Quiz — 259.._sujet 1.pdf

Question 1 sur 10 20:00

[{"id":12578,"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 de 3x² est 6x, celle de 2x est 2, et celle de -5 est 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\": \"6x² + 2x\", \"d\": \"3x + 2\"}}","_debug_options_count":4},{"id":12579,"question":"Une suite géométrique de raison q = 2 et de premier terme u₀ = 3 a pour terme général :","option_a":"uₙ = 3 * 2ⁿ","option_b":"uₙ = 3 + 2n","option_c":"uₙ = 3 * n²","option_d":"uₙ = 2 * 3ⁿ","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour une suite géométrique, uₙ = u₀ * qⁿ. Ici, u₀ = 3 et q = 2, donc uₙ = 3 * 2ⁿ.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"uₙ = 3 * 2ⁿ\", \"b\": \"uₙ = 3 + 2n\", \"c\": \"uₙ = 3 * n²\", \"d","_debug_options_count":4},{"id":12580,"question":"Vrai ou Faux : La fonction exponentielle est toujours croissante.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction exponentielle, définie par exp(x) = eˣ, est strictement croissante sur ℝ car sa dérivée exp'(x) = eˣ \u003E 0 pour tout 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":12581,"question":"Quelle est la limite de la suite uₙ = (2n + 1)\/(n + 3) lorsque n tend vers l'infini ?","option_a":"0","option_b":"1","option_c":"2","option_d":"∞","option_e":"","option_f":"","bonne_reponse":"b","explication":"En divisant numérateur et dénominateur par n, on obtient uₙ = (2 + 1\/n)\/(1 + 3\/n). Lorsque n → ∞, 1\/n → 0, donc uₙ → 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\": \"0\", \"b\": \"1\", \"c\": \"2\", \"d\": \"∞\"}}","_debug_options_count":4},{"id":12582,"question":"Vrai ou Faux : Si une fonction est dérivable en un point, alors elle est continue en ce point.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Théorème fondamental : toute fonction dérivable en un point est nécessairement continue en ce point.","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":12583,"question":"Quelle est la solution de l'équation différentielle y' = 2y avec y(0) = 5 ?","option_a":"y = 5e²ˣ","option_b":"y = 5eˣ","option_c":"y = 2e⁵ˣ","option_d":"y = 5x + 2","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'équation y' = 2y a pour solution générale y = Ce²ˣ. Avec y(0) = 5, on trouve C = 5, donc y = 5e²ˣ.","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 = 5e²ˣ\", \"b\": \"y = 5eˣ\", \"c\": \"y = 2e⁵ˣ\", \"d\": \"y = 5x + ","_debug_options_count":4},{"id":12584,"question":"Vrai ou Faux : La somme de deux suites convergentes est une suite convergente.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Si (uₙ) et (vₙ) convergent respectivement vers L et M, alors (uₙ + vₙ) converge vers L + M.","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":12585,"question":"Quelle est la valeur de l'intégrale ∫₀¹ (3x² + 2x) dx ?","option_a":"2","option_b":"3","option_c":"1","option_d":"4","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'intégrale de 3x² + 2x est x³ + x². Évaluée entre 0 et 1, cela donne (1 + 1) - (0 + 0) = 2. L'option correcte est donc 2, mais elle n'est pas présente. Correction : la bonne réponse est 2, mais l'option 1 est incorrecte. Réponse attendue : 2, mais ici l'option 1 est incorrecte. Réponse correcte : 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\": \"3\", \"c\": \"1\", \"d\": \"4\"}}","_debug_options_count":4},{"id":12586,"question":"Vrai ou Faux : Toute suite bornée est convergente.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Contre-exemple : la suite uₙ = (-1)ⁿ est bornée mais ne converge pas (elle oscille entre -1 et 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":12587,"question":"Quelle est la probabilité d'obtenir un double 6 en lançant deux dés équilibrés ?","option_a":"1\/6","option_b":"1\/12","option_c":"1\/36","option_d":"1\/18","option_e":"","option_f":"","bonne_reponse":"c","explication":"Il y a 6 × 6 = 36 issues possibles. Seul le couple (6,6) donne un double 6, donc la probabilité est 1\/36.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"1\/6\", \"b\": \"1\/12\", \"c\": \"1\/36\", \"d\": \"1\/18\"}}","_debug_options_count":4}]