{"id":9218,"date":"2012-05-21T19:19:35","date_gmt":"2012-05-21T18:19:35","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=9218"},"modified":"2021-12-29T13:56:32","modified_gmt":"2021-12-29T13:56:32","slug":"qual-e-a-resposta-correta","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=9218","title":{"rendered":"Qual \u00e9 a resposta correta"},"content":{"rendered":"<p><ul id='GTTabs_ul_9218' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_9218' class='GTTabs_curr'><a  id=\"9218_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_9218' ><a  id=\"9218_1\" onMouseOver=\"GTTabsShowLinks('Resolu\u00e7\u00e3o'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Resolu\u00e7\u00e3o<\/a><\/li>\n<\/ul>\n\n<div class='GTTabs_divs GTTabs_curr_div' id='GTTabs_0_9218'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Para os exerc\u00edcios seguintes, s\u00f3 uma das respostas est\u00e1 correta. Indique qual.<\/p>\n<p><span style=\"text-decoration: underline;\"><strong>Exerc\u00edcio 1<\/strong><\/span><\/p>\n<p>No plano complexo os afixos ${M_1}$, ${M_2}$ e ${M_3}$ dos n\u00fameros complexos $0$, $z$ e $\\frac{1}{z}$ $\\left( {z \\ne 0} \\right)$:<\/p>\n<p><strong>[A]<\/strong> s\u00e3o colineares;<\/p>\n<p><strong>[B]<\/strong> s\u00e3o colineares para alguns n\u00fameros complexos;<\/p>\n<p><strong>[C]<\/strong> nunca s\u00e3o colineares.<\/p>\n<p><span style=\"text-decoration: underline;\"><strong>Exerc\u00edcio 2<\/strong><\/span><\/p>\n<p>Se $\\rho\u00a0 \\in {\\mathbb{R}^ + }$ e $\\theta\u00a0 \\in \\mathbb{R}$, o conjugado de $\\rho \\operatorname{cis} \\theta $ \u00e9:<\/p>\n<p><strong>[A]<\/strong> $ &#8211; \\rho \\operatorname{cis} \\theta $;<\/p>\n<p><strong>[B]<\/strong> $ &#8211; \\rho \\operatorname{cis} \\left( { &#8211; \\theta } \\right)$;<\/p>\n<p><strong>[C]<\/strong> $\\rho \\operatorname{cis} \\left( { &#8211; \\theta } \\right)$.<\/p>\n<p><span style=\"text-decoration: underline;\"><strong>Exerc\u00edcio 3<\/strong><\/span><\/p>\n<p>Se $z$ \u00e9 um n\u00famero complexo n\u00e3o nulo e $\\pi $ \u00e9 um dos seus argumentos, ent\u00e3o:<\/p>\n<p><strong>[A]<\/strong> $z$ \u00e9 um n\u00famero real negativo;<\/p>\n<p><strong>[B]<\/strong> $z$ \u00e9 um n\u00famero real positivo;<\/p>\n<p><strong>[C]<\/strong> $z$ n\u00e3o \u00e9 necessariamente um real.<\/p>\n<p><span style=\"text-decoration: underline;\"><strong>Exerc\u00edcio 4<\/strong><\/span><\/p>\n<p>Se o conjugado \u00e9 igual ao inverso de um n\u00famero complexo $z$ $\\left( {z \\ne 0} \\right)$, ent\u00e3o:<\/p>\n<p><strong>[A]<\/strong> $z$ \u00e9 real;<\/p>\n<p><strong>[B]<\/strong> $z$ \u00e9 imagin\u00e1rio puro;<\/p>\n<p><strong>[C]<\/strong> $\\left| z \\right| = 1$.<\/p>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_9218' onClick='GTTabs_show(1,9218)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_9218'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<p><span style=\"text-decoration: underline;\"><strong>Exerc\u00edcio 1<\/strong><\/span><\/p>\n<p>No plano complexo os afixos ${M_1}$, ${M_2}$ e ${M_3}$ dos n\u00fameros complexos $0$, $z$ e $\\frac{1}{z}$ $\\left( {z \\ne 0} \\right)$:<\/p>\n<p><strong>[A]<\/strong> s\u00e3o colineares;<\/p>\n<p><strong>[B]<\/strong> s\u00e3o colineares para alguns n\u00fameros complexos;<\/p>\n<p><strong>[C]<\/strong> nunca s\u00e3o colineares.<\/p>\n<blockquote>\n<p><span class=\"alignright\">\u00a0<script src=\"https:\/\/tube.geogebra.org\/scripts\/deployggb.js\"><\/script>\r\n<div id=\"ggbApplet\" align=\"center\"><\/div>\r\n<script>\r\nvar parameters = {\r\n\"id\": \"ggbApplet\",\r\n\"width\":380,\r\n\"height\":371,\r\n\"showMenuBar\":false,\r\n\"showAlgebraInput\":false,\r\n\"showToolBar\":false,\r\n\"customToolBar\":\"0 39 59 || 1 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is3D=is 3D applet using 3D view, AV=Algebra View, SV=Spreadsheet View, CV=CAS View, EV2=Graphics View 2, CP=Construction Protocol, PC=Probability Calculator, DA=Data Analysis, FI=Function Inspector, PV=Python, macro=Macro View\r\nvar views = {'is3D': 0,'AV': 0,'SV': 0,'CV': 0,'EV2': 0,'CP': 0,'PC': 0,'DA': 0,'FI': 0,'PV': 0,'macro': 0};\r\nvar applet = new GGBApplet(parameters, '5.0', views);\r\nwindow.onload = function() {applet.inject('ggbApplet')};\r\n<\/script><\/span>A resposta correta \u00e9 <strong>B<\/strong>.