{"id":37337,"date":"2020-01-07T12:08:31","date_gmt":"2020-01-07T15:08:31","guid":{"rendered":"https:\/\/blog.estrategiavestibulares.com.br\/?p=37337"},"modified":"2021-03-11T17:25:59","modified_gmt":"2021-03-11T20:25:59","slug":"matematica-da-2-fase-da-fuvest-2020","status":"publish","type":"post","link":"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/matematica-da-2-fase-da-fuvest-2020\/","title":{"rendered":"Resolu\u00e7\u00e3o da prova de Matem\u00e1tica da 2\u00aa Fase da Fuvest 2020"},"content":{"rendered":"<p>Ol&aacute;, pessoal&hellip; Tudo bem? Sou o prof. Mar&ccedil;al Ferreira, professor de Matem&aacute;tica do Estrat&eacute;gia Vestibulares. Escrevo este artigo para resolver as quest&otilde;es da prova da 2&ordf; Fase do vestibular da FUVET 2020. Nesta p&aacute;gina, voc&ecirc; vai conferir a resolu&ccedil;&atilde;o completa e ainda vai poder baixar gratuitamente os coment&aacute;rios em PDF.<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_76 counter-hierarchy ez-toc-counter ez-toc-transparent ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\"><p class=\"ez-toc-title\" style=\"cursor:inherit\">Navegue pelo conte\u00fado<\/p>\n<\/div><nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/matematica-da-2-fase-da-fuvest-2020\/#Questao-01\" >Quest&atilde;o 01<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/matematica-da-2-fase-da-fuvest-2020\/#Questao-02\" >Quest&atilde;o 02<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/matematica-da-2-fase-da-fuvest-2020\/#Questao-03\" >Quest&atilde;o 03<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/matematica-da-2-fase-da-fuvest-2020\/#Questao-04\" >Quest&atilde;o 04<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/matematica-da-2-fase-da-fuvest-2020\/#Questao-05\" >Quest&atilde;o 05<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/matematica-da-2-fase-da-fuvest-2020\/#Questao-06\" >Quest&atilde;o 06<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Questao-01\"><\/span>Quest&atilde;o 01<span class=\"ez-toc-section-end\"><\/span><\/h2><p>A figura apresenta uma parte de uma tabela na qual cada linha e cada coluna seguem de acordo com o padr&atilde;o representado.<\/p><div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/cdn.blog.estrategiavestibulares.com.br\/vestibulares\/wp-content\/uploads\/2020\/01\/image16.png\" alt=\"\" class=\"wp-image-37339\"><\/figure><\/div><p>Com rela&ccedil;&atilde;o a essa tabela de n&uacute;meros:<\/p><p>a) Escolha um quadrado 3 x 3 e, exibindo a soma de\nseus 9 n&uacute;meros, verifique que o resultado &eacute; m&uacute;ltiplo de 9.<\/p><p>b) Um quadrado com 16 n&uacute;meros tem por soma de todos\nesses n&uacute;meros o valor de 1.056 (mil e cinquenta e seis). Descubra o menor\nn&uacute;mero desse quadrado.<\/p><p>c) A soma de todos os n&uacute;meros de um quadrado n x n, com menor n&uacute;mero igual a 4, &eacute; de 108.000 (cento e oito mil). Qual &eacute; o valor de n?<\/p><h3 class=\"wp-block-heading\">Gabarito<\/h3><p>a) Iniciando a contagem em qualquer posi&ccedil;&atilde;o, temos um quadrado de 3&times;3 gen&eacute;rico da forma:<\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5Cbegin%7Bmatrix%7Dx&amp;x+1&amp;x+2%5C%5Cx+7&amp;x+8&amp;x+9%5C%5Cx+14&amp;x+15&amp;x+16%5C%5C%5Cend%7Bmatrix%7D\" alt=\"\\large \\begin{matrix}x&amp;x+1&amp;x+2\\\\x+7&amp;x+8&amp;x+9\\\\x+14&amp;x+15&amp;x+16\\\\\\end{matrix}\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;Soma=9x+1+2+7+8+9+14+15+16\" alt=\"\\large Soma=9x+1+2+7+8+9+14+15+16\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;Soma=9x+72\" alt=\"\\large Soma=9x+72\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;Soma=9%5Ccdot%5Cleft(x+8%5Cright)\" alt=\"\\large Soma=9\\cdot\\left(x+8\\right)\" align=\"absmiddle\"><\/p><p>Dessa forma, a soma sempre ser&aacute; um m&uacute;ltiplo de nove.<\/p><p>b)&nbsp;De maneira similar ao anterior, temos:<\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5Cbegin%7Bmatrix%7Dx&amp;x+1&amp;x+2&amp;x+3%5C%5Cx+7&amp;x+8&amp;x+9&amp;x+10%5C%5Cx+14&amp;x+15&amp;x+16&amp;x+17%5C%5Cx+21&amp;x+22&amp;x+23&amp;x+24%5C%5C%5Cend%7Bmatrix%7D\" alt=\"\\large \\begin{matrix}x&amp;x+1&amp;x+2&amp;x+3\\\\x+7&amp;x+8&amp;x+9&amp;x+10\\\\x+14&amp;x+15&amp;x+16&amp;x+17\\\\x+21&amp;x+22&amp;x+23&amp;x+24\\\\\\end{matrix}\" align=\"absmiddle\"><\/p><p><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;16x+192=1056\" alt=\"\\large 16x+192=1056\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;16x=864\" alt=\"\\large 16x=864\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;x=54\" alt=\"\\large x=54\" align=\"absmiddle\"><\/p><p>c) O quadrado em quest&atilde;o &eacute; do tipo:<\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5Cbegin%7Bmatrix%7D4&amp;5&amp;6&amp;7&amp;%5Ccdots&amp;4+n-1%5C%5C11&amp;12&amp;13&amp;14&amp;%5Ccdots&amp;11+n-1%5C%5C18&amp;19&amp;20&amp;21&amp;%5Ccdots&amp;18+n-1%5C%5C25&amp;26&amp;27&amp;28&amp;%5Ccdots&amp;25+n-1%5C%5C%5Cvdots&amp;%5Ccdot%5Cvdots&amp;%5Cvdots&amp;%5Cvdots&amp;%5Cvdots&amp;%5Cddots%5C%5C%5Cend%7Bmatrix%7D\" alt=\"\\large \\begin{matrix}4&amp;5&amp;6&amp;7&amp;\\cdots&amp;4+n-1\\\\11&amp;12&amp;13&amp;14&amp;\\cdots&amp;11+n-1\\\\18&amp;19&amp;20&amp;21&amp;\\cdots&amp;18+n-1\\\\25&amp;26&amp;27&amp;28&amp;\\cdots&amp;25+n-1\\\\\\vdots&amp;\\cdot\\vdots&amp;\\vdots&amp;\\vdots&amp;\\vdots&amp;\\ddots\\\\\\end{matrix}\" align=\"absmiddle\"><\/p><p><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5Cbegin%7Bmatrix%7D4&amp;4+1&amp;4+2&amp;4+3&amp;%5Ccdots&amp;4+n-1%5C%5C4+7&amp;4+8&amp;4+9&amp;4+10&amp;%5Ccdots&amp;4+7+n-1%5C%5C4+14&amp;4+15&amp;4+16&amp;4+17&amp;%5Ccdots&amp;4+14+n-1%5C%5C4+21&amp;4+22&amp;4+23&amp;4+24&amp;%5Ccdots&amp;4+21+n-1%5C%5C%5Cvdots&amp;%5Ccdot%5Cvdots&amp;%5Cvdots&amp;%5Cvdots&amp;%5Cvdots&amp;%5Cddots%5C%5C%5Cend%7Bmatrix%7D\" alt=\"\\large \\begin{matrix}4&amp;4+1&amp;4+2&amp;4+3&amp;\\cdots&amp;4+n-1\\\\4+7&amp;4+8&amp;4+9&amp;4+10&amp;\\cdots&amp;4+7+n-1\\\\4+14&amp;4+15&amp;4+16&amp;4+17&amp;\\cdots&amp;4+14+n-1\\\\4+21&amp;4+22&amp;4+23&amp;4+24&amp;\\cdots&amp;4+21+n-1\\\\\\vdots&amp;\\cdot\\vdots&amp;\\vdots&amp;\\vdots&amp;\\vdots&amp;\\ddots\\\\\\end{matrix}\" align=\"absmiddle\"><\/p><p><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;4n%5E2+%5Cbegin%7Bmatrix%7D0&amp;1&amp;2&amp;3&amp;%5Ccdots&amp;n-1%5C%5C7&amp;7+1&amp;7+2&amp;7+3&amp;%5Ccdots&amp;7+n-1%5C%5C7%5Ccdot2&amp;7%5Ccdot2+1&amp;7%5Ccdot2+2&amp;7%5Ccdot2+3&amp;%5Ccdots&amp;7%5Ccdot2+n-1%5C%5C7%5Ccdot3&amp;7%5Ccdot3+1&amp;7%5Ccdot3+2&amp;7%5Ccdot3+3&amp;%5Ccdots&amp;7%5Ccdot3+n-1%5C%5C%5Cvdots&amp;%5Ccdot%5Cvdots&amp;%5Cvdots&amp;%5Cvdots&amp;%5Cvdots&amp;%5Cddots%5C%5C%5Cend%7Bmatrix%7D\" alt=\"\\large 4n^2+\\begin{matrix}0&amp;1&amp;2&amp;3&amp;\\cdots&amp;n-1\\\\7&amp;7+1&amp;7+2&amp;7+3&amp;\\cdots&amp;7+n-1\\\\7\\cdot2&amp;7\\cdot2+1&amp;7\\cdot2+2&amp;7\\cdot2+3&amp;\\cdots&amp;7\\cdot2+n-1\\\\7\\cdot3&amp;7\\cdot3+1&amp;7\\cdot3+2&amp;7\\cdot3+3&amp;\\cdots&amp;7\\cdot3+n-1\\\\\\vdots&amp;\\cdot\\vdots&amp;\\vdots&amp;\\vdots&amp;\\vdots&amp;\\ddots\\\\\\end{matrix}\" align=\"absmiddle\"><\/p><p><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;4n%5E2+%5Cleft(0+1+2+3+%5Cldots+n-1%5Cright)%5Ccdot&amp;space;n+\" alt=\"\\large 4n^2+\\left(0+1+2+3+\\ldots+n-1\\right)\\cdot n+\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;+%5Cleft(0+7+7%5Ccdot1+7%5Ccdot2+7%5Ccdot3+%5Cldots+7%5Ccdot%5Cleft(n-1%5Cright)%5Cright)%5Ccdot&amp;space;n=108000\" alt=\"\\large +\\left(0+7+7\\cdot1+7\\cdot2+7\\cdot3+\\ldots+7\\cdot\\left(n-1\\right)\\right)\\cdot n=108000\" align=\"absmiddle\"><\/p><p><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;4n%5E2+%5Cleft(0+1+2+3+%5Cldots+n-1%5Cright)%5Ccdot&amp;space;n+\" alt=\"\\large 4n^2+\\left(0+1+2+3+\\ldots+n-1\\right)\\cdot n+\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;+7%5Ccdot%5Cleft(0+1+2+3+%5Cldots+n-1%5Cright)%5Ccdot&amp;space;n=108000\" alt=\"\\large +7\\cdot\\left(0+1+2+3+\\ldots+n-1\\right)\\cdot n=108000\" align=\"absmiddle\"><\/p><p><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;4n%5E2+8%5Ccdot%5Cleft(0+1+2+3+%5Cldots+n-1%5Cright)%5Ccdot&amp;space;n=108000\" alt=\"\\large 4n^2+8\\cdot\\left(0+1+2+3+\\ldots+n-1\\right)\\cdot n=108000\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;4n%5E2+8%5Ccdot%5Cleft(n-1%5Cright)%5Ccdot%5Cfrac%7Bn%7D%7B2%7D%5Ccdot&amp;space;n=108000\" alt=\"\\large 4n^2+8\\cdot\\left(n-1\\right)\\cdot\\frac{n}{2}\\cdot n=108000\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;4n%5E2+4n%5E3-4n%5E2=108000\" alt=\"\\large 4n^2+4n^3-4n^2=108000\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;4n%5E3=108000\" alt=\"\\large 4n^3=108000\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;n%5E3=27000\" alt=\"\\large n^3=27000\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;n=30\" alt=\"\\large n=30\" align=\"absmiddle\"><\/p><h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Questao-02\"><\/span>Quest&atilde;o 02<span class=\"ez-toc-section-end\"><\/span><\/h2><p>O Floco de Neve de Koch (ou Estrela de Koch) &eacute; uma constru&ccedil;&atilde;o geom&eacute;trica recursiva cujos primeiros passos se desenvolvem da seguinte forma:<\/p><figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/cdn.blog.estrategiavestibulares.com.br\/vestibulares\/wp-content\/uploads\/2020\/01\/image17.png\" alt=\"\" class=\"wp-image-37347\"><\/figure><p>Os passos seguintes (Passo 3, Passo 4, Passo 5, &hellip;)\nseguem o mesmo procedimento descrito no Passo 1, em cada lado da figura obtida\nno passo anterior. Considerando os passos descritos e os pr&oacute;ximos passos,\nresponda:<\/p><p>a) Qual &eacute; o n&uacute;mero de lados da figura no Passo 3?<\/p><p>b) Qual &eacute; o per&iacute;metro da figura no Passo 5?<\/p><p>c) A partir de qual Passo o n&uacute;mero de lados da figura supera 6.000.000.000.000 (seis trilh&otilde;es)?<\/p><h3 class=\"wp-block-heading\">Gabarito<\/h3><p>a) <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;N%5Cleft(p%5Cright)=3%5Ccdot%5Cleft(4%5Cright)%5Ep\" alt=\"\\large N\\left(p\\right)=3\\cdot\\left(4\\right)^p\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;N%5Cleft(3%5Cright)=3%5Ccdot%5Cleft(4%5Cright)%5E3\" alt=\"\\large N\\left(3\\right)=3\\cdot\\left(4\\right)^3\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;N%5Cleft(3%5Cright)=3%5Ccdot2%5E6\" alt=\"\\large N\\left(3\\right)=3\\cdot2^6\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;N%5Cleft(3%5Cright)=192\" alt=\"\\large N\\left(3\\right)=192\" align=\"absmiddle\"><\/p><p><\/p><p>b) <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;P%5Cleft(p%5Cright)=3%5Ccdot%5Cleft(%5Cfrac%7B4%7D%7B3%7D%5Cright)%5Ep\" alt=\"\\large P\\left(p\\right)=3\\cdot\\left(\\frac{4}{3}\\right)^p\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;P%5Cleft(5%5Cright)=3%5Ccdot%5Cleft(%5Cfrac%7B4%7D%7B3%7D%5Cright)%5E5\" alt=\"\\large P\\left(5\\right)=3\\cdot\\left(\\frac{4}{3}\\right)^5\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;P%5Cleft(5%5Cright)=3%5Ccdot%5Cfrac%7B2%5E%7B10%7D%7D%7B3%5E5%7D\" alt=\"\\large P\\left(5\\right)=3\\cdot\\frac{2^{10}}{3^5}\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;P%5Cleft(5%5Cright)=%5Cfrac%7B2%5E%7B10%7D%7D%7B3%5E4%7D\" alt=\"\\large P\\left(5\\right)=\\frac{2^{10}}{3^4}\" align=\"absmiddle\"><\/p><p><\/p><p>c) <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;N%5Cleft(p%5Cright)=3%5Ccdot%5Cleft(4%5Cright)%5Ep\" alt=\"\\large N\\left(p\\right)=3\\cdot\\left(4\\right)^p\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;3%5Ccdot%5Cleft(4%5Cright)%5Ep&gt;6%5Ccdot%7B10%7D%5E%7B12%7D%E2%80%9D%20alt=%E2%80%9D%5Clarge%203%5Ccdot%5Cleft(4%5Cright)%5Ep&gt;6%5Ccdot%7B10%7D%5E%7B12%7D%E2%80%9D%20align=%E2%80%9Dabsmiddle%E2%80%9D&gt;&lt;\/p&gt;%0A&lt;p&gt;&lt;img%20src=\" https:>2\\cdot{10}^{12}&rdquo; alt=&rdquo;\\large \\left(4\\right)^p&gt;2\\cdot{10}^{12}&rdquo; align=&rdquo;absmiddle&rdquo;&gt;<\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;2%5E%7B2p%7D&gt;2%5Ccdot%7B10%7D%5E%7B12%7D%E2%80%9D%20alt=%E2%80%9D%5Clarge%202%5E%7B2p%7D&gt;2%5Ccdot%7B10%7D%5E%7B12%7D%E2%80%9D%20align=%E2%80%9Dabsmiddle%E2%80%9D&gt;&lt;\/p&gt;%0A&lt;p&gt;&lt;img%20src=\" https:>{10}^{12}&rdquo; alt=&rdquo;\\large 2^{2p-1}&gt;{10}^{12}&rdquo; align=&rdquo;absmiddle&rdquo;&gt;<\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5Clog%7B2%5E%7B2p-1%7D%7D&gt;%5Clog%7B%7B10%7D%5E%7B12%7D%7D%E2%80%9D%20alt=%E2%80%9D%5Clarge%20%5Clog%7B2%5E%7B2p-1%7D%7D&gt;%5Clog%7B%7B10%7D%5E%7B12%7D%7D%E2%80%9D%20align=%E2%80%9Dabsmiddle%E2%80%9D&gt;&lt;\/p&gt;%0A&lt;p&gt;&lt;img%20src=\" https:>12\\cdot\\log{10}&rdquo; alt=&rdquo;\\large \\left(2p-1\\right)\\cdot\\log{2}&gt;12\\cdot\\log{10}&rdquo; align=&rdquo;absmiddle&rdquo;&gt;<\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;2p-1&gt;%5Cfrac%7B12%7D%7B0,301%7D%E2%80%9D%20alt=%E2%80%9D%5Clarge%202p-1&gt;%5Cfrac%7B12%7D%7B0,301%7D%E2%80%9D%20align=%E2%80%9Dabsmiddle%E2%80%9D&gt;&lt;\/p&gt;%0A&lt;p&gt;&lt;img%20src=\" https:>\\frac{12}{0,301}+1&Prime; alt=&rdquo;\\large 2p&gt;\\frac{12}{0,301}+1&Prime; align=&rdquo;absmiddle&rdquo;&gt;<\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;p&gt;%5Cfrac%7B6%7D%7B0,301%7D+%5Cfrac%7B1%7D%7B2%7D%E2%80%9D%20alt=%E2%80%9D%5Clarge%20p&gt;%5Cfrac%7B6%7D%7B0,301%7D+%5Cfrac%7B1%7D%7B2%7D%E2%80%9D%20align=%E2%80%9Dabsmiddle%E2%80%9D&gt;&lt;\/p&gt;%0A&lt;p&gt;&lt;img%20src=\" https:>\\frac{6}{0,301}+\\frac{1}{2}&rdquo; alt=&rdquo;\\large p&gt;\\frac{6}{0,301}+\\frac{1}{2}&rdquo; align=&rdquo;absmiddle&rdquo;&gt;<\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;p&gt;20,43%5Cldots%E2%80%9D%20alt=%E2%80%9D%5Clarge%20p&gt;20,43%5Cldots%E2%80%9D%20align=%E2%80%9Dabsmiddle%E2%80%9D&gt;&lt;\/p&gt;%0A&lt;p&gt;&lt;img%20src=\" https:>21&Prime; alt=&rdquo;\\large p&gt;21&Prime; align=&rdquo;absmiddle&rdquo;&gt;<\/p><p>A partir do 21&deg; passo.<\/p><h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Questao-03\"><\/span>Quest&atilde;o 03<span class=\"ez-toc-section-end\"><\/span><\/h2><p>Um jogo educativo possui 16 pe&ccedil;as nos formatos:\nc&iacute;rculo, tri&acirc;ngulo, quadrado e estrela, e cada formato &eacute; apresentado em 4 cores:\namarelo, branco, laranja e verde. Dois jogadores distribuem entre si\nquantidades iguais dessas pe&ccedil;as, de forma aleat&oacute;ria.<\/p><p>O conjunto de 8 pe&ccedil;as que cada jogador recebe &eacute;\nchamado de cole&ccedil;&atilde;o.<\/p><p>a) Quantas s&atilde;o as poss&iacute;veis cole&ccedil;&otilde;es que um jogador\npode receber?<\/p><p>b) Qual &eacute; a probabilidade de que os dois jogadores\nrecebam a mesma quantidade de pe&ccedil;as amarelas?<\/p><p>c) A regra do jogo estabelece pontua&ccedil;&otilde;es para as pe&ccedil;as, da seguinte forma: c&iacute;rculo = 1 ponto, tri&acirc;ngulo = 2 pontos, quadrado = 3 pontos e estrela = 4 pontos. Quantas s&atilde;o as poss&iacute;veis cole&ccedil;&otilde;es que valem 26 pontos ou mais?<\/p><h3 class=\"wp-block-heading\">Gabarito<\/h3><p>a) <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;C_%7B16,8%7D=%5Cleft(%5Cbegin%7Bmatrix%7D16%5C%5C8%5C%5C%5Cend%7Bmatrix%7D%5Cright)=%5Cfrac%7B16!%7D%7B8!%5Ccdot8!%7D\" alt=\"\\large C_{16,8}=\\left(\\begin{matrix}16\\\\8\\\\\\end{matrix}\\right)=\\frac{16!}{8!\\cdot8!}\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;C_%7B16,8%7D=%5Cfrac%7B16%5Ccdot15%5Ccdot14%5Ccdot13%5Ccdot12%5Ccdot11%5Ccdot10%5Ccdot9%5Ccdot8!%7D%7B8!%5Ccdot8%5Ccdot7%5Ccdot6%5Ccdot5%5Ccdot4%5Ccdot3%5Ccdot2%5Ccdot1%7D\" alt=\"\\large C_{16,8}=\\frac{16\\cdot15\\cdot14\\cdot13\\cdot12\\cdot11\\cdot10\\cdot9\\cdot8!}{8!\\cdot8\\cdot7\\cdot6\\cdot5\\cdot4\\cdot3\\cdot2\\cdot1}\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;C_%7B16,8%7D=13%5Ccdot11%5Ccdot10%5Ccdot9\" alt=\"\\large C_{16,8}=13\\cdot11\\cdot10\\cdot9\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;C_%7B16,8%7D=12870\" alt=\"\\large C_{16,8}=12870\" align=\"absmiddle\"><\/p><p><\/p><p>b)&nbsp;<img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;P=%5Cfrac%7B%5Cleft(%5Cbegin%7Bmatrix%7D12%5C%5C2%5C%5C%5Cend%7Bmatrix%7D%5Cright)%5Ccdot%5Cleft(%5Cbegin%7Bmatrix%7D4%5C%5C2%5C%5C%5Cend%7Bmatrix%7D%5Cright)%7D%7B%5Cleft(%5Cbegin%7Bmatrix%7D16%5C%5C8%5C%5C%5Cend%7Bmatrix%7D%5Cright)%7D\" alt=\"\\large P=\\frac{\\left(\\begin{matrix}12\\\\2\\\\\\end{matrix}\\right)\\cdot\\left(\\begin{matrix}4\\\\2\\\\\\end{matrix}\\right)}{\\left(\\begin{matrix}16\\\\8\\\\\\end{matrix}\\right)}\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;P=%5Cfrac%7B924%5Ccdot6%7D%7B12870%7D\" alt=\"\\large P=\\frac{924\\cdot6}{12870}\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;P=%5Cfrac%7B28%7D%7B65%7D\" alt=\"\\large P=\\frac{28}{65}\" align=\"absmiddle\"><\/p><p><\/p><p>c) <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;P_1+P_2+P_3+P_4+P_5+P_6+P_7+P_8%5Cgeq26\" alt=\"\\large P_1+P_2+P_3+P_4+P_5+P_6+P_7+P_8\\geq26\" align=\"absmiddle\"><\/p><p>As cole&ccedil;&otilde;es poss&iacute;veis s&atilde;o:<\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;28%5C&amp;space;pontos\" alt=\"\\large 28\\ pontos\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;4%5C&amp;space;estrelas+4%5C&amp;space;quadrados\" alt=\"\\large 4\\ estrelas+4\\ quadrados\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;C_%7B4,4%7D%5Ccdot&amp;space;C_%7B4,4%7D\" alt=\"\\large C_{4,4}\\cdot C_{4,4}\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;1%5Ccdot1\" alt=\"\\large 1\\cdot1\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;1\" alt=\"\\large 1\" align=\"absmiddle\"><\/p><p><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;27%5C&amp;space;pontos\" alt=\"\\large 27\\ pontos\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;4%5C&amp;space;estrelas+3%5C&amp;space;quadrados+1%5C&amp;space;triangulo\" alt=\"\\large 4\\ estrelas+3\\ quadrados+1\\ triangulo\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;C_%7B4,4%7D%5Ccdot&amp;space;C_%7B4,3%7D%5Ccdot&amp;space;C_%7B4,1%7D\" alt=\"\\large C_{4,4}\\cdot C_{4,3}\\cdot C_{4,1}\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;1%5Ccdot4%5Ccdot4\" alt=\"\\large 1\\cdot4\\cdot4\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;16\" alt=\"\\large 16\" align=\"absmiddle\"><\/p><p><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;26%5C&amp;space;pontos\" alt=\"\\large 26\\ pontos\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;4%5C&amp;space;estrelas+3%5C&amp;space;quadrados+1%5C&amp;space;circulo\" alt=\"\\large 4\\ estrelas+3\\ quadrados+1\\ circulo\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;C_%7B4,4%7D%5Ccdot&amp;space;C_%7B4,3%7D%5Ccdot&amp;space;C_%7B4,1%7D\" alt=\"\\large C_{4,4}\\cdot C_{4,3}\\cdot C_{4,1}\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;1%5Ccdot4%5Ccdot4\" alt=\"\\large 1\\cdot4\\cdot4\" align=\"absmiddle\"><\/p><p><del><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;16\" alt=\"\\large 16\" align=\"absmiddle\"><\/del><\/p><p><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;26%5C&amp;space;pontos\" alt=\"\\large 26\\ pontos\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;4%5C&amp;space;estrelas+2%5C&amp;space;quadrados+2%5C&amp;space;triangulos\" alt=\"\\large 4\\ estrelas+2\\ quadrados+2\\ triangulos\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;C_%7B4,4%7D%5Ccdot&amp;space;C_%7B4,2%7D%5Ccdot&amp;space;C_%7B4,2%7D\" alt=\"\\large C_{4,4}\\cdot C_{4,2}\\cdot C_{4,2}\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;1%5Ccdot6%5Ccdot6\" alt=\"\\large 1\\cdot6\\cdot6\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;36\" alt=\"\\large 36\" align=\"absmiddle\"><\/p><p><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;26%5C&amp;space;pontos\" alt=\"\\large 26\\ pontos\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;3%5C&amp;space;estrelas+4%5C&amp;space;quadrados+1%5C&amp;space;triangulos\" alt=\"\\large 3\\ estrelas+4\\ quadrados+1\\ triangulos\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;C_%7B4,3%7D%5Ccdot&amp;space;C_%7B4,4%7D%5Ccdot&amp;space;C_%7B4,1%7D\" alt=\"\\large C_{4,3}\\cdot C_{4,4}\\cdot C_{4,1}\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;4%5Ccdot1%5Ccdot4\" alt=\"\\large 4\\cdot1\\cdot4\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;16\" alt=\"\\large 16\" align=\"absmiddle\"><\/p><p><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;N=1+16+16+36+16\" alt=\"\\large N=1+16+16+36+16\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;N=85\" alt=\"\\large N=85\" align=\"absmiddle\"><\/p><h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Questao-04\"><\/span>Quest&atilde;o 04<span class=\"ez-toc-section-end\"><\/span><\/h2><p>S&atilde;o dados:<\/p><p>&bull; uma circunfer&ecirc;ncia S de centro 0 e raio 5;<\/p><p>&bull; quatro pontos X, Y, Z e W em 5 de tal forma que as retas tangentes a S nesses pontos formam um trap&eacute;zio ABCD, como na figura;<\/p><p>&bull; <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;sen%5C&amp;space;(B%C3%82W)=&amp;space;%5Cfrac%7B3%7D%7B5%7D%5C&amp;space;e%5C&amp;space;CD=15\" alt=\"\\large sen\\ (B&Acirc;W)= \\frac{3}{5}\\ e\\ CD=15\" align=\"absmiddle\">.<\/p><div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/cdn.blog.estrategiavestibulares.com.br\/vestibulares\/wp-content\/uploads\/2020\/01\/image18.png\" alt=\"\" class=\"wp-image-37354\"><\/figure><\/div><p>a) a medida de <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5Coverline%7BAB%7D\" alt=\"\\large \\overline{AB}\" align=\"absmiddle\">;<\/p><p>b) a medida de <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5Coverline%7BAW%7D\" alt=\"\\large \\overline{AW}\" align=\"absmiddle\"> e <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5Coverline%7BAX%7D\" alt=\"\\large \\overline{AX}\" align=\"absmiddle\">;<\/p><p>c) a &aacute;rea da regi&atilde;o delimitada pelo trap&eacute;zio ABCD.<\/p><h3>Gabarito<\/h3><p>a) <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;sen%5Calpha=%5Cfrac%7B3%7D%7B5%7D\" alt=\"\\large sen\\alpha=\\frac{3}{5}\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;sen%5Calpha=%5Cfrac%7Bco%7D%7Bhip%7D=%5Cfrac%7B2R%7D%7BAB%7D\" alt=\"\\large sen\\alpha=\\frac{co}{hip}=\\frac{2R}{AB}\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5Cfrac%7B3%7D%7B5%7D=%5Cfrac%7B2R%7D%7BAB%7D\" alt=\"\\large \\frac{3}{5}=\\frac{2R}{AB}\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;AB=%5Cfrac%7B50%7D%7B3%7D\" alt=\"\\large AB=\\frac{50}{3}\" align=\"absmiddle\"><\/p><p><\/p><p>b) <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;sen%5Cleft(%5Cfrac%7B%5Calpha%7D%7B2%7D%5Cright)=%5Cfrac%7BR%7D%7BAW%7D\" alt=\"\\large sen\\left(\\frac{\\alpha}{2}\\right)=\\frac{R}{AW}\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;AW=%5Cfrac%7BR%7D%7Bsen%5Cleft(%5Cfrac%7B%5Calpha%7D%7B2%7D%5Cright)%7D\" alt=\"\\large AW=\\frac{R}{sen\\left(\\frac{\\alpha}{2}\\right)}\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;sen%5E2%5Calpha+%5Ccos%5E2%7B%5Calpha%7D=1\" alt=\"\\large sen^2\\alpha+\\cos^2{\\alpha}=1\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;sen%5Calpha=%5Cfrac%7B3%7D%7B5%7D\" alt=\"\\large sen\\alpha=\\frac{3}{5}\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;cos%5C&amp;space;%5Calpha=%5Cfrac%7B4%7D%7B5%7D\" alt=\"\\large cos\\ \\alpha=\\frac{4}{5}\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;tg%5Cleft(%5Calpha%5Cright)=%5Cfrac%7Bsen%5Calpha%7D%7Bcos%5C&amp;space;%5Calpha%7D=%5Cfrac%7B%5Cfrac%7B3%7D%7B5%7D%7D%7B%5Cfrac%7B4%7D%7B5%7D%7D=%5Cfrac%7B3%7D%7B%5Ccancel&amp;space;5%7D%5Ccdot%5Cfrac%7B%5Ccancel&amp;space;5%7D%7B4%7D=%5Cfrac%7B3%7D%7B4%7D\" alt=\"\\large