{"id":37742,"date":"2020-01-14T14:15:40","date_gmt":"2020-01-14T17:15:40","guid":{"rendered":"https:\/\/blog.estrategiavestibulares.com.br\/?p=37742"},"modified":"2022-04-25T17:43:23","modified_gmt":"2022-04-25T20:43:23","slug":"prova-matematica-2-fase-unicamp-2020","status":"publish","type":"post","link":"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/prova-matematica-2-fase-unicamp-2020\/","title":{"rendered":"Resolu\u00e7\u00e3o da prova de Matem\u00e1tica da 2\u00aa Fase da Unicamp 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 Unicamp 2020.<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\/prova-matematica-2-fase-unicamp-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-2\" href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/prova-matematica-2-fase-unicamp-2020\/#Questao-06\" >Quest&atilde;o 06<\/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\/prova-matematica-2-fase-unicamp-2020\/#Questao-07\" >Quest&atilde;o 07<\/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\/prova-matematica-2-fase-unicamp-2020\/#Questao-08\" >Quest&atilde;o 08<\/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\/prova-matematica-2-fase-unicamp-2020\/#Questao-09\" >Quest&atilde;o 09<\/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\/prova-matematica-2-fase-unicamp-2020\/#Questao-10\" >Quest&atilde;o 10<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/prova-matematica-2-fase-unicamp-2020\/#Veja-tambem\" >Veja tamb&eacute;m:<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\" id=\"h-questao-05\"><span class=\"ez-toc-section\" id=\"Questao-05\"><\/span>Quest&atilde;o 05<span class=\"ez-toc-section-end\"><\/span><\/h2><p>Dois tipos de exames para a detec&ccedil;&atilde;o de certo v&iacute;rus foram aplicados em um grupo de 80 pacientes, dos quais, com certeza, 60 s&atilde;o portadores desse v&iacute;rus e 20 n&atilde;o s&atilde;o. Os resultados dos exames est&atilde;o organizados nas tabelas abaixo. <\/p><figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/cdn.blog.estrategiavestibulares.com.br\/vestibulares\/wp-content\/uploads\/2020\/01\/matematica-unicamp-2020-2-fase-questao-05.jpg\" alt=\"\" class=\"wp-image-37747\"><\/figure><p>Note que em cada exame ocorrem tanto falsos\npositivos (pacientes n&atilde;o portadores do v&iacute;rus com resultado positivo no exame)\nquanto falsos negativos (pacientes portadores do v&iacute;rus com resultado negativo\nno exame).<\/p><p>a) Calcule a porcentagem de pacientes portadores do v&iacute;rus no grupo em estudo. b) Considerando os resultados positivos em cada exame, qual dos dois exames tem a menor porcentagem de falsos positivos? Justifique sua resposta.<\/p><h3 class=\"wp-block-heading\" id=\"h-gabarito\">Gabarito<\/h3><p>a) Como s&atilde;o 60 portadores do v&iacute;rus em um grupo de 80 pacientes, temos:<\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;%5Cfrac%7B60%7D%7B80%7D=%5Cfrac%7B6%7D%7B8%7D=%5Cfrac%7B3%7D%7B4%7D=75%5C%\" alt=\"\\dpi{100} \\large \\frac{60}{80}=\\frac{6}{8}=\\frac{3}{4}=75\\%\" align=\"absmiddle\"><\/p><p>b) O n&uacute;mero de falsos positivos para o exame 1 &eacute;:<\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;%5Cfrac%7B6%7D%7B48%7D=%5Cfrac%7B1%7D%7B8%7D=12,5%5C%\" alt=\"\\dpi{100} \\large \\frac{6}{48}=\\frac{1}{8}=12,5\\%\" align=\"absmiddle\"><\/p><p>O n&uacute;mero de falsos positivos para o exame 2 &eacute;:<\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;%5Cfrac%7B7%7D%7B63%7D=%5Cfrac%7B1%7D%7B9%7D=11,%5Cbar%7B1%7D%5C%\" alt=\"\\dpi{100} \\large \\frac{7}{63}=\\frac{1}{9}=11,\\bar{1}\\%\" align=\"absmiddle\"><\/p><p>Dessa forma, o exame 2 apresenta a menor porcentagem de falsos positivos.<\/p><h2 class=\"wp-block-heading\" id=\"h-questao-06\"><span class=\"ez-toc-section\" id=\"Questao-06\"><\/span>Quest&atilde;o 06<span class=\"ez-toc-section-end\"><\/span><\/h2><p>A figura abaixo exibe um tri&acirc;ngulo is&oacute;sceles com dois lados de comprimento <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;a=5cm\" alt=\"\\dpi{100} \\large a=5cm\" align=\"absmiddle\"> e um dos &acirc;ngulos internos igual a <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;%5Ctheta\" alt=\"\\dpi{100} \\large \\theta\" align=\"absmiddle\">, em que <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;cos%5C&amp;space;%5Ctheta=3\/5\" alt=\"\\dpi{100} \\large cos\\ \\theta=3\/5\" 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\/matematica-unicamp-2020-2-fase-questC3A3o-6.jpg\" alt=\"\" class=\"wp-image-37750\"><\/figure><\/div><p>a) Calcule a &aacute;rea desse tri&acirc;ngulo.