Quiz — 1576612895Exercice 7.pdf.pdf

Question 1 sur 10 20:00

[{"id":50883,"question":"Quelle est la dérivée de la fonction f(x) = x² * ln(x) pour x \u003E 0 ?","option_a":"2x * ln(x) + x","option_b":"2x * ln(x) + 1","option_c":"x * ln(x) + x","option_d":"x * ln(x) + 2x","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'un produit u*v est u'v + uv'. Ici, u = x² (u' = 2x) et v = ln(x) (v' = 1\/x). Donc f'(x) = 2x * ln(x) + x² * (1\/x) = 2x * ln(x) + 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\": \"2x * ln(x) + x\", \"b\": \"2x * ln(x) + 1\", \"c\": \"x * ln(x) + x\", \"d\"","_debug_options_count":4},{"id":50884,"question":"La fonction f(x) = (x² + 1)\/x est-elle définie en x = 0 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction f(x) = (x² + 1)\/x n'est pas définie en x = 0 car le dénominateur s'annule. Le domaine de définition est ℝ*.","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":50885,"question":"Soit une suite géométrique de premier terme u₀ = 3 et de raison q = 2. Quelle est la valeur de u₅ ?","option_a":"32","option_b":"64","option_c":"96","option_d":"128","option_e":"","option_f":"","bonne_reponse":"d","explication":"La formule d'une suite géométrique est uₙ = u₀ * qⁿ. Donc u₅ = 3 * 2⁵ = 3 * 32 = 96. Attention à ne pas confondre avec 2⁶ = 64.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"d\", \"options\": {\"a\": \"32\", \"b\": \"64\", \"c\": \"96\", \"d\": \"128\"}}","_debug_options_count":4},{"id":50886,"question":"Dans une classe de 30 élèves, 18 sont des filles. Quelle est la probabilité de choisir au hasard une fille ?","option_a":"0.4","option_b":"0.6","option_c":"0.3","option_d":"0.5","option_e":"","option_f":"","bonne_reponse":"b","explication":"La probabilité est le rapport du nombre de cas favorables sur le nombre total de cas. Ici, 18 filles \/ 30 élèves = 0.6.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"0.4\", \"b\": \"0.6\", \"c\": \"0.3\", \"d\": \"0.5\"}}","_debug_options_count":4},{"id":50887,"question":"L'équation x² + 4x + 5 = 0 admet-elle des solutions réelles ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le discriminant Δ = b² - 4ac = 16 - 20 = -4 \u003C 0. L'équation n'a donc pas de solutions réelles (solutions complexes uniquement).","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":50888,"question":"Quelle est la limite de la fonction f(x) = (3x² + 2x - 1)\/(x² - 4) quand x tend vers +∞ ?","option_a":"0","option_b":"3","option_c":"1","option_d":"∞","option_e":"","option_f":"","bonne_reponse":"b","explication":"Pour x → ∞, on garde uniquement les termes de plus haut degré : f(x) ≈ 3x²\/x² = 3. La limite est donc 3.","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\": \"3\", \"c\": \"1\", \"d\": \"∞\"}}","_debug_options_count":4},{"id":50889,"question":"Un dé équilibré à 6 faces est lancé deux fois. Quelle est la probabilité d'obtenir deux fois le chiffre 6 ?","option_a":"1\/6","option_b":"1\/12","option_c":"1\/36","option_d":"1\/18","option_e":"","option_f":"","bonne_reponse":"c","explication":"La probabilité d'obtenir un 6 en un lancer est 1\/6. Pour deux lancers indépendants, la probabilité est (1\/6) * (1\/6) = 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},{"id":50890,"question":"La fonction f(x) = x³ - 3x est-elle croissante sur l'intervalle [-2 ; 2] ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée f'(x) = 3x² - 3 = 3(x² - 1). f'(x) \u003C 0 pour x ∈ ]-1 ; 1[, donc la fonction est décroissante sur cet intervalle. Elle n'est donc pas croissante sur [-2 ; 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":50891,"question":"Soit un vecteur u(2 ; -1 ; 3) dans l'espace. Quelle est la norme de u ?","option_a":"√14","option_b":"√13","option_c":"√12","option_d":"√11","option_e":"","option_f":"","bonne_reponse":"a","explication":"La norme d'un vecteur u(a ; b ; c) est √(a² + b² + c²) = √(4 + 1 + 9) = √14.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"√14\", \"b\": \"√13\", \"c\": \"√12\", \"d\": \"√11\"}}","_debug_options_count":4},{"id":50892,"question":"La fonction f(x) = e^(2x) vérifie-t-elle f'(x) = 2f(x) ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de e^(u(x)) est u'(x) * e^(u(x)). Ici, u(x) = 2x, donc u'(x) = 2. Ainsi, f'(x) = 2 * e^(2x) = 2f(x). L'affirmation est 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}]