Quiz — Exercice type devoir 8.pdf

Question 1 sur 10 20:00

[{"id":64776,"question":"Quelle est la solution de l'équation 2x² - 5x + 2 = 0 ?","option_a":"x = 1 ou x = 2","option_b":"x = 0,5 ou x = 2","option_c":"x = -1 ou x = 2","option_d":"x = 1 ou x = -2","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'équation 2x² - 5x + 2 = 0 se résout en utilisant le discriminant Δ = b² - 4ac = 25 - 16 = 9. Les solutions sont x = (5 ± 3)\/4, soit x = 2 ou x = 0,5.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"x = 1 ou x = 2\", \"b\": \"x = 0,5 ou x = 2\", \"c\": \"x = -1 ou x = 2\",","_debug_options_count":4},{"id":64777,"question":"La fonction f(x) = x³ - 3x + 2 admet-elle un extremum local en x = 1 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée f'(x) = 3x² - 3 s'annule en x = 1. En étudiant le signe de f'(x), on observe un changement de signe, donc f admet un extremum local en x = 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":64778,"question":"Quelle est la limite de la fonction f(x) = (2x² + 3x - 5)\/(x² - 4) lorsque x tend vers +∞ ?","option_a":"2","option_b":"1","option_c":"0","option_d":"+∞","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour x tendant vers l'infini, on compare les termes dominants : f(x) ≈ 2x²\/x² = 2. La limite est donc 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\": \"2\", \"b\": \"1\", \"c\": \"0\", \"d\": \"+∞\"}}","_debug_options_count":4},{"id":64779,"question":"Dans une classe de 30 élèves, 12 sont des filles. Quelle est la probabilité de choisir une fille au hasard ?","option_a":"0,3","option_b":"0,4","option_c":"0,5","option_d":"0,6","option_e":"","option_f":"","bonne_reponse":"b","explication":"La probabilité est le rapport du nombre de cas favorables (12 filles) sur le nombre total (30 élèves), soit 12\/30 = 0,4.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"0,3\", \"b\": \"0,4\", \"c\": \"0,5\", \"d\": \"0,6\"}}","_debug_options_count":4},{"id":64780,"question":"La fonction exponentielle est-elle toujours croissante sur ℝ ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction exponentielle f(x) = eˣ a pour dérivée f'(x) = eˣ, qui est toujours positive sur ℝ. Elle est donc strictement 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},{"id":64781,"question":"Quelle est la dérivée de la fonction f(x) = ln(3x² + 1) ?","option_a":"f'(x) = 6x\/(3x² + 1)","option_b":"f'(x) = 3x\/(3x² + 1)","option_c":"f'(x) = 6x²\/(3x² + 1)","option_d":"f'(x) = (3x² + 1)\/6x","option_e":"","option_f":"","bonne_reponse":"a","explication":"En utilisant la dérivée d'une fonction composée, f'(x) = (1\/(3x² + 1)) * 6x = 6x\/(3x² + 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\": \"f'(x) = 6x\/(3x² + 1)\", \"b\": \"f'(x) = 3x\/(3x² + 1)\", \"c\": \"f'(x)","_debug_options_count":4},{"id":64782,"question":"Un dé équilibré à 6 faces est lancé deux fois. Quelle est la probabilité d'obtenir deux fois le chiffre 3 ?","option_a":"1\/6","option_b":"1\/12","option_c":"1\/36","option_d":"1\/18","option_e":"","option_f":"","bonne_reponse":"c","explication":"La probabilité d'obtenir un 3 en un lancer est 1\/6. Pour deux lancers indépendants, la probabilité est (1\/6) * (1\/6) = 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\": \"1\/6\", \"b\": \"1\/12\", \"c\": \"1\/36\", \"d\": \"1\/18\"}}","_debug_options_count":4},{"id":64783,"question":"La fonction f(x) = √(x² + 1) est-elle définie pour tout x réel ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction √(x² + 1) est définie pour tout x réel car x² + 1 ≥ 1 \u003E 0. Elle est donc définie sur ℝ.","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":64784,"question":"Quelle est la solution de l'inéquation (x - 2)(x + 3) \u003E 0 ?","option_a":"x ∈ ]-3 ; 2[","option_b":"x ∈ ]-∞ ; -3[ ∪ ]2 ; +∞[","option_c":"x ∈ ]-∞ ; -3[","option_d":"x ∈ ]2 ; +∞[","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le produit (x - 2)(x + 3) est positif lorsque les deux facteurs sont de même signe. Les solutions sont donc x \u003C -3 ou x \u003E 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\": \"x ∈ ]-3 ; 2[\", \"b\": \"x ∈ ]-∞ ; -3[ ∪ ]2 ; +∞[\", \"c\": \"x","_debug_options_count":4},{"id":64785,"question":"La suite uₙ définie par uₙ = n²\/(n + 1) est-elle convergente ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"En étudiant la limite de uₙ lorsque n tend vers l'infini, on obtient lim(n²\/(n + 1)) = lim(n) = +∞. La suite n'est donc pas convergente.","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}]