{"id":18444,"date":"2020-10-19T19:44:01","date_gmt":"2020-10-19T22:44:01","guid":{"rendered":"https:\/\/beduka.com\/blog\/?p=18444"},"modified":"2023-04-03T17:06:02","modified_gmt":"2023-04-03T20:06:02","slug":"exercicios-sobre-numeros-complexos","status":"publish","type":"post","link":"https:\/\/beduka.com\/blog\/exercicios\/matematica-exercicios\/exercicios-sobre-numeros-complexos\/","title":{"rendered":"Os 7 Melhores Exerc\u00edcios sobre N\u00fameros Complexos com Gabarito"},"content":{"rendered":"\n<blockquote class=\"wp-block-quote\"><p><strong><em>Os n\u00fameros complexos s\u00e3o representados de 3 formas: a forma geom\u00e9trica (representada no plano complexo conhecido como plano de Argand-Gauss, a forma trigonom\u00e9trica, conhecida como forma polar e a forma alg\u00e9brica (z = a + b), composta por uma parte real \u201ca\u201d e uma parte imagin\u00e1ria \u201cb\u201d. Leia nosso resumo e fa\u00e7a os 8 Melhores Exerc\u00edcios sobre N\u00fameros Complexos. <\/em><\/strong><\/p><\/blockquote>\n\n\n\n<p>Quando voc\u00ea terminar os <strong>8 Melhores Exerc\u00edcios sobre N\u00fameros Complexos<\/strong>, coloque em pr\u00e1tica todo seu conhecimento com <a href=\"https:\/\/beduka.com\/simulado-enem\/#\/\" target=\"_blank\" rel=\"noreferrer noopener\" aria-label=\" (abre numa nova aba)\">O Melhor Simulado Enem do Brasil.&nbsp;<\/a><\/p>\n\n\n\n<p><strong>Estudar n\u00fameros complexos n\u00e3o \u00e9 uma tarefa f\u00e1cil. <\/strong>Eles <strong>n\u00e3o<\/strong> s\u00e3o complexos apenas no nome\u2026 Calma!<\/p>\n\n\n\n<p><strong>N\u00f3s sabemos que voc\u00ea n\u00e3o tem tempo para enrola\u00e7\u00f5es, certo?<\/strong> Ent\u00e3o vamos direto ao ponto:<\/p>\n\n\n\n<p>Escrevemos um resumo que lhe passar\u00e1 <strong>uma mat\u00e9ria dif\u00edcil de maneira mais simples<\/strong> e, ao final, ainda lhe daremos a oportunidade de fazer <strong>Os 10 Melhores Exerc\u00edcios sobre N\u00fameros Complexos.<\/strong><\/p>\n\n\n\n<p>N\u00e3o \u00e9 um milagre, mas <strong>j\u00e1<\/strong> <strong>\u00e9 uma m\u00e3o na roda, n\u00e9?<\/strong><\/p>\n\n\n\n<p><strong>Ent\u00e3o se gostou da ideia,<\/strong> leia este artigo at\u00e9 o final.<\/p>\n\n\n\n<ul><li>Sabia que os n\u00fameros complexos \u00e9 um dos <a href=\"https:\/\/beduka.com\/blog\/dicas\/vestibulares\/assuntos-que-mais-caem-na-fuvest\/\" target=\"_blank\" rel=\"noreferrer noopener\" aria-label=\" (abre numa nova aba)\">assuntos que mais caem na prova da Fuvest?<\/a><\/li><\/ul>\n\n\n\n<h2 class=\"wp-block-heading\">O que \u00e9 um N\u00famero Complexo?<\/h2>\n\n\n\n<p>Os n\u00fameros complexos s\u00e3o representados de 3 formas: <strong>a forma geom\u00e9trica<\/strong> (representada no plano complexo conhecido como plano de Argand-Gauss), <strong>a forma trigonom\u00e9trica<\/strong>, conhecida como forma polar e <strong>a forma alg\u00e9brica <\/strong>(z = a + b), composta por uma <strong>parte real \u201ca\u201d<\/strong> e uma <strong>parte imagin\u00e1ria \u201cb\u201d.<\/strong><\/p>\n\n\n\n<p><strong>Os n\u00fameros complexos <\/strong>surgiram a partir da necessidade de resolver equa\u00e7\u00f5es que possuem ra\u00edzes de n\u00fameros negativos. Podendo ser <strong>expresso de diferentes formas.