Quiz — Corrige_TD.pdf

Question 1 sur 10 20:00

[{"id":129803,"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. 6x - 5","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'une fonction polynôme s'obtient en appliquant la formule (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. 6x - 5","_debug_options_count":4},{"id":129804,"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 = b² - 4ac = 16 - 12 = 4 \u003E 0, donc l'équation admet deux solutions réelles distinctes.","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":129805,"question":"Quelle est la limite de la fonction f(x) = (2x² + 3x) \/ (x² - 1) quand x tend vers l'infini ?","option_a":"A. 2","option_b":"B. 0","option_c":"C. -2","option_d":"D. +∞","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour les limites à l'infini de fonctions rationnelles, on compare les degrés du numérateur et du dénominateur. Ici, les degrés sont égaux (2), donc la limite est le rapport des coefficients dominants : 2\/1 = 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. 2\", \"b\": \"B. 0\", \"c\": \"C. -2\", \"d\": \"D. +∞\"}}","_debug_options_count":4},{"id":129806,"question":"La fonction f(x) = x³ - 3x² est croissante sur l'intervalle [0 ; 2].","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée f'(x) = 3x² - 6x = 3x(x - 2). Sur [0 ; 2], f'(x) ≤ 0, donc la fonction est 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":129807,"question":"Quelle est la solution de l'inéquation 2x - 5 \u003E 3x + 1 ?","option_a":"A. x \u003C -6","option_b":"B. x \u003E -6","option_c":"C. x \u003C 6","option_d":"D. x \u003E 6","option_e":"","option_f":"","bonne_reponse":"a","explication":"2x - 5 \u003E 3x + 1 ⇒ -5 - 1 \u003E 3x - 2x ⇒ -6 \u003E x ⇒ x \u003C -6.","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 \u003C -6\", \"b\": \"B. x \u003E -6\", \"c\": \"C. x \u003C 6\", \"d\": \"D. x \u003E 6\"}}","_debug_options_count":4},{"id":129808,"question":"La fonction exponentielle est toujours positive sur ℝ.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction exponentielle, notée exp(x) ou e^x, est strictement positive pour tout réel 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":129809,"question":"Quelle est l'intégrale de la fonction f(x) = 4x³ entre 0 et 2 ?","option_a":"A. 16","option_b":"B. 8","option_c":"C. 32","option_d":"D. 4","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'intégrale de 4x³ est x⁴. Évaluée entre 0 et 2 : (2⁴) - (0⁴) = 16 - 0 = 16.","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. 16\", \"b\": \"B. 8\", \"c\": \"C. 32\", \"d\": \"D. 4\"}}","_debug_options_count":4},{"id":129810,"question":"La probabilité d'un événement impossible est égale à 1.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La probabilité d'un événement impossible est égale à 0. La probabilité d'un événement certain est égale à 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":129811,"question":"Quelle est la valeur de la dérivée seconde de f(x) = x⁴ - 2x³ + x² en x = 1 ?","option_a":"A. 0","option_b":"B. 2","option_c":"C. 4","option_d":"D. 6","option_e":"","option_f":"","bonne_reponse":"c","explication":"f'(x) = 4x³ - 6x² + 2x, f''(x) = 12x² - 12x + 2. En x = 1 : f''(1) = 12 - 12 + 2 = 2.","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. 2\", \"c\": \"C. 4\", \"d\": \"D. 6\"}}","_debug_options_count":4},{"id":129812,"question":"L'équation de la tangente à la courbe de f(x) = x² au point d'abscisse 1 est y = 2x - 1.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La tangente en x = 1 a pour équation y = f'(1)(x - 1) + f(1) = 2(x - 1) + 1 = 2x - 1. L'affirmation est donc 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}]