{"id":61551,"date":"2023-09-07T15:11:00","date_gmt":"2023-09-07T09:41:00","guid":{"rendered":"https:\/\/www.kopykitab.com\/blog\/?p=61551"},"modified":"2023-11-03T10:01:59","modified_gmt":"2023-11-03T04:31:59","slug":"rd-sharma-solutions-class-9-maths-chapter-4-algebraic-identities","status":"publish","type":"post","link":"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-9-maths-chapter-4-algebraic-identities\/","title":{"rendered":"RD Sharma Solutions Class 9 Maths Chapter 4 &#8211; Algebraic Identities (Updated for 2024)"},"content":{"rendered":"\n<p><img class=\"alignnone size-full wp-image-124413\" src=\"https:\/\/www.kopykitab.com\/blog\/wp-content\/uploads\/2021\/09\/RD-Sharma-Solutions-Class-9-Maths-Chapter-4-Algebraic-Identities.png\" alt=\"RD-Sharma-Solutions-Class-9-Maths-Chapter-4-Algebraic-Identities\" width=\"1200\" height=\"675\" srcset=\"https:\/\/www.kopykitab.com\/blog\/wp-content\/uploads\/2021\/09\/RD-Sharma-Solutions-Class-9-Maths-Chapter-4-Algebraic-Identities.png 1200w, https:\/\/www.kopykitab.com\/blog\/wp-content\/uploads\/2021\/09\/RD-Sharma-Solutions-Class-9-Maths-Chapter-4-Algebraic-Identities-768x432.png 768w\" sizes=\"(max-width: 1200px) 100vw, 1200px\" \/><\/p>\n<p><span style=\"font-weight: 400;\"><strong>RD Sharma Solutions Class 9 Maths Chapter 4 &#8211; Algebraic Identities:<\/strong> Have doubts with Mathematics exam preparation? Have you started looking for some good study material yet? If not, don&#8217;t worry, we bring you <a href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions\/\" target=\"_blank\" rel=\"noopener\">RD Sharma Solutions Class 9 Maths<\/a> Chapter 4 Algebraic Identities for solving the confusion with Algebraic identities. To know more, read the whole blog.<\/span><\/p>\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_47_1 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"ez-toc-toggle-icon-1\"><label for=\"item-69e5f17a727f4\" aria-label=\"Table of Content\"><span style=\"display: flex;align-items: center;width: 35px;height: 30px;justify-content: center;direction:ltr;\"><svg style=\"fill: #000000;color:#000000\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #000000;color:#000000\" 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Chapter 4 Algebraic Identities: PDF<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-9-maths-chapter-4-algebraic-identities\/#rd-sharma-solutions-class-9-maths-chapter-4-algebraic-identities-exercise-wise-solutions\" title=\"RD Sharma Solutions Class 9 Maths Chapter 4 Algebraic Identities: Exercise-wise Solutions\">RD Sharma Solutions Class 9 Maths Chapter 4 Algebraic Identities: Exercise-wise Solutions<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-9-maths-chapter-4-algebraic-identities\/#access-answers-of-rd-sharma-solutions-class-9-maths-chapter-4-%e2%80%93-algebraic-identities\" title=\"Access