{"id":26455,"date":"2022-02-17T16:37:37","date_gmt":"2022-02-17T11:07:37","guid":{"rendered":"https:\/\/yanamtakshashila.com\/?p=26455"},"modified":"2024-05-25T11:49:53","modified_gmt":"2024-05-25T06:19:53","slug":"roots-of-complex-numbers-excercise-problems","status":"publish","type":"post","link":"https:\/\/yanamtakshashila.com\/?p=26455","title":{"rendered":"ROOTS OF COMPLEX NUMBERS (Excercise Problems)"},"content":{"rendered":"\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\LARGE{\\color {purple} {PART- B}}\\]<script src=\"https:\/\/yanamtakshashila.com\/wp-includes\/js\/dist\/hooks.min.js?ver=dd5603f07f9220ed27f1\" id=\"wp-hooks-js\"><\/script>\n<script src=\"https:\/\/yanamtakshashila.com\/wp-includes\/js\/dist\/i18n.min.js?ver=c26c3dc7bed366793375\" id=\"wp-i18n-js\"><\/script>\n<script id=\"wp-i18n-js-after\">\nwp.i18n.setLocaleData( { 'text direction\\u0004ltr': [ 'ltr' ] } );\n\/\/# sourceURL=wp-i18n-js-after\n<\/script>\n<script  async src=\"https:\/\/cdnjs.cloudflare.com\/ajax\/libs\/mathjax\/2.7.7\/MathJax.js?config=TeX-MML-AM_CHTML\" id=\"mathjax-js\"><\/script>\n<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {1.}\\ \\color {red} {If \\ \u03c9\\ is\\ the\\ cube\\ roots\\ of unity,\\ what\\ is\\ the\\ value\\ of}\\ \u03c9( \u03c9\\ +\\ 1)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\  Given\\ \u03c9\\ is\\ the\\ cube\\ roots\\ of unity\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[i.\\ e\\ 1\\ +\\ \u03c9\\  +\\ \u03c9 ^2\\ =\\ 0\\ &#8212;&#8212;\\ (1)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\u03c9( \u03c9\\ +\\ 1)\\ =\\ \u03c9 ^2\\ +\\ \u03c9\\ =\\ -\\ 1\\ (using\\ (1))\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js?client=ca-pub-9453835310745500\" crossorigin=\"anonymous\"><\/script>\n<!-- display h -->\n<ins class=\"adsbygoogle\" style=\"display:block\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"3583972194\" data-ad-format=\"auto\" data-full-width-responsive=\"true\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\LARGE{\\color {purple} {PART- C}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {2\\ .}\\ \\color {red} {Solve}\\  x^4 -\\ 1\\  =\\ 0\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ x^4\\   =\\ 1\\  \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\ x\\  =\\ (1)^\\frac{1}{4}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[  =\\ (cos\\ 0\\ +\\ i\\ sin\\ 0)^\\frac{1}{4}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[  =\\ (cos\\ 2k\u03c0\\ +\\ i\\ sin\\ 2k\u03c0)^\\frac{1}{4}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[  =\\ cos\\ (\\frac{2k\u03c0}{4})\\ +\\ i\\ sin\\ (\\frac{2k\u03c0}{4})\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[  =\\ cos\\ (\\frac{k\u03c0}{2})\\ +\\ i\\ sin\\ (\\frac{k\u03c0}{2})\\ where\\ k\\ =\\ 0,\\ 1,\\ 2,\\  3\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[The\\ roots\\ are\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[When\\ k = 0,\\ \\hspace{2cm}\\    x\\ =\\ cos\\ 0\\   +\\ i\\ sin\\ 0\\ =\\ 1\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[When\\ k = 1,\\ \\hspace{2cm}\\    x\\ =\\ cos\\ \\frac{\u03c0}{2}\\   +\\ i\\ sin\\ \\frac{\u03c0}{2}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[When\\ k = 2,\\ \\hspace{2cm}\\    x\\ =\\ cos\\ \\frac{2\u03c0}{2}\\   +\\ i\\ sin\\ \\frac{2\u03c0}{2}\\ =\\ cos\\ \u03c0\\   +\\  i\\ sin\\ \u03c0\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[When\\ k = 3,\\ \\hspace{2cm}\\    x\\ =\\ cos\\ \\frac{3\u03c0}{2}\\   +\\ i\\ sin\\ \\frac{3\u03c0}{2}\\]<\/div>\n\n\n<p><iframe width=\"1366\" height=\"768\" src=\"https:\/\/www.youtube.com\/embed\/9Zz8K-Kly5Q\" title=\"ROOTS OF COMPLEX NUMBERS - PART - 1\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {3\\ .