<\/p><p>b) Determine o comprimento do raio da circunfer&ecirc;ncia circunscrita a esse tri&acirc;ngulo.<\/p><h3 class=\"wp-block-heading\" id=\"h-resolucao\">Resolu&ccedil;&atilde;o<\/h3><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;%7B%5Crm&amp;space;sen%7D%5E2%7B%5Cleft(x%5Cright)%7D+%5Ccos%5E2%7B%5Cleft(x%5Cright)%7D=1\" alt=\"\\dpi{100} \\large {\\rm sen}^2{\\left(x\\right)}+\\cos^2{\\left(x\\right)}=1\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;%7B%5Crm&amp;space;sen%7D%5E2%7B%5Cleft(x%5Cright)%7D+%5Cleft(%5Cfrac%7B3%7D%7B5%7D%5Cright)%5E2=1\" alt=\"\\dpi{100} \\large {\\rm sen}^2{\\left(x\\right)}+\\left(\\frac{3}{5}\\right)^2=1\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;%7B%5Crm&amp;space;sen%7D%5E2%7B%5Cleft(x%5Cright)%7D+%5Cfrac%7B9%7D%7B25%7D=1\" alt=\"\\dpi{100} \\large {\\rm sen}^2{\\left(x\\right)}+\\frac{9}{25}=1\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;%7B%5Crm&amp;space;sen%7D%5E2%7B%5Cleft(x%5Cright)%7D=%5Cfrac%7B16%7D%7B25%7D\" alt=\"\\dpi{100} \\large {\\rm sen}^2{\\left(x\\right)}=\\frac{16}{25}\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;sen%5Cleft(x%5Cright)=%5Cfrac%7B4%7D%7B5%7D\" alt=\"\\dpi{100} \\large sen\\left(x\\right)=\\frac{4}{5}\" align=\"absmiddle\"><\/p><p class=\"Questes-enunciado\">a) Calcule a &aacute;rea desse tri&acirc;ngulo.<\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;A=%5Cfrac%7B1%7D%7B2%7D%5Ccdot&amp;space;a%5Ccdot&amp;space;b%5Ccdot&amp;space;sen%5Cleft(%5Ctheta%5Cright)\" alt=\"\\dpi{100} \\large A=\\frac{1}{2}\\cdot a\\cdot b\\cdot sen\\left(\\theta\\right)\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;A=%5Cfrac%7B1%7D%7B2%7D%5Ccdot5%5Ccdot5%5Ccdot%5Cfrac%7B4%7D%7B5%7D\" alt=\"\\dpi{100} \\large A=\\frac{1}{2}\\cdot5\\cdot5\\cdot\\frac{4}{5}\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;A=10%5C&amp;space;cm%5E2\" alt=\"\\dpi{100} \\large A=10\\ cm^2\" align=\"absmiddle\"><\/p><p class=\"Questes-enunciado\">b) Determine o comprimento do raio da circunfer&ecirc;ncia circunscrita a esse tri&acirc;ngulo.<\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;x%5E2=a%5E2+a%5E2-2%5Ccdot&amp;space;a%5Ccdot&amp;space;b%5Ccdot%5Ccos%7B%5Cleft(%5Ctheta%5Cright)%7D\" alt=\"\\dpi{100} \\large x^2=a^2+a^2-2\\cdot a\\cdot b\\cdot\\cos{\\left(\\theta\\right)}\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;x%5E2=5%5E2+5%5E2-2%5Ccdot5%5Ccdot5%5Ccdot%5Cfrac%7B3%7D%7B5%7D\" alt=\"\\dpi{100} \\large x^2=5^2+5^2-2\\cdot5\\cdot5\\cdot\\frac{3}{5}\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;x%5E2=50-30\" alt=\"\\dpi{100} \\large x^2=50-30\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;x=%5Csqrt%7B20%7D\" alt=\"\\dpi{100} \\large x=\\sqrt{20}\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;x=2%5Csqrt5%5C&amp;space;cm\" alt=\"\\dpi{100} \\large x=2\\sqrt5\\ cm\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;%5Cfrac%7Bx%7D%7Bsen%5Cleft(%5Ctheta%5Cright)%7D=2%5Ccdot&amp;space;R\" alt=\"\\dpi{100} \\large \\frac{x}{sen\\left(\\theta\\right)}=2\\cdot R\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;%5Cfrac%7B2%5Csqrt5%7D%7B%5Cfrac%7B4%7D%7B5%7D%7D=2%5Ccdot&amp;space;R\" alt=\"\\dpi{100} \\large \\frac{2\\sqrt5}{\\frac{4}{5}}=2\\cdot R\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;%5Cfrac%7B5%5Csqrt5%7D%7B4%7D%5C&amp;space;cm=R\" alt=\"\\dpi{100} \\large \\frac{5\\sqrt5}{4}\\ cm=R\" align=\"absmiddle\"><\/p><h2 class=\"wp-block-heading\" id=\"h-questao-07\"><span class=\"ez-toc-section\" id=\"Questao-07\"><\/span>Quest&atilde;o 07<span class=\"ez-toc-section-end\"><\/span><\/h2><p>Seja a matriz de ordem <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;2%5Ctimes3\" alt=\"\\dpi{100} \\large 2\\times3\" align=\"absmiddle\">, dada por <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;A=%5Cleft%5B%5Cbegin%7Bmatrix%7D1&amp;1&amp;1%5C%5C1&amp;2&amp;3%5C%5C%5Cend%7Bmatrix%7D%5Cright%5D\" alt=\"\\dpi{100} \\large A=\\left[\\begin{matrix}1&amp;1&amp;1\\\\1&amp;2&amp;3\\\\\\end{matrix}\\right]\" align=\"absmiddle\">.<\/p><p>a) Seja C a matriz de ordem <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;3%5Ctimes2\" alt=\"\\dpi{100} \\large 3\\times2\" align=\"absmiddle\">, cujos elementos s&atilde;o dados por <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;c_%7Bij%7D=%5Cleft(-1%5Cright)%5E%7Bi+j%7D\" alt=\"\\dpi{100} \\large c_{ij}=\\left(-1\\right)^{i+j}\" align=\"absmiddle\">, para <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;i=1,2,3\" alt=\"\\dpi{100} \\large i=1,2,3\" align=\"absmiddle\"> e <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;j=1,2\" alt=\"\\dpi{100} \\large j=1,2\" align=\"absmiddle\">. Determine o produto AC.