{"id":22197,"date":"2021-09-16T20:03:03","date_gmt":"2021-09-16T14:33:03","guid":{"rendered":"https:\/\/yanamtakshashila.com\/?p=22197"},"modified":"2024-06-30T20:21:09","modified_gmt":"2024-06-30T14:51:09","slug":"unit-i-algebra-2","status":"publish","type":"post","link":"https:\/\/yanamtakshashila.com\/?p=22197","title":{"rendered":"BINOMIAL THEOREM (Text)"},"content":{"rendered":"\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {royalblue} {Definition\\ of\\ Factorial\\ Notation}:\\ \\hspace{20cm}\\] <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\">\\[The\\  continued\\ product\\ of\\ first\\ \u2018n\u2019 \\ natural\\ numbers\\ is\\ called\\ n\\ factorial\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[and\\ is\\ denoted\\ by\\ n!\\ and\\ defined\\ by\\ n!\\ = 1\u00d72\u00d73\u00d7\u2026\u2026\u2026..\u00d7n\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Ex:\\ 5!\\  =\\ 1\u00d72\u00d73\u00d74\u00d75\\ = 120\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Note:\\ 0!\\  =\\ 1\\ \\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {royalblue} {Binomial\\ theorem\\ for\\ a\\ positive\\ integral\\ index}:\\ \\hspace{20cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[If\\ \u2018n\u2019\\  is\\ any\\ positive\\ integer,\\  then\\  \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(X\\ +\\ a)^n\\ =\\ X^n\\ +\\ nc_1 X^{n-1}a\\ +\\ \\ nc_2 X^{n-2}a^2\\ +\\ &#8230;&#8230;&#8230;&#8230;&#8230;&#8230;&#8230;..\\ +\\ \\ nc_r X^{n-r}a^r\\ +\\ &#8230;&#8230;.\\ a^n\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Note:\\ nc_r\\ =\\ \\frac{n!}{(n-r)!\\ r!}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<script async=\"\" src=\"https:\/\/cse.google.com\/cse.js?cx=2f7b32630c6dfd4fb\"><\/script>\n<div class=\"gcse-search\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {royalblue} {General\\ term\\ of\\ the\\ expansion\\ of\\ (X\\ +\\ a)^n}:\\ \\hspace{20cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[In\\ the\\ binomial\\ expansion\\ for\\ (X\\ +\\ a)^n ,\\ we\\ observe\\ that\\ the\\ first\\ term\\ is\\ nC_0x^n,\\ the\\ second\\ term\\ is\\  nC_1 x^{n-1} a,\\  the\\ third\\ term\\ is\\ nC_2x^{n-2},\\ and\\ so\\ on.\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Looking\\ at\\ the\\ pattern\\ of\\ the\\ successive\\ terms\\ we\\ can\\ say\\ that\\ the\\ (r +1 )^{th}\\ term\\ is\\ nC_rx^{n-r} a^r.\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[The\\ (r +1 )^{th}\\ term\\ is\\ also\\ called\\ the\\ general\\ term\\ of\\ the\\ expansion\\ ( x +a )^n .\\ It\\ is\\ denoted\\ by\\ T_{r + 1}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[T_{r + 1} = nC_rx^{n-r} a^r.\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js?client=ca-pub-9453835310745500\" crossorigin=\"anonymous\"><\/script>\n<ins class=\"adsbygoogle\" style=\"display:block\" data-ad-format=\"autorelaxed\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"4869133702\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 1:}\\ \\color{red}{Find\\ the\\ general\\ term\\ of\\ the\\ expansion}\\ (x\\ +\\ a)^n\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\ October\\ 2023\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ T_{r + 1} = nC_rx^{n-r} a^r \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 2:}\\ \\color{red}{Write\\ down\\ the\\ general\\ term\\ of\\ the\\ expansion\\ of}\\ (x\\ +\\ \\frac{2}{x})^{12}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ T_{r + 1} = nC_rx^{n-r} a^r \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Here\\ X\\ =\\ x,\\  a\\ =\\ \\frac{2}{x},\\ n\\ =\\ 12\\ \\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[T_{r + 1} = 12C_r\\ x^{12-r}\\ (\\frac{2}{x})^r \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[T_{r + 1} = 12C_r\\ x^{12-r}\\ 2^r\\ x^{-r}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = 12C_r\\ 2^r\\ x^{12-r-r}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{T_{r + 1} = 12C_r\\ 2^r\\ x^{12-2r}}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 3:}\\ \\color {red}{Write\\ down\\ the\\ general\\ term\\ of\\ the\\ expansion\\ of}\\ (x^2\\ -\\ \\frac{2}{x})^{9}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ T_{r + 1} = nC_rx^{n-r} a^r \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Here\\ X\\ =\\ x^2,\\  a\\ =\\ -\\frac{2}{x},\\ n\\ =\\ 9\\ \\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[T_{r + 1} = 9C_r\\ (x^2)^{9-r}\\ (-\\frac{2}{x})^r \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = 9C_r\\ x^{18-2r}\\ (-2)^r\\ x^{-r}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = 9C_r\\ (-2)^r\\ x^{18-2r-r}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{T_{r + 1} = 9C_r\\ (-2)^r\\ x^{18-3r}}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 4:}\\ \\color {red}{Find\\ the\\ 6^{th}\\ term\\ in\\ the\\ expansion\\ of}\\ (x\\ +\\ \\frac{1}{x})^{10}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ T_{r + 1} = nC_rx^{n-r} a^r \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Here\\ X\\ =\\ x,\\  a\\ =\\ \\frac{1}{x},\\ n\\ =\\ 10,\\  r\\ =\\ 5\\ \\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[T_{5 + 1} = 10C_5\\ x^{10-5}\\ (\\frac{1}{x})^5 \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = 10C_5\\ x^5\\ 1^5\\ x^{-5}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = 10C_5\\ x^{5 &#8211; 5}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[T_6 = 10C_5\\ x^0\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\therefore\\ \\boxed{The\\ 6^{th}\\ term\\ is\\ 10C_5}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 5:}\\ \\color {red}{Find\\ the\\ 4^{th}\\ term\\ in\\ the\\ expansion\\ of}\\ (x^2\\ +\\ \\frac{1}{x})^8\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{18cm}\\ Feb\\ 2022,\\ April\\ 2024\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ T_{r + 1} = nC_rx^{n-r} a^r \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Here\\ X\\ =\\ x^2,\\  a\\ =\\ \\frac{1}{x},\\ n\\ =\\ 8,\\  r\\ =\\ 3\\ \\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[T_{3 + 1} = 8C_3\\ {(x^2)}^{8 &#8211; 3}\\ (\\frac{1}{x})^3 \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[T_{4} = 8C_3\\ {(x^2)}^{5}\\ \\frac{1^3}{x^3} \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[T_4 = 8C_3\\ x^{10}\\ 1^3\\ x^{-3}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = 8C_3\\ x^{10 &#8211; 3}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[T_4 = 8C_3\\ x^7\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js?client=ca-pub-9453835310745500\" crossorigin=\"anonymous\"><\/script>\n<!-- billboard -->\n<ins class=\"adsbygoogle\" style=\"display:inline-block;width:970px;height:250px\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"9933277607\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {royalblue} {To\\ find\\ the\\ Middle\\ term\\ of\\ the\\ expansion\\ of\\ (X\\ +\\ a)^n}:\\ \\hspace{20cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {brown} {cases}:\\ \\hspace{18cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[1.\\ If\\ n\\ is\\ an\\ even\\ number,\\ there\\ is\\ one\\ middle\\ term\\ =\\ (\\frac{n+2}{2})^{th}\\ term\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2.\\ If\\ n\\ is\\ odd\\ number,\\ there\\ are\\ two\\ middle\\ terms\\ (\\frac{n+1}{2})^{th}\\ and\\ \\ (\\frac{n+1}{2})^{th}\\ term\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 6\\ .