Quiz — item 47 sousse 2018 insuf surr.pdf

Question 1 sur 10 20:00

[{"id":4146,"question":"Quelle est la dérivée de la fonction f(x) = x³ + 2x² - 5x + 1 ?","option_a":"3x² + 4x - 5","option_b":"x³ + 4x - 5","option_c":"3x² + 4x + 1","option_d":"x² + 2x - 5","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'une fonction polynôme s'obtient en appliquant la règle de dérivation terme à terme : (x^n)' = n*x^(n-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\": \"3x² + 4x - 5\", \"b\": \"x³ + 4x - 5\", \"c\": \"3x² + 4x + 1\", \"d\": \"","_debug_options_count":4},{"id":4147,"question":"Soit une suite géométrique de premier terme u₀ = 2 et de raison q = 3. Quel est le terme u₃ ?","option_a":"18","option_b":"54","option_c":"162","option_d":"81","option_e":"","option_f":"","bonne_reponse":"b","explication":"Pour une suite géométrique, uₙ = u₀ * qⁿ. Ici, u₃ = 2 * 3³ = 2 * 27 = 54.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"18\", \"b\": \"54\", \"c\": \"162\", \"d\": \"81\"}}","_debug_options_count":4},{"id":4148,"question":"La fonction f(x) = x² est-elle convexe sur ℝ ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Une fonction est convexe si sa dérivée seconde est positive. Ici, f''(x) = 2 \u003E 0, donc f est convexe.","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":4149,"question":"Quelle est la limite de la suite uₙ = (2n + 1)\/(n + 3) quand n tend vers l'infini ?","option_a":"0","option_b":"1","option_c":"2","option_d":"∞","option_e":"","option_f":"","bonne_reponse":"b","explication":"En divisant numérateur et dénominateur par n, on obtient lim (2 + 1\/n)\/(1 + 3\/n) = 2\/1 = 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\": \"0\", \"b\": \"1\", \"c\": \"2\", \"d\": \"∞\"}}","_debug_options_count":4},{"id":4150,"question":"Si P(A) = 0.4 et P(B) = 0.5, quelle est la valeur maximale possible de P(A ∩ B) ?","option_a":"0.4","option_b":"0.5","option_c":"0.9","option_d":"0.2","option_e":"","option_f":"","bonne_reponse":"a","explication":"P(A ∩ B) ≤ min(P(A), P(B)) = min(0.4, 0.5) = 0.4. La valeur maximale est donc 0.4.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"0.4\", \"b\": \"0.5\", \"c\": \"0.9\", \"d\": \"0.2\"}}","_debug_options_count":4},{"id":4151,"question":"La dérivée de la fonction f(x) = ln(x) est f'(x) = 1\/x.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de ln(x) est bien 1\/x, c'est une propriété fondamentale des logarithmes.","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":4152,"question":"Quelle est la somme des 5 premiers termes d'une suite arithmétique de premier terme u₀ = 3 et de raison r = 2 ?","option_a":"25","option_b":"30","option_c":"35","option_d":"40","option_e":"","option_f":"","bonne_reponse":"b","explication":"La somme des n premiers termes d'une suite arithmétique est Sₙ = n\/2 * (2u₀ + (n-1)r). Ici, S₅ = 5\/2 * (6 + 8) = 5\/2 * 14 = 35.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"25\", \"b\": \"30\", \"c\": \"35\", \"d\": \"40\"}}","_debug_options_count":4},{"id":4153,"question":"Si deux événements A et B sont incompatibles, alors P(A ∪ B) = P(A) + P(B).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour des événements incompatibles, P(A ∪ B) = P(A) + P(B) car P(A ∩ B) = 0.","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":4154,"question":"Quelle est la dérivée seconde de la fonction f(x) = e^(2x) ?","option_a":"2e^(2x)","option_b":"4e^(2x)","option_c":"e^(2x)","option_d":"2xe^(2x)","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée première est f'(x) = 2e^(2x), et la dérivée seconde est f''(x) = 4e^(2x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"2e^(2x)\", \"b\": \"4e^(2x)\", \"c\": \"e^(2x)\", \"d\": \"2xe^(2x)\"}}","_debug_options_count":4},{"id":4155,"question":"La suite uₙ = (-1)^n est-elle convergente ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La suite alterne entre -1 et 1 et ne tend vers aucune limite finie, donc elle est divergente.","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}]