Quiz — Lycée Pilote Sousse - DC 2 2017 (+Corrigé).pdf

Question 1 sur 10 20:00

[{"id":24284,"question":"Quelle est la dérivée de la fonction f(x) = 3x² + 2x - 5 ?","option_a":"A. 6x + 2","option_b":"B. 3x + 2","option_c":"C. 6x² + 2","option_d":"D. 3x² + 2x","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'une fonction polynomiale se calcule en appliquant la règle : (ax^n)' = n*ax^(n-1). Ici, 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\": \"A. 6x + 2\", \"b\": \"B. 3x + 2\", \"c\": \"C. 6x² + 2\", \"d\": \"D. 3x² +","_debug_options_count":4},{"id":24285,"question":"La fonction f(x) = x³ - 3x est décroissante sur l'intervalle [-1, 1].","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée f'(x) = 3x² - 3 est négative sur [-1, 1] car 3x² ≤ 3. La fonction est donc décroissante sur cet intervalle.","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":24286,"question":"Quelle est l'équation de la tangente à la courbe de f(x) = x² + 1 au point d'abscisse x = 1 ?","option_a":"A. y = 2x + 1","option_b":"B. y = 2x - 1","option_c":"C. y = x + 2","option_d":"D. y = x² + 1","option_e":"","option_f":"","bonne_reponse":"b","explication":"La tangente en x = 1 a pour équation y = f'(1)(x - 1) + f(1). Ici, f'(1) = 2 et f(1) = 2, donc y = 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 = 2x + 1\", \"b\": \"B. y = 2x - 1\", \"c\": \"C. y = x + 2\", \"d\": \"","_debug_options_count":4},{"id":24287,"question":"L'équation x² - 4x + 3 = 0 admet deux solutions réelles distinctes.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le discriminant Δ = 16 - 12 = 4 \u003E 0, donc l'équation admet deux solutions réelles distinctes : x = 1 et x = 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":24288,"question":"Quelle est la dérivée de la fonction f(x) = (2x + 1)\/(x - 3) ?","option_a":"A. (2)\/(x - 3)","option_b":"B. (-7)\/(x - 3)²","option_c":"C. (2x - 5)\/(x - 3)²","option_d":"D. (2x + 1)\/(x - 3)","option_e":"","option_f":"","bonne_reponse":"b","explication":"En utilisant la formule de dérivation d'un quotient : (u\/v)' = (u'v - uv')\/v². Ici, u = 2x + 1, v = x - 3, donc f'(x) = (-7)\/(x - 3)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)\/(x - 3)\", \"b\": \"B. (-7)\/(x - 3)²\", \"c\": \"C. (2x - 5)\/(x -","_debug_options_count":4},{"id":24289,"question":"La fonction f(x) = x⁴ - 4x³ admet un extremum local en x = 3.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"f'(x) = 4x³ - 12x² = 4x²(x - 3). Le point critique x = 3 est un point d'inflexion, pas un extremum local.","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":24290,"question":"Quelle est la solution de l'inéquation x² - 5x + 6 \u003E 0 ?","option_a":"A. x ∈ ]2, 3[","option_b":"B. x ∈ ]-∞, 2[ ∪ ]3, +∞[","option_c":"C. x ∈ ]-∞, 2] ∪ [3, +∞[","option_d":"D. x ∈ ]2, +∞[","option_e":"","option_f":"","bonne_reponse":"b","explication":"Les racines de l'équation x² - 5x + 6 = 0 sont x = 2 et x = 3. Le polynôme est positif à l'extérieur de l'intervalle [2, 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\": \"A. x ∈ ]2, 3[\", \"b\": \"B. x ∈ ]-∞, 2[ ∪ ]3, +∞[\", \"c\": \"","_debug_options_count":4},{"id":24291,"question":"La dérivée seconde de f(x) = x³ - 3x² + 2x est f''(x) = 6x - 6.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"f'(x) = 3x² - 6x + 2, donc f''(x) = 6x - 6. La dérivée seconde est correcte.","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":24292,"question":"Quelle est l'équation de la droite passant par les points A(1, 2) et B(3, 4) ?","option_a":"A. y = x + 1","option_b":"B. y = 2x - 1","option_c":"C. y = x - 1","option_d":"D. y = 2x + 1","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le coefficient directeur est (4-2)\/(3-1) = 1. L'équation est y = x + 1 (car 2 = 1*1 + 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. y = x + 1\", \"b\": \"B. y = 2x - 1\", \"c\": \"C. y = x - 1\", \"d\": \"D","_debug_options_count":4},{"id":24293,"question":"La fonction f(x) = e^x est toujours 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) = e^x est toujours positive sur ℝ, donc la fonction est toujours 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}]