Quiz — Serie Math 23.12.2021 Revision 2 makhdouma.pdf

Question 1 sur 10 20:00

[{"id":47308,"question":"Quelle est la dérivée de la fonction f(x) = 3x² + 2x - 5 ?","option_a":"6x + 2","option_b":"3x² + 2","option_c":"6x - 5","option_d":"3x + 2","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'une fonction polynôme se calcule terme par terme : (3x²)' = 6x, (2x)' = 2, et (-5)' = 0. 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\": \"6x - 5\", \"d\": \"3x + 2\"}}","_debug_options_count":4},{"id":47309,"question":"La fonction f(x) = x³ - 3x est décroissante sur l'intervalle [-1, 1].","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée f'(x) = 3x² - 3 est négative sur [-1, 1] (car 3x² ≤ 3), donc la fonction est décroissante sur cet intervalle. L'affirmation est donc fausse.","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":47310,"question":"Quelle est la limite de (sin x)\/x lorsque x tend vers 0 ?","option_a":"0","option_b":"1","option_c":"∞","option_d":"Non définie","option_e":"","option_f":"","bonne_reponse":"b","explication":"C'est un résultat classique en analyse : lim(x→0) (sin x)\/x = 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\": \"0\", \"b\": \"1\", \"c\": \"∞\", \"d\": \"Non définie\"}}","_debug_options_count":4},{"id":47311,"question":"Soit une suite géométrique de premier terme u₀ = 2 et de raison q = 3. Quel est le terme u₅ ?","option_a":"486","option_b":"162","option_c":"243","option_d":"729","option_e":"","option_f":"","bonne_reponse":"a","explication":"La formule d'une suite géométrique est uₙ = u₀ × qⁿ. Donc u₅ = 2 × 3⁵ = 2 × 243 = 486.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"486\", \"b\": \"162\", \"c\": \"243\", \"d\": \"729\"}}","_debug_options_count":4},{"id":47312,"question":"Le produit scalaire de deux vecteurs orthogonaux est toujours égal à 0.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Par définition, deux vecteurs sont orthogonaux si et seulement si leur produit scalaire est nul.","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":47313,"question":"Quelle est la primitive de la fonction f(x) = 4x³ ?","option_a":"x⁴ + C","option_b":"x⁴","option_c":"12x² + C","option_d":"x³ + C","option_e":"","option_f":"","bonne_reponse":"a","explication":"La primitive d'une fonction polynôme se calcule en augmentant l'exposant de 1 et en divisant par le nouvel exposant : ∫4x³ dx = x⁴ + 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\": \"x⁴ + C\", \"b\": \"x⁴\", \"c\": \"12x² + C\", \"d\": \"x³ + C\"}}","_debug_options_count":4},{"id":47314,"question":"Dans une expérience aléatoire, la probabilité d'un événement A est P(A) = 0,3. Quelle est la probabilité de l'événement contraire A̅ ?","option_a":"0,3","option_b":"0,7","option_c":"1","option_d":"0","option_e":"","option_f":"","bonne_reponse":"b","explication":"La probabilité de l'événement contraire est P(A̅) = 1 - P(A) = 1 - 0,3 = 0,7.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"0,3\", \"b\": \"0,7\", \"c\": \"1\", \"d\": \"0\"}}","_debug_options_count":4},{"id":47315,"question":"La fonction exponentielle est toujours positive sur ℝ.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Par définition, eˣ \u003E 0 pour tout x réel. Donc la fonction exponentielle est toujours positive.","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":47316,"question":"Quelle est la solution de l'équation différentielle y' + 2y = 0 ?","option_a":"y = Ce⁻²ˣ","option_b":"y = Ce²ˣ","option_c":"y = C + e⁻²ˣ","option_d":"y = Cx²","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'équation différentielle linéaire du premier ordre y' + 2y = 0 a pour solution générale y = Ce⁻²ˣ, où C est une constante.","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 = Ce⁻²ˣ\", \"b\": \"y = Ce²ˣ\", \"c\": \"y = C + e⁻²ˣ\", \"d\": ","_debug_options_count":4},{"id":47317,"question":"Soit f une fonction continue sur [a, b]. Le théorème des valeurs intermédiaires garantit que f prend toutes les valeurs comprises entre f(a) et f(b).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le théorème des valeurs intermédiaires stipule que si f est continue sur [a, b], alors pour toute valeur k entre f(a) et f(b), il existe c ∈ [a, b] tel que f(c) = k.","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}]