tg\\left(\\alpha\\right)=\\frac{sen\\alpha}{cos\\ \\alpha}=\\frac{\\frac{3}{5}}{\\frac{4}{5}}=\\frac{3}{\\cancel 5}\\cdot\\frac{\\cancel 5}{4}=\\frac{3}{4}\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;tg%5Cleft(2x%5Cright)=%5Cfrac%7B2%5Ccdot&amp;space;t&amp;space;g%5Cleft(x%5Cright)%7D%7B1-tg%5E2%5Cleft(x%5Cright)%7D\" alt=\"\\large tg\\left(2x\\right)=\\frac{2\\cdot t g\\left(x\\right)}{1-tg^2\\left(x\\right)}\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;tg%5Cleft(%5Calpha%5Cright)=%5Cfrac%7B2%5Ccdot&amp;space;t&amp;space;g%5Cleft(%5Cfrac%7B%5Calpha%7D%7B2%7D%5Cright)%7D%7B1-tg%5E2%5Cleft(%5Cfrac%7B%5Calpha%7D%7B2%7D%5Cright)%7D\" alt=\"\\large tg\\left(\\alpha\\right)=\\frac{2\\cdot t g\\left(\\frac{\\alpha}{2}\\right)}{1-tg^2\\left(\\frac{\\alpha}{2}\\right)}\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5Cfrac%7B3%7D%7B4%7D=%5Cfrac%7B2%5Ccdot&amp;space;t&amp;space;g%5Cleft(%5Cfrac%7B%5Calpha%7D%7B2%7D%5Cright)%7D%7B1-tg%5E2%5Cleft(%5Cfrac%7B%5Calpha%7D%7B2%7D%5Cright)%7D\" alt=\"\\large \\frac{3}{4}=\\frac{2\\cdot t g\\left(\\frac{\\alpha}{2}\\right)}{1-tg^2\\left(\\frac{\\alpha}{2}\\right)}\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;3%5Ccdot%5Cleft(1-tg%5E2%5Cleft(%5Cfrac%7B%5Calpha%7D%7B2%7D%5Cright)%5Cright)=4%5Ccdot2%5Ccdot&amp;space;tg%5Cleft(%5Cfrac%7B%5Calpha%7D%7B2%7D%5Cright)\" alt=\"\\large 3\\cdot\\left(1-tg^2\\left(\\frac{\\alpha}{2}\\right)\\right)=4\\cdot2\\cdot tg\\left(\\frac{\\alpha}{2}\\right)\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;3-3%5Ccdot&amp;space;tg%5E2%5Cleft(%5Cfrac%7B%5Calpha%7D%7B2%7D%5Cright)=4%5Ccdot2%5Ccdot&amp;space;tg%5Cleft(%5Cfrac%7B%5Calpha%7D%7B2%7D%5Cright)\" alt=\"\\large 3-3\\cdot tg^2\\left(\\frac{\\alpha}{2}\\right)=4\\cdot2\\cdot tg\\left(\\frac{\\alpha}{2}\\right)\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;3%5Ccdot&amp;space;tg%5E2%5Cleft(%5Cfrac%7B%5Calpha%7D%7B2%7D%5Cright)+8%5Ccdot&amp;space;tg%5Cleft(%5Cfrac%7B%5Calpha%7D%7B2%7D%5Cright)-3=0\" alt=\"\\large 3\\cdot tg^2\\left(\\frac{\\alpha}{2}\\right)+8\\cdot tg\\left(\\frac{\\alpha}{2}\\right)-3=0\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5CDelta&amp;space;=&amp;space;8%5E2-4%5Ccdot&amp;space;3%5Ccdot&amp;space;%5Cleft(-3%5Cright)=64+36=100\" alt=\"\\large \\Delta = 8^2-4\\cdot 3\\cdot \\left(-3\\right)=64+36=100\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;tg%5Cleft(%5Cfrac%7B%5Calpha%7D%7B2%7D%5Cright)=%5Cfrac%7B-8%5Cpm%5Csqrt%7B100%7D%7D%7B2%5Ccdot3%7D==%5Cleft%5C%7B%5Cbegin%7Bmatrix%7D&amp;space;tg'%5Cleft&amp;space;(&amp;space;%5Cfrac%7B%5Calpha&amp;space;%7D%7B2%7D&amp;space;%5Cright&amp;space;)=%5Cfrac%7B-8+10%7D%7B6%7D=%5Cfrac%7B1%7D%7B3%7D&amp;space;&amp;&amp;space;%5C%5C&amp;space;&amp;&amp;space;%5C%5C&amp;space;&amp;&amp;space;%5C%5C&amp;space;tg''%5Cleft&amp;space;(&amp;space;%5Cfrac%7B%5Calpha&amp;space;%7D%7B2%7D&amp;space;%5Cright&amp;space;)==%5Cfrac%7B-8-10%7D%7B6%7D=%7B%5Ccancel%7B-3%7D%7D%5Cend%7Bmatrix%7D%5Cright.\" alt=\"\\large tg\\left(\\frac{\\alpha}{2}\\right)=\\frac{-8\\pm\\sqrt{100}}{2\\cdot3}==\\left\\{\\begin{matrix} tg'\\left ( \\frac{\\alpha }{2} \\right )=\\frac{-8+10}{6}=\\frac{1}{3} &amp; \\\\ &amp; \\\\ &amp; \\\\ tg''\\left ( \\frac{\\alpha }{2} \\right )==\\frac{-8-10}{6}={\\cancel{-3}}\\end{matrix}\\right.\" align=\"absmiddle\"><\/p><p>N&atilde;o h&aacute; tangente negativa.<\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;AW=%5Cfrac%7BR%7D%7Btg%5Cleft(%5Cfrac%7B%5Calpha%7D%7B2%7D%5Cright)%7D\" alt=\"\\large AW=\\frac{R}{tg\\left(\\frac{\\alpha}{2}\\right)}\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;AW=%5Cfrac%7B5%7D%7B%5Cfrac%7B1%7D%7B3%7D%7D\" alt=\"\\large AW=\\frac{5}{\\frac{1}{3}}\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;AW=15\" alt=\"\\large AW=15\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;AX=15\" alt=\"\\large AX=15\" align=\"absmiddle\"><\/p><p><\/p><p>c) <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;S=%5Cfrac%7B%5Cleft(B+b%5Cright)%5Ccdot&amp;space;h%7D%7B2%7D\" alt=\"\\large S=\\frac{\\left(B+b\\right)\\cdot h}{2}\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;S=%5Cfrac%7B%5Cleft(AD+BC%5Cright)%5Ccdot2R%7D%7B2%7D\" alt=\"\\large S=\\frac{\\left(AD+BC\\right)\\cdot2R}{2}\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;S=%5Cleft(AB+CD%5Cright)%5Ccdot&amp;space;R\" alt=\"\\large S=\\left(AB+CD\\right)\\cdot R\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;S=%5Cleft(%5Cfrac%7B50%7D%7B3%7D+15%5Cright)%5Ccdot5\" alt=\"\\large S=\\left(\\frac{50}{3}+15\\right)\\cdot5\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;S=%5Cfrac%7B475%7D%7B3%7D\" alt=\"\\large S=\\frac{475}{3}\" align=\"absmiddle\"><\/p><h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Questao-05\"><\/span>Quest&atilde;o 05<span class=\"ez-toc-section-end\"><\/span><\/h2><p>&Eacute; dada a fun&ccedil;&atilde;o <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;f:%5Cleft%5B0;%5Cpi%5Cright%5D%5Crightarrow%5Cmathbb%7BR%7D\" alt=\"\\large f:\\left[0;\\pi\\right]\\rightarrow\\mathbb{R}\" align=\"absmiddle\"> definida por <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;f%5Cleft(x%5Cright)=sen%5E4x+%5Ccos%5E4%7Bx%7D\" alt=\"\\large f\\left(x\\right)=sen^4x+\\cos^4{x}\" align=\"absmiddle\"> para todo <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;x%5Cin%5Cleft%5B0;%5Cpi%5Cright%5D\" alt=\"\\large x\\in\\left[0;\\pi\\right]\" align=\"absmiddle\">.<\/p><p>a) Apresente tr&ecirc;s valores <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;x%5Cin%5Cleft%5B0;%5Cpi%5Cright%5D\" alt=\"\\large x\\in\\left[0;\\pi\\right]\" align=\"absmiddle\"> para os quais <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;f%5Cleft(x%5Cright)=1\" alt=\"\\large f\\left(x\\right)=1\" align=\"absmiddle\">.