Quiz — 6387345ae0ed2_corrigé_Complx.pdf

Question 1 sur 10 20:00

[{"id":115474,"question":"Quelle est la forme algébrique du nombre complexe z = (3 + 2i)(1 - i) ?","option_a":"A) 5 + i","option_b":"B) 1 + 5i","option_c":"C) 3 - 2i","option_d":"D) 1 - i","option_e":"","option_f":"","bonne_reponse":"a","explication":"On développe le produit : (3 + 2i)(1 - i) = 3*1 + 3*(-i) + 2i*1 + 2i*(-i) = 3 - 3i + 2i - 2i². Comme i² = -1, on obtient 3 - i + 2 = 5 - i. La forme algébrique est donc 5 + 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\": \"A) 5 + i\", \"b\": \"B) 1 + 5i\", \"c\": \"C) 3 - 2i\", \"d\": \"D) 1 - i\"}}","_debug_options_count":4},{"id":115475,"question":"La fonction f(x) = x³ - 3x + 2 admet-elle un extremum local en x = 1 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Calculons la dérivée : f'(x) = 3x² - 3. En x = 1, f'(1) = 0. La dérivée seconde est f''(x) = 6x, donc f''(1) = 6 \u003E 0. La fonction admet donc un minimum local en 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":115476,"question":"Quel est le module du nombre complexe z = 4 - 3i ?","option_a":"A) 5","option_b":"B) 7","option_c":"C) √7","option_d":"D) √13","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le module d'un nombre complexe z = a + bi est donné par |z| = √(a² + b²). Ici, |z| = √(4² + (-3)²) = √(16 + 9) = √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\": \"A) 5\", \"b\": \"B) 7\", \"c\": \"C) √7\", \"d\": \"D) √13\"}}","_debug_options_count":4},{"id":115477,"question":"La suite définie par uₙ = (n² + 1)\/n converge-t-elle vers 0 quand n tend vers l'infini ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La suite uₙ = (n² + 1)\/n = n + 1\/n. Quand n tend vers l'infini, n + 1\/n tend vers l'infini. La suite ne converge donc pas vers 0.","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":115478,"question":"Quelle est la dérivée de la fonction f(x) = ln(2x + 1) ?","option_a":"A) 1\/(2x + 1)","option_b":"B) 2\/(2x + 1)","option_c":"C) 1\/x","option_d":"D) 2x\/(2x + 1)","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée de ln(u) est u'\/u. Ici, u = 2x + 1, donc u' = 2. Ainsi, f'(x) = 2\/(2x + 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\": \"A) 1\/(2x + 1)\", \"b\": \"B) 2\/(2x + 1)\", \"c\": \"C) 1\/x\", \"d\": \"D) 2x\/","_debug_options_count":4},{"id":115479,"question":"Quel est l'argument principal du nombre complexe z = -1 + i√3 ?","option_a":"A) π\/3","option_b":"B) 2π\/3","option_c":"C) π\/6","option_d":"D) 5π\/6","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le nombre complexe z = -1 + i√3 est dans le deuxième quadrant. Son argument θ vérifie tan(θ) = √3\/(-1) = -√3. L'argument principal est donc 2π\/3.","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) π\/3\", \"b\": \"B) 2π\/3\", \"c\": \"C) π\/6\", \"d\": \"D) 5π\/6\"}}","_debug_options_count":4},{"id":115480,"question":"La fonction f(x) = x⁴ - 4x³ admet-elle un point d'inflexion en x = 2 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Calculons la dérivée seconde : f'(x) = 4x³ - 12x², f''(x) = 12x² - 24x. En x = 2, f''(2) = 48 - 48 = 0. La dérivée seconde change de signe autour de x = 2, donc il y a un point d'inflexion.","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":115481,"question":"Quelle est la solution de l'équation différentielle y' + 2y = 0 avec y(0) = 3 ?","option_a":"A) y = 3e^(2x)","option_b":"B) y = 3e^(-2x)","option_c":"C) y = -6e^(2x)","option_d":"D) y = 6e^(-2x)","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'équation différentielle est linéaire d'ordre 1. La solution générale est y = Ce^(-2x). Avec y(0) = 3, on obtient C = 3. La solution est donc y = 3e^(-2x).","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) y = 3e^(2x)\", \"b\": \"B) y = 3e^(-2x)\", \"c\": \"C) y = -6e^(2x)\", ","_debug_options_count":4},{"id":115482,"question":"Le nombre complexe z = 1 + i a-t-il un argument égal à π\/4 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le nombre complexe z = 1 + i est dans le premier quadrant. Son argument θ vérifie tan(θ) = 1\/1 = 1, donc θ = π\/4.","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":115483,"question":"Quelle est la limite de la fonction f(x) = (x² - 1)\/(x - 1) quand x tend vers 1 ?","option_a":"A) 0","option_b":"B) 1","option_c":"C) 2","option_d":"D) +∞","option_e":"","option_f":"","bonne_reponse":"d","explication":"La fonction f(x) = (x² - 1)\/(x - 1) peut se simplifier en f(x) = (x - 1)(x + 1)\/(x - 1) = x + 1 pour x ≠ 1. Quand x tend vers 1, f(x) tend vers 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\": \"A) 0\", \"b\": \"B) 1\", \"c\": \"C) 2\", \"d\": \"D) +∞\"}}","_debug_options_count":4}]