{"id":25438,"date":"2022-01-25T20:27:40","date_gmt":"2022-01-25T14:57:40","guid":{"rendered":"https:\/\/yanamtakshashila.com\/?p=25438"},"modified":"2024-06-30T20:23:01","modified_gmt":"2024-06-30T14:53:01","slug":"binomial-theorem-exercise-problems","status":"publish","type":"post","link":"https:\/\/yanamtakshashila.com\/?p=25438","title":{"rendered":"BINOMIAL THEOREM (Exercise Problems with Solutions)"},"content":{"rendered":"\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\LARGE{\\color {purple} {PART- A}}\\]<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\">\\[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\">\\[\\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\\ \\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[T_{r + 1} = 10C_r\\ x^{10-r}\\ (\\frac{1}{x})^r \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[T_{r + 1} = 10C_r\\ x^{10-r}\\ 1^r\\ x^{-r}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = 10C_r\\ 1^r\\ x^{10-r-r}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ T_{r + 1} = 10C_r\\ 1^r\\ x^{10-2r}\\ \\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\">\\[\\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{6+2}{2})^{th}\\ term\\ =\\ (\\frac{8}{2})^{th}\\ term\\ =\\ 4^{th}\\ term\\ \\hspace{16cm}\\]<\/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\">\\[\\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\\ =\\  x,\\   n\\ =\\ &#8211; 4\\  \\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(1\\ +\\ x)^{-4}\\ =\\ 1\\ +\\ (-4)(x)\\ +\\ \\frac{-4(-4-1)}{1.\\ 2}\\ (x)^2\\ +\\ \\frac{-4(-4-1)(-4-2)}{1.\\ 2.\\ 3}\\ (x)^3\\ &#8230;&#8230;.\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(1\\ +\\ x)^{-2}\\ =\\ 1\\ +\\ 2x\\ +\\ (-1) (-3) x^2\\ +\\ (-2) (-5)(-2) x^3\\ &#8230;&#8230;.\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(1\\ +\\ x)^{-4}\\ =\\ 1\\ -\\ 4x\\ +\\ 10 x^2\\ -\\ 20\\ x^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<!-- Ad1 -->\n<ins class=\"adsbygoogle\" style=\"display:block\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"8240817448\" 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- B}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[4.\\ \\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-group is-layout-constrained wp-block-group-is-layout-constrained\"><div class=\"wp-block-group__inner-container\">\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<\/div><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[5.\\ \\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<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^2},\\ n\\ =\\ 10,\\  r\\ =\\ 4\\ \\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[T_{4 + 1} = 10C_4\\ {(x^2)}^{10-4}\\ (\\frac{1}{x^2})^4 \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[T_{5} = 10C_4\\ {(x^2)}^{6}\\ \\frac{1^4}{x^8} \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[T_5 = 10C_4\\ x^{12}\\ 1^4\\ x^{-8}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = 10C_4\\ x^{12 &#8211; 8}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[T_5 = 10C_4\\ x^4\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[6.\\ \\color {red}{Find\\ the\\ 6^{th}\\ term\\ in\\ the\\ expansion\\ of}\\ (x^3\\ -\\ \\frac{1}{x^2})^{14}\\ \\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^3,\\  a\\ =\\ -\\frac{1}{x^2},\\ n\\ =\\ 14,\\  r\\ =\\ 5\\ \\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[T_{5 + 1} = 14C_5\\ {(x^3)}^{14-5}\\ (-\\frac{1}{x^2})^5 \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[T_{6} = 14C_5\\ {(x^3)}^{9}\\ \\frac{{-1}^5}{x^{10}} \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[T_6 = 14C_5\\ x^{27}\\ {(-1)}^5\\ x^{-10}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = -\\ 14C_5\\ x^{27 &#8211; 10}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[T_6 =\\ -\\ 14C_5\\ x^{17}\\ \\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\">\\[\\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\">\\[\\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{10+2}{2})^{th}\\ term\\ =\\ (\\frac{12}{2})^{th}\\ term\\ =\\ 6^{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\\ =\\ x,\\  a\\ =\\ \\frac{2}{x^3},\\ 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{2}{x^3})^5 \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[T_6 = 10C_5\\ (x)^5\\ \\frac{2^5}{x^15}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = 10C_5\\ \\ x^5\\ 2^5\\ x^{-15}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = 10C_5\\ \\ 2^5\\ \\ x^5\\ x^{-15}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = 10C_5\\ \\ 2^5\\ \\ x^{5-15}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{T_6 = 10C_5\\ 2^5\\ x^{-10}}\\ \\hspace{15cm}\\]<\/div>\n\n\n<p><iframe width=\"683\" height=\"384\" src=\"https:\/\/www.youtube.com\/embed\/vlAHERrT1xc\" title=\"BINOMIAL THEOREM - 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\">\\[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\">\\[\\color {blue}{Solution:}\\ Here\\ n\\ is\\ odd\\ number,\\ so\\ there\\ are\\ two\\ middle\\ terms\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(\\frac{n+1}{2})^{th}\\ term\\ =\\ (\\frac{11+1}{2})^{th}\\ term\\ =\\ (\\frac{12}{2})^{th}\\ term\\ =\\ 6^{th}\\ term\\ \\hspace{16cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(\\frac{n+3}{2})^{th}\\ term\\ =\\ (\\frac{11+3}{2})^{th}\\ term\\ =\\ (\\frac{14}{2})^{th}\\ term\\ =\\ 7^{th}\\ term\\ \\hspace{16cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\underline{To\\ find\\ T_6\\ :}\\ \\hspace{15cm}\\]<\/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\\ =\\ x^3,\\     a\\ =\\ \\frac{2}{x^3},\\ n\\ =\\ 11,\\  r\\ =\\ 5\\ \\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[T_{5 + 1} = 11C_5\\ (x^3)^{11-5}\\ (\\frac{2}{x^3})^5 \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[T_6 = 11C_5\\ (x^3)^6\\ \\frac{(2)^5}{(x^3)^5}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = 11C_5\\ \\ 2^5\\ \\ x^{18}\\ x^{-15}\\  \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = 11C_5\\ \\ 2^5\\ \\ x^{18\\ -\\ 15}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[T_6 = 11C_5\\ 2^5\\  x^3\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\underline{To\\ find\\ T_7\\ :}\\ \\hspace{15cm}\\]<\/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\\ =\\ x^3,\\     a\\ =\\ \\frac{2}{x^3},\\ n\\ =\\ 11,\\  r\\ =\\ 6\\ \\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[T_{6 + 1} = 11C_6\\ (x^3)^{11-6}\\ (\\frac{2}{x^3})^6 \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[T_7 = 11C_6\\ (x^3)^5\\ \\frac{(2)^6}{(x^3)^6}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = 11C_6\\ \\ 2^6\\ \\ x^{15}\\ x^{-18}\\  \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = 11C_6\\ \\ 2^6\\ \\ x^{15\\ -\\ 18}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[T_7 = 11C_6\\ 2^6\\  x^{-3}\\ \\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 h -->\n<ins 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[]).push({});\n<\/script>\n","protected":false},"excerpt":{"rendered":"","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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