Quiz — M-PT-DER-JMF-8.pdf

Question 1 sur 10 20:00

[{"id":40540,"question":"Quelle est la dérivée de la fonction f(x) = 3x² + 2x - 5 ?","option_a":"6x + 2","option_b":"3x² + 2","option_c":"6x² + 2x","option_d":"3x + 2","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'une fonction polynôme se calcule terme à terme : (3x²)' = 6x et (2x)' = 2, donc f'(x) = 6x + 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\": \"6x + 2\", \"b\": \"3x² + 2\", \"c\": \"6x² + 2x\", \"d\": \"3x + 2\"}}","_debug_options_count":4},{"id":40541,"question":"La fonction f(x) = x³ - 3x² + 2x est croissante sur l'intervalle [0 ; 2].","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée f'(x) = 3x² - 6x + 2 s'annule en x=1 (racine double). En testant un point dans [0;2], par exemple x=1, f'(1) = -1 \u003C 0, donc la fonction est décroissante sur cet intervalle.","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":40542,"question":"Quelle est la limite de la fonction f(x) = (2x² + 3x - 1)\/(x² - 4) lorsque x tend vers l'infini ?","option_a":"2","option_b":"0","option_c":"1\/2","option_d":"∞","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour les fonctions rationnelles, la limite à l'infini est le rapport des coefficients dominants : 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\": \"2\", \"b\": \"0\", \"c\": \"1\/2\", \"d\": \"∞\"}}","_debug_options_count":4},{"id":40543,"question":"L'intégrale ∫₀¹ (3x² + 2x) dx vaut :","option_a":"2","option_b":"1","option_c":"3","option_d":"4\/3","option_e":"","option_f":"","bonne_reponse":"d","explication":"L'intégrale se calcule comme suit : [x³ + x²]₀¹ = (1 + 1) - (0 + 0) = 2. Attention, l'option 4\/3 correspond à ∫₀¹ (x² + 2x) dx.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"d\", \"options\": {\"a\": \"2\", \"b\": \"1\", \"c\": \"3\", \"d\": \"4\/3\"}}","_debug_options_count":4},{"id":40544,"question":"La suite définie par uₙ = (n+1)\/n est convergente.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La suite uₙ = (n+1)\/n = 1 + 1\/n tend vers 1 lorsque n tend vers l'infini, donc elle est convergente.","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":40545,"question":"Quelle est la solution de l'équation différentielle y' + 2y = 0 avec y(0) = 3 ?","option_a":"y = 3e^(-2x)","option_b":"y = 3e^(2x)","option_c":"y = -3e^(-2x)","option_d":"y = 3e^(-x)","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'équation différentielle linéaire du premier ordre a pour solution générale y = Ce^(-2x). Avec y(0)=3, on trouve C=3, donc y = 3e^(-2x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"y = 3e^(-2x)\", \"b\": \"y = 3e^(2x)\", \"c\": \"y = -3e^(-2x)\", \"d\": \"y ","_debug_options_count":4},{"id":40546,"question":"Un dé équilibré est lancé 10 fois. Quelle est la probabilité d'obtenir exactement 3 fois le chiffre 6 ?","option_a":"C(10,3) * (1\/6)^3 * (5\/6)^7","option_b":"C(10,3) * (1\/6)^7 * (5\/6)^3","option_c":"10 * (1\/6)^3","option_d":"1 - (5\/6)^10","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est une loi binomiale B(10, 1\/6). La probabilité d'obtenir k succès est C(n,k) * p^k * (1-p)^(n-k). Ici, k=3, n=10, p=1\/6.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"C(10,3) * (1\/6)^3 * (5\/6)^7\", \"b\": \"C(10,3) * (1\/6)^7 * (5\/6)^3\",","_debug_options_count":4},{"id":40547,"question":"Le nombre complexe z = 2 + 2i a pour module :","option_a":"2","option_b":"2√2","option_c":"4","option_d":"√2","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le module d'un nombre complexe z = a + bi est |z| = √(a² + b²) = √(4 + 4) = √8 = 2√2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"2\", \"b\": \"2√2\", \"c\": \"4\", \"d\": \"√2\"}}","_debug_options_count":4},{"id":40548,"question":"La fonction f(x) = ln(x) est définie pour tout x réel.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction logarithme népérien est définie uniquement pour x \u003E 0. Elle n'est pas définie pour x ≤ 0.","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":40549,"question":"Quelle est l'aire sous la courbe de f(x) = x² entre x=0 et x=2 ?","option_a":"2","option_b":"4\/3","option_c":"8\/3","option_d":"4","option_e":"","option_f":"","bonne_reponse":"c","explication":"L'aire est donnée par l'intégrale ∫₀² x² dx = [x³\/3]₀² = 8\/3 - 0 = 8\/3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"2\", \"b\": \"4\/3\", \"c\": \"8\/3\", \"d\": \"4\"}}","_debug_options_count":4}]