Quiz — item 45 fmm 2018 IU.pdf

Question 1 sur 10 20:00

[{"id":80209,"question":"Quelle est la dérivée de la fonction f(x) = 3x² - 5x + 2 ?","option_a":"A : 6x - 5","option_b":"B : 3x - 5","option_c":"C : 6x² - 5x","option_d":"D : 6x - 5x + 2","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'une fonction polynôme se calcule terme à terme : la dérivée de 3x² est 6x, celle de -5x est -5, et la dérivée d'une constante (2) est 0.","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 - 5\", \"b\": \"B : 3x - 5\", \"c\": \"C : 6x² - 5x\", \"d\": \"D : 6","_debug_options_count":4},{"id":80210,"question":"L'équation différentielle y' = 2y admet pour solution générale y = Ce^(2x), où C est une constante réelle.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La solution générale de l'équation différentielle y' = ky est y = Ce^(kx). Ici, k = 2, donc la solution est bien y = Ce^(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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":80211,"question":"Soit une suite géométrique de premier terme u₀ = 4 et de raison q = 3. Quelle est la valeur de u₃ ?","option_a":"A : 12","option_b":"B : 36","option_c":"C : 108","option_d":"D : 216","option_e":"","option_f":"","bonne_reponse":"c","explication":"La formule d'une suite géométrique est uₙ = u₀ × qⁿ. Donc u₃ = 4 × 3³ = 4 × 27 = 108.","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 : 12\", \"b\": \"B : 36\", \"c\": \"C : 108\", \"d\": \"D : 216\"}}","_debug_options_count":4},{"id":80212,"question":"La probabilité d'un événement A est P(A) = 0,7. Quelle est la probabilité de l'événement contraire A̅ ?","option_a":"A : 0,3","option_b":"B : 0,7","option_c":"C : 0,1","option_d":"D : 0,4","option_e":"","option_f":"","bonne_reponse":"a","explication":"La probabilité de l'événement contraire est P(A̅) = 1 - P(A) = 1 - 0,7 = 0,3.","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 : 0,3\", \"b\": \"B : 0,7\", \"c\": \"C : 0,1\", \"d\": \"D : 0,4\"}}","_debug_options_count":4},{"id":80213,"question":"La fonction 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 ln(x) est définie uniquement pour x \u003E 0. Elle n'est pas définie pour les valeurs négatives ou nulles de x.","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":80214,"question":"Quelle est la limite de la fonction f(x) = (2x² + 3x - 1)\/(x² - 5) lorsque x tend vers l'infini ?","option_a":"A : 0","option_b":"B : 2","option_c":"C : +∞","option_d":"D : -∞","option_e":"","option_f":"","bonne_reponse":"b","explication":"Pour les limites à l'infini de fonctions rationnelles, on compare les degrés du numérateur et du dénominateur. Ici, les degrés sont égaux (2), donc la limite est le rapport des coefficients dominants : 2\/1 = 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\": \"A : 0\", \"b\": \"B : 2\", \"c\": \"C : +∞\", \"d\": \"D : -∞\"}}","_debug_options_count":4},{"id":80215,"question":"Un dé équilibré à 6 faces est lancé. Quelle est la probabilité d'obtenir un nombre pair ?","option_a":"A : 1\/6","option_b":"B : 1\/3","option_c":"C : 1\/2","option_d":"D : 2\/3","option_e":"","option_f":"","bonne_reponse":"c","explication":"Les nombres pairs sur un dé sont 2, 4 et 6, soit 3 issues favorables sur 6 possibles. Donc P(pair) = 3\/6 = 1\/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\/6\", \"b\": \"B : 1\/3\", \"c\": \"C : 1\/2\", \"d\": \"D : 2\/3\"}}","_debug_options_count":4},{"id":80216,"question":"La fonction exponentielle est toujours positive.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour tout x réel, e^x \u003E 0. La fonction exponentielle ne prend donc que des valeurs strictement positives.","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":80217,"question":"Résoudre l'inéquation : 2x - 5 \u003E 3x + 1.","option_a":"A : x \u003C -6","option_b":"B : x \u003E -6","option_c":"C : x \u003C 6","option_d":"D : x \u003E 6","option_e":"","option_f":"","bonne_reponse":"a","explication":"En simplifiant l'inéquation : 2x - 5 \u003E 3x + 1 → -5 - 1 \u003E 3x - 2x → -6 \u003E x → x \u003C -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\": \"A : x \u003C -6\", \"b\": \"B : x \u003E -6\", \"c\": \"C : x \u003C 6\", \"d\": \"D : x \u003E 6","_debug_options_count":4},{"id":80218,"question":"La dérivée de la fonction f(x) = ln(x² + 1) est f'(x) = 2x\/(x² + 1).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"En utilisant la formule de dérivation d'une fonction composée : f'(x) = (1\/(x² + 1)) × 2x = 2x\/(x² + 1). L'affirmation est donc 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}]