<\/strong><\/p>\n\n\n\n<p>Eles possuem opera\u00e7\u00f5es bem definidas:&nbsp;<\/p>\n\n\n\n<ul><li>subtra\u00e7\u00e3o;<\/li><li>multiplica\u00e7\u00e3o;&nbsp;<\/li><li>divis\u00e3o;<\/li><li>potencializa\u00e7\u00e3o.<\/li><\/ul>\n\n\n\n<h2 class=\"wp-block-heading\">Forma alg\u00e9brica de um n\u00famero complexo (z = a + b)<\/h2>\n\n\n\n<p>Antigamente, quando tent\u00e1vamos resolver uma opera\u00e7\u00e3o do segundo grau, como<strong> X\u00b2 = -16<\/strong>, consider\u00e1vamos ela sem solu\u00e7\u00e3o. <strong>Isso mudou\u2026<\/strong><\/p>\n\n\n\n<p>Com o surgimento dos n\u00fameros complexos e da possibilidade de represent\u00e1-los algebricamente,<strong> tornou-se poss\u00edvel a realiza\u00e7\u00e3o desse tipo de opera\u00e7\u00e3o.<\/strong><\/p>\n\n\n\n<p><strong>Para possibilitar<\/strong> a resolu\u00e7\u00e3o das situa\u00e7\u00f5es em que se trabalha com a raiz quadrada de um n\u00famero negativo, <strong>foi determinada a unidade imagin\u00e1ria:<\/strong><\/p>\n\n\n\n<p class=\"has-text-align-center\">i =<strong> <\/strong><strong>-1<\/strong><\/p>\n\n\n\n<p><strong>Ent\u00e3o<\/strong>, analisando- a equa\u00e7\u00e3o apresentada em <strong>x\u00b2 = -16<\/strong>, temos que:<\/p>\n\n\n\n<p><strong>X\u00b2 =<\/strong> -16<\/p>\n\n\n\n<p><strong>X =<\/strong> +\/- -16<\/p>\n\n\n\n<p><strong>X =<\/strong> +\/- 16 . (-1)<\/p>\n\n\n\n<p><strong>X =<\/strong> +\/- (16. -1)<\/p>\n\n\n\n<p><strong>X =<\/strong> +\/- 4i<\/p>\n\n\n\n<p>Sendo assim, as solu\u00e7\u00f5es para esta equa\u00e7\u00e3o s\u00e3o <strong>-4i<\/strong> e <strong>4i<\/strong><\/p>\n\n\n\n<ul><li>Relembre tudo sobre <a href=\"https:\/\/beduka.com\/blog\/materias\/matematica\/equacao-do-segundo-grau-2\/\" target=\"_blank\" rel=\"noreferrer noopener\" aria-label=\" (abre numa nova aba)\">Equa\u00e7\u00e3o do 2\u00b0 Grau<\/a><\/li><\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Estrutura de um N\u00famero Complexo&nbsp;<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" loading=\"lazy\" width=\"600\" height=\"300\" src=\"https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/10\/N\u00fameros-Complexos-imagem-1.jpg\" alt=\"Estrutura de um N\u00famero Complexo \" class=\"wp-image-18446\" srcset=\"https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/10\/N\u00fameros-Complexos-imagem-1.jpg 600w, https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/10\/N\u00fameros-Complexos-imagem-1-300x150.jpg 300w, https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/10\/N\u00fameros-Complexos-imagem-1-320x160.jpg 320w\" sizes=\"(max-width: 600px) 100vw, 600px\" \/><\/figure><\/div>\n\n\n<p>Sendo que:<\/p>\n\n\n\n<ul><li><strong><em>a <\/em><\/strong><strong>e<\/strong><strong><em> b<\/em><\/strong><em> <\/em>s\u00e3o n\u00fameros reais.<\/li><li><strong><em>a:<\/em><\/strong> parte real, indicada por <strong>a = Re(z);<\/strong><\/li><li><strong><em>b<\/em><\/strong><strong>:<\/strong> parte imagin\u00e1ria, indicada por <strong>Im(z);<\/strong><\/li><li><strong><em>i<\/em><\/strong><strong>:<\/strong> unidade imagin\u00e1ria.