Quiz — Série d'exercices de Révision N°1 - Math barycentre et polynômes - 2ème Sciences (2011-2012) Mr HammiMoez.pdf

Question 1 sur 10 20:00

[{"id":25394,"question":"Quel est le barycentre G des points A(1;2) et B(3;4) pondérés respectivement par 1 et 3 ?","option_a":"G(2;3)","option_b":"G(2.5;3.5)","option_c":"G(11\/4;14\/4)","option_d":"G(3;4)","option_e":"","option_f":"","bonne_reponse":"c","explication":"Le barycentre G vérifie la relation vectorielle : 1*GA + 3*GB = 0. En coordonnées, cela donne G((1*1 + 3*3)\/(1+3) ; (1*2 + 3*4)\/(1+3)) = (10\/4 ; 14\/4) = (5\/2 ; 7\/2).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"G(2;3)\", \"b\": \"G(2.5;3.5)\", \"c\": \"G(11\/4;14\/4)\", \"d\": \"G(3;4)\"}}","_debug_options_count":4},{"id":25395,"question":"Un polynôme P(x) = x² - 5x + 6 admet pour racines 2 et 3.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"En substituant x=2 et x=3 dans P(x), on obtient P(2)=0 et P(3)=0. Les racines sont bien 2 et 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":25396,"question":"Quelle est la factorisation correcte de P(x) = x² - 9 ?","option_a":"(x-3)(x+3)","option_b":"(x-9)(x+1)","option_c":"(x-3)²","option_d":"(x+3)²","option_e":"","option_f":"","bonne_reponse":"a","explication":"La différence de carrés se factorise en (x-3)(x+3), car x² - 9 = x² - 3².","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-3)(x+3)\", \"b\": \"(x-9)(x+1)\", \"c\": \"(x-3)²\", \"d\": \"(x+3)²\"}}","_debug_options_count":4},{"id":25397,"question":"Si G est le barycentre des points A, B et C pondérés par 2, 3 et 5, alors G est aussi le barycentre des points A, B et C pondérés par 4, 6 et 10.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Les coefficients sont proportionnels (multipliés par 2), donc le barycentre reste le même.","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":25398,"question":"Quel est le coefficient dominant du polynôme P(x) = 3x³ - 2x² + x - 5 ?","option_a":"3","option_b":"-2","option_c":"1","option_d":"-5","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le coefficient dominant est celui de la puissance la plus élevée, ici 3 pour x³.","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\", \"b\": \"-2\", \"c\": \"1\", \"d\": \"-5\"}}","_debug_options_count":4},{"id":25399,"question":"Le barycentre de deux points A et B est toujours situé entre A et B.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le barycentre peut être situé en dehors du segment [AB] si les coefficients sont de signes opposés.","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":25400,"question":"Quelle est la somme des racines du polynôme P(x) = x² - 4x + 3 ?","option_a":"4","option_b":"-4","option_c":"3","option_d":"-3","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour un polynôme du second degré ax² + bx + c, la somme des racines est -b\/a. Ici, -(-4)\/1 = 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\": \"4\", \"b\": \"-4\", \"c\": \"3\", \"d\": \"-3\"}}","_debug_options_count":4},{"id":25401,"question":"Si G est le barycentre de A et B avec des coefficients positifs, alors G appartient au segment [AB].","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Avec des coefficients positifs, le barycentre est une combinaison convexe des points, donc il appartient au segment [AB].","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":25402,"question":"Quel est le produit des racines du polynôme P(x) = 2x² - 8x + 6 ?","option_a":"3","option_b":"-3","option_c":"4","option_d":"-4","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour un polynôme du second degré ax² + bx + c, le produit des racines est c\/a. Ici, 6\/2 = 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\": \"3\", \"b\": \"-3\", \"c\": \"4\", \"d\": \"-4\"}}","_debug_options_count":4},{"id":25403,"question":"Le barycentre de trois points non alignés est toujours situé à l'intérieur du triangle formé par ces points.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le barycentre peut être situé à l'extérieur du triangle si les coefficients ne sont pas tous positifs ou si leur somme n'est pas égale à 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4}]