}\\ \\color {red}{Find\\ the\\ middle\\ term\\ in\\ the\\ expansion\\ of}\\ (2x^4\\ +\\ \\frac{3}{x})^{16}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ Here\\ n\\ is\\ even\\ number,\\ so\\ there\\ is\\ one\\ middle\\ term\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(\\frac{n+2}{2})^{th}\\ term\\ =\\ (\\frac{16+2}{2})^{th}\\ term\\ =\\ (\\frac{18}{2})^{th}\\ term\\ =\\ 9^{th}\\ term\\ \\hspace{16cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[T_{r + 1} = nC_rx^{n-r} a^r \\hspace{19cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Here\\ X\\ =\\ 2x^4,\\  a\\ =\\ \\frac{3}{x},\\ n\\ =\\ 16,\\  r\\ =\\ 8\\ \\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[T_{8 + 1} = 16C_8\\ (2x^4)^{16-8}\\ (\\frac{3}{x})^8 \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[T_9 = 16C_8\\ (2x^4)^8\\ \\frac{3^8}{x^8}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = 16C_8\\ \\ 2^8\\ \\ 3^8\\ x^{32}\\ x^{-8}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = 16C_8\\ \\ 2^8\\ \\ 3^8\\ x^{32 &#8211; 8}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{T_9 = 16C_8\\ 2^8\\ 3^8\\ x^{24}}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js?client=ca-pub-9453835310745500\" crossorigin=\"anonymous\"><\/script>\n<ins class=\"adsbygoogle\" style=\"display:block\" data-ad-format=\"fluid\" data-ad-layout-key=\"-6x+db-3g-5s+t1\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"3292591486\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {royalblue} {To\\ find\\ the\\ coefficient\\ of\\ x^n\\ for\\ particular\\ value\\ of\\ n\\ and\\ the\\ term\\ independent\\ of\\ x\\ in\\ the\\ expansion\\ of\\ (X\\ +\\ a)^n}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 7 :}\\ \\color {red}{Find\\ the\\ coefficient\\ of\\ x^{32}\\ in\\ the\\ expansion\\ of}\\ (x^4\\ +\\ \\frac{1}{x^3})^{15}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ T_{r + 1} = nC_rx^{n-r} a^r \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Here\\ X\\ =\\ x^4,\\  a\\ =\\ \\frac{1}{x^3},\\ n\\ =\\ 15,\\  r\\ =\\ r\\ \\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[T_{r + 1} = 15C_r\\ (x^4)^{15-r}\\ (\\frac{1}{x^3})^r \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = 15C_r\\ x^{60 &#8211; 4r}\\ \\frac{1^r}{x^{3r}} \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = 15C_r\\ x^{60 &#8211; 4r &#8211; 3r} \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[T_{r + 1}  = 15C_r\\ x^{60 &#8211; 7r}\\ &#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212; (1)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[We\\ assume\\ that\\ x^{32}\\ occurs\\ in\\ 15C_r\\ x^{60 &#8211; 7r}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\therefore\\  x^{60 &#8211; 7r}\\ =\\ x^{32}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Equating\\ the\\ indices\\ 60 &#8211; 7r\\ =\\ 32\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[-7r\\ =\\ 32\\ -\\ 60\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[-7r\\ = -\\ 28\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{r\\ = 4}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Put\\ r\\ =\\ 4\\ in\\ (1)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[T_{4 + 1}  = 15C_4\\ 1^4\\ x^{60\\ -\\ 28}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[T_5  = 15C_4\\ x^{32}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{The\\ coefficient\\ of\\ x^{32}\\ =\\ 15\\ C_4}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 8 :}\\ \\color {red}{Find\\ the\\ coefficient\\ of\\ x^5\\ in\\ the\\ expansion\\ of}\\ (ax\\ +\\ \\frac{b}{x})^{11}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ T_{r + 1} = nC_rx^{n-r} a^r \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Here\\ X\\ =\\ ax,\\  a\\ =\\ \\frac{b}{x},\\ n\\ =\\ 11,\\  r\\ =\\ r\\ \\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[T_{r + 1} = 11C_r\\ (ax)^{11-r}\\ (\\frac{b}{x})^r \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = 11C_r\\ a^{11\\ -\\ r}\\ x^{11 &#8211; r}\\ b^r\\ x^{-r}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = 11C_r\\  a^{11\\ -\\ r}\\ b^r\\ x^{11\\ -\\ r}\\ x^{-r}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ 11C_r\\  a^{11\\ -\\ r}\\ b^r\\ x^{11\\ -\\ 2r}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[T_{r + 1}\\  =\\ =\\ 11C_r\\  a^{11\\ -\\ r}\\ b^r\\ x^{11\\ -\\ 2r}\\ &#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212; (1)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[We\\ assume\\ that\\ x^5\\ occurs\\ in\\ 11C_r\\  a^{11\\ -\\ r}\\ b^r\\ x^{11\\ -\\ 2r}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\therefore\\  x^{11\\ -\\ 2r}\\ =\\ x^5\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Equating\\ the\\ indices\\ 11\\ -\\ 2r\\ =\\ 5\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[-2r\\ =\\ 5\\ -\\ 11\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[-2r\\ = -\\ 6\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{r\\ = 3}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Put\\ r\\ =\\ 3\\ in\\ (1)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[T_{3 + 1}  = 11C_3\\ a^{11\\ -\\ 3}\\ b^3\\ x^{11\\ -\\ 6}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[T_4  =\\ 11C_3\\ a^8\\ b^3\\ x^5\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{The\\ coefficient\\ of\\ x^5\\ =\\ 11C_3\\ a^8\\ b^3}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 9 :}\\ \\color {red}{Find\\ the\\ term\\ independent\\ of\\ x\\ in\\ the\\ expansion\\ of}\\ (x^2\\ +\\ \\frac{1}{x})^{12}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\ October\\ 2023\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ T_{r + 1} = nC_rx^{n-r} a^r \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Here\\ X\\ =\\ x^2,\\  a\\ =\\ \\frac{1}{x},\\ n\\ =\\ 12,\\  r\\ =\\ r\\ \\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[T_{r + 1} = 12C_r\\ (x^2)^{12-r}\\ (\\frac{1}{x})^r \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = 12C_r\\ (x^2)^{12 &#8211; r}\\ \\frac{1^r}{x^{r}} \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = 12C_r\\  x^{24\\ -\\ 2r}\\ 1^r\\ x^{-r}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = 12C_r\\  1^r\\ x^{24 &#8211; 2r &#8211; r} \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[T_{r + 1}  = 12C_r\\  1^r\\ x^{24 &#8211; 3r}\\ &#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212; (1)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[To\\ find\\ the\\ independent\\ term\\ of\\ x,\\ find\\ the\\ coefficient\\ of\\ x^0.\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\therefore\\  x^{24\\ -\\ 3r}\\ =\\ x^0\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[24 &#8211; 3r\\ =\\ 0\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[-3r\\ = -\\ 24\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[r\\ = 8\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Put\\ r\\ =\\ 8\\ in\\ (1)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\therefore\\  independent\\ term\\ of\\ x =\\ 12C_8\\  1^8\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 12C_8\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 10 :}\\ \\color {red}{Find\\ the\\ term\\ independent\\ of\\ x\\ in\\ the\\ expansion\\ of}\\ (\\sqrt{x} &#8211; \\frac{1}{x^2})^{20}\\  \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ T_{r + 1} = nC_rx^{n-r} a^r \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Here\\ X\\ =\\ \\sqrt{x},\\  a\\ =\\ \\frac{-1}{x^2},\\ n\\ =\\ 20,\\  r\\ =\\ r\\ \\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[T_{r + 1} = 20C_r\\ (\\sqrt{x})^{20-r}\\ (-\\frac{1}{x^2})^r \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = 20C_r\\ x^(\\frac{20-r}{2})\\ \\frac{-1^r}{x^{2r}} \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = 20C_r\\ x^(\\frac{20-r}{2})\\  {-1}^r\\ \\ x^{-2r}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = 20C_r\\ x^(\\frac{20-r}{2}\\ -\\ 2r)\\ {-1}^r\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = 20C_r\\ x^(\\frac{20-r &#8211; 4r}{2})\\ {-1}^r\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[T_{r + 1} = 20C_r\\ x^(\\frac{20-5r}{2})\\ {-1}^r\\ &#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212; (1)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[To\\ find\\ the\\ independent\\ term\\ of\\ x,\\ find\\ the\\ coefficient\\ of\\ x^0.\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\therefore\\  x^{\\frac{20-5r}{2}}\\ =\\ x^0\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{20-5r}{2}\\ =\\ 0\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[20 &#8211; 5r\\ =\\ 0\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[-5r\\ = -\\ 20\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{r\\ = 4}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Put\\ r\\ =\\ 4\\ in\\ (1)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\therefore\\  independent\\ term\\ of\\ x =\\ 20C_4\\ (-1)^4\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 20C_4\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js?client=ca-pub-9453835310745500\" crossorigin=\"anonymous\"><\/script>\n<!-- Ad1 -->\n<ins class=\"adsbygoogle\" style=\"display:inline-block;width:300px;height:600px\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"8240817448\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {royalblue} {Binomial\\ theorem\\ for\\ rational\\ index}:\\ \\hspace{20cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[If\\ \u2018n\u2019\\  is\\ any\\ rational\\ number\\  then\\  \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(1\\ +\\ x)^n\\ =\\ 1\\ +\\ nx\\ +\\ \\frac{n(n-1)}{1.\\ 2}\\ x^2\\ +\\ \\frac{n(n-1)(n-2)}{1.\\ 2.\\ 3}\\ x^3\\ +\\ &#8230;&#8230;.\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 11:}\\ \\color {red}{Write\\ down\\ the\\ first\\ three\\ terms\\ in\\ the\\ expansion\\ of\\ (1\\ -\\ x)^{-2}\\ when\\ \\mid x\\ \\mid \\lt\\ 1}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ W.\\ K.\\ T\\  (1\\ +\\ x)^n\\ =\\ 1\\ +\\ nx\\ +\\ \\frac{n(n-1)}{1.\\ 2}\\ x^2\\ +\\ \\frac{n(n-1)(n-2)}{1.\\ 2.\\ 3}\\ x^3\\ +\\ &#8230;&#8230;.\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Here\\ X\\ =\\ &#8211; x,\\   n\\ =\\ &#8211; 2\\  \\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(1\\ +\\ x)^{-2}\\ =\\ 1\\ +\\ (-2)(-x)\\ +\\ \\frac{-2(-2-1)}{1.\\ 2}\\ (-x)^2\\ +\\  &#8230;&#8230;.\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(1\\ +\\ x)^{-2}\\ =\\ 1\\ +\\ 2x\\ +\\ (-1) (-3) x^2\\ +\\  &#8230;&#8230;.\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(1\\ +\\ x)^{-2}\\ =\\ 1\\ +\\ 2x\\ +\\ 3 x^2\\ +\\  &#8230;&#8230;.\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 12:}\\ \\color {red}{Expand\\ using\\ (1\\ -\\ 2x)^{-3}\\ Binomial\\ Theorem}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ W.\\ K.\\ T\\  (1\\ +\\ x)^n\\ =\\ 1\\ +\\ nx\\ +\\ \\frac{n(n-1)}{1.\\ 2}\\ x^2\\ +\\ \\frac{n(n-1)(n-2)}{1.\\ 2.\\ 3}\\ x^3\\ +\\ &#8230;&#8230;.\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Here\\ X\\ =\\ &#8211; 2x,\\   n\\ =\\ &#8211; 3\\  \\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ W.\\ K.\\ T\\  (1\\ -\\ 2x)^{-3}\\ =\\ 1\\ +\\ (-3)(-2x)\\ +\\ \\frac{-3(-3-1)}{1.\\ 2}\\ (-2x)^2\\  +\\ \\frac{-3(-3-1)(-3-2)}{1.\\ 2.\\ 3}\\ (-2x)^3\\ +\\  &#8230;&#8230;.\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ 1\\ +\\ 6x\\ +\\ 6(4x^2)\\  +\\ 8(10x^3) +\\  &#8230;&#8230;.