<\/li><\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Adi\u00e7\u00e3o entre N\u00fameros Complexos<\/h3>\n\n\n\n<p>Tendo em mente dois n\u00fameros complexos <strong>(T<sub>1 <\/sub>e T<sub>2<\/sub>)<\/strong>, faremos a adi\u00e7\u00e3o da parte imagin\u00e1ria e depois a adi\u00e7\u00e3o da parte real entre eles, <strong>observe:<\/strong><\/p>\n\n\n\n<p><strong>T<\/strong><strong><sub>1 <\/sub><\/strong>= a + bi<\/p>\n\n\n\n<p><strong>T<\/strong><strong><sub>2 <\/sub><\/strong>= c + di<\/p>\n\n\n\n<p><strong>T<\/strong><strong><sub>1 <\/sub><\/strong><strong>+<\/strong><strong><sub> <\/sub><\/strong><strong>T<\/strong><strong><sub>2 <\/sub><\/strong>= (a + c) + (b + d)i<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Subtra\u00e7\u00e3o entre N\u00fameros Complexos<\/h3>\n\n\n\n<p>Tendo em mente dois n\u00fameros complexos <strong>(T<sub>1 <\/sub>e T<sub>2<\/sub>)<\/strong>, faremos a subtra\u00e7\u00e3o da parte imagin\u00e1ria e depois a subtra\u00e7\u00e3o da parte real entre eles, <strong>observe:<\/strong><\/p>\n\n\n\n<p>Realiza\u00e7\u00e3o da subtra\u00e7\u00e3o de <strong>T<\/strong><strong><sub>1 <\/sub><\/strong><strong>e T<\/strong><strong><sub>2<\/sub><\/strong><strong>.<\/strong><\/p>\n\n\n\n<p><strong>T<\/strong><strong><sub>1 <\/sub><\/strong><strong>=<\/strong> 2 + 3i<\/p>\n\n\n\n<p><strong>T<\/strong><strong><sub>2<\/sub><\/strong><strong> =<\/strong> 1 + 2i<\/p>\n\n\n\n<p><strong>T<\/strong><strong><sub>1 <\/sub><\/strong><strong>\u2013<\/strong><strong><sub> <\/sub><\/strong><strong>T<\/strong><strong><sub>2 <\/sub><\/strong><strong>=<\/strong> (2 \u2013 1) + (3 \u2013 2)i<\/p>\n\n\n\n<p><strong>T<\/strong><strong><sub>1 <\/sub><\/strong><strong>\u2013<\/strong><strong><sub> <\/sub><\/strong><strong>T<\/strong><strong><sub>2 <\/sub><\/strong><strong>=<\/strong> 1 + 1i = 1+ i<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Pot\u00eancia da Unidade Imagin\u00e1ria<\/h3>\n\n\n\n<p>Para encontrarmos uma pot\u00eancia da unidade <strong><em>i<\/em><\/strong><strong> <\/strong><strong><sup>n<\/sup><\/strong>, f<strong>aremos a divis\u00e3o de n (o expoente) por 4<\/strong>, e<strong> o resto<\/strong><strong><em> <\/em><\/strong><strong>dessa divis\u00e3o<\/strong> (<em>r <\/em>= { 0, 1, 2, 3}) <strong>ser\u00e1 o novo expoente<\/strong> de <strong><em>i<\/em><\/strong><strong>. Observe:<\/strong><\/p>\n\n\n\n<p>C\u00e1lculo de i<sup>25<\/sup><\/p>\n\n\n\n<p>Ao resolvermos <strong>a divis\u00e3o de 25 por 4<\/strong>, o quociente ser\u00e1 <strong>6<\/strong> e o resto ser\u00e1 igual a <strong>1<\/strong>. Logo:<\/p>\n\n\n\n<p class=\"has-text-align-center\"><strong>i <sup>25 <\/sup>= i<sup>1 <\/sup>= i<\/strong><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Multiplica\u00e7\u00e3o de N\u00fameros Complexos&nbsp;<\/h3>\n\n\n\n<p>Antes de resolvermos a multiplica\u00e7\u00e3o \u00e9 preciso entender a propriedade distributiva. <strong>Observe:<\/strong><\/p>\n\n\n\n<p class=\"has-text-align-left\"><strong>T<\/strong><strong><sub>1 <\/sub><\/strong><strong>=<\/strong> a + bi<\/p>\n\n\n\n<p class=\"has-text-align-left\"><strong>T<\/strong><strong><sub>2<\/sub><\/strong><strong> =<\/strong> c +di, ent\u00e3o o produto:<\/p>\n\n\n\n<p