<\/p><p>b) Determine a solu&ccedil;&atilde;o do sistema linear <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;A%5Cleft%5B%5Cbegin%7Bmatrix%7Dx%5C%5Cy%5C%5Cz%5C%5C%5Cend%7Bmatrix%7D%5Cright%5D=%5Cleft%5B%5Cbegin%7Bmatrix%7D6%5C%5C6%5C%5C%5Cend%7Bmatrix%7D%5Cright%5D\" alt=\"\\dpi{100} \\large A\\left[\\begin{matrix}x\\\\y\\\\z\\\\\\end{matrix}\\right]=\\left[\\begin{matrix}6\\\\6\\\\\\end{matrix}\\right]\" align=\"absmiddle\">, nas vari&aacute;veis reais x,y, e z, em que <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;%5Cleft(x,y,z%5Cright)\" alt=\"\\dpi{100} \\large \\left(x,y,z\\right)\" align=\"absmiddle\"> &eacute; uma progress&atilde;o aritm&eacute;tica.<\/p><h3 class=\"wp-block-heading\" id=\"h-resolucao-1\">Resolu&ccedil;&atilde;o<\/h3><p>a)<\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;c_%7B11%7D=%5Cleft(-1%5Cright)%5E%7B1+1%7D=1\" alt=\"\\dpi{100} \\large c_{11}=\\left(-1\\right)^{1+1}=1\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;c_%7B12%7D=%5Cleft(-1%5Cright)%5E%7B1+2%7D=-1\" alt=\"\\dpi{100} \\large c_{12}=\\left(-1\\right)^{1+2}=-1\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;c_%7B21%7D=%5Cleft(-1%5Cright)%5E%7B2+1%7D=-1\" alt=\"\\dpi{100} \\large c_{21}=\\left(-1\\right)^{2+1}=-1\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;c_%7B22%7D=%5Cleft(-1%5Cright)%5E%7B2+2%7D=1\" alt=\"\\dpi{100} \\large c_{22}=\\left(-1\\right)^{2+2}=1\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;c_%7B31%7D=%5Cleft(-1%5Cright)%5E%7B3+1%7D=1\" alt=\"\\dpi{100} \\large c_{31}=\\left(-1\\right)^{3+1}=1\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;c_%7B32%7D=%5Cleft(-1%5Cright)%5E%7B3+2%7D=-1\" alt=\"\\dpi{100} \\large c_{32}=\\left(-1\\right)^{3+2}=-1\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;C=%5Cleft%5B%5Cbegin%7Bmatrix%7D1&amp;-1%5C%5C-1&amp;1%5C%5C1&amp;-1%5C%5C%5Cend%7Bmatrix%7D%5Cright%5D\" alt=\"\\dpi{100} \\large C=\\left[\\begin{matrix}1&amp;-1\\\\-1&amp;1\\\\1&amp;-1\\\\\\end{matrix}\\right]\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;A%5Ccdot&amp;space;C=%5Cleft%5B%5Cbegin%7Bmatrix%7D1&amp;1&amp;1%5C%5C1&amp;2&amp;3%5C%5C%5Cend%7Bmatrix%7D%5Cright%5D%5Ccdot%5Cleft%5B%5Cbegin%7Bmatrix%7D1&amp;-1%5C%5C-1&amp;1%5C%5C1&amp;-1%5C%5C%5Cend%7Bmatrix%7D%5Cright%5D\" alt=\"\\dpi{100} \\large A\\cdot C=\\left[\\begin{matrix}1&amp;1&amp;1\\\\1&amp;2&amp;3\\\\\\end{matrix}\\right]\\cdot\\left[\\begin{matrix}1&amp;-1\\\\-1&amp;1\\\\1&amp;-1\\\\\\end{matrix}\\right]\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;A%5Ccdot&amp;space;C=%5Cleft%5B%5Cbegin%7Bmatrix%7D1-1+1&amp;-1+1-1%5C%5C1-2+3&amp;-1+2-3%5C%5C%5Cend%7Bmatrix%7D%5Cright%5D\" alt=\"\\dpi{100} \\large A\\cdot C=\\left[\\begin{matrix}1-1+1&amp;-1+1-1\\\\1-2+3&amp;-1+2-3\\\\\\end{matrix}\\right]\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;A%5Ccdot&amp;space;C=%5Cleft%5B%5Cbegin%7Bmatrix%7D1&amp;-1%5C%5C2&amp;-2%5C%5C%5Cend%7Bmatrix%7D%5Cright%5D\" alt=\"\\dpi{100} \\large A\\cdot C=\\left[\\begin{matrix}1&amp;-1\\\\2&amp;-2\\\\\\end{matrix}\\right]\" align=\"absmiddle\"><\/p><p><\/p><p>b)<\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;A%5Cleft%5B%5Cbegin%7Bmatrix%7Dx%5C%5Cy%5C%5Cz%5C%5C%5Cend%7Bmatrix%7D%5Cright%5D=%5Cleft%5B%5Cbegin%7Bmatrix%7D6%5C%5C6%5C%5C%5Cend%7Bmatrix%7D%5Cright%5D\" alt=\"\\dpi{100} \\large A\\left[\\begin{matrix}x\\\\y\\\\z\\\\\\end{matrix}\\right]=\\left[\\begin{matrix}6\\\\6\\\\\\end{matrix}\\right]\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;%5Cleft%5B%5Cbegin%7Bmatrix%7D1&amp;1&amp;1%5C%5C1&amp;2&amp;3%5C%5C%5Cend%7Bmatrix%7D%5Cright%5D%5Cleft%5B%5Cbegin%7Bmatrix%7Dx%5C%5Cy%5C%5Cz%5C%5C%5Cend%7Bmatrix%7D%5Cright%5D=%5Cleft%5B%5Cbegin%7Bmatrix%7D6%5C%5C6%5C%5C%5Cend%7Bmatrix%7D%5Cright%5D\" alt=\"\\dpi{100} \\large \\left[\\begin{matrix}1&amp;1&amp;1\\\\1&amp;2&amp;3\\\\\\end{matrix}\\right]\\left[\\begin{matrix}x\\\\y\\\\z\\\\\\end{matrix}\\right]=\\left[\\begin{matrix}6\\\\6\\\\\\end{matrix}\\right]\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;%5Cleft%5C%7B%5Cbegin%7Bmatrix%7Dx+y+z=6%5C%5C%5C&amp;space;%5C%5Cx+2y+3z=6%5C%5C%5Cend%7Bmatrix%7D%5Cright.\" alt=\"\\dpi{100} \\large \\left\\{\\begin{matrix}x+y+z=6\\\\\\ \\\\x+2y+3z=6\\\\\\end{matrix}\\right.