Quiz — serie 14 p2.pdf

Question 1 sur 10 20:00

[{"id":13098,"question":"Quelle est la limite de la fonction f(x) = (3x² - 2x + 1)\/(x² + 1) quand x tend vers +∞ ?","option_a":"A. 0","option_b":"B. 1","option_c":"C. 3","option_d":"D. +∞","option_e":"","option_f":"","bonne_reponse":"c","explication":"La limite d'un quotient de polynômes de même degré est égale au rapport des coefficients dominants. Ici, 3x²\/x² = 3.","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. 0\", \"b\": \"B. 1\", \"c\": \"C. 3\", \"d\": \"D. +∞\"}}","_debug_options_count":4},{"id":13099,"question":"Soit une suite définie par u₀ = 2 et uₙ₊₁ = 0.5uₙ + 3. Cette suite est :","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La suite est croissante car uₙ₊₁ - uₙ = -0.5uₙ + 3 \u003E 0 pour uₙ \u003C 6. Or u₀ = 2 \u003C 6, donc la suite est 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":13100,"question":"Quelle est la dérivée de la fonction f(x) = ln(3x² + 1) ?","option_a":"A. 6x\/(3x² + 1)","option_b":"B. 3x\/(3x² + 1)","option_c":"C. 6x","option_d":"D. 1\/(3x² + 1)","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de ln(u) est u'\/u. Ici, u = 3x² + 1, donc u' = 6x. Ainsi, f'(x) = 6x\/(3x² + 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\": \"A. 6x\/(3x² + 1)\", \"b\": \"B. 3x\/(3x² + 1)\", \"c\": \"C. 6x\", \"d\": \"D","_debug_options_count":4},{"id":13101,"question":"L’équation différentielle y' + 2y = 0 admet pour solution générale :","option_a":"A. y = Ce^(-2x)","option_b":"B. y = Ce^(2x)","option_c":"C. y = Cx^2","option_d":"D. y = C\/x","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est une équation différentielle linéaire du 1er ordre. La solution générale est y = Ce^(-2x), 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\": \"a\", \"options\": {\"a\": \"A. y = Ce^(-2x)\", \"b\": \"B. y = Ce^(2x)\", \"c\": \"C. y = Cx^2\", \"d\":","_debug_options_count":4},{"id":13102,"question":"Le produit scalaire de deux vecteurs u et v est nul si et seulement si :","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le produit scalaire u·v = 0 si et seulement si les vecteurs u et v sont orthogonaux (ou l'un des deux est nul).","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":13103,"question":"Quelle est la valeur de l’intégrale ∫₀¹ (2x + 1) dx ?","option_a":"A. 1","option_b":"B. 2","option_c":"C. 3","option_d":"D. 4","option_e":"","option_f":"","bonne_reponse":"b","explication":"L’intégrale se calcule comme [x² + x]₀¹ = (1 + 1) - (0 + 0) = 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. 1\", \"b\": \"B. 2\", \"c\": \"C. 3\", \"d\": \"D. 4\"}}","_debug_options_count":4},{"id":13104,"question":"Soit A une matrice carrée d’ordre 2. Si det(A) = 0, alors :","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Si le déterminant d’une matrice est nul, celle-ci n’est pas inversible et son rang est inférieur à 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":13105,"question":"La fonction f(x) = e^(-x²) est :","option_a":"A. Croissante sur ℝ","option_b":"B. Décroissante sur ℝ","option_c":"C. Admet un maximum en x = 0","option_d":"D. Admet un minimum en x = 0","option_e":"","option_f":"","bonne_reponse":"c","explication":"La dérivée f'(x) = -2x e^(-x²) s’annule en x = 0. La fonction admet un maximum en ce point car f''(0) \u003C 0.","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. Croissante sur ℝ\", \"b\": \"B. Décroissante sur ℝ\", \"c\": \"C.","_debug_options_count":4},{"id":13106,"question":"Quelle est la solution de l’équation différentielle y'' + 4y = 0 ?","option_a":"A. y = Ce^(-4x)","option_b":"B. y = C₁cos(2x) + C₂sin(2x)","option_c":"C. y = Cx","option_d":"D. y = Ce^(2x)","option_e":"","option_f":"","bonne_reponse":"b","explication":"L’équation caractéristique est r² + 4 = 0, soit r = ±2i. La solution générale est donc y = C₁cos(2x) + C₂sin(2x).","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 = Ce^(-4x)\", \"b\": \"B. y = C₁cos(2x) + C₂sin(2x)\", \"c\": \"","_debug_options_count":4},{"id":13107,"question":"Le nombre complexe z = 3 + 4i a pour module :","option_a":"A. 5","option_b":"B. 7","option_c":"C. √7","option_d":"D. 25","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le module de z = a + bi est √(a² + b²) = √(9 + 16) = 5.","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. 5\", \"b\": \"B. 7\", \"c\": \"C. √7\", \"d\": \"D. 25\"}}","_debug_options_count":4}]