Quiz — daniel-fredon-algebre-et-geometrie-en-30-fiches.pdf

Question 1 sur 10 20:00

[{"id":11303,"question":"Quelle est la dimension d’un espace vectoriel de base composée de 3 vecteurs non nuls et linéairement indépendants ?","option_a":"0","option_b":"1","option_c":"2","option_d":"3","option_e":"","option_f":"","bonne_reponse":"d","explication":"La dimension d’un espace vectoriel est égale au nombre de vecteurs dans une base de cet espace. Ici, la base compte 3 vecteurs, donc la dimension 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\": \"0\", \"b\": \"1\", \"c\": \"2\", \"d\": \"3\"}}","_debug_options_count":4},{"id":11304,"question":"Une application linéaire entre deux espaces vectoriels de même dimension est toujours bijective.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Une application linéaire entre deux espaces de même dimension n’est pas nécessairement bijective. Par exemple, une projection sur un sous-espace n’est pas injective.","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":11305,"question":"Quel est le déterminant d’une matrice triangulaire supérieure dont tous les éléments diagonaux valent 2 ?","option_a":"0","option_b":"2","option_c":"4","option_d":"8","option_e":"","option_f":"","bonne_reponse":"d","explication":"Le déterminant d’une matrice triangulaire est le produit des éléments diagonaux. Ici, 2 × 2 × 2 = 8.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"d\", \"options\": {\"a\": \"0\", \"b\": \"2\", \"c\": \"4\", \"d\": \"8\"}}","_debug_options_count":4},{"id":11306,"question":"Soit une matrice A de taille 2×2. Si det(A) = 0, alors la matrice A est inversible.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Une matrice est inversible si et seulement si son déterminant est non nul. Si det(A) = 0, la matrice n’est pas inversible.","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":11307,"question":"Quelle est la solution du système d’équations suivant ? { 2x + y = 5 ; x - y = 1 }","option_a":"(x=2, y=1)","option_b":"(x=1, y=2)","option_c":"(x=3, y=-1)","option_d":"(x=0, y=5)","option_e":"","option_f":"","bonne_reponse":"a","explication":"En résolvant le système, on trouve x = 2 et y = 1. Vérification : 2(2) + 1 = 5 et 2 - 1 = 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\": \"(x=2, y=1)\", \"b\": \"(x=1, y=2)\", \"c\": \"(x=3, y=-1)\", \"d\": \"(x=0, y","_debug_options_count":4},{"id":11308,"question":"Les vecteurs u = (1, 2) et v = (2, 4) sont linéairement indépendants.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Les vecteurs u et v sont colinéaires (v = 2u), donc ils sont linéairement dépendants.","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":11309,"question":"Quelle est la valeur propre d’une matrice diagonale dont les éléments diagonaux sont 3, 3 et 3 ?","option_a":"0","option_b":"1","option_c":"3","option_d":"9","option_e":"","option_f":"","bonne_reponse":"c","explication":"Les valeurs propres d’une matrice diagonale sont les éléments diagonaux. Ici, la seule valeur propre est 3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"0\", \"b\": \"1\", \"c\": \"3\", \"d\": \"9\"}}","_debug_options_count":4},{"id":11310,"question":"Soit une application linéaire f : R² → R² définie par f(x, y) = (x + y, x - y). Quelle est la matrice de f dans la base canonique ?","option_a":"[[1, 1], [1, -1]]","option_b":"[[1, 0], [0, 1]]","option_c":"[[0, 1], [1, 0]]","option_d":"[[1, 1], [-1, 1]]","option_e":"","option_f":"","bonne_reponse":"a","explication":"La matrice de f est obtenue en appliquant f aux vecteurs de la base canonique : f(1,0) = (1,1) et f(0,1) = (1,-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\": \"[[1, 1], [1, -1]]\", \"b\": \"[[1, 0], [0, 1]]\", \"c\": \"[[0, 1], [1, 0","_debug_options_count":4},{"id":11311,"question":"Le noyau d’une application linéaire est toujours un sous-espace vectoriel.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le noyau d’une application linéaire est l’ensemble des vecteurs dont l’image est le vecteur nul. C’est un sous-espace vectoriel par définition.","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":11312,"question":"Quelle est la solution générale de l’équation différentielle y' = 2y ?","option_a":"y = Ce^(2x)","option_b":"y = 2x + C","option_c":"y = Ce^(-2x)","option_d":"y = x² + C","option_e":"","option_f":"","bonne_reponse":"a","explication":"L’équation y' = 2y est une équation différentielle linéaire d’ordre 1. Sa solution générale est y = Ce^(2x), où C est une constante.","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 = Ce^(2x)\", \"b\": \"y = 2x + C\", \"c\": \"y = Ce^(-2x)\", \"d\": \"y = x","_debug_options_count":4}]