\" align=\"absmiddle\"><\/p><p>Como <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;%5Cleft(x,y,z%5Cright)\" alt=\"\\dpi{100} \\large \\left(x,y,z\\right)\" align=\"absmiddle\"> &eacute; uma progress&atilde;o aritm&eacute;tica, temos:<\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;%5Cfrac%7Bx+z%7D%7B2%7D=y\" alt=\"\\dpi{100} \\large \\frac{x+z}{2}=y\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;x+z=2y\" alt=\"\\dpi{100} \\large x+z=2y\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;x-2y+z=0\" alt=\"\\dpi{100} \\large x-2y+z=0\" align=\"absmiddle\"><\/p><p>Assim, nosso sistema passa a ser:<\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;%5Cleft%5C%7B%5Cbegin%7Bmatrix%7Dx+y+z=6%5C%5C%5C&amp;space;%5C%5Cx+2y+3z=6%5C%5C%5C&amp;space;%5C%5Cx-2y+z=0%5C%5C%5Cend%7Bmatrix%7D%5Cright.\" alt=\"\\dpi{100} \\large \\left\\{\\begin{matrix}x+y+z=6\\\\\\ \\\\x+2y+3z=6\\\\\\ \\\\x-2y+z=0\\\\\\end{matrix}\\right.\" align=\"absmiddle\"><\/p><p>Escrevendo na forma matricial estendida:<\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;%5Cleft%5B%5Cbegin%7Bmatrix%7D1&amp;1&amp;1&amp;6%5C%5C1&amp;2&amp;3&amp;6%5C%5C1&amp;-2&amp;1&amp;0%5C%5C%5Cend%7Bmatrix%7D%5Cright%5D%5Cbegin%7Bmatrix%7D%5Ccdot%5Cleft(-1%5Cright)&amp;%5Ccdot%5Cleft(-1%5Cright)%5C%5C+&amp;%5C&amp;space;%5C%5C%5C&amp;space;&amp;+%5C%5C%5Cend%7Bmatrix%7D\" alt=\"\\dpi{100} \\large \\left[\\begin{matrix}1&amp;1&amp;1&amp;6\\\\1&amp;2&amp;3&amp;6\\\\1&amp;-2&amp;1&amp;0\\\\\\end{matrix}\\right]\\begin{matrix}\\cdot\\left(-1\\right)&amp;\\cdot\\left(-1\\right)\\\\+&amp;\\ \\\\\\ &amp;+\\\\\\end{matrix}\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;%5Cleft%5B%5Cbegin%7Bmatrix%7D1&amp;1&amp;1&amp;6%5C%5C0&amp;1&amp;2&amp;0%5C%5C0&amp;-3&amp;0&amp;-6%5C%5C%5Cend%7Bmatrix%7D%5Cright%5D\" alt=\"\\dpi{100} \\large \\left[\\begin{matrix}1&amp;1&amp;1&amp;6\\\\0&amp;1&amp;2&amp;0\\\\0&amp;-3&amp;0&amp;-6\\\\\\end{matrix}\\right]\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;%5Cleft%5C%7B%5Cbegin%7Bmatrix%7Dx+y+z=6%5C%5C%5C&amp;space;%5C%5Cx+2y+3z=6%5C%5C%5C&amp;space;%5C%5Cx-2y+z=0%5C%5C%5Cend%7Bmatrix%7D%5Cright.%5Crightarrow%5Cleft%5C%7B%5Cbegin%7Bmatrix%7Dx+y+z=6%5C%5C%5C&amp;space;%5C%5Cy+2z=0%5C%5C%5C&amp;space;%5C%5C-3y=-6%5C%5C%5Cend%7Bmatrix%7D%5Cright.%5Crightarrow\" alt=\"\\dpi{100} \\large \\left\\{\\begin{matrix}x+y+z=6\\\\\\ \\\\x+2y+3z=6\\\\\\ \\\\x-2y+z=0\\\\\\end{matrix}\\right.\\rightarrow\\left\\{\\begin{matrix}x+y+z=6\\\\\\ \\\\y+2z=0\\\\\\ \\\\-3y=-6\\\\\\end{matrix}\\right.\\rightarrow\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;%5Crightarrow%5Cleft%5C%7B%5Cbegin%7Bmatrix%7Dx+y+z=6%5C%5C%5C&amp;space;%5C%5Cy+2z=0%5C%5C%5C&amp;space;%5C%5Cy=2%5C%5C%5Cend%7Bmatrix%7D%5Cright.%5Crightarrow%5Cleft%5C%7B%5Cbegin%7Bmatrix%7Dx+y+z=6%5C%5C%5C&amp;space;%5C%5Cz=-1%5C%5C%5C&amp;space;%5C%5Cy=2%5C%5C%5Cend%7Bmatrix%7D%5Cright.%5Crightarrow%5Cleft%5C%7B%5Cbegin%7Bmatrix%7Dx=5%5C%5C%5C&amp;space;%5C%5Cz=-1%5C%5C%5C&amp;space;%5C%5Cy=2%5C%5C%5Cend%7Bmatrix%7D%5Cright.\" alt=\"\\dpi{100} \\large \\rightarrow\\left\\{\\begin{matrix}x+y+z=6\\\\\\ \\\\y+2z=0\\\\\\ \\\\y=2\\\\\\end{matrix}\\right.\\rightarrow\\left\\{\\begin{matrix}x+y+z=6\\\\\\ \\\\z=-1\\\\\\ \\\\y=2\\\\\\end{matrix}\\right.\\rightarrow\\left\\{\\begin{matrix}x=5\\\\\\ \\\\z=-1\\\\\\ \\\\y=2\\\\\\end{matrix}\\right.\" align=\"absmiddle\"><\/p><p>Assim, a solu&ccedil;&atilde;o para o sistema solicitado &eacute;<\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;%5Cleft%5B%5Cbegin%7Bmatrix%7Dx%5C%5Cy%5C%5Cz%5C%5C%5Cend%7Bmatrix%7D%5Cright%5D=%5Cleft%5B%5Cbegin%7Bmatrix%7D5%5C%5C2%5C%5C-1%5C%5C%5Cend%7Bmatrix%7D%5Cright%5D\" alt=\"\\dpi{100} \\large \\left[\\begin{matrix}x\\\\y\\\\z\\\\\\end{matrix}\\right]=\\left[\\begin{matrix}5\\\\2\\\\-1\\\\\\end{matrix}\\right]\" align=\"absmiddle\"><\/p><h2 class=\"wp-block-heading\" id=\"h-questao-08\"><span class=\"ez-toc-section\" id=\"Questao-08\"><\/span>Quest&atilde;o 08<span class=\"ez-toc-section-end\"><\/span><\/h2><p>A figura abaixo exibe, no plano cartesiano, o gr&aacute;fico de <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;y=%5Csqrt&amp;space;x\" alt=\"\\dpi{100} \\large y=\\sqrt x\" align=\"absmiddle\"> para <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;x%5Cgeq0\" alt=\"\\dpi{100} \\large x\\geq0\" align=\"absmiddle\">, em que os pontos A e B t&ecirc;m abscissas <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;x_A=a&gt;0%E2%80%B3%20alt=%E2%80%9D%5Cdpi%7B100%7D%20%5Clarge%20x_A=a&gt;0%E2%80%B3%20align=%E2%80%9Dabsmiddle%E2%80%9D&gt;%20e%20&lt;img%20src=\" https:>0&Prime; alt=&rdquo;\\dpi{100} \\large x_B=b&gt;0&Prime; align=&rdquo;absmiddle&rdquo;&gt;, e O &eacute; a origem do sistema de coordenadas.