\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ 1\\ +\\ 6x\\ +\\ 24x^2\\  +\\ 80x^3 +\\  &#8230;&#8230;.\\]<\/div>\n\n\n\n<script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js?client=ca-pub-9453835310745500\" crossorigin=\"anonymous\"><\/script>\n<!-- billboard -->\n<ins class=\"adsbygoogle\" style=\"display:inline-block;width:970px;height:250px\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"9933277607\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n\n\n\n<h3 class=\"wp-block-heading has-text-align-center has-vivid-purple-color has-text-color\">Exercise Problems<\/h3>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\LARGE{\\color {purple} {PART- A}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[1. \\ \\color {red}{Find\\  the\\ general\\ term\\ of\\ the\\ expansion\\ of}\\ (x\\ +\\ \\frac{1}{x})^{10}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2.\\ \\color {red}{which\\ term\\ in\\ (3\\ x\\ -\\ y)^6\\ is\\  the\\ middle\\ term\\ ?}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[3.\\ \\color {red}{Write\\ down\\ the\\ first\\ three\\ terms\\ in\\ the\\ expansion\\ of\\ (1\\ +\\ x)^{-4}\\ when\\ \\mid x\\ \\mid \\lt\\ 1}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[4.\\ \\color {red}{Write\\ down\\ the\\ first\\ two\\ terms\\ in\\ the\\ expansion\\ of\\ (1\\ -\\ 2x)^{-3}\\ when\\ \\mid x\\ \\mid \\lt\\ 1}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js?client=ca-pub-9453835310745500\" crossorigin=\"anonymous\"><\/script>\n<ins class=\"adsbygoogle\" style=\"display:block; text-align:center;\" data-ad-layout=\"in-article\" data-ad-format=\"fluid\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"6565358754\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\LARGE{\\color {purple} {PART- B}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[5.\\ \\color {red}{Find\\ the\\ 4^{th}\\ term\\ in\\ the\\ expansion\\ of}\\ (x^2\\ +\\ \\frac{1}{x})^8\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[6.\\ \\color {red}{Find\\ the\\ 5^{th}\\ term\\ in\\ the\\ expansion\\ of}\\ (x^2\\ +\\ \\frac{1}{x^2})^{10}\\ \\hspace{15cm}\\]<\/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 s -->\n<ins class=\"adsbygoogle\" style=\"display:block\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"4398623749\" 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\">\\[7 .\\ \\color {red}{Find\\ the\\ middle\\ terms\\ in\\ the\\ expansion}\\ of\\ (x\\ +\\ \\frac{2}{x^3})^{10}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[8 .\\ \\color {red}{Find\\ the\\ middle\\ terms\\ in\\ the\\ expansion\\ of}\\ (x^3\\ +\\ \\frac{2}{x^3})^{11}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[9.\\ \\color {red}{Find\\ the\\ term\\ independent\\ of\\ x\\ in\\ the\\ expansion\\ of}\\ (x^2\\ +\\ \\frac{1}{x})^{12}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<script async=\"\" src=\"https:\/\/cse.google.com\/cse.js?cx=2f7b32630c6dfd4fb\"><\/script>\n<div class=\"gcse-search\"><\/div>\n","protected":false},"excerpt":{"rendered":"<p>Exercise Problems<\/p>\n","protected":false},"author":187055548,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"set","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center 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center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2},"_wpas_customize_per_network":false,"jetpack_post_was_ever_published":false},"categories":[711788228,711787789,711787788,711788223],"tags":[],"class_list":["post-22197","post","type-post","status-publish","format-standard","hentry","category-binomial-theorem-text","category-n-engineering-mathematics-1","category-n-scheme","category-n-unit-i-algebra"],"yoast_head":"<!-- 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