Quiz — 651328f84bc83_Corrigé_Série d'exercices n 5.pdf

Question 1 sur 5 10:00

[{"id":1249,"question":"Quelle est la solution générale de l'équation sin(x) = √2\/2 dans ℝ ?","option_a":"x = π\/4 + 2kπ ou x = 3π\/4 + 2kπ, k ∈ ℤ","option_b":"x = π\/4 + kπ, k ∈ ℤ","option_c":"x = π\/4 + kπ\/2, k ∈ ℤ","option_d":"x = π\/2 + 2kπ, k ∈ ℤ","option_e":"","option_f":"","bonne_reponse":"a","explication":"Les solutions de sin(x) = √2\/2 sont x = π\/4 + 2kπ et x = 3π\/4 + 2kπ pour tout k ∈ ℤ, car sin(π\/4) = sin(3π\/4) = √2\/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\": \"x = π\/4 + 2kπ ou x = 3π\/4 + 2kπ, k ∈ ℤ\", \"b\": \"x = π\/4 +","_debug_options_count":4},{"id":1250,"question":"Dans un triangle ABC, si (DE) \/\/ (BC) avec D ∈ [AB] et E ∈ [AC], quelle égalité est vérifiée selon le théorème de Thalès ?","option_a":"AD\/AB = AE\/AC = DE\/BC","option_b":"AD\/DB = AE\/EC","option_c":"AD\/AC = AB\/AE","option_d":"AD\/DB = DE\/BC","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le théorème de Thalès énonce que si (DE) est parallèle à (BC), alors AD\/AB = AE\/AC = DE\/BC.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"AD\/AB = AE\/AC = DE\/BC\", \"b\": \"AD\/DB = AE\/EC\", \"c\": \"AD\/AC = AB\/AE","_debug_options_count":4},{"id":1251,"question":"Quelle est l'équation réduite de la droite passant par les points A(1, 2) et B(3, 4) ?","option_a":"y = x + 1","option_b":"y = 2x - 1","option_c":"y = x - 1","option_d":"y = 2x + 1","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le coefficient directeur est (4-2)\/(3-1) = 1. L'équation est y = x + b. En remplaçant A(1,2), on trouve b = 1. Donc 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 = 2x - 1\", \"c\": \"y = x - 1\", \"d\": \"y = 2x + 1","_debug_options_count":4},{"id":1252,"question":"Si cos(x) = -1\/2, quelle est la valeur de sin(x) pour x ∈ [0, π] ?","option_a":"√3\/2","option_b":"-√3\/2","option_c":"1\/2","option_d":"-1\/2","option_e":"","option_f":"","bonne_reponse":"b","explication":"Pour x ∈ [0, π], sin(x) est positif. Avec cos(x) = -1\/2, on a sin²(x) = 1 - cos²(x) = 3\/4, donc sin(x) = √3\/2. Mais comme x ∈ [π\/2, π], sin(x) est positif, mais cos(x) négatif, donc sin(x) = √3\/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\": \"√3\/2\", \"b\": \"-√3\/2\", \"c\": \"1\/2\", \"d\": \"-1\/2\"}}","_debug_options_count":4},{"id":1253,"question":"Quel est le rayon d'un cercle de centre O(2, 3) tangent à la droite d'équation 3x + 4y - 10 = 0 ?","option_a":"1","option_b":"2","option_c":"3","option_d":"4","option_e":"","option_f":"","bonne_reponse":"a","explication":"La distance du centre O(2,3) à la droite 3x + 4y - 10 = 0 est |3*2 + 4*3 - 10| \/ √(3² + 4²) = |14| \/ 5 = 14\/5 = 2.8. Le rayon est égal à cette distance, donc 2.8 ≈ 3 (arrondi).","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\": \"2\", \"c\": \"3\", \"d\": \"4\"}}","_debug_options_count":4}]