Quiz — corrigé série 4.pdf

Question 1 sur 10 20:00

[{"id":30670,"question":"Quelle est la solution de l'équation x² - 5x + 6 = 0 ?","option_a":"A : x = 2 ou x = 3","option_b":"B : x = -2 ou x = -3","option_c":"C : x = 1 ou x = 6","option_d":"D : x = 0 ou x = 5","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'équation x² - 5x + 6 = 0 se factorise en (x-2)(x-3) = 0, donc les solutions sont x = 2 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\": \"A : x = 2 ou x = 3\", \"b\": \"B : x = -2 ou x = -3\", \"c\": \"C : x = 1","_debug_options_count":4},{"id":30671,"question":"La fonction f(x) = x³ - 3x est-elle croissante sur ℝ ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée f'(x) = 3x² - 3 s'annule en x = ±1. La fonction n'est donc pas croissante sur tout ℝ (elle décroît entre -1 et 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":30672,"question":"Quelle est la limite de (2x² + 3x - 5)\/(x² - 1) quand x tend vers +∞ ?","option_a":"A : 0","option_b":"B : 2","option_c":"C : +∞","option_d":"D : -∞","option_e":"","option_f":"","bonne_reponse":"b","explication":"En divisant numérateur et dénominateur par x², on obtient (2 + 3\/x - 5\/x²)\/(1 - 1\/x²) → 2\/1 = 2 quand x → +∞.","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 : 2\", \"c\": \"C : +∞\", \"d\": \"D : -∞\"}}","_debug_options_count":4},{"id":30673,"question":"Une suite géométrique de raison q = -2 et de premier terme u₀ = 3 est-elle 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 ±∞ selon le signe de q).","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":30674,"question":"Quelle est l'équation de la tangente à la courbe y = x² + 2x au point d'abscisse x = 1 ?","option_a":"A : y = 4x - 1","option_b":"B : y = 2x + 1","option_c":"C : y = 4x + 1","option_d":"D : y = x + 2","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée est f'(x) = 2x + 2. En x = 1, f(1) = 3 et f'(1) = 4. L'équation de la tangente est y = 4(x-1) + 3 = 4x - 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 = 4x - 1\", \"b\": \"B : y = 2x + 1\", \"c\": \"C : y = 4x + 1\", \"d","_debug_options_count":4},{"id":30675,"question":"Le discriminant d'une équation du second degré ax² + bx + c = 0 est donné par Δ = b² - 4ac.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Cette formule est exacte et définit le discriminant Δ utilisé pour déterminer le nombre de solutions réelles de l'équation.","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":30676,"question":"Quelle est la somme des 5 premiers termes d'une suite arithmétique de premier terme u₀ = 2 et de raison r = 3 ?","option_a":"A : 15","option_b":"B : 35","option_c":"C : 40","option_d":"D : 50","option_e":"","option_f":"","bonne_reponse":"b","explication":"La somme S₅ = (5\/2)(u₀ + u₄) = (5\/2)(2 + 14) = (5\/2)(16) = 40.","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 : 15\", \"b\": \"B : 35\", \"c\": \"C : 40\", \"d\": \"D : 50\"}}","_debug_options_count":4},{"id":30677,"question":"La fonction f(x) = 1\/x est-elle définie en x = 0 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction f(x) = 1\/x n'est pas définie en x = 0 car la division par zéro est impossible.","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":30678,"question":"Quelle est la dérivée de la fonction f(x) = (3x² + 2x)(x - 1) ?","option_a":"A : 9x² - 2x + 2","option_b":"B : 6x + 2","option_c":"C : 3x² - 4x - 2","option_d":"D : 9x² - 4x - 2","option_e":"","option_f":"","bonne_reponse":"d","explication":"En développant f(x) = 3x³ - x² - 2x, on obtient f'(x) = 9x² - 2x - 2. Alternativement, en utilisant la formule (uv)' = u'v + uv', on retrouve le même résultat.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"d\", \"options\": {\"a\": \"A : 9x² - 2x + 2\", \"b\": \"B : 6x + 2\", \"c\": \"C : 3x² - 4x - 2\", ","_debug_options_count":4},{"id":30679,"question":"L'équation d'un cercle de centre Ω(2; -3) et de rayon 5 est (x-2)² + (y+3)² = 25.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Cette équation correspond bien à un cercle de centre Ω(2; -3) et de rayon √25 = 5, donc l'affirmation est 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}]