Quiz — مواضيع أولمبياد الرياضيات 2023- بالعربية، الإنجليزية والفرنسية - موقع الأستاذ راحيس عمر.pdf

Question 1 sur 10 20:00

[{"id":159841,"question":"Quel est le résultat de l'inégalité suivante : x² - 5x + 6 \u003C 0 ?","option_a":"A. x ∈ ]2, 3[","option_b":"B. x ∈ ]-∞, 2[ ∪ ]3, +∞[","option_c":"C. x ∈ ]-∞, +∞[","option_d":"D. x ∈ {2, 3}","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'inégalité x² - 5x + 6 \u003C 0 se résout en factorisant : (x-2)(x-3) \u003C 0. Le produit est négatif lorsque x est entre 2 et 3.","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. x ∈ ]2, 3[\", \"b\": \"B. x ∈ ]-∞, 2[ ∪ ]3, +∞[\", \"c\": \"","_debug_options_count":4},{"id":159842,"question":"La somme des angles intérieurs d'un pentagone convexe est égale à 540°.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La formule générale pour un polygone à n côtés est (n-2) × 180°. Pour n=5, on obtient 540°.","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":159843,"question":"Quel est le dernier chiffre de 7^2023 ?","option_a":"A. 1","option_b":"B. 3","option_c":"C. 7","option_d":"D. 9","option_e":"","option_f":"","bonne_reponse":"d","explication":"Les puissances de 7 cyclent tous les 4 : 7^1=7, 7^2=9, 7^3=3, 7^4=1. 2023 mod 4 = 3, donc le dernier chiffre est 3.","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\", \"b\": \"B. 3\", \"c\": \"C. 7\", \"d\": \"D. 9\"}}","_debug_options_count":4},{"id":159844,"question":"Soit f(x) = 3x² - 2x + 1. Quelle est la dérivée de f en x=2 ?","option_a":"A. 10","option_b":"B. 12","option_c":"C. 14","option_d":"D. 16","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée de f(x) est f'(x) = 6x - 2. En x=2, f'(2) = 6×2 - 2 = 10.","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. 10\", \"b\": \"B. 12\", \"c\": \"C. 14\", \"d\": \"D. 16\"}}","_debug_options_count":4},{"id":159845,"question":"Un nombre premier est toujours impair.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le nombre 2 est le seul nombre premier pair, tous les autres sont impairs.","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":159846,"question":"Quelle est la solution de l'équation 2^(x+1) = 8 ?","option_a":"A. x = 1","option_b":"B. x = 2","option_c":"C. x = 3","option_d":"D. x = 4","option_e":"","option_f":"","bonne_reponse":"b","explication":"8 = 2³, donc 2^(x+1) = 2³ ⇒ x+1 = 3 ⇒ x = 2. (Correction : x=2 est la réponse correcte, l'option C correspond à x=3 qui est incorrect.)","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. x = 1\", \"b\": \"B. x = 2\", \"c\": \"C. x = 3\", \"d\": \"D. x = 4\"}}","_debug_options_count":4},{"id":159847,"question":"Dans un triangle rectangle, le carré de l'hypoténuse est égal à la somme des carrés des deux autres côtés.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est le théorème de Pythagore, une propriété fondamentale des triangles rectangles.","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":159848,"question":"Quel est le PGCD de 120 et 180 ?","option_a":"A. 30","option_b":"B. 40","option_c":"C. 60","option_d":"D. 120","option_e":"","option_f":"","bonne_reponse":"a","explication":"La décomposition en facteurs premiers donne 120 = 2³×3×5 et 180 = 2²×3²×5. Le PGCD est 2²×3×5 = 60.","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. 30\", \"b\": \"B. 40\", \"c\": \"C. 60\", \"d\": \"D. 120\"}}","_debug_options_count":4},{"id":159849,"question":"La fonction f(x) = x³ - 3x est croissante sur l'intervalle ]-∞, -1[.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée f'(x) = 3x² - 3 est négative sur ]-1, 1[, donc la fonction est décroissante sur cet intervalle. Elle est croissante sur ]-∞, -1[ et ]1, +∞[.","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":159850,"question":"Quel est le résultat de la somme 1 + 2 + 3 + ... + 100 ?","option_a":"A. 5050","option_b":"B. 5000","option_c":"C. 4950","option_d":"D. 5100","option_e":"","option_f":"","bonne_reponse":"a","explication":"La somme des n premiers entiers est n(n+1)\/2. Pour n=100, on obtient 100×101\/2 = 5050.","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. 5050\", \"b\": \"B. 5000\", \"c\": \"C. 4950\", \"d\": \"D. 5100\"}}","_debug_options_count":4}]