<\/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\/matem%C3%A1tica-unicamo-2020-2-fase-questC3A3o-08.jpg\" alt=\"\" class=\"wp-image-37758\"><\/figure><\/div><p>a) Prove que os pontos A, B e <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;C=%5Cleft(-%5Csqrt%7Bab%7D,0%5Cright)\" alt=\"\\dpi{100} \\large C=\\left(-\\sqrt{ab},0\\right)\" align=\"absmiddle\"> s&atilde;o colineares.<\/p><p>b) Para b=3, determine o valor de a para o qual a dist&acirc;ncia da origem ao ponto A &eacute; igual &agrave; dist&acirc;ncia do ponto A ao ponto B.<\/p><h3 class=\"wp-block-heading\" id=\"h-resolucao-2\">Resolu&ccedil;&atilde;o<\/h3><p>a) Os pontos em quest&atilde;o s&atilde;o<\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;A=%5Cleft(a,%5Csqrt&amp;space;a%5Cright)\" alt=\"\\dpi{100} \\large A=\\left(a,\\sqrt a\\right)\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;B=%5Cleft(b,%5Csqrt&amp;space;b%5Cright)\" alt=\"\\dpi{100} \\large B=\\left(b,\\sqrt b\\right)\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;C=%5Cleft(-%5Csqrt%7Bab%7D,0%5Cright)\" alt=\"\\dpi{100} \\large C=\\left(-\\sqrt{ab},0\\right)\" align=\"absmiddle\"><\/p><p>Para serem colineares, a seguinte equa&ccedil;&atilde;o deve ser verdadeira:<\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;%5Cleft%7C%5Cbegin%7Bmatrix%7Da&amp;%5Csqrt&amp;space;a&amp;1%5C%5Cb&amp;%5Csqrt&amp;space;b&amp;1%5C%5C-%5Csqrt%7Bab%7D&amp;0&amp;1%5C%5C%5Cend%7Bmatrix%7D%5Cright%7C=0\" alt=\"\\dpi{100} \\large \\left|\\begin{matrix}a&amp;\\sqrt a&amp;1\\\\b&amp;\\sqrt b&amp;1\\\\-\\sqrt{ab}&amp;0&amp;1\\\\\\end{matrix}\\right|=0\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;a%5Csqrt&amp;space;b+%5Csqrt&amp;space;a%5Ccdot%5Cleft(-%5Csqrt%7Bab%7D%5Cright)-%5Cleft(-%5Csqrt%7Bab%7D%5Cright)%5Ccdot%5Csqrt&amp;space;b-b%5Csqrt&amp;space;a=0\" alt=\"\\dpi{100} \\large a\\sqrt b+\\sqrt a\\cdot\\left(-\\sqrt{ab}\\right)-\\left(-\\sqrt{ab}\\right)\\cdot\\sqrt b-b\\sqrt a=0\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;a%5Csqrt&amp;space;b-a%5Csqrt&amp;space;b+b%5Csqrt&amp;space;a-b%5Csqrt&amp;space;a=0\" alt=\"\\dpi{100} \\large a\\sqrt b-a\\sqrt b+b\\sqrt a-b\\sqrt a=0\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;0=0\" alt=\"\\dpi{100} \\large 0=0\" align=\"absmiddle\"><\/p><p>Assim, os tr&ecirc;s pontos, A, B e C est&atilde;o alinhados.<\/p><p><\/p><p>b )Para b=3, temos os pontos:<\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;O=%5Cleft(0,0%5Cright)\" alt=\"\\dpi{100} \\large O=\\left(0,0\\right)\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;A=%5Cleft(a,%5Csqrt&amp;space;a%5Cright)\" alt=\"\\dpi{100} \\large A=\\left(a,\\sqrt a\\right)\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;B=%5Cleft(3,%5Csqrt3%5Cright)\" alt=\"\\dpi{100} \\large B=\\left(3,\\sqrt3\\right)\" align=\"absmiddle\"><\/p><p>Pelo enunciado, devemos ter:<\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;d_%7BOA%7D=d_%7BAB%7D\" alt=\"\\dpi{100} \\large d_{OA}=d_{AB}\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;%5Csqrt%7B%5Cleft(a-0%5Cright)%5E2+%5Cleft(%5Csqrt&amp;space;a-0%5Cright)%5E2%7D=%5Csqrt%7B%5Cleft(3-a%5Cright)%5E2+%5Cleft(%5Csqrt3-%5Csqrt&amp;space;a%5Cright)%5E2%7D\" alt=\"\\dpi{100} \\large \\sqrt{\\left(a-0\\right)^2+\\left(\\sqrt a-0\\right)^2}=\\sqrt{\\left(3-a\\right)^2+\\left(\\sqrt3-\\sqrt a\\right)^2}\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;%5Cleft(a-0%5Cright)%5E2+%5Cleft(%5Csqrt&amp;space;a-0%5Cright)%5E2=%5Cleft(3-a%5Cright)%5E2+%5Cleft(%5Csqrt3-%5Csqrt&amp;space;a%5Cright)%5E2\" alt=\"\\dpi{100} \\large \\left(a-0\\right)^2+\\left(\\sqrt a-0\\right)^2=\\left(3-a\\right)^2+\\left(\\sqrt3-\\sqrt a\\right)^2\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;a%5E2+a=9-6a+a%5E2+3-2%5Ccdot%5Csqrt3%5Ccdot%5Csqrt&amp;space;a+a\" alt=\"\\dpi{100} \\large a^2+a=9-6a+a^2+3-2\\cdot\\sqrt3\\cdot\\sqrt a+a\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;0=12-6a-2%5Ccdot%5Csqrt3%5Ccdot%5Csqrt&amp;space;a\" alt=\"\\dpi{100} \\large 0=12-6a-2\\cdot\\sqrt3\\cdot\\sqrt a\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;0=-6a-2%5Ccdot%5Csqrt3%5Ccdot%5Csqrt&amp;space;a+12\" alt=\"\\dpi{100} \\large 0=-6a-2\\cdot\\sqrt3\\cdot\\sqrt a+12\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;%5Csqrt&amp;space;a=x\" alt=\"\\dpi{100} \\large \\sqrt a=x\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;0=-6x%5E2-2%5Ccdot%5Csqrt3%5Ccdot&amp;space;x+12\" alt=\"\\dpi{100} \\large 0=-6x^2-2\\cdot\\sqrt3\\cdot x+12\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;%5CDelta&amp;space;=%5Cleft(-2%5Ccdot%5Csqrt3%5Cright)%5E2-4%5Ccdot%5Cleft(-6%5Cright)%5Ccdot12=12+288=300\" alt=\"\\dpi{100} \\large \\Delta =\\left(-2\\cdot\\sqrt3\\right)^2-4\\cdot\\left(-6\\right)\\cdot12=12+288=300\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;x=%5Cfrac%7B-%5Cleft(-2%5Ccdot%5Csqrt3%5Cright)%5Cpm%5Csqrt%7B300%7D%7D%7B2%5Ccdot%5Cleft(-6%5Cright)%7D%5Cleft%5C%7B%5Cbegin%7Bmatrix%7Dx%5E%5Cprime=%5Cfrac%7B2%5Csqrt3+10%5Csqrt3%7D%7B-12%7D=%5Cfrac%7B12%5Csqrt3%7D%7B-12%7D=-%5Csqrt3%5C%5C%5C&amp;space;%5C%5Cx%5E%7B%5Cprime%5Cprime%7D=%5Cfrac%7B2%5Csqrt3-10%5Csqrt3%7D%7B-12%7D=%5Cfrac%7B-8%5Csqrt3%7D%7B-12%7D=%5Cfrac%7B2%5Csqrt3%7D%7B3%7D%5C%5C%5Cend%7Bmatrix%7D%5Cright.