Quiz — 4. Complexes 0-maths et Sc-exp.pdf

Question 1 sur 10 20:00

[{"id":31660,"question":"Quel est le module du nombre complexe z = 3 - 4i ?","option_a":"5","option_b":"7","option_c":"√7","option_d":"25","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le module se calcule par |z| = √(a² + b²) = √(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\": \"√7\", \"d\": \"25\"}}","_debug_options_count":4},{"id":31661,"question":"L'argument principal d'un nombre complexe est toujours compris entre 0 et π.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'argument principal est généralement défini dans l'intervalle ]-π, π] ou [0, 2π[ selon les conventions.","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":31662,"question":"Quel est le conjugué du nombre complexe z = 2 + 5i ?","option_a":"2 - 5i","option_b":"-2 + 5i","option_c":"5 + 2i","option_d":"-2 - 5i","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le conjugué d'un complexe z = a + ib est z̄ = a - ib.","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 - 5i\", \"b\": \"-2 + 5i\", \"c\": \"5 + 2i\", \"d\": \"-2 - 5i\"}}","_debug_options_count":4},{"id":31663,"question":"La forme trigonométrique de z = -1 + i√3 est :","option_a":"2(cos(2π\/3) + i sin(2π\/3))","option_b":"2(cos(π\/3) + i sin(π\/3))","option_c":"√2(cos(3π\/4) + i sin(3π\/4))","option_d":"1(cos(π) + i sin(π))","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le module est √((-1)² + (√3)²) = 2, et l'argument θ vérifie cosθ = -1\/2 et sinθ = √3\/2, donc θ = 2π\/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\": \"2(cos(2π\/3) + i sin(2π\/3))\", \"b\": \"2(cos(π\/3) + i sin(π\/3))\",","_debug_options_count":4},{"id":31664,"question":"L'équation z² = -4 admet deux solutions complexes conjuguées.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Les solutions sont z = 2i et z = -2i, qui sont bien conjuguées.","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":31665,"question":"Quel est le résultat de (1 + i)² ?","option_a":"2i","option_b":"1 + 2i","option_c":"2 + 2i","option_d":"0","option_e":"","option_f":"","bonne_reponse":"a","explication":"(1 + i)² = 1 + 2i + i² = 1 + 2i - 1 = 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\": \"2i\", \"b\": \"1 + 2i\", \"c\": \"2 + 2i\", \"d\": \"0\"}}","_debug_options_count":4},{"id":31666,"question":"L'affixe d'un point M dans le plan complexe est z = -3 - 2i. Quelle est la distance OM où O est l'origine ?","option_a":"√13","option_b":"5","option_c":"√5","option_d":"13","option_e":"","option_f":"","bonne_reponse":"a","explication":"La distance OM est le module de z : |z| = √((-3)² + (-2)²) = √(9 + 4) = √13.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"√13\", \"b\": \"5\", \"c\": \"√5\", \"d\": \"13\"}}","_debug_options_count":4},{"id":31667,"question":"Pour tout nombre complexe z, on a |z²| = |z|².","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"En effet, |z²| = |z|² car |z²| = |z * z| = |z| * |z| = |z|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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":31668,"question":"La forme exponentielle d'un nombre complexe z = r(cosθ + i sinθ) est :","option_a":"z = re^(iθ)","option_b":"z = r²e^(iθ)","option_c":"z = re^(-iθ)","option_d":"z = e^(irθ)","option_e":"","option_f":"","bonne_reponse":"a","explication":"La formule d'Euler donne z = r(cosθ + i sinθ) = re^(iθ).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"z = re^(iθ)\", \"b\": \"z = r²e^(iθ)\", \"c\": \"z = re^(-iθ)\", \"d\": ","_debug_options_count":4},{"id":31669,"question":"Si z₁ = 2e^(iπ\/4) et z₂ = 3e^(iπ\/6), quel est le module de z₁ * z₂ ?","option_a":"5","option_b":"6","option_c":"√6","option_d":"1","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le module de z₁ * z₂ est |z₁| * |z₂| = 2 * 3 = 6.","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\", \"b\": \"6\", \"c\": \"√6\", \"d\": \"1\"}}","_debug_options_count":4}]