Quiz — serie32_arith-recurrence_corrigé.pdf

Question 1 sur 10 20:00

[{"id":43350,"question":"Soit la propriété P(n) : « 2^n \u003E n^2 ». Pour n = 5, la propriété est-elle vraie ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour n=5, 2^5 = 32 et 5^2 = 25. Comme 32 \u003E 25, la propriété est vraie.","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":43351,"question":"Quelle est la valeur de x telle que 3x ≡ 2 [5] ?","option_a":"x ≡ 1 [5]","option_b":"x ≡ 2 [5]","option_c":"x ≡ 3 [5]","option_d":"x ≡ 4 [5]","option_e":"","option_f":"","bonne_reponse":"a","explication":"3*4 = 12 ≡ 2 [5]. Les autres options ne satisfont pas l'équation.","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 ≡ 1 [5]\", \"b\": \"x ≡ 2 [5]\", \"c\": \"x ≡ 3 [5]\", \"d\": \"x ≡","_debug_options_count":4},{"id":43352,"question":"Pour démontrer par récurrence une propriété P(n), il faut vérifier :","option_a":"P(0) est vraie et P(n) implique P(n+1)","option_b":"P(1) est vraie et P(n) implique P(n-1)","option_c":"P(0) est vraie et P(n) implique P(2n)","option_d":"P(1) est vraie et P(n) implique P(n^2)","option_e":"","option_f":"","bonne_reponse":"a","explication":"La récurrence nécessite une initialisation (P(0) ou P(1) vraie) et une hérédité (P(n) implique P(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\": \"P(0) est vraie et P(n) implique P(n+1)\", \"b\": \"P(1) est vraie et ","_debug_options_count":4},{"id":43353,"question":"Le nombre 123456789 est-il divisible par 9 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La somme des chiffres est 1+2+3+4+5+6+7+8+9 = 45, qui est divisible par 9. Donc 123456789 est divisible par 9.","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":43354,"question":"Quelle est la solution de l'équation diophantienne 5x + 3y = 1 ?","option_a":"x=2, y=-3","option_b":"x=-1, y=2","option_c":"x=1, y=-1","option_d":"x=0, y=1\/3","option_e":"","option_f":"","bonne_reponse":"b","explication":"Pour x=-1 et y=2, 5*(-1) + 3*2 = -5 + 6 = 1. Les autres options ne satisfont pas l'équation.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"x=2, y=-3\", \"b\": \"x=-1, y=2\", \"c\": \"x=1, y=-1\", \"d\": \"x=0, y=1\/3\"","_debug_options_count":4},{"id":43355,"question":"Si a ≡ b [n] et c ≡ d [n], alors a + c ≡ b + d [n]. Cette affirmation est-elle vraie ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est une propriété fondamentale des congruences : la somme de deux congruences est congruente à la somme des congruences.","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":43356,"question":"Pour n ≥ 1, la suite définie par u_{n+1} = 2u_n + 1 avec u_1 = 1 est-elle arithmétique ?","option_a":"Oui, car u_{n+1} - u_n est constant","option_b":"Non, car la différence n'est pas constante","option_c":"Oui, car elle est définie par une relation de récurrence","option_d":"Non, car elle n'est pas définie pour n=0","option_e":"","option_f":"","bonne_reponse":"b","explication":"La différence u_{n+1} - u_n = u_n + 1 n'est pas constante, donc la suite n'est pas 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\": \"Oui, car u_{n+1} - u_n est constant\", \"b\": \"Non, car la différen","_debug_options_count":4},{"id":43357,"question":"Quel est le reste de la division de 10^2024 par 7 ?","option_a":"1","option_b":"2","option_c":"3","option_d":"4","option_e":"","option_f":"","bonne_reponse":"d","explication":"10 ≡ 3 [7], donc 10^2024 ≡ 3^2024 [7]. Or 3^6 ≡ 1 [7], donc 3^2024 = 3^(6*337 + 2) ≡ 3^2 ≡ 2 [7]. Le reste est donc 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"d\", \"options\": {\"a\": \"1\", \"b\": \"2\", \"c\": \"3\", \"d\": \"4\"}}","_debug_options_count":4},{"id":43358,"question":"La propriété P(n) : « n^2 + n + 41 est premier pour tout n ∈ ℕ » est-elle vraie ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Pour n=40, 40^2 + 40 + 41 = 1681 = 41^2, qui n'est pas premier. La propriété est donc fausse.","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":43359,"question":"Quelle est la valeur de la somme S = 1 + 3 + 5 + ... + (2n-1) ?","option_a":"n^2","option_b":"2n^2","option_c":"n(n+1)","option_d":"(2n-1)^2","option_e":"","option_f":"","bonne_reponse":"a","explication":"La somme des n premiers nombres impairs est égale à n^2. C'est une propriété classique des suites arithmétiques.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"n^2\", \"b\": \"2n^2\", \"c\": \"n(n+1)\", \"d\": \"(2n-1)^2\"}}","_debug_options_count":4}]