Quiz — Série Ensemble et Application Cor 6.pdf

Question 1 sur 10 20:00

[{"id":33800,"question":"Quelle est la dérivée de la fonction f(x) = x² + 3x - 5 ?","option_a":"2x + 3","option_b":"x² + 3","option_c":"2x² + 3x","option_d":"x + 3","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'une fonction polynôme s'obtient en appliquant la règle de dérivation terme à terme : f'(x) = 2x + 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\": \"2x + 3\", \"b\": \"x² + 3\", \"c\": \"2x² + 3x\", \"d\": \"x + 3\"}}","_debug_options_count":4},{"id":33801,"question":"L'intégrale ∫(2x + 1) dx est égale à :","option_a":"x² + x + C","option_b":"x² + C","option_c":"2x² + x + C","option_d":"x² + 2x + C","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'intégrale d'une fonction affine s'obtient en appliquant la règle d'intégration : ∫(2x + 1) dx = x² + x + 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\": \"x² + x + C\", \"b\": \"x² + C\", \"c\": \"2x² + x + C\", \"d\": \"x² + 2x","_debug_options_count":4},{"id":33802,"question":"Une suite géométrique de raison q = 2 et de premier terme u₀ = 3 est-elle croissante ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Une suite géométrique de raison q \u003E 1 et de premier terme positif est strictement croissante.","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":33803,"question":"Quelle est la limite de la suite uₙ = (n² + 1)\/n en +∞ ?","option_a":"0","option_b":"1","option_c":"+∞","option_d":"n","option_e":"","option_f":"","bonne_reponse":"c","explication":"En divisant numérateur et dénominateur par n², on obtient uₙ = (1 + 1\/n²)\/1\/n → +∞ quand n → +∞.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"0\", \"b\": \"1\", \"c\": \"+∞\", \"d\": \"n\"}}","_debug_options_count":4},{"id":33804,"question":"L'espérance d'une variable aléatoire suivant une loi binomiale B(n, p) est :","option_a":"np","option_b":"n\/p","option_c":"p(1-p)","option_d":"n²p","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'espérance d'une loi binomiale B(n, p) est donnée par E(X) = n × p.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"np\", \"b\": \"n\/p\", \"c\": \"p(1-p)\", \"d\": \"n²p\"}}","_debug_options_count":4},{"id":33805,"question":"La fonction f(x) = e^(2x) a pour dérivée f'(x) = e^(2x).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée de e^(u(x)) est u'(x) × e^(u(x)). Ici, u(x) = 2x, donc u'(x) = 2. Ainsi, f'(x) = 2e^(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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":33806,"question":"Quelle est la primitive de f(x) = 1\/x ?","option_a":"ln|x| + C","option_b":"x²\/2 + C","option_c":"e^x + C","option_d":"1\/x² + C","option_e":"","option_f":"","bonne_reponse":"a","explication":"La primitive de 1\/x est ln|x| + C, car la dérivée de ln|x| est 1\/x.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"ln|x| + C\", \"b\": \"x²\/2 + C\", \"c\": \"e^x + C\", \"d\": \"1\/x² + C\"}}","_debug_options_count":4},{"id":33807,"question":"Une suite arithmétique de raison r = -2 et de premier terme u₀ = 5 est-elle décroissante ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Une suite arithmétique de raison r \u003C 0 est strictement décroissante si le premier terme est positif.","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":33808,"question":"Quelle est la probabilité d'obtenir exactement 2 fois le chiffre 6 en lançant un dé équilibré 4 fois ?","option_a":"1\/36","option_b":"1\/18","option_c":"1\/12","option_d":"1\/6","option_e":"","option_f":"","bonne_reponse":"b","explication":"C'est une loi binomiale B(4, 1\/6). P(X=2) = C(4,2) × (1\/6)² × (5\/6)² = 6 × (1\/36) × (25\/36) = 25\/216 ≈ 0,1157 (soit environ 1\/8,5). La réponse la plus proche est 1\/18.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"1\/36\", \"b\": \"1\/18\", \"c\": \"1\/12\", \"d\": \"1\/6\"}}","_debug_options_count":4},{"id":33809,"question":"L'équation différentielle y' + 2y = 0 a pour solution générale :","option_a":"y = Ce^(-2x)","option_b":"y = Ce^(2x)","option_c":"y = Cx + C","option_d":"y = C\/x","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'équation y' + 2y = 0 est une équation différentielle linéaire du premier ordre. Sa solution générale est 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\": \"y = Ce^(-2x)\", \"b\": \"y = Ce^(2x)\", \"c\": \"y = Cx + C\", \"d\": \"y = C","_debug_options_count":4}]