Quiz — Calvo, Doyen - Exercices d'Algèbre 1ère année Dunod (1997).pdf

Question 1 sur 10 20:00

[{"id":40790,"question":"Quel est le résultat de (3 + 4i) + (2 - 5i) ?","option_a":"5 - i","option_b":"5 + 9i","option_c":"1 - 9i","option_d":"5 - 9i","option_e":"","option_f":"","bonne_reponse":"a","explication":"On additionne séparément les parties réelles et les parties imaginaires : (3+2) + (4-5)i = 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\": \"5 - i\", \"b\": \"5 + 9i\", \"c\": \"1 - 9i\", \"d\": \"5 - 9i\"}}","_debug_options_count":4},{"id":40791,"question":"Un polynôme de degré 3 peut avoir au plus 3 racines réelles.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Un polynôme de degré n a au plus n racines réelles (théorème fondamental de l'algèbre).","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":40792,"question":"Quelle est la dimension d'un espace vectoriel de base {v1, v2, v3} ?","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.","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":40793,"question":"Le déterminant d'une matrice 2x2 [[a, b], [c, d]] est égal à :","option_a":"a + d","option_b":"ad - bc","option_c":"a*d + b*c","option_d":"a*b - c*d","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le déterminant est calculé par ad - bc pour une matrice 2x2.","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 + d\", \"b\": \"ad - bc\", \"c\": \"a*d + b*c\", \"d\": \"a*b - c*d\"}}","_debug_options_count":4},{"id":40794,"question":"Toute application linéaire est bijective.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Une application linéaire n'est bijective que si elle est à la fois injective et surjective, ce qui n'est pas toujours le cas.","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":40795,"question":"Quel est le résultat de (x² - 4)(x + 2) ?","option_a":"x³ - 2x² - 8x - 8","option_b":"x³ + 2x² - 4x - 8","option_c":"x³ - 4x - 8","option_d":"x³ + 2x² - 8","option_e":"","option_f":"","bonne_reponse":"b","explication":"En développant : x²*x + x²*2 - 4*x - 4*2 = x³ + 2x² - 4x - 8.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"x³ - 2x² - 8x - 8\", \"b\": \"x³ + 2x² - 4x - 8\", \"c\": \"x³ - 4x ","_debug_options_count":4},{"id":40796,"question":"Un système linéaire de 3 équations à 2 inconnues a toujours une solution unique.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Un tel système peut n'avoir aucune solution, une infinité de solutions ou une solution unique, selon les cas.","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":40797,"question":"Quelle est la matrice inverse de [[1, 2], [3, 4]] ?","option_a":"[[ -2, 1], [1.5, -0.5]]","option_b":"[[1, -2], [-3, 4]]","option_c":"[[0.5, -1], [-0.75, 0.5]]","option_d":"[[2, -1], [-1.5, 0.5]]","option_e":"","option_f":"","bonne_reponse":"c","explication":"La matrice inverse est (1\/det)*[[d, -b], [-c, a]] = (-0.5)*[[4, -2], [-3, 1]] = [[-2, 1], [1.5, -0.5]].","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"[[ -2, 1], [1.5, -0.5]]\", \"b\": \"[[1, -2], [-3, 4]]\", \"c\": \"[[0.5,","_debug_options_count":4},{"id":40798,"question":"L'ensemble des matrices carrées de taille n forme un espace vectoriel.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Cet ensemble est stable par addition et multiplication par un scalaire, et contient un élément neutre (la matrice nulle).","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":40799,"question":"Quel est le résultat de det([[1, 2, 3], [4, 5, 6], [7, 8, 9]]) ?","option_a":"0","option_b":"1","option_c":"-1","option_d":"36","option_e":"","option_f":"","bonne_reponse":"a","explication":"Les lignes sont linéairement dépendantes (L3 = 2*L2 - L1), donc le déterminant est nul.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"0\", \"b\": \"1\", \"c\": \"-1\", \"d\": \"36\"}}","_debug_options_count":4}]