Quiz — سلسلة الاعداد المركبة و التحويلات النقطية.pdf

Question 1 sur 10 20:00

[{"id":20199,"question":"Quel est le module du nombre complexe z = 3 - 4i ?","option_a":"5","option_b":"7","option_c":"1","option_d":"√7","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le module de z = a + bi est donné par |z| = √(a² + b²). Ici, |z| = √(3² + (-4)²) = √(9 + 16) = √25 = 5.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"5\", \"b\": \"7\", \"c\": \"1\", \"d\": \"√7\"}}","_debug_options_count":4},{"id":20200,"question":"L'argument d'un nombre complexe est toujours mesuré par rapport à l'axe des abscisses.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'argument d'un nombre complexe z = a + bi est l'angle θ entre l'axe des abscisses et le vecteur OM représentant z dans le plan complexe, mesuré dans le sens trigonométrique.","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":20201,"question":"Quelle transformation transforme le point M d'affixe z en un point M' d'affixe z' = 2z ?","option_a":"Une rotation d'angle π\/2","option_b":"Une homothétie de rapport 2","option_c":"Une translation de vecteur 2","option_d":"Une symétrie par rapport à l'axe des réels","option_e":"","option_f":"","bonne_reponse":"b","explication":"La transformation z' = 2z correspond à une homothétie de centre O et de rapport 2, qui multiplie les distances par 2 sans changer les angles.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Une rotation d'angle π\/2\", \"b\": \"Une homothétie de rapport 2\", ","_debug_options_count":4},{"id":20202,"question":"Si z = 1 + i√3, quel est son argument principal ?","option_a":"π\/6","option_b":"π\/3","option_c":"π\/4","option_d":"π\/2","option_e":"","option_f":"","bonne_reponse":"b","explication":"Pour z = 1 + i√3, on a a = 1 et b = √3. L'argument θ vérifie tan(θ) = b\/a = √3, donc θ = π\/3 (60°).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"π\/6\", \"b\": \"π\/3\", \"c\": \"π\/4\", \"d\": \"π\/2\"}}","_debug_options_count":4},{"id":20203,"question":"Une rotation d'angle π et de centre O transforme le point M d'affixe z en un point M' d'affixe z' = -z.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Une rotation d'angle π (180°) de centre O transforme z en z' = -z, car elle inverse les coordonnées (a, b) en (-a, -b).","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":20204,"question":"Quel est l'affixe du point M' image de M d'affixe z = 2 + 3i par une translation de vecteur u d'affixe 1 - i ?","option_a":"3 + 2i","option_b":"1 + 2i","option_c":"3 + 4i","option_d":"2 + 2i","option_e":"","option_f":"","bonne_reponse":"a","explication":"Une translation de vecteur u d'affixe u = 1 - i transforme z en z' = z + u = (2 + 3i) + (1 - i) = 3 + 2i.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"3 + 2i\", \"b\": \"1 + 2i\", \"c\": \"3 + 4i\", \"d\": \"2 + 2i\"}}","_debug_options_count":4},{"id":20205,"question":"Le conjugué d'un nombre complexe z = a + bi est donné par z̄ = -a + bi.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le conjugué de z = a + bi est z̄ = a - bi, où seul le signe de la partie imaginaire change.","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":20206,"question":"Quelle est l'image du point M d'affixe z = 1 - i par une rotation d'angle π\/2 autour de l'origine ?","option_a":"-1 + i","option_b":"1 + i","option_c":"-1 - i","option_d":"i - 1","option_e":"","option_f":"","bonne_reponse":"a","explication":"Une rotation d'angle π\/2 transforme z = a + bi en z' = -b + ai. Ici, z' = -(-1) + 1i = 1 + i.","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 + i\", \"b\": \"1 + i\", \"c\": \"-1 - i\", \"d\": \"i - 1\"}}","_debug_options_count":4},{"id":20207,"question":"Si |z| = 5 et arg(z) = π\/4, quelle est la forme algébrique de z ?","option_a":"5 + 5i","option_b":"5√2\/2 + 5√2\/2 i","option_c":"5\/√2 + 5\/√2 i","option_d":"2,5 + 2,5i","option_e":"","option_f":"","bonne_reponse":"b","explication":"Pour z = r(cosθ + i sinθ), avec r = 5 et θ = π\/4, on a z = 5(√2\/2 + i√2\/2) = 5√2\/2 + 5√2\/2 i.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"5 + 5i\", \"b\": \"5√2\/2 + 5√2\/2 i\", \"c\": \"5\/√2 + 5\/√2 i\", \"d","_debug_options_count":4},{"id":20208,"question":"Une homothétie de rapport k = -1 est équivalente à une symétrie centrale par rapport à l'origine.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Une homothétie de rapport k = -1 transforme z en z' = -z, ce qui correspond à une symétrie centrale par rapport à l'origine (inversion des coordonnées).","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}]