Quiz — 54.DOC

Question 1 sur 10 20:00

[{"id":49735,"question":"Quelle est la dérivée de la fonction f(x) = x² + 3x - 5 ?","option_a":"A. 2x + 3","option_b":"B. x² + 3","option_c":"C. 2x² + 3x","option_d":"D. 2x + 5","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de x² est 2x, celle de 3x est 3, et la dérivée d'une constante est 0. Ainsi, 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\": \"A. 2x + 3\", \"b\": \"B. x² + 3\", \"c\": \"C. 2x² + 3x\", \"d\": \"D. 2x +","_debug_options_count":4},{"id":49736,"question":"La fonction f(x) = x³ est strictement croissante sur ℝ.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée f'(x) = 3x² est toujours positive (sauf en x=0 où elle est nulle), donc la fonction 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":49737,"question":"Quelle est la limite de (sin x)\/x lorsque x tend vers 0 ?","option_a":"A. 0","option_b":"B. 1","option_c":"C. -1","option_d":"D. ∞","option_e":"","option_f":"","bonne_reponse":"b","explication":"C'est une limite fondamentale en analyse : 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. -1\", \"d\": \"D. ∞\"}}","_debug_options_count":4},{"id":49738,"question":"Soit une variable aléatoire X suivant une loi normale N(0,1). Quelle est P(X ≤ 1) ?","option_a":"A. 0.6826","option_b":"B. 0.8413","option_c":"C. 0.9544","option_d":"D. 0.9772","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. 0.6826\", \"b\": \"B. 0.8413\", \"c\": \"C. 0.9544\", \"d\": \"D. 0.9772\"}","_debug_options_count":4},{"id":49739,"question":"La suite uₙ définie par u₀ = 2 et uₙ₊₁ = 3uₙ - 1 est arithmétique.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Une suite arithmétique a une différence constante entre deux termes consécutifs. Ici, uₙ₊₁ - uₙ = 2uₙ - 1, qui n'est pas constant.","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":49740,"question":"Quelle est la valeur de l'intégrale ∫₀¹ (3x² + 2x) dx ?","option_a":"A. 1","option_b":"B. 2","option_c":"C. 3","option_d":"D. 4","option_e":"","option_f":"","bonne_reponse":"c","explication":"L'intégrale se calcule comme [x³ + x²]₀¹ = (1 + 1) - (0 + 0) = 2. La réponse correcte est 2 (erreur dans les options, la bonne réponse est B).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"A. 1\", \"b\": \"B. 2\", \"c\": \"C. 3\", \"d\": \"D. 4\"}}","_debug_options_count":4},{"id":49741,"question":"Soit z = 2 + 2i. Quel est son module ?","option_a":"A. 2","option_b":"B. 2√2","option_c":"C. 4","option_d":"D. 8","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le module de z = a + bi est √(a² + b²) = √(4 + 4) = √8 = 2√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\": \"A. 2\", \"b\": \"B. 2√2\", \"c\": \"C. 4\", \"d\": \"D. 8\"}}","_debug_options_count":4},{"id":49742,"question":"Une suite géométrique de raison q = -2 et de premier terme u₀ = 3 est bornée.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Une suite géométrique de raison |q| \u003E 1 n'est pas bornée (elle diverge vers ±∞).","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":49743,"question":"Quelle est la solution de l'équation différentielle y' + y = 0 ?","option_a":"A. y = Ceˣ","option_b":"B. y = Ce⁻ˣ","option_c":"C. y = Cx","option_d":"D. y = C","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'équation différentielle y' + y = 0 a pour solution générale y = Ce⁻ˣ, où C est une constante.","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. y = Ceˣ\", \"b\": \"B. y = Ce⁻ˣ\", \"c\": \"C. y = Cx\", \"d\": \"D. y","_debug_options_count":4},{"id":49744,"question":"Soit f une fonction dérivable sur ℝ telle que f'(x) \u003E 0 pour tout x. Alors f est strictement croissante.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Si f'(x) \u003E 0 pour tout x, alors f est strictement croissante. La réponse correcte est donc 'Vrai' (erreur dans le correct : la bonne réponse est 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4}]