class=\"has-text-align-left\"><strong>T<\/strong><strong><sub>1 <\/sub><\/strong><strong>\u00b7 T<\/strong><strong><sub>2<\/sub><\/strong><strong> =<\/strong> (a + bi) (c + di), aplicando a propriedade distributiva,<\/p>\n\n\n\n<p class=\"has-text-align-left\"><strong>T<\/strong><strong><sub>1 <\/sub><\/strong><strong>\u00b7 T<\/strong><strong><sub>2 <\/sub><\/strong><strong>=<\/strong> ac + adi + cbi + bdi <sup>2<\/sup>, mas, como vimos, i \u00b2 = -1<\/p>\n\n\n\n<p class=\"has-text-align-left\"><strong>T<\/strong><strong><sub>1 <\/sub><\/strong><strong>\u00b7 T<\/strong><strong><sub>2 <\/sub><\/strong><strong>=<\/strong> ac + adi + cbi \u2013 bd<\/p>\n\n\n\n<p class=\"has-text-align-left\"><strong>T<\/strong><strong><sub>1 <\/sub><\/strong><strong>\u00b7 T<\/strong><strong><sub>2 <\/sub><\/strong><strong>=<\/strong> (ac \u2013 bd) + (ad + cb)i<\/p>\n\n\n\n<p>A partir disso, \u00e9 poss\u00edvel encontrarmos o produto de quaisquer n\u00fameros complexos.&nbsp;<\/p>\n\n\n\n<p>N\u00e3o \u00e9 preciso decorar essa f\u00f3rmula, j\u00e1 que, para o c\u00e1lculo em quest\u00e3o, <strong>basta aplicarmos a propriedade distributiva. <\/strong>Entenda melhor com um exemplo:<\/p>\n\n\n\n<p><strong>Calcule o produto<\/strong> de (2+3i) (1 \u2013 4i):<\/p>\n\n\n\n<p>(2+3i) (1 \u2013 4i) = 2 \u2013 8i + 3i \u2013 12i \u00b2, lembrando que i\u00b2 = -1:<\/p>\n\n\n\n<p>(2 + 3i) (1 \u2013 4i) = 2 \u2013 8i + 3i + 12<\/p>\n\n\n\n<p>(2 + 3i) (1 \u2013 4i) = (2 + 12) + (\u2013 8 + 3)i<\/p>\n\n\n\n<p><strong>(2+3i) (1 \u2013 4i) = 14 \u2013 5i<\/strong><\/p>\n\n\n\n<p>Divis\u00e3o de N\u00fameros Complexos<\/p>\n\n\n\n<p><strong>Multiplique a fra\u00e7\u00e3o pelo conjugado do denominador<\/strong> para que fique bem definido o que \u00e9 a parte real e o que \u00e9 a parte imagin\u00e1ria.<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" loading=\"lazy\" width=\"600\" height=\"300\" src=\"https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/10\/N\u00fameros-Complexos-imagem-2.jpg\" alt=\"Multiplica\u00e7\u00e3o de N\u00fameros Complexos\" class=\"wp-image-18447\" srcset=\"https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/10\/N\u00fameros-Complexos-imagem-2.jpg 600w, https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/10\/N\u00fameros-Complexos-imagem-2-300x150.jpg 300w, https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/10\/N\u00fameros-Complexos-imagem-2-320x160.jpg 320w\" sizes=\"(max-width: 600px) 100vw, 600px\" \/><\/figure><\/div>\n\n\n<p>Para encontrarmos o conjugado de um n\u00famero complexo, <strong>trocamos o sinal da parte imagin\u00e1ria<\/strong>. Sendo assim:<\/p>\n\n\n\n<p class=\"has-text-align-center\"><strong>T =<\/strong> a + bi <strong>\u2192 <\/strong><strong>T<\/strong><strong> =<\/strong> a &#8211; bi<\/p>\n\n\n\n<p>Calcule a divis\u00e3o de (6 &#8211; 4<em>i<\/em>) : (4 + 2<em>i<\/em>)<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" loading=\"lazy\" width=\"600\" height=\"300\" src=\"https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/10\/N\u00fameros-Complexos-imagem-3.jpg\" alt=\"Exemplo de Multiplica\u00e7\u00e3o de N\u00fameros