Quiz — 65e0b20d715c6_corrigé_Série 43 (1).pdf

Question 1 sur 10 20:00

[{"id":24064,"question":"Quelle est la dérivée de la fonction f(x) = 3x² - 5x + 2 ?","option_a":"A) 6x - 5","option_b":"B) 3x - 5","option_c":"C) 6x² - 5","option_d":"D) 3x² - 5","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'une fonction polynôme se calcule terme à terme : (3x²)' = 6x, (-5x)' = -5, et 2' = 0. Donc f'(x) = 6x - 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) 6x - 5\", \"b\": \"B) 3x - 5\", \"c\": \"C) 6x² - 5\", \"d\": \"D) 3x² -","_debug_options_count":4},{"id":24065,"question":"La suite définie par uₙ = (n² + 1)\/n a pour limite +∞ quand n tend vers +∞.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"En simplifiant uₙ = n + 1\/n, on voit que la limite est +∞, car n domine 1\/n. La réponse est donc Vrai.","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":24066,"question":"Quelle est la solution générale de l'équation différentielle y' + 2y = 0 ?","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":"b","explication":"L'équation y' + 2y = 0 est une équation différentielle linéaire du premier ordre. Sa solution générale est y = Ce^(-2x), où C est une constante réelle.","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^(2x)\", \"b\": \"B) y = Ce^(-2x)\", \"c\": \"C) y = Cx + 2\", \"d","_debug_options_count":4},{"id":24067,"question":"Soit X une variable aléatoire suivant une loi normale N(0,1). Que vaut P(X ≤ 1) ?","option_a":"A) 0,6826","option_b":"B) 0,8413","option_c":"C) 0,9544","option_d":"D) 0,9974","option_e":"","option_f":"","bonne_reponse":"b","explication":"Pour une loi normale centrée réduite, P(X ≤ 1) ≈ 0,8413 (valeur lue dans la table de la loi normale).","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) 0,6826\", \"b\": \"B) 0,8413\", \"c\": \"C) 0,9544\", \"d\": \"D) 0,9974\"}","_debug_options_count":4},{"id":24068,"question":"L'intersection de deux plans dans l'espace est toujours une droite.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'intersection de deux plans peut être une droite (si les plans ne sont pas parallèles) ou vide (si les plans sont strictement parallèles). La réponse est donc Vrai.","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":24069,"question":"Quelle est la valeur de la somme S = 1 + 2 + 3 + ... + n ?","option_a":"A) n(n+1)\/2","option_b":"B) n²","option_c":"C) (n+1)²\/2","option_d":"D) n(n-1)\/2","option_e":"","option_f":"","bonne_reponse":"a","explication":"La somme des n premiers entiers naturels est donnée par la formule S = n(n+1)\/2, appelée formule de Gauss.","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) n(n+1)\/2\", \"b\": \"B) n²\", \"c\": \"C) (n+1)²\/2\", \"d\": \"D) n(n-1)","_debug_options_count":4},{"id":24070,"question":"Soit f une fonction dérivable sur ℝ. Si f'(x) \u003E 0 pour tout x, alors f est croissante sur ℝ.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Si f est dérivable et f'(x) \u003E 0 pour tout x, alors f est strictement croissante sur ℝ. La réponse est donc Vrai.","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":24071,"question":"Quelle est la probabilité d'obtenir un double 6 en lançant deux dés équilibrés ?","option_a":"A) 1\/6","option_b":"B) 1\/12","option_c":"C) 1\/36","option_d":"D) 1\/18","option_e":"","option_f":"","bonne_reponse":"c","explication":"Il y a 6×6 = 36 issues possibles. Une seule issue correspond à un double 6 (6;6). La probabilité est donc 1\/36.","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) 1\/6\", \"b\": \"B) 1\/12\", \"c\": \"C) 1\/36\", \"d\": \"D) 1\/18\"}}","_debug_options_count":4},{"id":24072,"question":"L'équation x² + y² + z² = 1 représente une sphère de centre O et de rayon 1.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'équation x² + y² + z² = 1 est bien celle d'une sphère centrée à l'origine (0;0;0) avec un rayon de 1. La réponse est donc Vrai.","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":24073,"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 → +∞, 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}]