<\/p><p>b) Determine os valores de <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;x%5Cin%5Cleft%5B0;%5Cpi%5Cright%5D\" alt=\"\\large x\\in\\left[0;\\pi\\right]\" align=\"absmiddle\"> para os quais <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;f%5Cleft(x%5Cright)=%5Cfrac%7B5%7D%7B8%7D\" alt=\"\\large f\\left(x\\right)=\\frac{5}{8}\" align=\"absmiddle\">.<\/p><p>c) Determine os valores de <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;x%5Cin%5Cleft%5B0;%5Cpi%5Cright%5D\" alt=\"\\large x\\in\\left[0;\\pi\\right]\" align=\"absmiddle\"> para os quais <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5Cfrac%7B1%7D%7B2%7D%5Ccdot&amp;space;f%5Cleft(x%5Cright)+%5Cfrac%7B3%7D%7B8%7D%5Ccdot&amp;space;sen%5Cleft(2x%5Cright)%5Cgeq%5Cfrac%7B5%7D%7B8%7D\" alt=\"\\large \\frac{1}{2}\\cdot f\\left(x\\right)+\\frac{3}{8}\\cdot sen\\left(2x\\right)\\geq\\frac{5}{8}\" align=\"absmiddle\">.<\/p><h3>Gabarito<\/h3><p>a) Apresente tr&ecirc;s valores <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;x%5Cin%5Cleft%5B0;%5Cpi%5Cright%5D\" alt=\"\\large x\\in\\left[0;\\pi\\right]\" align=\"absmiddle\"> para os quais <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;f%5Cleft(x%5Cright)=1\" alt=\"\\large f\\left(x\\right)=1\" align=\"absmiddle\">.<\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;f%5Cleft(x%5Cright)=1\" alt=\"\\large f\\left(x\\right)=1\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;sen%5E4x+%5Ccos%5E4%7Bx%7D=1\" alt=\"\\large sen^4x+\\cos^4{x}=1\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;sen%5E2x+%5Ccos%5E2%7Bx%7D=1\" alt=\"\\large sen^2x+\\cos^2{x}=1\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5Cleft(sen%5E2x+%5Ccos%5E2%7Bx%7D%5Cright)%5E2=1%5E2\" alt=\"\\large \\left(sen^2x+\\cos^2{x}\\right)^2=1^2\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;sen%5E4x+2%5Ccdot&amp;space;sen%5E2x%5Ccdot%5Ccos%5E2%7Bx%7D+%5Ccos%5E4%7Bx%7D=1\" alt=\"\\large sen^4x+2\\cdot sen^2x\\cdot\\cos^2{x}+\\cos^4{x}=1\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;f%5Cleft(x%5Cright)+2%5Ccdot&amp;space;sen%5E2x%5Ccdot%5Ccos%5E2%7Bx%7D=1\" alt=\"\\large f\\left(x\\right)+2\\cdot sen^2x\\cdot\\cos^2{x}=1\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;f%5Cleft(x%5Cright)+2%5Ccdot&amp;space;sen%5C&amp;space;x%5Ccdot%5Ccos%7Bx%7D%5Ccdot&amp;space;sen%5C&amp;space;x%5Ccdot%5Ccos%7Bx%7D=1\" alt=\"\\large f\\left(x\\right)+2\\cdot sen\\ x\\cdot\\cos{x}\\cdot sen\\ x\\cdot\\cos{x}=1\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;f%5Cleft(x%5Cright)+sen%5C&amp;space;2x%5Ccdot%5Cfrac%7Bsen%5C&amp;space;2x%7D%7B2%7D=1\" alt=\"\\large f\\left(x\\right)+sen\\ 2x\\cdot\\frac{sen\\ 2x}{2}=1\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;f%5Cleft(x%5Cright)+%5Cfrac%7Bsen%5E2%5C&amp;space;2x%7D%7B2%7D=1\" alt=\"\\large f\\left(x\\right)+\\frac{sen^2\\ 2x}{2}=1\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;1+%5Cfrac%7Bsen%5E2%5C&amp;space;2x%7D%7B2%7D=1\" alt=\"\\large 1+\\frac{sen^2\\ 2x}{2}=1\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5Cfrac%7Bsen%5E2%5C&amp;space;2x%7D%7B2%7D=0\" alt=\"\\large \\frac{sen^2\\ 2x}{2}=0\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;sen%5E2%5C&amp;space;2x=0\" alt=\"\\large sen^2\\ 2x=0\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;sen%5C&amp;space;2x=0\" alt=\"\\large sen\\ 2x=0\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;2x=k%5Ccdot%5Cpi\" alt=\"\\large 2x=k\\cdot\\pi\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;x=%5Cfrac%7Bk%5Ccdot%5Cpi%7D%7B2%7D\" alt=\"\\large x=\\frac{k\\cdot\\pi}{2}\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;x=0;%5Cfrac%7B%5Cpi%7D%7B2%7D;%5Cpi\" alt=\"\\large x=0;\\frac{\\pi}{2};\\pi\" align=\"absmiddle\"><\/p><p><\/p><p>b) Determine os valores de <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;x%5Cin%5Cleft%5B0;%5Cpi%5Cright%5D\" alt=\"\\large x\\in\\left[0;\\pi\\right]\" align=\"absmiddle\"> para os quais <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;f%5Cleft(x%5Cright)=%5Cfrac%7B5%7D%7B8%7D\" alt=\"\\large f\\left(x\\right)=\\frac{5}{8}\" align=\"absmiddle\">.<\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;f%5Cleft(x%5Cright)=%5Cfrac%7B5%7D%7B8%7D\" alt=\"\\large f\\left(x\\right)=\\frac{5}{8}\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;f%5Cleft(x%5Cright)+%5Cfrac%7Bsen%5E2%5C&amp;space;2x%7D%7B2%7D=1\" alt=\"\\large f\\left(x\\right)+\\frac{sen^2\\ 2x}{2}=1\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5Cfrac%7B5%7D%7B8%7D+%5Cfrac%7Bsen%5E2%5C&amp;space;2x%7D%7B2%7D=1\" alt=\"\\large \\frac{5}{8}+\\frac{sen^2\\ 2x}{2}=1\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5Cfrac%7Bsen%5E2%5C&amp;space;2x%7D%7B2%7D=%5Cfrac%7B3%7D%7B8%7D\" alt=\"\\large \\frac{sen^2\\ 2x}{2}=\\frac{3}{8}\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;sen%5E2%5C&amp;space;2x=%5Cfrac%7B3%7D%7B4%7D\" alt=\"\\large sen^2\\ 2x=\\frac{3}{4}\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;sen%5C&amp;space;2x=%5Cpm%5Cfrac%7B%5Csqrt3%7D%7B2%7D\" alt=\"\\large sen\\ 2x=\\pm\\frac{\\sqrt3}{2}\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;2x=%5Cpm%5Cfrac%7B%5Cpi%7D%7B3%7D+k%5Cpi\" alt=\"\\large 2x=\\pm\\frac{\\pi}{3}+k\\pi\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;x=%5Cpm%5Cfrac%7B%5Cpi%7D%7B6%7D+%5Cfrac%7Bk%5Cpi%7D%7B2%7D\" alt=\"\\large x=\\pm\\frac{\\pi}{6}+\\frac{k\\pi}{2}\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5Cleft%5C%7B%5Cfrac%7B%5Cpi%7D%7B6%7D;%5Cfrac%7B%5Cpi%7D%7B3%7D;%5Cfrac%7B2%5Cpi%7D%7B3%7D;%5Cfrac%7B5%5Cpi%7D%7B6%7D%5Cright%5C%7D\" alt=\"\\large \\left\\{\\frac{\\pi}{6};\\frac{\\pi}{3};\\frac{2\\pi}{3};\\frac{5\\pi}{6}\\right\\}\" align=\"absmiddle\"><\/p><p><\/p><p>c) Determine os valores de <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;x%5Cin%5Cleft%5B0;%5Cpi%5Cright%5D\" alt=\"\\large x\\in\\left[0;\\pi\\right]\" align=\"absmiddle\"> para os quais <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5Cfrac%7B1%7D%7B2%7D%5Ccdot&amp;space;f%5Cleft(x%5Cright)+%5Cfrac%7B3%7D%7B8%7D%5Ccdot&amp;space;sen%5Cleft(2x%5Cright)%5Cgeq%5Cfrac%7B5%7D%7B8%7D\" alt=\"\\large \\frac{1}{2}\\cdot f\\left(x\\right)+\\frac{3}{8}\\cdot sen\\left(2x\\right)\\geq\\frac{5}{8}\" align=\"absmiddle\">.