Complexos\" class=\"wp-image-18448\" srcset=\"https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/10\/N\u00fameros-Complexos-imagem-3.jpg 600w, https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/10\/N\u00fameros-Complexos-imagem-3-300x150.jpg 300w, https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/10\/N\u00fameros-Complexos-imagem-3-320x160.jpg 320w\" sizes=\"(max-width: 600px) 100vw, 600px\" \/><\/figure><\/div>\n\n\n<h2 class=\"wp-block-heading\">Plano complexo ou plano de Argand-Gauss (Forma Geom\u00e9trica)<\/h2>\n\n\n\n<p><strong>Esse plano \u00e9 uma adapta\u00e7\u00e3o do plano cartesiano<\/strong> com o objetivo de representar n\u00fameros complexos<\/p>\n\n\n\n<ul><li><strong>Re (z):<\/strong> eixo da parte real;<\/li><li><strong>Im (z):<\/strong> eixo da parte imagin\u00e1ria;<\/li><\/ul>\n\n\n\n<p>Representa\u00e7\u00e3o do n\u00famero <strong>4 + 3i<\/strong> na forma geom\u00e9trica T(4,3).<br><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" loading=\"lazy\" width=\"600\" height=\"300\" src=\"https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/10\/N\u00fameros-Complexos-imagem-4.jpg\" alt=\"Plano complexo ou plano de Argand-Gauss (Forma Geom\u00e9trica)\" class=\"wp-image-18449\" srcset=\"https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/10\/N\u00fameros-Complexos-imagem-4.jpg 600w, https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/10\/N\u00fameros-Complexos-imagem-4-300x150.jpg 300w, https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/10\/N\u00fameros-Complexos-imagem-4-320x160.jpg 320w\" sizes=\"(max-width: 600px) 100vw, 600px\" \/><\/figure><\/div>\n\n\n<ul><li><a href=\"https:\/\/beduka.com\/blog\/materias\/matematica\/geometria-espacial\/\" target=\"_blank\" rel=\"noreferrer noopener\" aria-label=\" (abre numa nova aba)\">ENTENDA TUDO SOBRE GEOMETRIA!<\/a><\/li><\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Representa\u00e7\u00e3o Trigonom\u00e9trica dos N\u00fameros Complexos (Forma Polar)<\/h3>\n\n\n\n<p>A partir da forma alg\u00e9brica <strong>z = a + bi<\/strong>, iremos analisar o plano complexo:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" loading=\"lazy\" width=\"600\" height=\"300\" src=\"https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/10\/N\u00fameros-Complexos-imagem-5.jpg\" alt=\"Representa\u00e7\u00e3o Trigonom\u00e9trica dos N\u00fameros Complexos (Forma Polar)\" class=\"wp-image-18450\" srcset=\"https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/10\/N\u00fameros-Complexos-imagem-5.jpg 600w, https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/10\/N\u00fameros-Complexos-imagem-5-300x150.jpg 300w, https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/10\/N\u00fameros-Complexos-imagem-5-320x160.jpg 320w\" sizes=\"(max-width: 600px) 100vw, 600px\" \/><\/figure><\/div>\n\n\n<p>Ao substituirmos, na forma alg\u00e9brica, os valores de<strong> a = |T| <\/strong>cos \u03b8 e <strong>b = |T|<\/strong> sen \u03b8, temos que:<\/p>\n\n\n\n<p class=\"has-text-align-center\"><strong>T =<\/strong> a + bi<\/p>\n\n\n\n<p>Com <strong>T<\/strong> = <strong>|T|<\/strong> cos \u03b8 + <strong>|T|<\/strong> sen \u03b8 \u00b7 