answers of RD Sharma Solutions Class 9 Maths Chapter 4 &#8211; Algebraic Identities\">Access answers of RD Sharma Solutions Class 9 Maths Chapter 4 &#8211; Algebraic Identities<\/a><ul class='ez-toc-list-level-3'><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-9-maths-chapter-4-algebraic-identities\/#exercise-41-page-no-46\" title=\"Exercise 4.1 Page No: 4.6\">Exercise 4.1 Page No: 4.6<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-9-maths-chapter-4-algebraic-identities\/#exercise-42-page-no-411\" title=\"Exercise 4.2 Page No: 4.11\">Exercise 4.2 Page No: 4.11<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-9-maths-chapter-4-algebraic-identities\/#exercise-43-page-no-419\" title=\"Exercise 4.3 Page No: 4.19\">Exercise 4.3 Page No: 4.19<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-9-maths-chapter-4-algebraic-identities\/#exercise-44-page-no-423\" title=\"Exercise 4.4 Page No: 4.23\">Exercise 4.4 Page No: 4.23<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-9-maths-chapter-4-algebraic-identities\/#exercise-45-page-no-428\" title=\"Exercise 4.5 Page No: 4.28\">Exercise 4.5 Page No: 4.28<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-9-maths-chapter-4-algebraic-identities\/#exercise-vsaqs-page-no-428\" title=\"Exercise VSAQs Page No: 4.28\">Exercise VSAQs Page No: 4.28<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-9-maths-chapter-4-algebraic-identities\/#rd-sharma-solutions-for-class-9-maths-chapter-4-algebraic-identities\" title=\"RD Sharma Solutions for Class 9 Maths Chapter 4 Algebraic Identities\">RD Sharma Solutions for Class 9 Maths Chapter 4 Algebraic Identities<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-11\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-9-maths-chapter-4-algebraic-identities\/#important-topics-rd-sharma-solutions-class-9-maths-chapter-4\" title=\"Important Topics: RD Sharma Solutions Class 9 Maths Chapter 4&nbsp;\">Important Topics: RD Sharma Solutions Class 9 Maths Chapter 4&nbsp;<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-12\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-9-maths-chapter-4-algebraic-identities\/#faqs-on-rd-sharma-solutions-class-9-maths-chapter-4-%e2%80%93-algebraic-identities\" title=\"FAQs on RD Sharma Solutions Class 9 Maths Chapter 4 &#8211; Algebraic Identities\">FAQs on RD Sharma Solutions Class 9 Maths Chapter 4 &#8211; Algebraic Identities<\/a><ul class='ez-toc-list-level-3'><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-13\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-9-maths-chapter-4-algebraic-identities\/#can-i-access-the-rd-sharma-solutions-for-class-9-maths-chapter-4-pdf-offline\" title=\"Can I access the RD Sharma Solutions for Class 9 Maths Chapter 4\u00a0PDF offline?