\" alt=\"\\dpi{100} \\large x=\\frac{-\\left(-2\\cdot\\sqrt3\\right)\\pm\\sqrt{300}}{2\\cdot\\left(-6\\right)}\\left\\{\\begin{matrix}x^\\prime=\\frac{2\\sqrt3+10\\sqrt3}{-12}=\\frac{12\\sqrt3}{-12}=-\\sqrt3\\\\\\ \\\\x^{\\prime\\prime}=\\frac{2\\sqrt3-10\\sqrt3}{-12}=\\frac{-8\\sqrt3}{-12}=\\frac{2\\sqrt3}{3}\\\\\\end{matrix}\\right.\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;%5Csqrt&amp;space;a=x\" alt=\"\\dpi{100} \\large \\sqrt a=x\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;%5Csqrt&amp;space;a=%5Cfrac%7B2%5Csqrt3%7D%7B3%7D\" alt=\"\\dpi{100} \\large \\sqrt a=\\frac{2\\sqrt3}{3}\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;a=%5Cleft(%5Cfrac%7B2%5Csqrt3%7D%7B3%7D%5Cright)%5E2\" alt=\"\\dpi{100} \\large a=\\left(\\frac{2\\sqrt3}{3}\\right)^2\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;a=%5Cfrac%7B4%5Ccdot3%7D%7B9%7D\" alt=\"\\dpi{100} \\large a=\\frac{4\\cdot3}{9}\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;a=%5Cfrac%7B4%7D%7B3%7D\" alt=\"\\dpi{100} \\large a=\\frac{4}{3}\" align=\"absmiddle\"><\/p><h2 class=\"wp-block-heading\" id=\"h-questao-09\"><span class=\"ez-toc-section\" id=\"Questao-09\"><\/span>Quest&atilde;o 09<span class=\"ez-toc-section-end\"><\/span><\/h2><p>Seja a fun&ccedil;&atilde;o <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;f%5Cleft(x%5Cright)=%5Cfrac%7B2+senx%7D%7B2+cosx%7D\" alt=\"\\dpi{100} \\large f\\left(x\\right)=\\frac{2+senx}{2+cosx}\" align=\"absmiddle\">, definida para todo n&uacute;mero real x.<\/p><p>a) Mostre que <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;f%5Cleft(%5Cfrac%7B%5Cpi%7D%7B2%7D%5Cright)+f%5Cleft(-%5Cfrac%7B%5Cpi%7D%7B2%7D%5Cright)=f%5Cleft(%5Cpi%5Cright)%5Ccdot&amp;space;f%5Cleft(%5Cfrac%7B%5Cpi%7D%7B4%7D%5Cright)\" alt=\"\\dpi{100} \\large f\\left(\\frac{\\pi}{2}\\right)+f\\left(-\\frac{\\pi}{2}\\right)=f\\left(\\pi\\right)\\cdot f\\left(\\frac{\\pi}{4}\\right)\" align=\"absmiddle\">.<\/p><p>b) Seja <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;%5Ctheta\" alt=\"\\dpi{100} \\large \\theta\" align=\"absmiddle\"> um n&uacute;mero real tal que <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;f%5Cleft(%5Ctheta%5Cright)=2\" alt=\"\\dpi{100} \\large f\\left(\\theta\\right)=2\" align=\"absmiddle\">. Determine os poss&iacute;veis valores para <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;sen%5C&amp;space;%5Ctheta\" alt=\"\\dpi{100} \\large sen\\ \\theta\" align=\"absmiddle\">.<\/p><h3>Resolu&ccedil;&atilde;o<\/h3><p>a) <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;f%5Cleft(%5Cfrac%7B%5Cpi%7D%7B2%7D%5Cright)+f%5Cleft(-%5Cfrac%7B%5Cpi%7D%7B2%7D%5Cright)=f%5Cleft(%5Cpi%5Cright)%5Ccdot&amp;space;f%5Cleft(%5Cfrac%7B%5Cpi%7D%7B4%7D%5Cright)\" alt=\"\\dpi{100} \\large f\\left(\\frac{\\pi}{2}\\right)+f\\left(-\\frac{\\pi}{2}\\right)=f\\left(\\pi\\right)\\cdot f\\left(\\frac{\\pi}{4}\\right)\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;%5Cfrac%7B2+sen%5Cleft(%5Cfrac%7B%5Cpi%7D%7B2%7D%5Cright)%7D%7B2+cos%5Cleft(%5Cfrac%7B%5Cpi%7D%7B2%7D%5Cright)%7D+%5Cfrac%7B2+sen%5Cleft(-%5Cfrac%7B%5Cpi%7D%7B2%7D%5Cright)%7D%7B2+cos%5Cleft(-%5Cfrac%7B%5Cpi%7D%7B2%7D%5Cright)%7D=%5Cfrac%7B2+sen%5Cleft(%5Cpi%5Cright)%7D%7B2+cos%5Cleft(%5Cpi%5Cright)%7D%5Ccdot%5Cfrac%7B2+sen%5Cleft(%5Cfrac%7B%5Cpi%7D%7B4%7D%5Cright)%7D%7B2+cos%5Cleft(%5Cfrac%7B%5Cpi%7D%7B4%7D%5Cright)%7D\" alt=\"\\dpi{100} \\large \\frac{2+sen\\left(\\frac{\\pi}{2}\\right)}{2+cos\\left(\\frac{\\pi}{2}\\right)}+\\frac{2+sen\\left(-\\frac{\\pi}{2}\\right)}{2+cos\\left(-\\frac{\\pi}{2}\\right)}=\\frac{2+sen\\left(\\pi\\right)}{2+cos\\left(\\pi\\right)}\\cdot\\frac{2+sen\\left(\\frac{\\pi}{4}\\right)}{2+cos\\left(\\frac{\\pi}{4}\\right)}\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;%5Cfrac%7B2+1%7D%7B2+0%7D+%5Cfrac%7B2+%5Cleft(-1%5Cright)%7D%7B2+0%7D=%5Cfrac%7B2+0%7D%7B2+%5Cleft(-1%5Cright)%7D%5Ccdot%5Cfrac%7B2+%5Cfrac%7B%5Csqrt2%7D%7B2%7D%7D%7B2+%5Cfrac%7B%5Csqrt2%7D%7B2%7D%7D\" alt=\"\\dpi{100} \\large \\frac{2+1}{2+0}+\\frac{2+\\left(-1\\right)}{2+0}=\\frac{2+0}{2+\\left(-1\\right)}\\cdot\\frac{2+\\frac{\\sqrt2}{2}}{2+\\frac{\\sqrt2}{2}}\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;%5Cfrac%7B3%7D%7B2%7D+%5Cfrac%7B1%7D%7B2%7D=%5Cfrac%7B2%7D%7B1%7D%5Ccdot1\" alt=\"\\dpi{100} \\large \\frac{3}{2}+\\frac{1}{2}=\\frac{2}{1}\\cdot1\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;%5Cfrac%7B4%7D%7B2%7D=2\" alt=\"\\dpi{100} \\large \\frac{4}{2}=2\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;2=2\" alt=\"\\dpi{100} \\large 2=2\" align=\"absmiddle\"><\/p><p>Assim, pode-se afirmar que<\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;f%5Cleft(%5Cfrac%7B%5Cpi%7D%7B2%7D%5Cright)+f%5Cleft(-%5Cfrac%7B%5Cpi%7D%7B2%7D%5Cright)=f%5Cleft(%5Cpi%5Cright)%5Ccdot&amp;space;f%5Cleft(%5Cfrac%7B%5Cpi%7D%7B4%7D%5Cright)\" alt=\"\\dpi{100} \\large f\\left(\\frac{\\pi}{2}\\right)+f\\left(-\\frac{\\pi}{2}\\right)=f\\left(\\pi\\right)\\cdot f\\left(\\frac{\\pi}{4}\\right)\" align=\"absmiddle\"><\/p><p><\/p><p>b) <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;f%5Cleft(%5Ctheta%5Cright)=%5Cfrac%7B2+sen%5Ctheta%7D%7B2+cos%5Ctheta%7D=2\" alt=\"\\dpi{100} \\large f\\left(\\theta\\right)=\\frac{2+sen\\theta}{2+cos\\theta}=2\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;2+sen%5Ctheta=4+2%5Ccdot&amp;space;cos%5Ctheta\" alt=\"\\dpi{100} \\large 2+sen\\theta=4+2\\cdot cos\\theta\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;sen%5Ctheta-2=2%5Ccdot&amp;space;cos%5Ctheta\" alt=\"\\dpi{100} \\large sen\\theta-2=2\\cdot cos\\theta\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;sen%5E2%5Ctheta-4sen%5Ctheta+4=4%5Ccdot%5Ccos%5E2%7B%5Ctheta%7D\" alt=\"\\dpi{100} \\large sen^2\\theta-4sen\\theta+4=4\\cdot\\cos^2{\\theta}\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;sen%5E2%5Ctheta-4sen%5Ctheta+4=4%5Ccdot%5Cleft(1-sen%5E2%5Ctheta%5Cright)\" alt=\"\\dpi{100} \\large sen^2\\theta-4sen\\theta+4=4\\cdot\\left(1-sen^2\\theta\\right)\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;sen%5E2%5Ctheta-4sen%5Ctheta+4=4-4%5Ccdot&amp;space;sen%5E2%5Ctheta\" alt=\"\\dpi{100} \\large sen^2\\theta-4sen\\theta+4=4-4\\cdot sen^2\\theta\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;5%5Ccdot&amp;space;sen%5E2%5Ctheta-4sen%5Ctheta=0\" alt=\"\\dpi{100} \\large 5\\cdot sen^2\\theta-4sen\\theta=0\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;sen%5Ctheta%5Ccdot%5Cleft(5%5Ccdot&amp;space;s&amp;space;e&amp;space;n%5Ctheta-4%5Cright)=0\" alt=\"\\dpi{100} \\large sen\\theta\\cdot\\left(5\\cdot s e n\\theta-4\\right)=0\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;sen%5Ctheta=0%5C&amp;space;%5C&amp;space;%5C&amp;space;%5C&amp;space;%5C&amp;space;ou%5C&amp;space;%5C&amp;space;%5C&amp;space;%5C&amp;space;%5C&amp;space;%5C&amp;space;5%5Ccdot&amp;space;sen%5Ctheta-4=0\" alt=\"\\dpi{100} \\large sen\\theta=0\\ \\ \\ \\ \\ ou\\ \\ \\ \\ \\ \\ 5\\cdot sen\\theta-4=0\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;sen%5Ctheta=0%5C&amp;space;%5C&amp;space;%5C&amp;space;%5C&amp;space;%5C&amp;space;ou%5C&amp;space;%5C&amp;space;%5C&amp;space;%5C&amp;space;%5C&amp;space;%5C&amp;space;sen%5Ctheta=%5Cfrac%7B4%7D%7B5%7D\" alt=\"\\dpi{100} \\large sen\\theta=0\\ \\ \\ \\ \\ ou\\ \\ \\ \\ \\ \\ sen\\theta=\\frac{4}{5}\" align=\"absmiddle\"><\/p><h2 class=\"wp-block-heading\" id=\"h-questao-10\"><span class=\"ez-toc-section\" id=\"Questao-10\"><\/span>Quest&atilde;o 10<span class=\"ez-toc-section-end\"><\/span><\/h2><p>A figura abaixo exibe a planifica&ccedil;&atilde;o de um poliedro convexo, com faces triangulares congruentes e faces retangulares, em que s&atilde;o indicados os comprimentos a, b e 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\/matem%C3%A1tica-unicamo-2020-2-fase-questC3A3o-10.jpg\" alt=\"\" class=\"wp-image-37763\"><\/figure><\/div><p>a) Determine o n&uacute;mero de v&eacute;rtices e de arestas desse poliedro.<\/p><p>b) Para a = 13 cm, b = 1 cm e c = 10 cm, calcule o volume desse poliedro.<\/p><h3 class=\"wp-block-heading\" id=\"h-resolucao-3\">Resolu&ccedil;&atilde;o<\/h3><p>O poliedro tem o seguinte formato:<\/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\/matem%C3%A1tica-unicamo-2020-2-fase-questC3A3o-10-gabarito.jpg\" alt=\"\" class=\"wp-image-37765\"><\/figure><\/div><p>Assim, temos 16 arestas e 9 v&eacute;rtices.<\/p><p>b) Calculando a altura H da pir&acirc;mide de base quadrada da parte superior do poliedro.