i, colocando <strong>|T|<\/strong> em evid\u00eancia, <strong>chegamos \u00e0 f\u00f3rmula da forma trigonom\u00e9trica:<\/strong><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" loading=\"lazy\" width=\"600\" height=\"300\" src=\"https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/10\/N\u00fameros-Complexos-imagem-6.jpg\" alt=\"Forma Polar de N\u00fameros Complexos\" class=\"wp-image-18451\" srcset=\"https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/10\/N\u00fameros-Complexos-imagem-6.jpg 600w, https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/10\/N\u00fameros-Complexos-imagem-6-300x150.jpg 300w, https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/10\/N\u00fameros-Complexos-imagem-6-320x160.jpg 320w\" sizes=\"(max-width: 600px) 100vw, 600px\" \/><\/figure><\/div>\n\n\n<ul><li><strong>\u00c9 essencia<\/strong>l conhecer bastante sobre <a href=\"https:\/\/beduka.com\/blog\/materias\/matematica\/trigonometria-no-triangulo-retangulo\/\" target=\"_blank\" rel=\"noreferrer noopener\" aria-label=\" (abre numa nova aba)\">Trigonometria<\/a> para mandar bem nos Exerc\u00edcios sobre N\u00fameros Complexos<\/li><\/ul>\n\n\n\n<h2 class=\"wp-block-heading\">Exerc\u00edcios sobre N\u00fameros Complexos com Gabarito<\/h2>\n\n\n\n<p>Esperamos que, com esse resumo, <strong>tudo tenha ficado mais claro para voc\u00ea.&nbsp;<\/strong><\/p>\n\n\n\n<p><strong>Parab\u00e9ns<\/strong> por ter lido at\u00e9 aqui!<\/p>\n\n\n\n<p><strong>Quest\u00e3o 1 &#8211;<\/strong> <a rel=\"noreferrer noopener\" aria-label=\" (abre numa nova aba)\" href=\"https:\/\/www.unesp.br\/\" target=\"_blank\">(Unes<\/a><a href=\"https:\/\/www.unesp.br\/\" target=\"_blank\" rel=\"noreferrer noopener\" aria-label=\"p (abre numa nova aba)\">p<\/a><a rel=\"noreferrer noopener\" aria-label=\" (abre numa nova aba)\" href=\"https:\/\/www.unesp.br\/\" target=\"_blank\">-SP)<\/a> Se z = (2 + i) \u2219 (1 + i) \u2219 i, ent\u00e3o z, o conjugado de z, ser\u00e1 dado por:<\/p>\n\n\n\n<p><strong>a)<\/strong> \u22123 \u2212 i<\/p>\n\n\n\n<p><strong>b)<\/strong> 1 \u2212 3i<\/p>\n\n\n\n<p><strong>c)<\/strong> 3 \u2212 i<\/p>\n\n\n\n<p><strong>d) <\/strong>\u22123 + i<\/p>\n\n\n\n<p><strong>e)<\/strong> 3 + i<\/p>\n\n\n\n<p><strong>Quest\u00e3o 2 &#8211; <\/strong><a rel=\"noreferrer noopener\" aria-label=\" (abre numa nova aba)\" href=\"https:\/\/www.pucminas.br\/main\/Paginas\/default.aspx\" target=\"_blank\">(PUC)<\/a> Na soma S = 1 + i + i<sup>2 <\/sup>+ i<sup>3 <\/sup>+ i<sup>4 <\/sup>+ i<sup>5<\/sup>, onde i = \u221a \u20131, o valor de S \u00e9:<\/p>\n\n\n\n<p><strong>a)<\/strong> 2 \u2013 i<\/p>\n\n\n\n<p><strong>b)<\/strong> 1 \u2013 i<\/p>\n\n\n\n<p><strong>c)<\/strong> 2 + i<\/p>\n\n\n\n<p><strong>d) <\/strong>1 + i<\/p>\n\n\n\n<p><strong>Quest\u00e3o 3 &#8211;<\/strong> <a rel=\"noreferrer noopener\" aria-label=\" (abre numa nova aba)\" href=\"http:\/\/www.ufu.br\/\" target=\"_blank\">(UFU 2008)<\/a> Considere o tri\u00e2ngulo cujos v\u00e9rtices correspondem aos n\u00fameros complexos z1= 3, z2= 6 e z3=8+3i, em que i \u00e9 a unidade imagin\u00e1ria. Sabe-se que outro tri\u00e2ngulo de v\u00e9rtices correspondentes a w1 = \u0096iz1, w2 = \u0096iz2 e w3 = \u0096ihz3, sendo h um n\u00famero real positivo, possui \u00e1rea igual a 18. Ent\u00e3o, o valor de h \u00e9 igual a:<\/p>\n\n\n\n<p><strong>a)<\/strong> 10<\/p>\n\n\n\n<p><strong>b)<\/strong> 6<\/p>\n\n\n\n<p><strong>c)<\/strong> 8<\/p>\n\n\n\n<p><strong>d)<\/strong> 4<\/p>\n\n\n\n<ul><li><strong>Voc\u00ea est\u00e1 indo muito bem! <\/strong>Chegamos \u00e0 metade <strong>dos Exerc\u00edcios sobre N\u00fameros Complexos.