question":"La fonction f(x) = x³ - 3x est croissante sur l'intervalle [-2 ; 2].","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. La fonction est décroissante sur [-1 ; 1] et croissante ailleurs.","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":101210,"question":"Quel est le résultat de l'intégrale ∫₀¹ (2x + 1) 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 d'une fonction affine se calcule avec la formule [x² + x]₀¹ = (1 + 1) - (0 + 0) = 2. Mais ici, ∫(2x + 1)dx = [x² + x]₀¹ = 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":101211,"question":"L'espace est muni d'un repère orthonormé. Les points A(1, 2, 3) et B(-1, 0, 5) sont tels que AB = √14. Vrai ou faux ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"AB = √[(-1-1)² + (0-2)² + (5-3)²] = √(4 + 4 + 4) = √12 ≠ √14.","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}]