<\/p>\n<p>Repare que, por exemplo, para $z = 2$ \u00e9 $\\frac{1}{z} = \\frac{1}{2}$, pelos que os afixos ${M_1}$, ${M_2}$ e ${M_3}$ s\u00e3o colineares.<\/p>\n<p>Contudo, para $z = 1 + i$ \u00e9 $\\frac{1}{z} = \\frac{1}{2} &#8211; \\frac{1}{2}i$, pelo que os afixos ${M_1}$, ${M_2}$ e ${M_3}$ n\u00e3o s\u00e3o colineares.<\/p>\n<\/blockquote>\n<p><span style=\"text-decoration: underline;\"><strong>Exerc\u00edcio 2<\/strong><\/span><\/p>\n<p>Se $\\rho\u00a0 \\in {\\mathbb{R}^ + }$ e $\\theta\u00a0 \\in \\mathbb{R}$, o conjugado de $\\rho \\operatorname{cis} \\theta $ \u00e9:<\/p>\n<p><strong>[A]<\/strong> $ &#8211; \\rho \\operatorname{cis} \\theta $;<\/p>\n<p><strong>[B]<\/strong> $ &#8211; \\rho \\operatorname{cis} \\left( { &#8211; \\theta } \\right)$;<\/p>\n<p><strong>[C]<\/strong> $\\rho \\operatorname{cis} \\left( { &#8211; \\theta } \\right)$.<\/p>\n<blockquote>\n<p>A resposta correta \u00e9 <strong>C<\/strong>.<\/p>\n<p>Se $z = \\rho \\operatorname{cis} \\theta $, ent\u00e3o $\\overline z\u00a0 = \\rho \\operatorname{cis} \\left( { &#8211; \\theta } \\right)$.<\/p>\n<\/blockquote>\n<p><span style=\"text-decoration: underline;\"><strong>Exerc\u00edcio 3<\/strong><\/span><\/p>\n<p>Se $z$ \u00e9 um n\u00famero complexo n\u00e3o nulo e $\\pi $ \u00e9 um dos seus argumentos, ent\u00e3o:<\/p>\n<p><strong>[A]<\/strong> $z$ \u00e9 um n\u00famero real negativo;<\/p>\n<p><strong>[B]<\/strong> $z$ \u00e9 um n\u00famero real positivo;<\/p>\n<p><strong>[C]<\/strong> $z$ n\u00e3o \u00e9 necessariamente um real.<\/p>\n<blockquote>\n<p>A resposta correta \u00e9 <strong>A<\/strong>.<\/p>\n<p>Se $z = \\rho \\operatorname{cis} \\pi $, com $\\rho\u00a0 \\in {\\mathbb{R}^ + }$, ent\u00e3o $z = \\rho \\operatorname{cis} \\pi\u00a0 = \\rho \\left( { &#8211; 1 + 0i} \\right) =\u00a0 &#8211; \\rho $.<\/p>\n<\/blockquote>\n<p><span style=\"text-decoration: underline;\"><strong>Exerc\u00edcio 4<\/strong><\/span><\/p>\n<p>Se o conjugado \u00e9 igual ao inverso de um n\u00famero complexo $z$ $\\left( {z \\ne 0} \\right)$, ent\u00e3o:<\/p>\n<p><strong>[A]<\/strong> $z$ \u00e9 real;<\/p>\n<p><strong>[B]<\/strong> $z$ \u00e9 imagin\u00e1rio puro;<\/p>\n<p><strong>[C]<\/strong> $\\left| z \\right| = 1$.<\/p>\n<blockquote>\n<p>A resposta correta \u00e9 <strong>C<\/strong>.<\/p>\n<p>Se $z = \\rho \\operatorname{cis} \\theta $, ent\u00e3o $\\overline z\u00a0 = \\rho \\operatorname{cis} \\left( { &#8211; \\theta } \\right)$ e $\\frac{1}{z} = \\frac{{\\overline z }}{{{{\\left| {\\text{z}} \\right|}^2}}} = \\frac{{\\rho \\operatorname{cis} \\left( { &#8211; \\theta } \\right)}}{{{\\rho ^2}}} = \\frac{1}{\\rho }\\operatorname{cis} \\left( { &#8211; \\theta } \\right)$.<br \/>\nLogo, se $\\overline z\u00a0 = \\frac{1}{z}$, ent\u00e3o $\\left| z \\right| = 1 = \\rho $.<\/p>\n<\/p>\n<\/blockquote>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_9218' onClick='GTTabs_show(0,9218)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Para os exerc\u00edcios seguintes, s\u00f3 uma das respostas est\u00e1 correta. Indique qual. Exerc\u00edcio 1 No plano complexo os afixos ${M_1}$, ${M_2}$ e ${M_3}$ dos n\u00fameros complexos $0$, $z$ e $\\frac{1}{z}$ $\\left(&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":19530,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[226,97,310],"tags":[427,18],"series":[],"class_list":["post-9218","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-12--ano","category-aplicando","category-numeros-complexos-12--ano","tag-12-o-ano","tag-numeros-complexos"],"views":2147,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2021\/12\/Mat149.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/9218","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=9218"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/9218\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=9218"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=9218"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=9218"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=9218"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}