<\/strong><\/li><\/ul>\n\n\n\n<p><strong>Quest\u00e3o 4 &#8211;<\/strong> <a rel=\"noreferrer noopener\" aria-label=\" (abre numa nova aba)\" href=\"http:\/\/www1.uea.edu.br\/\" target=\"_blank\">(UEA 2005)<\/a> Qual \u00e9 o conjunto das imagens dos complexos z tais que |z + 1| = |z &#8211; 1|?<\/p>\n\n\n\n<p><strong>a)<\/strong> reta<\/p>\n\n\n\n<p><strong>b)<\/strong> circunfer\u00eancia<\/p>\n\n\n\n<p><strong>c) <\/strong>elipse<\/p>\n\n\n\n<p><strong>d)<\/strong> hip\u00e9rbole<\/p>\n\n\n\n<p><strong>e)<\/strong> par\u00e1bola<\/p>\n\n\n\n<p><strong>Quest\u00e3o 5 &#8211; <\/strong><a rel=\"noreferrer noopener\" aria-label=\" (abre numa nova aba)\" href=\"https:\/\/www.furg.br\/\" target=\"_blank\">(FURG)<\/a> Se u = 1 \u2013 2i \u00e9 um n\u00famero complexo e U, seu conjugado, ent\u00e3o z = u2 + 3U \u00e9 igual a: <\/p>\n\n\n\n<p><strong>a)<\/strong> \u2013 6 \u2013 2i<\/p>\n\n\n\n<p><strong>b)<\/strong> 2i<\/p>\n\n\n\n<p><strong>c) <\/strong>\u2013 6<\/p>\n\n\n\n<p><strong>d)<\/strong> 8 + 2i<\/p>\n\n\n\n<p><strong>e)<\/strong> \u2013 6 + 2i<\/p>\n\n\n\n<ul><li><strong>Ufa!<\/strong> Estamos quase no fim, continue e fa\u00e7a <strong>o \u00faltimo exerc\u00edcios sobre N\u00fameros Complexos<\/strong><\/li><\/ul>\n\n\n\n<p><strong>Quest\u00e3o 6 &#8211; <\/strong><a rel=\"noreferrer noopener\" aria-label=\" (abre numa nova aba)\" href=\"http:\/\/www.uefs.br\/\" target=\"_blank\">(UEFS)<\/a> Se m &#8211; 1 + ni = (3 + i).(1 + 3i), ent\u00e3o m e n s\u00e3o respectivamente:<\/p>\n\n\n\n<p><strong>a)<\/strong> 1 e 10&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<\/p>\n\n\n\n<p><strong>b)<\/strong> 5 e 10&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<\/p>\n\n\n\n<p><strong>c)<\/strong> 7 e 9&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<\/p>\n\n\n\n<p><strong>d)<\/strong> 5 e 9&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<\/p>\n\n\n\n<p><strong>e)<\/strong> 0 e -9<\/p>\n\n\n\n<p><strong>Quest\u00e3o 7 &#8211; <\/strong><a rel=\"noreferrer noopener\" aria-label=\" (abre numa nova aba)\" href=\"http:\/\/www.uefs.br\/\" target=\"_blank\">(UEFS)<\/a> A soma de um n\u00famero complexo z com o triplo do seu conjugado \u00e9 igual a -8 &#8211; 6i. O m\u00f3dulo de z \u00e9:<\/p>\n\n\n\n<p><strong>a)<\/strong> \u00d613&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<\/p>\n\n\n\n<p><strong>b)<\/strong> \u00d67&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<\/p>\n\n\n\n<p><strong>c) <\/strong>13&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<\/p>\n\n\n\n<p><strong>d) <\/strong>7&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<\/p>\n\n\n\n<p><strong>e)<\/strong> 5<\/p>\n\n\n\n<ul><li><strong>Parab\u00e9ns<\/strong>, chegou ao fim dos