\">Can I access the RD Sharma Solutions for Class 9 Maths Chapter 4\u00a0PDF offline?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-14\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-9-maths-chapter-4-algebraic-identities\/#from-where-can-i-download-the-pdf-of-rd-sharma-solutions-class-9-maths-chapter-4\" title=\"From where can I download the PDF of RD Sharma Solutions Class 9 Maths Chapter 4?\">From where can I download the PDF of RD Sharma Solutions Class 9 Maths Chapter 4?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-15\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-9-maths-chapter-4-algebraic-identities\/#how-much-does-it-cost-to-download-the-pdf-of-rd-sharma-solutions-for-class-9-maths-chapter-4\" title=\"How much does it cost to download the PDF of RD Sharma Solutions for Class 9 Maths Chapter 4?\">How much does it cost to download the PDF of RD Sharma Solutions for Class 9 Maths Chapter 4?<\/a><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n<h2><span class=\"ez-toc-section\" id=\"download-rd-sharma-solutions-class-9-maths-chapter-4-algebraic-identities-pdf\"><\/span><strong>Download RD Sharma Solutions Class 9 Maths Chapter 4 Algebraic Identities: PDF<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><a href=\"https:\/\/www.kopykitab.com\/blog\/wp-content\/uploads\/2021\/09\/rd-4-1.pdf\">RD Sharma Solutions Class 9 Maths Chapter 4<\/a><\/p>\n<div id=\"example1\" style=\"text-align: justify;\">&nbsp;<\/div>\n<p style=\"text-align: justify;\"><style>\n.pdfobject-container { height: 500px;}<br \/>\n.pdfobject { border: 1px solid #666; }<br \/>\n<\/style><\/p>\n<p style=\"text-align: justify;\"><script src=\"https:\/\/www.kopykitab.com\/_utility\/js\/pdfobject.min.js\"><\/script><br><script>PDFObject.embed(\"https:\/\/www.kopykitab.com\/blog\/wp-content\/uploads\/2021\/09\/rd-4-1.pdf\", \"#example1\");<\/script><\/p>\n<h2><span class=\"ez-toc-section\" id=\"rd-sharma-solutions-class-9-maths-chapter-4-algebraic-identities-exercise-wise-solutions\"><\/span><strong>RD Sharma Solutions Class 9 Maths Chapter 4 Algebraic Identities: Exercise-wise Solutions<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td style=\"width: 100%;\"><a href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-chapter-4-class-9-maths-exercise-4-1-solutions\/\" target=\"_blank\" rel=\"noopener\">RD Sharma Solutions Class 9 Chapter 4 Exercise 4.1<\/a><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 100%;\"><a href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-chapter-4-class-9-maths-exercise-4-2-solutions\/\" target=\"_blank\" rel=\"noopener\">RD Sharma Solutions Class 9 Chapter 4 Exercise 4.2<\/a><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 100%;\"><a