Quiz — serie 29 p2.pdf

Question 1 sur 10 20:00

[{"id":26660,"question":"Quelle est la solution de l'équation différentielle y' = 2y avec y(0) = 3 ?","option_a":"A. y = 3e^(2x)","option_b":"B. y = 2e^(3x)","option_c":"C. y = 3e^(-2x)","option_d":"D. y = e^(2x) + 2","option_e":"","option_f":"","bonne_reponse":"a","explication":"La solution générale de y' = 2y est y = Ce^(2x). Avec y(0) = 3, on trouve C = 3, d'où y = 3e^(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\": \"A. y = 3e^(2x)\", \"b\": \"B. y = 2e^(3x)\", \"c\": \"C. y = 3e^(-2x)\", \"","_debug_options_count":4},{"id":26661,"question":"La suite définie par u_n = (n+1)\/n est-elle convergente ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La suite u_n = (n+1)\/n tend vers 1 quand n tend vers l'infini, donc elle est convergente.","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":26662,"question":"Quel est le coefficient directeur de la tangente à la courbe de f(x) = ln(x) au point d'abscisse 1 ?","option_a":"A. 0","option_b":"B. 1","option_c":"C. e","option_d":"D. -1","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée de f(x) = ln(x) est f'(x) = 1\/x. Au point x=1, f'(1) = 1, donc le coefficient directeur est 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\": \"A. 0\", \"b\": \"B. 1\", \"c\": \"C. e\", \"d\": \"D. -1\"}}","_debug_options_count":4},{"id":26663,"question":"Soit une variable aléatoire X suivant une loi normale N(0,1). Quelle est la probabilité P(X ≤ 1) ?","option_a":"A. 0.6827","option_b":"B. 0.8413","option_c":"C. 0.9772","option_d":"D. 0.3413","option_e":"","option_f":"","bonne_reponse":"b","explication":"Pour une loi normale centrée réduite, P(X ≤ 1) ≈ 0.8413 (valeur lue dans la table de la loi normale).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"A. 0.6827\", \"b\": \"B. 0.8413\", \"c\": \"C. 0.9772\", \"d\": \"D. 0.3413\"}","_debug_options_count":4},{"id":26664,"question":"L'équation x^3 - 3x + 2 = 0 admet-elle une solution réelle ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'équation peut s'écrire (x-1)^2(x+2) = 0, donc elle admet deux solutions réelles : x=1 (double) et x=-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":26665,"question":"Quelle est la limite de la fonction f(x) = (sin(x))\/x quand x tend vers 0 ?","option_a":"A. 0","option_b":"B. 1","option_c":"C. +∞","option_d":"D. -∞","option_e":"","option_f":"","bonne_reponse":"b","explication":"Par le théorème des gendarmes ou le développement limité, on sait que 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\": \"A. 0\", \"b\": \"B. 1\", \"c\": \"C. +∞\", \"d\": \"D. -∞\"}}","_debug_options_count":4},{"id":26666,"question":"Dans un repère orthonormé, quelle est l'équation du plan passant par les points A(1,0,0), B(0,1,0) et C(0,0,1) ?","option_a":"A. x + y + z = 1","option_b":"B. x - y + z = 0","option_c":"C. 2x + y - z = 1","option_d":"D. x + 2y - 3z = 0","option_e":"","option_f":"","bonne_reponse":"a","explication":"Les trois points vérifient l'équation x + y + z = 1, qui est donc l'équation du plan.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"A. x + y + z = 1\", \"b\": \"B. x - y + z = 0\", \"c\": \"C. 2x + y - z =","_debug_options_count":4},{"id":26667,"question":"La fonction f(x) = x^2 - 4x + 5 est-elle toujours positive ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le discriminant de f(x) est Δ = (-4)^2 - 4*1*5 = -4 \u003C 0, et le coefficient de x^2 est positif, donc f(x) \u003E 0 pour tout x réel.","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":26668,"question":"Quel est le résultat de l'intégrale ∫(0 à 1) e^(-x) dx ?","option_a":"A. 1 - 1\/e","option_b":"B. e - 1","option_c":"C. 1\/e","option_d":"D. 1","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'intégrale de e^(-x) est -e^(-x). Évaluée entre 0 et 1, on obtient -e^(-1) - (-e^(0)) = 1 - 1\/e.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"A. 1 - 1\/e\", \"b\": \"B. e - 1\", \"c\": \"C. 1\/e\", \"d\": \"D. 1\"}}","_debug_options_count":4},{"id":26669,"question":"Soit une suite géométrique de premier terme u_0 = 2 et de raison q = -1\/2. Quelle est la limite de u_n quand n tend vers l'infini ?","option_a":"A. 0","option_b":"B. +∞","option_c":"C. -∞","option_d":"D. 2","option_e":"","option_f":"","bonne_reponse":"a","explication":"Une suite géométrique de raison |q| \u003C 1 converge vers 0. Ici, q = -1\/2, donc lim(u_n) = 0.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"A. 0\", \"b\": \"B. +∞\", \"c\": \"C. -∞\", \"d\": \"D. 2\"}}","_debug_options_count":4}]