<\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;f%5Cleft(x%5Cright)+%5Cfrac%7Bsen%5E2%5C&amp;space;2x%7D%7B2%7D=1\" alt=\"\\large f\\left(x\\right)+\\frac{sen^2\\ 2x}{2}=1\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;f%5Cleft(x%5Cright)=1-%5Cfrac%7Bsen%5E2%5C&amp;space;2x%7D%7B2%7D\" alt=\"\\large f\\left(x\\right)=1-\\frac{sen^2\\ 2x}{2}\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5Cfrac%7B1%7D%7B2%7D%5Ccdot&amp;space;f%5Cleft(x%5Cright)+%5Cfrac%7B3%7D%7B8%7D%5Ccdot&amp;space;sen%5Cleft(2x%5Cright)%5Cgeq%5Cfrac%7B5%7D%7B8%7D\" alt=\"\\large \\frac{1}{2}\\cdot f\\left(x\\right)+\\frac{3}{8}\\cdot sen\\left(2x\\right)\\geq\\frac{5}{8}\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5Cfrac%7B1%7D%7B2%7D%5Ccdot%5Cleft(1-%5Cfrac%7Bsen%5E2%5C&amp;space;2x%7D%7B2%7D%5Cright)+%5Cfrac%7B3%7D%7B8%7D%5Ccdot&amp;space;sen%5Cleft(2x%5Cright)%5Cgeq%5Cfrac%7B5%7D%7B8%7D\" alt=\"\\large \\frac{1}{2}\\cdot\\left(1-\\frac{sen^2\\ 2x}{2}\\right)+\\frac{3}{8}\\cdot sen\\left(2x\\right)\\geq\\frac{5}{8}\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5Cfrac%7B1%7D%7B2%7D-%5Cfrac%7Bsen%5E2%5C&amp;space;2x%7D%7B4%7D+%5Cfrac%7B3%7D%7B8%7D%5Ccdot&amp;space;sen%5Cleft(2x%5Cright)%5Cgeq%5Cfrac%7B5%7D%7B8%7D\" alt=\"\\large \\frac{1}{2}-\\frac{sen^2\\ 2x}{4}+\\frac{3}{8}\\cdot sen\\left(2x\\right)\\geq\\frac{5}{8}\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;4-2%5Ccdot&amp;space;sen%5E2%5C&amp;space;2x+3%5Ccdot&amp;space;sen%5Cleft(2x%5Cright)%5Cgeq5\" alt=\"\\large 4-2\\cdot sen^2\\ 2x+3\\cdot sen\\left(2x\\right)\\geq5\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;-4+2%5Ccdot&amp;space;sen%5E2%5C&amp;space;2x-3%5Ccdot&amp;space;sen%5Cleft(2x%5Cright)%5Cle-5\" alt=\"\\large -4+2\\cdot sen^2\\ 2x-3\\cdot sen\\left(2x\\right)\\le-5\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;2%5Ccdot&amp;space;sen%5E2%5C&amp;space;2x-3%5Ccdot&amp;space;sen%5Cleft(2x%5Cright)+1%5Cle0\" alt=\"\\large 2\\cdot sen^2\\ 2x-3\\cdot sen\\left(2x\\right)+1\\le0\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5CDelta&amp;space;%5Cleft(-3%5Cright)%5E2-4%5Ccdot&amp;space;2%5Ccdot&amp;space;1=9-8=1\" alt=\"\\large \\Delta \\left(-3\\right)^2-4\\cdot 2\\cdot 1=9-8=1\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;sen%5Cleft(2x%5Cright)=%5Cfrac%7B-%5Cleft(-3%5Cright)%5Cpm%5Csqrt1%7D%7B2%5Ccdot2%7D=%5Cleft%5C%7B%5Cbegin%7Bmatrix%7Dsen'%5Cleft(2x%5Cright)=%5Cfrac%7B3+1%7D%7B4%7D=1%5C%5C%5C&amp;space;%5C%5Csen''%5Cleft(2x%5Cright)=%5Cfrac%7B3-1%7D%7B4%7D=%5Cfrac%7B1%7D%7B2%7D%5C%5C%5Cend%7Bmatrix%7D%5Cright.\" alt=\"\\large sen\\left(2x\\right)=\\frac{-\\left(-3\\right)\\pm\\sqrt1}{2\\cdot2}=\\left\\{\\begin{matrix}sen'\\left(2x\\right)=\\frac{3+1}{4}=1\\\\\\ \\\\sen''\\left(2x\\right)=\\frac{3-1}{4}=\\frac{1}{2}\\\\\\end{matrix}\\right.\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5Cfrac%7B1%7D%7B2%7D%5Cle&amp;space;sen%5C&amp;space;%5Cleft(2x%5Cright)%5Cle1\" alt=\"\\large \\frac{1}{2}\\le sen\\ \\left(2x\\right)\\le1\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5Cfrac%7B%5Cpi%7D%7B6%7D%5Cle2x%5Cle%5Cfrac%7B5%5Cpi%7D%7B6%7D\" alt=\"\\large \\frac{\\pi}{6}\\le2x\\le\\frac{5\\pi}{6}\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5Cfrac%7B%5Cpi%7D%7B12%7D%5Cle&amp;space;x%5Cle%5Cfrac%7B5%5Cpi%7D%7B12%7D\" alt=\"\\large \\frac{\\pi}{12}\\le x\\le\\frac{5\\pi}{12}\" align=\"absmiddle\"><\/p><h2><span class=\"ez-toc-section\" id=\"Questao-06\"><\/span>Quest&atilde;o 06<span class=\"ez-toc-section-end\"><\/span><\/h2><p>Resolva os tr&ecirc;s itens abaixo:<\/p><p>a) Considere o conjunto formado pelos n&uacute;meros complexos z que cumprem a condi&ccedil;&atilde;o <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;Re(z)=Im(z)\" alt=\"\\large Re(z)=Im(z)\" align=\"absmiddle\">. Cada elemento desse conjunto ser&aacute; objeto da transforma&ccedil;&atilde;o que leva um n&uacute;mero complexo em seu conjugado. Represente no plano complexo (ou plano de Argand-Gauss) da folha de respostas o conjunto resultante ap&oacute;s essa transforma&ccedil;&atilde;o.<\/p><p>b) Determine o lugar geom&eacute;trico dos pontos z do plano complexo tais que <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;z%5C&amp;space;%5Cneq-1\" alt=\"\\large z\\ \\neq-1\" align=\"absmiddle\"> e para os quais <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5Cfrac%7Bz-1%7D%7Bz+1%7D\" alt=\"\\large \\frac{z-1}{z+1}\" align=\"absmiddle\"> &eacute; um n&uacute;mero imagin&aacute;rio puro.<\/p><p>c) Determine as partes reais de todos os n&uacute;meros complexos z tais que as representa&ccedil;&otilde;es de z, i e 1 no plano complexo sejam v&eacute;rtices de um tri&acirc;ngulo equil&aacute;tero.<\/p><h3>Gabarito<\/h3><p>a)<\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;z=a+b%5Ccdot&amp;space;i\" alt=\"\\large z=a+b\\cdot i\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;Re%5Cleft(z%5Cright)=Im%5Cleft(z%5Cright)\" alt=\"\\large Re\\left(z\\right)=Im\\left(z\\right)\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;a=b\" alt=\"\\large a=b\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;z=a+a%5Ccdot&amp;space;i\" alt=\"\\large z=a+a\\cdot i\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5Cbar%7Bz%7D=a-a%5Ccdot&amp;space;i\" alt=\"\\large \\bar{z}=a-a\\cdot i\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5Calpha=-45%5Cdegree\" alt=\"\\large \\alpha=-45\\degree\" align=\"absmiddle\"><\/p><div class=\"wp-block-image\"><figure class=\"aligncenter is-resized\"><img