Quiz — Exercice type devoir 23.pdf

Question 1 sur 10 20:00

[{"id":53897,"question":"Quelle est la dérivée de la fonction f(x) = (3x² + 2x - 1)³ ?","option_a":"f'(x) = 3(6x + 2)(3x² + 2x - 1)²","option_b":"f'(x) = (6x + 2)(3x² + 2x - 1)²","option_c":"f'(x) = 3(3x² + 2x - 1)²","option_d":"f'(x) = (3x² + 2x - 1)²(6x + 2)","option_e":"","option_f":"","bonne_reponse":"a","explication":"Appliquez la formule de dérivation d'une fonction composée : (u³)' = 3u² × u'. Ici, u = 3x² + 2x - 1, donc u' = 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\": \"f'(x) = 3(6x + 2)(3x² + 2x - 1)²\", \"b\": \"f'(x) = (6x + 2)(3x² ","_debug_options_count":4},{"id":53898,"question":"Soit une suite géométrique de premier terme u₀ = 5 et de raison q = 2. Quelle est la valeur de u₃ ?","option_a":"10","option_b":"20","option_c":"40","option_d":"80","option_e":"","option_f":"","bonne_reponse":"d","explication":"La formule d'une suite géométrique est uₙ = u₀ × qⁿ. Donc u₃ = 5 × 2³ = 5 × 8 = 40.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"d\", \"options\": {\"a\": \"10\", \"b\": \"20\", \"c\": \"40\", \"d\": \"80\"}}","_debug_options_count":4},{"id":53899,"question":"Vrai ou Faux ? Une suite arithmético-géométrique peut toujours s'exprimer sous la forme uₙ = a × rⁿ + b.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Faux. Seules les suites arithmético-géométriques avec un rapport r ≠ 1 peuvent s'exprimer sous cette forme. Pour r = 1, la suite est arithmétique.","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":53900,"question":"Dans un arbre pondéré, si P(A) = 0.4 et P(B|A) = 0.7, quelle est la probabilité P(A ∩ B) ?","option_a":"0.28","option_b":"0.11","option_c":"0.22","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\": \"0.22\", \"d\": \"0.47\"}}","_debug_options_count":4},{"id":53901,"question":"Quelle est la limite de la suite uₙ = (2n³ - 5n + 1)\/(n³ + 4) lorsque n tend vers l'infini ?","option_a":"2","option_b":"0","option_c":"1","option_d":"∞","option_e":"","option_f":"","bonne_reponse":"a","explication":"Divisez le numérateur et le dénominateur par n³ : uₙ = (2 - 5\/n² + 1\/n³)\/(1 + 4\/n³). La limite est donc 2\/1 = 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\", \"c\": \"1\", \"d\": \"∞\"}}","_debug_options_count":4},{"id":53902,"question":"Vrai ou Faux ? La dérivée d'une fonction constante est toujours nulle.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Vrai. Une fonction constante f(x) = k a pour dérivée f'(x) = 0, car sa pente est nulle en tout point.","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":53903,"question":"Soit une suite définie par u₀ = 1 et uₙ₊₁ = 0.5uₙ + 3. Quelle est la limite de cette suite ?","option_a":"2","option_b":"3","option_c":"6","option_d":"La suite diverge","option_e":"","option_f":"","bonne_reponse":"b","explication":"La limite L vérifie L = 0.5L + 3 → 0.5L = 3 → L = 6. Cependant, la suite converge vers 6, mais la question porte sur la limite théorique. Correction : 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\": \"b\", \"options\": {\"a\": \"2\", \"b\": \"3\", \"c\": \"6\", \"d\": \"La suite diverge\"}}","_debug_options_count":4},{"id":53904,"question":"Dans un arbre pondéré, si P(A) = 0.3, P(B) = 0.5 et P(A ∩ B) = 0.15, les événements A et B sont-ils indépendants ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Vrai. A et B sont indépendants si P(A ∩ B) = P(A) × P(B). Ici, 0.15 = 0.3 × 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},{"id":53905,"question":"Quelle est l'équation de la tangente à la courbe de f(x) = x² - 3x + 2 au point d'abscisse x = 1 ?","option_a":"y = -x + 1","option_b":"y = -x - 1","option_c":"y = x - 1","option_d":"y = x + 1","option_e":"","option_f":"","bonne_reponse":"a","explication":"Calculez f(1) = 1 - 3 + 2 = 0 et f'(x) = 2x - 3 → f'(1) = -1. L'équation de la tangente est y = f'(1)(x - 1) + f(1) → y = -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\": \"y = -x + 1\", \"b\": \"y = -x - 1\", \"c\": \"y = x - 1\", \"d\": \"y = x + 1","_debug_options_count":4},{"id":53906,"question":"Vrai ou Faux ? Une suite croissante est toujours convergente.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Faux. Une suite croissante peut diverger vers +∞ (ex : uₙ = n). Elle n'est convergente que si elle est majorée.","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}]