href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-chapter-4-class-9-maths-exercise-4-3-solutions\/\" target=\"_blank\" rel=\"noopener\">RD Sharma Solutions Class 9 Chapter 4 Exercise 4.3<\/a><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 100%;\"><a href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-chapter-4-class-9-maths-exercise-4-4-solutions\/\" target=\"_blank\" rel=\"noopener\">RD Sharma Solutions Class 9 Chapter 4 Exercise 4.4<\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2><span class=\"ez-toc-section\" id=\"access-answers-of-rd-sharma-solutions-class-9-maths-chapter-4-%e2%80%93-algebraic-identities\"><\/span><strong>Access answers of RD Sharma Solutions Class 9 Maths Chapter 4 &#8211; Algebraic Identities<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3><span class=\"ez-toc-section\" id=\"exercise-41-page-no-46\"><\/span>Exercise 4.1 Page No: 4.6<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Question 1: Evaluate each of the following using identities:<\/strong><\/p>\n<p><strong>(i) (2x \u2013 1\/x)<sup>2<\/sup><\/strong><\/p>\n<p><strong>(ii) (2x + y) (2x \u2013 y)<\/strong><\/p>\n<p><strong>(iii) (a<sup>2<\/sup>b \u2013 b<sup>2<\/sup>a)<sup>2<\/sup><\/strong><\/p>\n<p><strong>(iv) (a \u2013 0.1) (a + 0.1)<\/strong><\/p>\n<p><strong>(v) (1.5.x<sup>2<\/sup>&nbsp;\u2013 0.3y<sup>2<\/sup>) (1.5x<sup>2<\/sup>&nbsp;+ 0.3y<sup>2<\/sup>)<\/strong><\/p>\n<p><strong>Solution<\/strong>:<\/p>\n<p><strong>(i)&nbsp;<\/strong>(2x \u2013 1\/x)<sup>2<\/sup><\/p>\n<p><em>[Use identity: (a \u2013 b)<sup>2<\/sup>&nbsp;= a<sup>2<\/sup>&nbsp;+ b<sup>2<\/sup>&nbsp;\u2013 2ab ]<\/em><\/p>\n<p>(2x \u2013 1\/x)<sup>2&nbsp;<\/sup>= (2x)<sup>2<\/sup>&nbsp;+ (1\/x)<sup>2&nbsp;<\/sup>\u2013 2 (2x)(1\/x)<\/p>\n<p><em>= 4x<sup>2&nbsp;<\/sup>+ 1\/x<sup>2&nbsp;<\/sup>\u2013 4<\/em><\/p>\n<p><strong>(ii)&nbsp;<\/strong>(2x + y) (2x \u2013 y)<\/p>\n<p><em>[Use identity: (a \u2013 b)(a + b) = a<sup>2<\/sup>&nbsp;\u2013 b<sup>2<\/sup>&nbsp;]<\/em><\/p>\n<p>(2x + y) (2x \u2013 y) = (2x )<sup>2&nbsp;<\/sup>\u2013 (y)<sup>2<\/sup><\/p>\n<p><em>= 4x<sup>2&nbsp;<\/sup>\u2013 y<strong><sup>2<\/sup><\/strong><\/em><\/p>\n<p><strong>(iii)&nbsp;<\/strong>(a<sup>2<\/sup>b \u2013 b<sup>2<\/sup>a)<sup>2<\/sup><\/p>\n<p><em>[Use identity: (a \u2013 b)<sup>2<\/sup>&nbsp;= a<sup>2<\/sup>&nbsp;+ b<sup>2<\/sup>&nbsp;\u2013 2ab ]<\/em><\/p>\n<p>(a<sup>2<\/sup>b \u2013 b<sup>2<\/sup>a)<sup>2&nbsp;<\/sup>= (a<sup>2<\/sup>b)<sup>&nbsp;2&nbsp;<\/sup>+ (b<sup>2<\/sup>a)<sup>2&nbsp;<\/sup>\u2013 2 (a<sup>2<\/sup>b)( b<sup>2<\/sup>a)<\/p>\n<p><em>= a<sup>4<\/sup>b<sup>&nbsp;2&nbsp;<\/sup>+ b<sup>4<\/sup>a<sup>2&nbsp;<\/sup>\u2013 2 a<sup>3<\/sup>b<sup>3<\/sup><\/em><\/p>\n<p><strong>(iv)&nbsp;<\/strong>(a \u2013 0.1) (a + 0.1)<\/p>\n<p><em>[Use identity: (a \u2013 b)(a + b) = a<sup>2<\/sup>&nbsp;\u2013 b<sup>2<\/sup>&nbsp;]<\/em><\/p>\n<p>(a \u2013 0.1) (a + 0.1) = (a)<sup>2<\/sup>&nbsp;\u2013 (0.1)<sup>2<\/sup><\/p>\n<p><em>= (a)<sup>2<\/sup>&nbsp;\u2013 0.01<\/em><\/p>\n<p><strong>(v)&nbsp;<\/strong>(1.5 x<sup>2<\/sup>&nbsp;\u2013 0.3y<sup>2<\/sup>) (1.5 x<sup>2<\/sup>&nbsp;+ 0.3y<sup>2<\/sup>)<\/p>\n<p><em>[Use identity: (a \u2013 b)(a + b) = a<sup>2<\/sup>&nbsp;\u2013 b<sup>2<\/sup>&nbsp;]<\/em><\/p>\n<p>(1.5 x<sup>2<\/sup>&nbsp;\u2013 0.3y<sup>2<\/sup>) (1.5x<sup>2<\/sup>&nbsp;+ 0.3y<sup>2<\/sup>) = (1.5 x<sup>2<\/sup>&nbsp;)&nbsp;<sup>2&nbsp;<\/sup>\u2013 (0.3y<sup>2<\/sup>)<sup>2<\/sup><\/p>\n<p><em>= 2.25 x<sup>4&nbsp;<\/sup>\u2013 0.09y<sup>4<\/sup><\/em><\/p>\n<p><strong>Question 2: Evaluate each of the following using identities:<\/strong><\/p>\n<p><strong>(i) (399)<sup>2<\/sup><\/strong><\/p>\n<p><strong>(ii) (0.98)<sup>2<\/sup><\/strong><\/p>\n<p><strong>(iii) 991 x 1009<\/strong><\/p>\n<p><strong>(iv) 117 x 83<\/strong><\/p>\n<p><strong>Solution<\/strong>:<\/p>\n<p><strong>(i)<\/strong><\/p>\n<p><img title=\"RD sharma class 9 maths chapter 4 ex 4.1 question 2 part 1 Solution\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2022\/05\/rd-sharma-solutions-class-9-maths-chapter-4-ex-4-1-question.png\" alt=\"RD sharma class 9 maths chapter 4 ex 4.1 question 2 part 1 Solution\"><\/p>\n<p><strong>(ii)<\/strong><\/p>\n<p><strong><img title=\"RD sharma class 9 maths chapter 4 ex 4.1 question 2 part 2 Solution\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/10\/rd-sharma-class-9-maths-chapter-4-ex-4-1-question-1.png\" alt=\"RD sharma class 9 maths chapter 4 ex 4.1 question 2 part 2 Solution\"><\/strong><\/p>\n<p><strong>(iii)<\/strong><\/p>\n<p><strong><img title=\"RD sharma class 9 maths chapter 4 ex 4.1 question 2 part 3 Solution\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/10\/rd-sharma-class-9-maths-chapter-4-ex-4-1-question-2.png\" alt=\"RD sharma class 9 maths chapter 4 ex 4.1 question 2 part 3 Solution\"><\/strong><\/p>\n<p><strong>(iv)<\/strong><\/p>\n<p><img title=\"RD sharma class 9 maths chapter 4 ex 4.1 question 2 part 4 Solution\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/10\/rd-sharma-class-9-maths-chapter-4-ex-4-1-question-3.png\" alt=\"RD sharma class 9 maths chapter 4 ex 4.1 question 2 part 4 Solution\"><\/p>\n<p><strong>Question 3: Simplify each of the following:<\/strong><\/p>\n<p><strong>(i) 175 x 175 +2 x 175 x 25 + 25 x 25<\/strong><\/p>\n<p><strong>(ii) 322 x 322 \u2013 2 x 322 x 22 + 22 x 22<\/strong><\/p>\n<p><strong>(iii) 0.76 x 0.76 + 2 x 0.76 x 0.24 + 0.24 x 0.24<\/strong><\/p>\n<p><strong>(iv)<\/strong><\/p>\n<p><strong><img title=\"RD sharma class 9 maths chapter 4 ex 4.1 question 3\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/10\/rd-sharma-class-9-maths-chapter-4-ex-4-1-question-4.png\" alt=\"RD sharma class 9 maths