Quiz — devoir 05.docx

Question 1 sur 10 20:00

[{"id":32800,"question":"Quelle est la dérivée de la fonction f(x) = 3x² + 2x - 5 ?","option_a":"A) 6x + 2","option_b":"B) 3x² + 2","option_c":"C) 6x - 5","option_d":"D) 6x + 2x","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de 3x² est 6x, celle de 2x est 2, et celle de -5 est 0. Donc 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) 3x² + 2\", \"c\": \"C) 6x - 5\", \"d\": \"D) 6x + 2","_debug_options_count":4},{"id":32801,"question":"La fonction f(x) = x³ est-elle dérivable en x = 0 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction f(x) = x³ est dérivable en tout point de ℝ, donc aussi en x = 0. Sa dérivée est f'(x) = 3x².","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":32802,"question":"Quelle est la limite de la suite uₙ = (2n + 1)\/(n + 3) quand n tend vers +∞ ?","option_a":"A) 0","option_b":"B) 1","option_c":"C) 2","option_d":"D) +∞","option_e":"","option_f":"","bonne_reponse":"c","explication":"En divisant numérateur et dénominateur par n, on obtient uₙ = (2 + 1\/n)\/(1 + 3\/n). Quand n → +∞, 1\/n → 0, donc uₙ → 2\/1 = 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) 1\", \"c\": \"C) 2\", \"d\": \"D) +∞\"}}","_debug_options_count":4},{"id":32803,"question":"L'équation x² - 4x + 3 = 0 a-t-elle 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 = (-4)² - 4×1×3 = 16 - 12 = 4 \u003E 0. Donc l'équation a deux solutions réelles distinctes : x₁ = 1 et x₂ = 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":32804,"question":"Soit f une fonction continue sur [a, b]. Le théorème des valeurs intermédiaires garantit que :","option_a":"A) f est dérivable sur [a, b]","option_b":"B) Pour tout k entre f(a) et f(b), il existe c ∈ [a, b] tel que f(c) = k","option_c":"C) f est constante sur [a, b]","option_d":"D) f est strictement croissante sur [a, b]","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le théorème des valeurs intermédiaires stipule que pour toute valeur k entre f(a) et f(b), il existe un réel c dans [a, b] tel que f(c) = k.","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) f est dérivable sur [a, b]\", \"b\": \"B) Pour tout k entre f(a) ","_debug_options_count":4},{"id":32805,"question":"La probabilité d'un événement A est P(A) = 0,6. Quelle est la probabilité de l'événement contraire A̅ ?","option_a":"A) 0,4","option_b":"B) 0,6","option_c":"C) 1","option_d":"D) 0","option_e":"","option_f":"","bonne_reponse":"a","explication":"La probabilité de l'événement contraire est P(A̅) = 1 - P(A) = 1 - 0,6 = 0,4.","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) 0,4\", \"b\": \"B) 0,6\", \"c\": \"C) 1\", \"d\": \"D) 0\"}}","_debug_options_count":4},{"id":32806,"question":"Soit une variable aléatoire X suivant une loi normale N(μ, σ²). Que vaut P(X = μ) ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Pour une loi normale continue, la probabilité d'une valeur ponctuelle est toujours nulle : P(X = μ) = 0.","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":32807,"question":"Quelle est la solution de l'inéquation 2x - 3 \u003E 5 ?","option_a":"A) x \u003E 1","option_b":"B) x \u003E 4","option_c":"C) x \u003C 4","option_d":"D) x \u003E -1","option_e":"","option_f":"","bonne_reponse":"b","explication":"2x - 3 \u003E 5 ⇒ 2x \u003E 8 ⇒ x \u003E 4.","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 \u003E 1\", \"b\": \"B) x \u003E 4\", \"c\": \"C) x \u003C 4\", \"d\": \"D) x \u003E -1\"}}","_debug_options_count":4},{"id":32808,"question":"Le produit scalaire de deux vecteurs orthogonaux est toujours égal à :","option_a":"A) 0","option_b":"B) 1","option_c":"C) -1","option_d":"D) Leur norme","option_e":"","option_f":"","bonne_reponse":"a","explication":"Deux vecteurs orthogonaux ont un produit scalaire nul (par définition).","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) 1\", \"c\": \"C) -1\", \"d\": \"D) Leur norme\"}}","_debug_options_count":4},{"id":32809,"question":"Soit f(x) = ln(x). Quelle est la valeur de f'(1) ?","option_a":"A) 0","option_b":"B) 1","option_c":"C) -1","option_d":"D) e","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée de ln(x) est f'(x) = 1\/x. Donc f'(1) = 1\/1 = 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) e\"}}","_debug_options_count":4}]