decoding=\"async\" src=\"https:\/\/cdn.blog.estrategiavestibulares.com.br\/vestibulares\/wp-content\/uploads\/2020\/01\/image19.png\" alt=\"\" class=\"wp-image-37365\" width=\"258\" height=\"240\"><\/figure><\/div><p>b)&nbsp;<\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5Cfrac%7Bz-1%7D%7Bz+1%7D\" alt=\"\\large \\frac{z-1}{z+1}\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5Cfrac%7Ba+b%5Ccdot&amp;space;i-1%7D%7Ba+b%5Ccdot&amp;space;i+1%7D\" alt=\"\\large \\frac{a+b\\cdot i-1}{a+b\\cdot i+1}\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5Cfrac%7B%5Cleft(a-1%5Cright)+b%5Ccdot&amp;space;i%7D%7B%5Cleft(a+1%5Cright)+b%5Ccdot&amp;space;i%7D%5Ccdot%5Cfrac%7B%5Cleft(a+1%5Cright)-b%5Ccdot&amp;space;i%7D%7B%5Cleft(a+1%5Cright)-b%5Ccdot&amp;space;i%7D\" alt=\"\\large \\frac{\\left(a-1\\right)+b\\cdot i}{\\left(a+1\\right)+b\\cdot i}\\cdot\\frac{\\left(a+1\\right)-b\\cdot i}{\\left(a+1\\right)-b\\cdot i}\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5Cfrac%7Ba%5E2+b%5E2-1+2%5Ccdot&amp;space;b%5Ccdot&amp;space;i%7D%7B%5Cleft(a+1%5Cright)%5E2+b%5E2%7D\" alt=\"\\large \\frac{a^2+b^2-1+2\\cdot b\\cdot i}{\\left(a+1\\right)^2+b^2}\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5Cfrac%7Ba%5E2+b%5E2-1%7D%7B%5Cleft(a+1%5Cright)%5E2+b%5E2%7D+%5Cfrac%7B2%5Ccdot&amp;space;b%5Ccdot&amp;space;i%7D%7B%5Cleft(a+1%5Cright)%5E2+b%5E2%7D\" alt=\"\\large \\frac{a^2+b^2-1}{\\left(a+1\\right)^2+b^2}+\\frac{2\\cdot b\\cdot i}{\\left(a+1\\right)^2+b^2}\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5Cfrac%7Ba%5E2+b%5E2-1%7D%7B%5Cleft(a+1%5Cright)%5E2+b%5E2%7D=0\" alt=\"\\large \\frac{a^2+b^2-1}{\\left(a+1\\right)^2+b^2}=0\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;a%5E2+b%5E2-1=0\" alt=\"\\large a^2+b^2-1=0\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;a%5E2+b%5E2=1\" alt=\"\\large a^2+b^2=1\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;z%5Cneq-1\" alt=\"\\large z\\neq-1\" align=\"absmiddle\"><\/p><p>Lugar Geom&eacute;trico &eacute; uma circunfer&ecirc;ncia com centro na origem, exceto o ponto z=-1.<\/p><p>c)<\/p><div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/cdn.blog.estrategiavestibulares.com.br\/vestibulares\/wp-content\/uploads\/2020\/01\/image20.png\" alt=\"\" class=\"wp-image-37368\"><\/figure><\/div><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;d=%5Csqrt2\" alt=\"\\large d=\\sqrt2\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5Csqrt%7B%5Cleft(1-x%5Cright)%5E2+%5Cleft(0-x%5Cright)%5E2%7D=%5Csqrt2\" alt=\"\\large \\sqrt{\\left(1-x\\right)^2+\\left(0-x\\right)^2}=\\sqrt2\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5Cleft(1-x%5Cright)%5E2+x%5E2=2\" alt=\"\\large \\left(1-x\\right)^2+x^2=2\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;1-2x+x%5E2+x%5E2=2\" alt=\"\\large 1-2x+x^2+x^2=2\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;2x%5E2-2x-1=0\" alt=\"\\large 2x^2-2x-1=0\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5CDelta&amp;space;=%5Cleft(-2%5Cright)%5E2-4%5Ccdot&amp;space;2%5Ccdot&amp;space;%5Cleft(-1%5Cright)=4+8=12\" alt=\"\\large \\Delta =\\left(-2\\right)^2-4\\cdot 2\\cdot \\left(-1\\right)=4+8=12\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;x=%5Cfrac%7B-%5Cleft(-2%5Cright)%5Cpm%5Csqrt%7B12%7D%7D%7B2%5Ccdot2%7D=%5Cleft%5C%7B%5Cbegin%7Bmatrix%7Dx%5E%5Cprime=%5Cfrac%7B2+2%5Csqrt3%7D%7B4%7D=%5Cfrac%7B1+%5Csqrt3%7D%7B2%7D%5C%5C%5C&amp;space;%5C%5Cx%5E%7B%5Cprime%5Cprime%7D=%5Cfrac%7B2-2%5Csqrt3%7D%7B4%7D=%5Cfrac%7B1-%5Csqrt3%7D%7B2%7D%5C%5C%5Cend%7Bmatrix%7D%5Cright.\" alt=\"\\large x=\\frac{-\\left(-2\\right)\\pm\\sqrt{12}}{2\\cdot2}=\\left\\{\\begin{matrix}x^\\prime=\\frac{2+2\\sqrt3}{4}=\\frac{1+\\sqrt3}{2}\\\\\\ \\\\x^{\\prime\\prime}=\\frac{2-2\\sqrt3}{4}=\\frac{1-\\sqrt3}{2}\\\\\\end{matrix}\\right.\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5Cfrac%7B1+%5Csqrt3%7D%7B2%7D%5C&amp;space;e%5C&amp;space;%5Cfrac%7B1-%5Csqrt3%7D%7B2%7D\" alt=\"\\large \\frac{1+\\sqrt3}{2}\\ e\\ \\frac{1-\\sqrt3}{2}\" align=\"absmiddle\"><\/p><p>&Eacute; isso, pessoal! Espero que tenham curtido a resolu&ccedil;&atilde;o da prova de Matem&aacute;tica da prova do Vestibular FUVEST 2020, 2&ordf; Fase.&nbsp;Sigam-me nas redes sociais. T&ecirc;m muitas dicas l&aacute;. Mande uma mensagem, caso tenha tido alguma d&uacute;vida.<\/p><p>Voc&ecirc; tamb&eacute;m pode baixar essa resolu&ccedil;&atilde;o em PDF. Basta clicar no link a seguir. Abra&ccedil;os!<\/p><div class=\"wp-block-file\">\"&gt;Resolu&ccedil;&atilde;o FUVEST 2020 &ndash; 2&ordf; Fase &ndash; Matem&aacute;tica\" class=\"wp-block-file__button\" download&gt;Baixar<\/div><p>Abra&ccedil;os!<\/p><p><strong>Instagram:<\/strong>&nbsp;<a href=\"https:\/\/www.instagram.com\/professor.marcal\/\" target=\"_blank\">@professor.marcal<\/a><\/p><p class=\"has-text-align-center has-light-green-cyan-background-color has-background has-medium-font-size\"><a href=\"https:\/\/estrategiavestibulares.com.br\/vestibulares\/fuvest\/\" target=\"_blank\" aria-label=\"CURSOS PARA FUVEST (abre numa nova aba)\">CURSOS PARA FUVEST<\/a><\/p><\/p>\n","protected":false},"excerpt":{"rendered":"Ol&aacute;, pessoal&hellip; Tudo bem? Sou o prof. Mar&ccedil;al Ferreira, professor de Matem&aacute;tica do Estrat&eacute;gia Vestibulares. Escrevo este artigo&hellip;\n","protected":false},"author":18,"featured_media":37372,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"wl_entities_gutenberg":"","footnotes":""},"categories":[29],"tags":[],"wl_entity_type":[732],"class_list":{"0":"post-37337","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-matematica","8":"wl_entity_type-article"},"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v25.9 (Yoast SEO v25.9) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Resolu\u00e7\u00e3o da prova de Matem\u00e1tica da 2\u00aa Fase da Fuvest 2020<\/title>\n<meta name=\"description\" content=\"O Estrat\u00e9gia Vestibulares traz para voc\u00ea a resolu\u00e7\u00e3o comentada da prova de matem\u00e1tica da segunda fase da fuvest! 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