Quiz — (ebook-pdf) - Mathematics - Concrete Mathematics.pdf

Question 1 sur 10 20:00

[{"id":8978,"question":"Quelle est la somme de la série géométrique infinie ∑ (de n=0 à ∞) (1\/2)^n ?","option_a":"1","option_b":"2","option_c":"1\/2","option_d":"∞","option_e":"","option_f":"","bonne_reponse":"b","explication":"La somme d'une série géométrique infinie ∑ (de n=0 à ∞) r^n est 1\/(1-r) pour |r| \u003C 1. Ici, r = 1\/2, donc la somme est 1\/(1-1\/2) = 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\": \"1\", \"b\": \"2\", \"c\": \"1\/2\", \"d\": \"∞\"}}","_debug_options_count":4},{"id":8979,"question":"La fonction f(x) = x^2 est-elle intégrable sur l'intervalle [-1, 1] ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction f(x) = x^2 est continue sur [-1, 1], donc elle est intégrable sur cet intervalle.","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":8980,"question":"Quel est le résultat de l'intégrale ∫ (de 0 à π) sin(x) dx ?","option_a":"0","option_b":"1","option_c":"2","option_d":"π","option_e":"","option_f":"","bonne_reponse":"c","explication":"L'intégrale de sin(x) est -cos(x). Évaluée entre 0 et π, on obtient -cos(π) - (-cos(0)) = -(-1) - (-1) = 1 + 1 = 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"0\", \"b\": \"1\", \"c\": \"2\", \"d\": \"π\"}}","_debug_options_count":4},{"id":8981,"question":"La récurrence F(n) = F(n-1) + F(n-2) avec F(0) = 0 et F(1) = 1 définit quelle suite célèbre ?","option_a":"Suite arithmétique","option_b":"Suite géométrique","option_c":"Suite de Fibonacci","option_d":"Suite harmonique","option_e":"","option_f":"","bonne_reponse":"c","explication":"La récurrence donnée est la définition classique de la suite de Fibonacci, où chaque terme est la somme des deux précédents.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"Suite arithmétique\", \"b\": \"Suite géométrique\", \"c\": \"Suite de ","_debug_options_count":4},{"id":8982,"question":"L'intégrale impropre ∫ (de 1 à ∞) (1\/x^2) dx converge-t-elle ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'intégrale ∫ (de 1 à ∞) (1\/x^2) dx converge car la primitive -1\/x, évaluée entre 1 et ∞, donne une valeur finie (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":8983,"question":"Quelle est la valeur de la série ∑ (de n=1 à ∞) (1\/n(n+1)) ?","option_a":"1","option_b":"0","option_c":"∞","option_d":"1\/2","option_e":"","option_f":"","bonne_reponse":"a","explication":"La série se télescope : ∑ (de n=1 à ∞) (1\/n - 1\/(n+1)) = 1 - lim (n→∞) 1\/(n+1) = 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\": \"1\", \"b\": \"0\", \"c\": \"∞\", \"d\": \"1\/2\"}}","_debug_options_count":4},{"id":8984,"question":"La fonction f(x) = e^x est-elle solution de l'équation différentielle y'' - y = 0 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée seconde de e^x est e^x. En substituant dans l'équation, on obtient e^x - e^x = 0, donc la fonction est bien solution.","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":8985,"question":"Quel est le résultat de la somme ∑ (de k=0 à n) C(n,k) ?","option_a":"n","option_b":"2^n","option_c":"n^2","option_d":"0","option_e":"","option_f":"","bonne_reponse":"b","explication":"La somme des coefficients binomiaux C(n,k) pour k de 0 à n est égale à 2^n, d'après le binôme de Newton.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"n\", \"b\": \"2^n\", \"c\": \"n^2\", \"d\": \"0\"}}","_debug_options_count":4},{"id":8986,"question":"L'intégrale ∫ (de -∞ à ∞) e^(-x^2) dx est-elle finie ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Cette intégrale converge et vaut √π, car la fonction e^(-x^2) décroît rapidement vers 0 lorsque x tend vers ±∞.","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":8987,"question":"Quelle est la valeur de la récurrence F(n) = 2F(n-1) avec F(0) = 1 ?","option_a":"2^n","option_b":"n^2","option_c":"n!","option_d":"1\/n","option_e":"","option_f":"","bonne_reponse":"a","explication":"La solution de cette récurrence linéaire est F(n) = 2^n, car chaque terme est le double du précédent.","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^n\", \"b\": \"n^2\", \"c\": \"n!\", \"d\": \"1\/n\"}}","_debug_options_count":4}]