Quiz — corrigé série 4

Question 1 sur 10 20:00

[{"id":114464,"question":"Quelle est la dérivée de la fonction f(x) = x³ - 2x² + 5x - 1 ?","option_a":"3x² - 4x + 5","option_b":"3x² - 4x + 5x","option_c":"x² - 2x + 5","option_d":"3x² - 4x + 1","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'un polynôme s'obtient en appliquant la règle de dérivation terme à terme : (x^n)' = n*x^(n-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\": \"3x² - 4x + 5\", \"b\": \"3x² - 4x + 5x\", \"c\": \"x² - 2x + 5\", \"d\": ","_debug_options_count":4},{"id":114465,"question":"La limite de la fonction f(x) = (2x² + 3x - 1)\/(x² - 4) quand x tend vers l'infini est égale à 2.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"En divisant numérateur et dénominateur par x², on obtient (2 + 3\/x - 1\/x²)\/(1 - 4\/x²), dont la limite est 2\/1 = 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":114466,"question":"Quelle est la solution de l'inéquation 2x - 5 \u003E 3x + 1 ?","option_a":"x \u003C -6","option_b":"x \u003E -6","option_c":"x \u003C 6","option_d":"x \u003E 6","option_e":"","option_f":"","bonne_reponse":"a","explication":"En isolant x, on obtient -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\": \"x \u003C -6\", \"b\": \"x \u003E -6\", \"c\": \"x \u003C 6\", \"d\": \"x \u003E 6\"}}","_debug_options_count":4},{"id":114467,"question":"Soit une suite géométrique de premier terme u₀ = 3 et de raison q = 2. Quelle est la valeur de u₃ ?","option_a":"12","option_b":"24","option_c":"48","option_d":"96","option_e":"","option_f":"","bonne_reponse":"b","explication":"La formule d'une suite géométrique est uₙ = u₀ * qⁿ. Donc u₃ = 3 * 2³ = 3 * 8 = 24.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"12\", \"b\": \"24\", \"c\": \"48\", \"d\": \"96\"}}","_debug_options_count":4},{"id":114468,"question":"La probabilité d'obtenir un 6 en lançant un dé équilibré est de 1\/6.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Un dé équilibré a 6 faces numérotées de 1 à 6, donc la probabilité d'obtenir un 6 est bien 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":114469,"question":"Quelle est la limite de la suite uₙ = (n² + 1)\/(2n² - 3) quand n tend vers l'infini ?","option_a":"0","option_b":"1\/2","option_c":"1","option_d":"∞","option_e":"","option_f":"","bonne_reponse":"b","explication":"En divisant numérateur et dénominateur par n², on obtient (1 + 1\/n²)\/(2 - 3\/n²), dont la limite est 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\": \"0\", \"b\": \"1\/2\", \"c\": \"1\", \"d\": \"∞\"}}","_debug_options_count":4},{"id":114470,"question":"Soit f une fonction continue sur [a, b]. Si f(a) = -2 et f(b) = 3, alors l'équation f(x) = 0 admet au moins une solution dans [a, b].","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est le théorème des valeurs intermédiaires : si f est continue et prend des valeurs de signes opposés, elle s'annule au moins une fois.","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":114471,"question":"Quelle est la valeur de l'intégrale ∫(3x² + 2x - 1) dx entre 0 et 2 ?","option_a":"8","option_b":"10","option_c":"12","option_d":"14","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'intégrale d'un polynôme s'obtient en calculant F(2) - F(0), où F(x) = x³ + x² - x. F(2) = 8 + 4 - 2 = 10, F(0) = 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\": \"8\", \"b\": \"10\", \"c\": \"12\", \"d\": \"14\"}}","_debug_options_count":4},{"id":114472,"question":"Soit A et B deux événements indépendants avec P(A) = 0.4 et P(B) = 0.5. Quelle est la probabilité de A ∩ B ?","option_a":"0.2","option_b":"0.45","option_c":"0.5","option_d":"0.9","option_e":"","option_f":"","bonne_reponse":"a","explication":"Si A et B sont indépendants, P(A ∩ B) = P(A) * P(B) = 0.4 * 0.5 = 0.2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"0.2\", \"b\": \"0.45\", \"c\": \"0.5\", \"d\": \"0.9\"}}","_debug_options_count":4},{"id":114473,"question":"La fonction f(x) = e^(2x) est-elle dérivable sur ℝ ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction exponentielle est dérivable sur ℝ, et sa dérivée est 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\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4}]