chapter 4 ex 4.1 question 3\"><\/strong><\/p>\n<p><strong>Solution<\/strong>:<\/p>\n<p><strong>(i)<\/strong>&nbsp;175 x 175 +2 x 175 x 25 + 25 x 25 = (175)<sup>2<\/sup>&nbsp;+ 2 (175) (25) + (25)<sup>2<\/sup><\/p>\n<p>= (175 + 25)<sup>2<\/sup><\/p>\n<p><em>[Because a<sup>2<\/sup>+ b<sup>2<\/sup>+2ab = (a+b)<sup>2<\/sup>&nbsp;]<\/em><\/p>\n<p>= (200)<sup>2<\/sup><\/p>\n<p><em>= 40000<\/em><\/p>\n<p>So, 175 x 175 +2 x 175 x 25 + 25 x 25 = 40000.<\/p>\n<p><strong>(ii)<\/strong>&nbsp;322 x 322 \u2013 2 x 322 x 22 + 22 x 22<\/p>\n<p>= (322)<sup>2<\/sup>&nbsp;\u2013 2 x 322 x 22 + (22)<sup>2<\/sup><\/p>\n<p>= (322 \u2013 22)<sup>2<\/sup><\/p>\n<p><em>[Because a<sup>2<\/sup>+ b<sup>2<\/sup>-2ab = (a-b)<sup>2<\/sup>]<\/em><\/p>\n<p>= (300)<sup>2<\/sup><\/p>\n<p><em>= 90000<\/em><\/p>\n<p>So, 322 x 322 \u2013 2 x 322 x 22 + 22 x 22= 90000.<\/p>\n<p><strong>(iii)<\/strong>&nbsp;0.76 x 0.76 + 2 x 0.76 x 0.24 + 0.24 x 0.24<\/p>\n<p>= (0.76)<sup>&nbsp;2<\/sup>&nbsp;+ 2 x 0.76 x 0.24 + (0.24)<sup>&nbsp;2<\/sup><\/p>\n<p>= (0.76+0.24)<sup>&nbsp;2<\/sup><\/p>\n<p><em>[ Because a<sup>2<\/sup>+ b<sup>2<\/sup>+2ab = (a+b)<sup>2<\/sup>]<\/em><\/p>\n<p>= (1.00)<sup>2<\/sup><\/p>\n<p><em>= 1<\/em><\/p>\n<p>So, 0.76 x 0.76 + 2 x 0.76 x 0.24 + 0.24 x 0.24 = 1.<\/p>\n<p><strong>(iv)<\/strong><\/p>\n<p><img title=\"RD sharma class 9 maths chapter 4 ex 4.1 question 3 part 4 solution\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/10\/rd-sharma-class-9-maths-chapter-4-ex-4-1-question-5.png\" alt=\"RD sharma class 9 maths chapter 4 ex 4.1 question 3 part 4 solution\"><\/p>\n<p><strong>Question 4: If x + 1\/x = 11, find the value of x<sup>2<\/sup>&nbsp;+1\/x<sup>2<\/sup>.<\/strong><\/p>\n<p><strong>Solution:<\/strong><\/p>\n<p><img title=\"RD sharma class 9 maths chapter 4 ex 4.1 question 4\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/10\/rd-sharma-class-9-maths-chapter-4-ex-4-1-question-6.png\" alt=\"RD sharma class 9 maths chapter 4 ex 4.1 question 4\"><\/p>\n<p><strong>Question 5: If x \u2013 1\/x = -1, find the value of x<sup>2<\/sup>&nbsp;+1\/x<sup>2<\/sup>.<\/strong><\/p>\n<p><strong>Solution:<\/strong><\/p>\n<p><strong><img title=\"RD sharma class 9 maths chapter 4 ex 4.1 question 5 solution\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/10\/rd-sharma-class-9-maths-chapter-4-ex-4-1-question-7.png\" alt=\"RD sharma class 9 maths chapter 4 ex 4.1 question 5 solution\"><\/strong><\/p>\n<hr>\n<h3><span class=\"ez-toc-section\" id=\"exercise-42-page-no-411\"><\/span>Exercise 4.2 Page No: 4.11<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Question 1: Write the following in the expanded form:<\/strong><\/p>\n<p><strong>(i) (a + 2b + c)<sup>2<\/sup><\/strong><\/p>\n<p><strong>(ii) (2a \u2212 3b \u2212 