Quiz — sujet 5(corrigé)B.pdf

Question 1 sur 10 20:00

[{"id":11533,"question":"Quelle est la dérivée de la fonction f(x) = 3x² + 2x - 5 ?","option_a":"6x + 2","option_b":"6x² + 2","option_c":"3x + 2","option_d":"6x - 5","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'une fonction polynôme se calcule terme à terme : 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\": \"6x + 2\", \"b\": \"6x² + 2\", \"c\": \"3x + 2\", \"d\": \"6x - 5\"}}","_debug_options_count":4},{"id":11534,"question":"La suite définie par uₙ₊₁ = 0.5uₙ + 3 est-elle convergente ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Une suite définie par uₙ₊₁ = auₙ + b avec |a| \u003C 1 est convergente vers b\/(1-a). Ici, la limite est 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":11535,"question":"Soit P(A) = 0.4 et P(B|A) = 0.7. Que vaut P(A ∩ B) ?","option_a":"0.28","option_b":"0.11","option_c":"0.3","option_d":"0.47","option_e":"","option_f":"","bonne_reponse":"a","explication":"P(A ∩ B) = P(A) × P(B|A) = 0.4 × 0.7 = 0.28.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"0.28\", \"b\": \"0.11\", \"c\": \"0.3\", \"d\": \"0.47\"}}","_debug_options_count":4},{"id":11536,"question":"L’équation différentielle y' + 2y = 0 a pour solution générale :","option_a":"y = Ce^(-2x)","option_b":"y = Ce^(2x)","option_c":"y = C + e^(-2x)","option_d":"y = Cx + e^(-2x)","option_e":"","option_f":"","bonne_reponse":"a","explication":"La solution générale d'une équation différentielle linéaire du 1er ordre est y = Ce^(-2x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"y = Ce^(-2x)\", \"b\": \"y = Ce^(2x)\", \"c\": \"y = C + e^(-2x)\", \"d\": \"","_debug_options_count":4},{"id":11537,"question":"Soit f(x) = x³ - 3x + 1. Le point critique en x = 1 est-il un maximum local ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"f'(x) = 3x² - 3. En x=1, f'(1)=0. f''(x)=6x, donc f''(1)=6\u003E0 : c'est un minimum local.","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":11538,"question":"Quelle est la limite de la suite uₙ = (2n + 1)\/(n + 3) quand n tend vers l'infini ?","option_a":"2","option_b":"0.5","option_c":"1","option_d":"∞","option_e":"","option_f":"","bonne_reponse":"a","explication":"En divisant numérateur et dénominateur par n, on obtient lim uₙ = 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\": \"0.5\", \"c\": \"1\", \"d\": \"∞\"}}","_debug_options_count":4},{"id":11539,"question":"Soit A et B deux événements indépendants avec P(A) = 0.3 et P(B) = 0.5. Que vaut P(A ∪ B) ?","option_a":"0.65","option_b":"0.8","option_c":"0.2","option_d":"0.15","option_e":"","option_f":"","bonne_reponse":"a","explication":"P(A ∪ B) = P(A) + P(B) - P(A ∩ B) = 0.3 + 0.5 - (0.3×0.5) = 0.65.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"0.65\", \"b\": \"0.8\", \"c\": \"0.2\", \"d\": \"0.15\"}}","_debug_options_count":4},{"id":11540,"question":"La fonction f(x) = ln(x) est-elle définie sur ℝ⁻ ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction ln(x) est définie uniquement pour x \u003E 0. Elle n'est pas définie sur ℝ⁻.","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":11541,"question":"Quelle est la solution de l’inéquation 2x - 5 \u003E 3x + 1 ?","option_a":"x \u003C -6","option_b":"x \u003E -6","option_c":"x \u003C 6","option_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\": \"x \u003C -6\", \"b\": \"x \u003E -6\", \"c\": \"x \u003C 6\", \"d\": \"x \u003E 6\"}}","_debug_options_count":4},{"id":11542,"question":"Soit f une fonction dérivable sur ℝ avec f'(x) = 0 pour tout x. Que peut-on dire de f ?","option_a":"f est constante","option_b":"f est linéaire","option_c":"f est croissante","option_d":"f est décroissante","option_e":"","option_f":"","bonne_reponse":"a","explication":"Si f'(x) = 0 pour tout x, alors f est une fonction constante (théorème des accroissements finis).","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 est constante\", \"b\": \"f est linéaire\", \"c\": \"f est croissante\"","_debug_options_count":4}]