Quiz — 32_CourbesParametrees_Corrige.pdf

Question 1 sur 10 20:00

[{"id":125508,"question":"Soit une courbe paramétrée définie par x = t² et y = t³. Quelle est la dérivée dy\/dx en fonction de t ?","option_a":"dy\/dx = 3t²\/2t","option_b":"dy\/dx = 3t\/2","option_c":"dy\/dx = 2t\/3t²","option_d":"dy\/dx = t\/2","option_e":"","option_f":"","bonne_reponse":"b","explication":"dy\/dx = (dy\/dt)\/(dx\/dt) = (3t²)\/(2t) = 3t\/2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"dy\/dx = 3t²\/2t\", \"b\": \"dy\/dx = 3t\/2\", \"c\": \"dy\/dx = 2t\/3t²\", \"d","_debug_options_count":4},{"id":125509,"question":"Une courbe paramétrée admet une branche infinie lorsque le paramètre t tend vers l'infini.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Une branche infinie peut apparaître lorsque t tend vers une valeur finie ou infinie, selon les cas.","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":125510,"question":"Pour la courbe paramétrée x = cos(t), y = sin(t), quelle est la nature de la courbe ?","option_a":"Une droite","option_b":"Un cercle","option_c":"Une parabole","option_d":"Une hyperbole","option_e":"","option_f":"","bonne_reponse":"b","explication":"x² + y² = cos²(t) + sin²(t) = 1, donc la courbe est un cercle de rayon 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\": \"Une droite\", \"b\": \"Un cercle\", \"c\": \"Une parabole\", \"d\": \"Une hyp","_debug_options_count":4},{"id":125511,"question":"Soit une courbe paramétrée avec x = t² - 1 et y = t³ - t. Quel est le coefficient directeur de la tangente en t = 1 ?","option_a":"0","option_b":"1","option_c":"2","option_d":"3","option_e":"","option_f":"","bonne_reponse":"b","explication":"dy\/dx = (dy\/dt)\/(dx\/dt) = (3t² - 1)\/(2t). En t = 1, dy\/dx = (3 - 1)\/2 = 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\": \"0\", \"b\": \"1\", \"c\": \"2\", \"d\": \"3\"}}","_debug_options_count":4},{"id":125512,"question":"Une courbe paramétrée peut avoir plusieurs points singuliers.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Un point singulier est un point où la dérivée s'annule ou n'existe pas. Une courbe peut en avoir plusieurs.","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":125513,"question":"Pour la courbe paramétrée x = t³, y = t², quelle est l'équation de la tangente au point t = 2 ?","option_a":"y = 4x - 16","option_b":"y = 2x - 4","option_c":"y = x - 2","option_d":"y = 3x - 8","option_e":"","option_f":"","bonne_reponse":"a","explication":"dy\/dx = (2t)\/(3t²) = 2\/(3t). En t = 2, dy\/dx = 1\/3. L'équation est y - 4 = (1\/3)(x - 8) =\u003E y = (1\/3)x - 4\/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\": \"y = 4x - 16\", \"b\": \"y = 2x - 4\", \"c\": \"y = x - 2\", \"d\": \"y = 3x -","_debug_options_count":4},{"id":125514,"question":"Quelle est la condition pour qu'une courbe paramétrée soit symétrique par rapport à l'axe des ordonnées ?","option_a":"x(t) = -x(-t) et y(t) = y(-t)","option_b":"x(t) = x(-t) et y(t) = -y(-t)","option_c":"x(t) = -x(-t) et y(t) = -y(-t)","option_d":"x(t) = x(-t) et y(t) = y(-t)","option_e":"","option_f":"","bonne_reponse":"d","explication":"Pour une symétrie par rapport à l'axe des ordonnées, x doit être pair et y doit être pair.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"d\", \"options\": {\"a\": \"x(t) = -x(-t) et y(t) = y(-t)\", \"b\": \"x(t) = x(-t) et y(t) = -y(-","_debug_options_count":4},{"id":125515,"question":"Soit une courbe paramétrée avec x = e^t et y = e^(-t). Quelle est la nature de cette courbe ?","option_a":"Une droite","option_b":"Une hyperbole","option_c":"Une parabole","option_d":"Un cercle","option_e":"","option_f":"","bonne_reponse":"b","explication":"xy = e^t * e^(-t) = 1, donc la courbe est une hyperbole.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Une droite\", \"b\": \"Une hyperbole\", \"c\": \"Une parabole\", \"d\": \"Un ","_debug_options_count":4},{"id":125516,"question":"Une courbe paramétrée peut avoir une asymptote verticale.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Une asymptote verticale apparaît lorsque x(t) tend vers une constante et y(t) tend vers l'infini, ou inversement.","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":125517,"question":"Pour la courbe paramétrée x = t + 1\/t, y = t - 1\/t (t ≠ 0), quelle est l'équation cartésienne de cette courbe ?","option_a":"x² - y² = 4","option_b":"x² + y² = 4","option_c":"xy = 1","option_d":"x + y = 2","option_e":"","option_f":"","bonne_reponse":"a","explication":"x² - y² = (t + 1\/t)² - (t - 1\/t)² = 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\": \"x² - y² = 4\", \"b\": \"x² + y² = 4\", \"c\": \"xy = 1\", \"d\": \"x + y ","_debug_options_count":4}]