}\\ \\color {red} {Solve}\\  x^5 +\\  x^3\\ +\\ x^2\\ +\\ 1\\  =\\ 0\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ Given\\ x^5 +\\  x^3\\ +\\ x^2\\ +\\ 1\\  =\\ 0\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[x^3(x^2 +\\ 1)\\ +\\ 1( x^2\\ +\\ 1)\\  =\\ 0\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(x^3\\ +\\ 1)\\ ( x^2\\ +\\ 1)\\  =\\ 0\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {green} {Case\\ ( i ) :}\\ \\hspace{16cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[x^3\\ +1  =\\ 0\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ x^3\\   =\\ -\\ 1\\  \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\ x\\  =\\ (-\\ 1)^\\frac{1}{3}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[  =\\ (cos\\ \u03c0\\ +\\ i\\ sin\\ \u03c0)^\\frac{1}{3}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[  =\\ (cos\\ (\u03c0\\ + 2k\u03c0) +\\ i\\ sin\\ (\u03c0\\ + 2k\u03c0))^\\frac{1}{3}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[  =\\ cos\\ (\\frac{\u03c0\\ + 2k\u03c0}{3})\\ +\\ i\\ sin\\ (\\frac{\u03c0\\ + 2k\u03c0}{3})\\ where\\ k\\ =\\ 0,\\ 1,\\ 2\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[The\\ roots\\ are\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[When\\ k = 0,\\ \\hspace{2cm}\\    x\\ =\\ cos\\ \\frac{\u03c0}{3}\\   +\\ i\\ sin\\ \\frac{\u03c0}{3}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[When\\ k = 1,\\ \\hspace{2cm}\\    x\\ =\\ cos\\ \\frac{3\u03c0}{3}\\   +\\ i\\ sin\\ \\frac{3\u03c0}{3}\\ =\\ -1\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[When\\ k = 2,\\ \\hspace{2cm}\\    x\\ =\\ cos\\ \\frac{5\u03c0}{3}\\   +\\ i\\ sin\\ \\frac{5\u03c0}{3}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {green} {Case\\ ( ii ) :}\\ \\hspace{16cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[x^2\\ +1  =\\ 0\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ x^2\\   =\\ -\\ 1\\  \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\ x\\  =\\ (-\\ 1)^\\frac{1}{2}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[  =\\ (cos\\ \u03c0\\ +\\ i\\ sin\\ \u03c0)^\\frac{1}{2}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[  =\\ (cos\\ (\u03c0\\ + 2k\u03c0) +\\ i\\ sin\\ (\u03c0\\ + 2k\u03c0))^\\frac{1}{2}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[  =\\ cos\\ (\\frac{\u03c0\\ + 2k\u03c0}{2})\\ +\\ i\\ sin\\ (\\frac{\u03c0\\ + 2k\u03c0}{2})\\ where\\ k\\ =\\ 0,\\ 1\\  \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[The\\ roots\\ are\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[When\\ k = 0,\\ \\hspace{2cm}\\    x\\ =\\ cos\\ \\frac{\u03c0}{2}\\   +\\ i\\ sin\\ \\frac{\u03c0}{2}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[When\\ k = 1,\\ \\hspace{2cm}\\    x\\ =\\ cos\\ \\frac{3\u03c0}{2}\\   +\\ i\\ sin\\ \\frac{3\u03c0}{2}\\]<\/div>\n\n\n\n<script async=\"\" 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