c)<sup>2<\/sup><\/strong><\/p>\n<p><strong>(iii) (\u22123x+y+z)<sup>2<\/sup><\/strong><\/p>\n<p><strong>(iv) (m+2n\u22125p)<sup>2<\/sup><\/strong><\/p>\n<p><strong>(v) (2+x\u22122y)<sup>2<\/sup><\/strong><\/p>\n<p><strong>(vi) (a<sup>2&nbsp;<\/sup>+b<sup>2&nbsp;<\/sup>+c<sup>2<\/sup>)<sup>&nbsp;2<\/sup><\/strong><\/p>\n<p><strong>(vii) (ab+bc+ca)<sup>&nbsp;2<\/sup><\/strong><\/p>\n<p><strong>(viii) (x\/y+y\/z+z\/x)2<\/strong><\/p>\n<p><strong>(ix) (a\/bc + b\/ac + c\/ab)<sup>&nbsp;2<\/sup><\/strong><\/p>\n<p><strong>(x) (x+2y+4z)<sup>&nbsp;2<\/sup><\/strong><\/p>\n<p><strong>(xi) (2x\u2212y+z)<sup>&nbsp;2<\/sup><\/strong><\/p>\n<p><strong>(xii) (\u22122x+3y+2z)<sup>&nbsp;2<\/sup><\/strong><\/p>\n<p><strong>Solution<\/strong>:<\/p>\n<p>Using identities:<\/p>\n<p><em>(x + y + z)<sup>2<\/sup>&nbsp;= x<sup>2<\/sup>&nbsp;+ y<sup>2<\/sup>&nbsp;+ z<sup>2<\/sup>&nbsp;+ 2xy + 2yz + 2xz<\/em><\/p>\n<p><strong>(i)<\/strong>&nbsp;(a + 2b + c)<sup>2<\/sup><\/p>\n<p>= a<sup>2<\/sup>&nbsp;+ (2b)<sup>&nbsp;2<\/sup>&nbsp;+ c<sup>2<\/sup>&nbsp;+ 2a(2b) + 2ac + 2(2b)c<\/p>\n<p><em>= a<sup>2<\/sup>&nbsp;+ 4b<sup>2<\/sup>&nbsp;+ c<sup>2<\/sup>&nbsp;+ 4ab + 2ac + 4bc<\/em><\/p>\n<p><strong>(ii)<\/strong>&nbsp;(2a \u2212 3b \u2212 c)<sup>2<\/sup><\/p>\n<p>= [(2a) + (\u22123b) + (\u2212c)]<sup>2<\/sup><\/p>\n<p>= (2a)<sup>&nbsp;2&nbsp;<\/sup>+ (\u22123b)<sup>&nbsp;2&nbsp;<\/sup>+ (\u2212c)<sup>&nbsp;2&nbsp;<\/sup>+ 2(2a)(\u22123b) + 2(\u22123b)(\u2212c) + 2(2a)(\u2212c)<\/p>\n<p><em>= 4a<sup>2&nbsp;<\/sup>+ 9b<sup>2&nbsp;<\/sup>+ c<sup>2&nbsp;<\/sup>\u2212 12ab + 6bc \u2212 4ca<\/em><\/p>\n<p><strong>(iii)<\/strong>&nbsp;(\u22123x+y+z)<sup>2<\/sup><\/p>\n<p>= [(\u22123x)<sup>&nbsp;2&nbsp;<\/sup>+ y<sup>2&nbsp;<\/sup>+ z<sup>2&nbsp;<\/sup>+ 2(\u22123x)y + 2yz + 2(\u22123x)z<\/p>\n<p><em>= 9x<sup>2&nbsp;<\/sup>+ y<sup>2&nbsp;<\/sup>+ z<sup>2&nbsp;<\/sup>\u2212 6xy + 2yz \u2212 6xz<\/em><\/p>\n<p><strong>(iv)<\/strong>&nbsp;(m+2n\u22125p)<sup>2<\/sup><\/p>\n<p>= m<sup>2&nbsp;<\/sup>+ (2n)<sup>&nbsp;2&nbsp;<\/sup>+ (\u22125p)<sup>&nbsp;2&nbsp;<\/sup>+ 2m \u00d7 2n + (2\u00d72n\u00d7\u22125p) + 2m \u00d7 \u22125p<\/p>\n<p><em>= m<sup>2&nbsp;<\/sup>+ 4n<sup>2<\/sup>&nbsp;+ 25p<sup>2&nbsp;<\/sup>+ 4mn \u2212 20np \u2212 10pm<\/em><\/p>\n<p><strong>(v)<\/strong>&nbsp;(2+x\u22122y)<sup>2<\/sup><\/p>\n<p>= 2<sup>2&nbsp;<\/sup>+ x<sup>2&nbsp;<\/sup>+ (\u22122y)<sup>&nbsp;2&nbsp;<\/sup>+ 2(2)(x) + 2(x)(\u22122y) + 2(2)(\u22122y)<\/p>\n<p><em>= 4 + x<sup>2&nbsp;<\/sup>+ 4y<sup>2&nbsp;<\/sup>+ 4 x \u2212 4xy \u2212 8y<\/em><\/p>\n<p><strong>(vi)<\/strong>&nbsp;(a<sup>2&nbsp;<\/sup>+b<sup>2&nbsp;<\/sup>+c<sup>2<\/sup>)<sup>&nbsp;2<\/sup><\/p>\n<p>= (a<sup>2<\/sup>)<sup>&nbsp;2&nbsp;<\/sup>+ (b<sup>2<\/sup>)<sup>&nbsp;2&nbsp;<\/sup>+ (c<sup>2&nbsp;<\/sup>)<sup>&nbsp;2&nbsp;<\/sup>+ 2a<sup>2&nbsp;<\/sup>b<sup>2&nbsp;<\/sup>+ 2b<sup>2<\/sup>c<sup>2&nbsp;<\/sup>+ 