Exerc\u00edcios sobre N\u00fameros Complexos. <strong>Confira agora o Gabarito:<\/strong><\/li><\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Gabarito dos Exerc\u00edcios sobre N\u00fameros Complexos<\/h3>\n\n\n\n<p><strong>Exerc\u00edcio resolvido da quest\u00e3o 1 &#8211;<\/strong><\/p>\n\n\n\n<p>Alternativa correta: <strong>a)<\/strong> \u22123 \u2212 i&nbsp;<\/p>\n\n\n\n<p><strong>Exerc\u00edcio resolvido da quest\u00e3o 2 &#8211;<\/strong><\/p>\n\n\n\n<p>Alternativa correta: <strong>d) <\/strong>1 + i<\/p>\n\n\n\n<p><strong>Exerc\u00edcio resolvido da quest\u00e3o 3 &#8211;<\/strong><\/p>\n\n\n\n<p>Alternativa correta: <strong>d)<\/strong> 4<\/p>\n\n\n\n<p><strong>Exerc\u00edcio resolvido da quest\u00e3o 4 &#8211;<\/strong><\/p>\n\n\n\n<p>Alternativa correta: <strong>a)<\/strong> reta<\/p>\n\n\n\n<p><strong>Exerc\u00edcio resolvido da quest\u00e3o 5 &#8211;<\/strong><\/p>\n\n\n\n<p>Alternativa correta: <strong>b)<\/strong> 2i<\/p>\n\n\n\n<p><strong>Exerc\u00edcio resolvido da quest\u00e3o 6 &#8211;<\/strong><\/p>\n\n\n\n<p>Alternativa correta: <strong>a)<\/strong> 1 e 10&nbsp;&nbsp;&nbsp;<\/p>\n\n\n\n<p><strong>Exerc\u00edcio resolvido da quest\u00e3o 7 &#8211;<\/strong><\/p>\n\n\n\n<p>Alternativa correta: <strong>a)<\/strong> \u00d613&nbsp;<\/p>\n\n\n\n<p>Estude para o Enem com o <a href=\"https:\/\/beduka.com\/simulado\" target=\"_blank\" rel=\"noreferrer noopener\" aria-label=\" (abre numa nova aba)\">Simulado Beduka.<\/a> <strong>\u00c9 gratuito!<\/strong><\/p>\n\n\n\n<p>Gostou dos nossos <strong>Exerc\u00edcios sobre N\u00fameros Complexos?<\/strong> Compartilhe com os seus amigos e comente abaixo sobre as \u00e1reas que voc\u00ea deseja mais explica\u00e7\u00f5es.<\/p>\n\n\n\n<p>Queremos te ajudar a encontrar a <strong>FACULDADE IDEAL! <\/strong>Logo abaixo, fa\u00e7a uma pesquisa por curso e cidade que te mostraremos todas as faculdades que podem te atender. <strong>Informamos a nota de corte, valor de mensalidade, nota do MEC, avalia\u00e7\u00e3o dos alunos, modalidades de ensino e muito mais.<\/strong><br><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Os n\u00fameros complexos s\u00e3o representados de 3 formas: a forma geom\u00e9trica (representada no plano complexo conhecido como plano de Argand-Gauss, a forma trigonom\u00e9trica, conhecida como forma polar e a forma alg\u00e9brica (z = a + b), composta por uma parte real \u201ca\u201d e uma parte imagin\u00e1ria \u201cb\u201d. Leia nosso resumo e fa\u00e7a os 8 Melhores [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":18445,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_mi_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0},"categories":[610],"tags":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v20.10 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Os 7 Melhores Exerc\u00edcios sobre N\u00fameros Complexos (GABARITO)<\/title>\n<meta name=\"description\" content=\"Leia o resumo e coloque seu conhecimento em pr\u00e1tica com os 7 melhores Exerc\u00edcios sobre N\u00fameros Complexos selecionados para voc\u00ea.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" 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