Quiz — 662a2f8aa2382_corrigé_série 22.pdf

Question 1 sur 10 20:00

[{"id":54157,"question":"Soit f une fonction dérivable sur ℝ telle que f'(x) = 3x² - 2x + 1. Quelle est la dérivée de la fonction g(x) = f(x²) ?","option_a":"A. 6x³ - 4x² + 2x","option_b":"B. 3x⁴ - 2x³ + x²","option_c":"C. 6x² - 4x + 2","option_d":"D. 3x² - 2x + 1","option_e":"","option_f":"","bonne_reponse":"c","explication":"On utilise la règle de dérivation des fonctions composées : g'(x) = f'(x²) * 2x = (3(x²)² - 2(x²) + 1) * 2x = (3x⁴ - 2x² + 1) * 2x = 6x⁵ - 4x³ + 2x.","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. 6x³ - 4x² + 2x\", \"b\": \"B. 3x⁴ - 2x³ + x²\", \"c\": \"C. 6x²","_debug_options_count":4},{"id":54158,"question":"Une suite (uₙ) est définie par u₀ = 2 et uₙ₊₁ = 0.8uₙ + 1.5. Cette suite est-elle convergente ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La suite est convergente car c'est une suite arithmético-géométrique avec un coefficient multiplicateur |0.8| \u003C 1. Sa limite L vérifie L = 0.8L + 1.5, soit L = 7.5.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":54159,"question":"Soit X une variable aléatoire suivant une loi binomiale B(n=10, p=0.3). Quelle est P(X ≥ 2) ?","option_a":"A. 1 - P(X=0) - P(X=1)","option_b":"B. P(X=2) + P(X=3) + ... + P(X=10)","option_c":"C. 1 - P(X ≤ 1)","option_d":"D. Toutes les réponses sont correctes","option_e":"","option_f":"","bonne_reponse":"d","explication":"P(X ≥ 2) = 1 - P(X \u003C 2) = 1 - [P(X=0) + P(X=1)]. Les options A et C sont équivalentes, et l'option B est une autre façon de l'exprimer.","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. 1 - P(X=0) - P(X=1)\", \"b\": \"B. P(X=2) + P(X=3) + ... + P(X=10)","_debug_options_count":4},{"id":54160,"question":"La dérivée de la fonction f(x) = ln(3x + 1) est f'(x) = 3\/(3x + 1).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de ln(u) est u'\/u. Ici, u = 3x + 1, donc u' = 3. Ainsi, f'(x) = 3\/(3x + 1). L'affirmation est correcte.","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":54161,"question":"Soit (uₙ) une suite géométrique de raison q = -2 et de premier terme u₀ = 5. Quelle est la valeur de u₃ ?","option_a":"A. -40","option_b":"B. 40","option_c":"C. -20","option_d":"D. 20","option_e":"","option_f":"","bonne_reponse":"a","explication":"uₙ = u₀ * qⁿ. Donc u₃ = 5 * (-2)³ = 5 * (-8) = -40.","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. -40\", \"b\": \"B. 40\", \"c\": \"C. -20\", \"d\": \"D. 20\"}}","_debug_options_count":4},{"id":54162,"question":"Soit A et B deux événements indépendants tels que P(A) = 0.4 et P(B) = 0.5. Quelle est P(A ∩ B) ?","option_a":"A. 0.9","option_b":"B. 0.2","option_c":"C. 0.5","option_d":"D. 0.4","option_e":"","option_f":"","bonne_reponse":"b","explication":"Pour des événements indépendants, P(A ∩ B) = P(A) * P(B) = 0.4 * 0.5 = 0.2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"A. 0.9\", \"b\": \"B. 0.2\", \"c\": \"C. 0.5\", \"d\": \"D. 0.4\"}}","_debug_options_count":4},{"id":54163,"question":"La fonction f(x) = x³ - 3x² + 4 est 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 - 6x = 3x(x - 2). f'(x) est négative sur ]0; 2[, donc f n'est pas croissante 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":54164,"question":"Soit (uₙ) une suite définie par uₙ = n² - 5n + 6. Quel est le terme u₄ ?","option_a":"A. 2","option_b":"B. 6","option_c":"C. -2","option_d":"D. 0","option_e":"","option_f":"","bonne_reponse":"a","explication":"u₄ = 4² - 5*4 + 6 = 16 - 20 + 6 = 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. 2\", \"b\": \"B. 6\", \"c\": \"C. -2\", \"d\": \"D. 0\"}}","_debug_options_count":4},{"id":54165,"question":"Soit X une variable aléatoire continue de densité f(x) = 2x sur [0; 1]. Quelle est P(0.5 ≤ X ≤ 1) ?","option_a":"A. 0.75","option_b":"B. 0.5","option_c":"C. 0.25","option_d":"D. 1","option_e":"","option_f":"","bonne_reponse":"a","explication":"P(0.5 ≤ X ≤ 1) = ∫₀.₅¹ 2x dx = [x²]₀.₅¹ = 1 - 0.25 = 0.75.","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. 0.75\", \"b\": \"B. 0.5\", \"c\": \"C. 0.25\", \"d\": \"D. 1\"}}","_debug_options_count":4},{"id":54166,"question":"La limite de la suite (uₙ) définie par uₙ = (n² + 1)\/(2n² - 3) est 0.5.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"En divisant numérateur et dénominateur par n², on obtient uₙ = (1 + 1\/n²)\/(2 - 3\/n²). La limite quand n→∞ est 1\/2 = 0.5.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4}]