<\/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\/matem%C3%A1tica-unicamo-2020-2-fase-questC3A3o-10-gabarito-letra-b.jpg\" alt=\"\" class=\"wp-image-37766\"><\/figure><\/div><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;a%5E2=H%5E2+%5Cleft(%5Cfrac%7Bc%7D%7B2%7D%5Cright)%5E2\" alt=\"\\dpi{100} \\large a^2=H^2+\\left(\\frac{c}{2}\\right)^2\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;%7B13%7D%5E2=H%5E2+%5Cleft(%5Cfrac%7B10%7D%7B2%7D%5Cright)%5E2\" alt=\"\\dpi{100} \\large {13}^2=H^2+\\left(\\frac{10}{2}\\right)^2\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;169-25=H%5E2\" alt=\"\\dpi{100} \\large 169-25=H^2\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;144=H%5E2\" alt=\"\\dpi{100} \\large 144=H^2\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;12=H\" alt=\"\\dpi{100} \\large 12=H\" align=\"absmiddle\"><\/p><p>Assim, o volume V do poliedro &eacute; dado por:<\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;V=c%5E2%5Ccdot&amp;space;b+%5Cfrac%7B1%7D%7B3%7D%5Ccdot&amp;space;c%5E2%5Ccdot&amp;space;H\" alt=\"\\dpi{100} \\large V=c^2\\cdot b+\\frac{1}{3}\\cdot c^2\\cdot H\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;V=%7B10%7D%5E2%5Ccdot16+%5Cfrac%7B1%7D%7B3%7D%5Ccdot%7B10%7D%5E2%5Ccdot12\" alt=\"\\dpi{100} \\large V={10}^2\\cdot16+\\frac{1}{3}\\cdot{10}^2\\cdot12\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;V=1600+400\" alt=\"\\dpi{100} \\large V=1600+400\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Cdpi%7B100%7D&amp;space;%5Clarge&amp;space;V=2000%5C&amp;space;cm%5E3\" alt=\"\\dpi{100} \\large V=2000\\ cm^3\" 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 UNICAMP 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;Prova de Matem&aacute;tica da Unicamp &ndash; 2&ordf; fase\" 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>\n\n\n\n\n\n<a id=\"cta\" class=\"cta-imagem\" href=\"https:\/\/vestibulares.estrategia.com\/curso\/pacote-para-unicamp\/\" target=\"blank\">\n                <img decoding=\"async\" width=\"100%\" src=\"https:\/\/vestibulares.estrategia.com\/portal\/wp-content\/uploads\/2021\/12\/Unicamp.jpg\" alt=\"CTA Curso Vestibular Unicamp\" title=\"CTA Curso Vestibular Unicamp\">\n        <\/a>\n\n\n\n<\/p><h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Veja-tambem\"><\/span>Veja tamb&eacute;m:<span class=\"ez-toc-section-end\"><\/span><\/h2><ul class=\"wp-block-list\"><li><a href=\"https:\/\/vestibulares.estrategia.com\/portal\/matematica\/geometria\/\">Geometria: o que &eacute; e como &eacute; classificada<\/a>&nbsp;<\/li><li><a href=\"https:\/\/vestibulares.estrategia.com\/portal\/matematica\/figuras-geometricas\/\">Figuras Geom&eacute;tricas: planas, espaciais e como aparece nos vestibulares<\/a>&nbsp;<\/li><li><a href=\"https:\/\/vestibulares.estrategia.com\/portal\/matematica\/geometria-plana-no-enem\/\">Geometria Plana no Enem: conceito e defini&ccedil;&otilde;es<\/a><\/li><li><a href=\"https:\/\/vestibulares.estrategia.com\/portal\/matematica\/semelhanca-de-triangulos\/\">Semelhan&ccedil;a de tri&acirc;ngulos<\/a><\/li><li><a href=\"https:\/\/vestibulares.estrategia.com\/portal\/matematica\/circulo-e-circunferencia\/\">C&iacute;rculo e circunfer&ecirc;ncia: conceitos e exerc&iacute;cios<\/a><\/li><li><a href=\"https:\/\/vestibulares.estrategia.com\/portal\/matematica\/geometria-analitica\/\">Geometria anal&iacute;tica: f&oacute;rmulas, gr&aacute;ficos e equa&ccedil;&otilde;es<\/a>&nbsp;<\/li><li><a href=\"https:\/\/vestibulares.estrategia.com\/portal\/matematica\/geometria-espacial\/\">Geometria Espacial: conhe&ccedil;a os s&oacute;lidos e f&oacute;rmulas<\/a>&nbsp;<\/li><li><a href=\"https:\/\/vestibulares.estrategia.com\/portal\/matematica\/teorema-de-pitot\/\">Teorema de Pitot: o que diz o teorema e como &eacute; cobrado em prova<\/a><\/li><li><a href=\"https:\/\/vestibulares.estrategia.com\/portal\/matematica\/area-do-trapezio\/\">&Aacute;rea do trap&eacute;zio: conhe&ccedil;a a f&oacute;rmula e como calcular!<\/a>&nbsp;<\/li><li><a href=\"https:\/\/vestibulares.estrategia.com\/portal\/matematica\/prova-matematica-fuvest-2020\/\">Prova de matem&aacute;tica Fuvest 2020 com Resolu&ccedil;&atilde;o Comentada<\/a><\/li><li><a href=\"https:\/\/vestibulares.estrategia.com\/portal\/matematica\/prova-matematica-unesp-2020\/\">Prova de Matem&aacute;tica UNESP 2020 com Resolu&ccedil;&atilde;o Comentada<\/a><\/li><\/ul><\/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":37745,"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-37742","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 Unicamp 2020<\/title>\n<meta name=\"description\" content=\"Voc\u00ea participou do Vestibular Unicamp 2020? 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