2a<sup>2<\/sup>c<sup>2<\/sup><\/p>\n<p><em>= a<sup>4&nbsp;<\/sup>+ b<sup>4&nbsp;<\/sup>+ c<sup>4&nbsp;<\/sup>+ 2a<sup>2<\/sup>&nbsp;b<sup>2&nbsp;<\/sup>+ 2b<sup>2&nbsp;<\/sup>c<sup>2&nbsp;<\/sup>+ 2c<sup>2&nbsp;<\/sup>a<sup>2<\/sup><\/em><\/p>\n<p><strong>(vii)<\/strong>&nbsp;(ab+bc+ca)<sup>&nbsp;2<\/sup><\/p>\n<p>= (ab)<sup>2<\/sup>&nbsp;+ (bc)<sup>&nbsp;2&nbsp;<\/sup>+ (ca)<sup>&nbsp;2&nbsp;<\/sup>+ 2(ab)(bc) + 2(bc)(ca) + 2(ab)(ca)<\/p>\n<p><em>= a<sup>2&nbsp;<\/sup>b<sup>2&nbsp;<\/sup>+ b<sup>2<\/sup>c<sup>2&nbsp;<\/sup>+ c<sup>2&nbsp;<\/sup>a<sup>2&nbsp;<\/sup>+ 2(ac)b<sup>2&nbsp;<\/sup>+ 2(ab)(c)<sup>&nbsp;2&nbsp;<\/sup>+ 2(bc)(a)<sup>&nbsp;2<\/sup><\/em><\/p>\n<p><strong>(viii)<\/strong>&nbsp;(x\/y+y\/z+z\/x)<sup>2<\/sup><\/p>\n<p><img title=\"RD sharma class 9 maths chapter 4 ex 4.2 question 1 solution\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2020\/10\/rd-sharma-class-9-maths-chapter-4-ex-4-2-question.png\" alt=\"RD sharma class 9 maths chapter 4 ex 4.2 question 1 solution\"><\/p>\n<p><strong>(ix)<\/strong>&nbsp;(a\/bc + b\/ac + c\/ab)<sup>&nbsp;2<\/sup><\/p>\n<p><img title=\"RD sharma class 9 maths chapter 4 ex 4.2 question 1 solution\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/10\/rd-sharma-class-9-maths-chapter-4-ex-4-2-question-1.png\" alt=\"RD sharma class 9 maths chapter 4 ex 4.2 question 1 solution\"><\/p>\n<p><strong>(x)<\/strong>&nbsp;(x+2y+4z)<sup>&nbsp;2<\/sup><\/p>\n<p>= x<sup>2&nbsp;<\/sup>+ (2y)<sup>&nbsp;2&nbsp;<\/sup>+ (4z)<sup>&nbsp;2&nbsp;<\/sup>+ (2x)(2y) + 2(2y)(4z) + 2x(4z)<\/p>\n<p><em>= x<sup>2&nbsp;<\/sup>+ 4y<sup>2&nbsp;<\/sup>+ 16z<sup>2&nbsp;<\/sup>+ 4xy + 16yz + 8xz<\/em><\/p>\n<p><strong>(xi)<\/strong>&nbsp;(2x\u2212y+z)<sup>&nbsp;2<\/sup><\/p>\n<p>= (2x)<sup>&nbsp;2&nbsp;<\/sup>+ (\u2212y)<sup>&nbsp;2&nbsp;<\/sup>+ (z)<sup>&nbsp;2&nbsp;<\/sup>+ 2(2x)(\u2212y) + 2(\u2212y)(z) + 2(2x)(z)<\/p>\n<p><em>= 4x<sup>2&nbsp;<\/sup>+ y<sup>2&nbsp;<\/sup>+ z<sup>2&nbsp;<\/sup>\u2212 4xy\u22122yz+4xz<\/em><\/p>\n<p><strong>(xii)<\/strong>&nbsp;(\u22122x+3y+2z)<sup>&nbsp;2<\/sup><\/p>\n<p>= (\u22122x)<sup>&nbsp;2&nbsp;<\/sup>+ (3y)<sup>&nbsp;2&nbsp;<\/sup>+ ( 2z)<sup>&nbsp;2&nbsp;<\/sup>+ 2(\u22122x)(3y)+2(3y)(2z)+2(\u22122x)(2z)<\/p>\n<p><em>= 4x<sup>2&nbsp;<\/sup>+ 9y<sup>2&nbsp;<\/sup>+ 4z<sup>2&nbsp;<\/sup>\u221212xy+12yz\u22128xz<\/em><\/p>\n<p><strong>Question 2: Simplify<\/strong><\/p>\n<p><strong>(i) (a + b + c)<sup>2<\/sup>&nbsp;+ (a \u2212 b + c)<sup>&nbsp;2<\/sup><\/strong><\/p>\n<p><strong>(ii) (a + b + c)<sup>2<\/sup>&nbsp;\u2212 (a \u2212 b + c)<sup>&nbsp;2<\/sup><\/strong><\/p>\n<p><strong>(iii) (a + b + c)<sup>2<\/sup>&nbsp;+ (a \u2013 b + c)<sup>&nbsp;2<\/sup>&nbsp;+ (a + b \u2212 c)<sup>&nbsp;2<\/sup><\/strong><\/p>\n<p><strong>(iv) (2x + p \u2212 c)<sup>2<\/sup>&nbsp;\u2212 (2x \u2212 p + c)<sup>&nbsp;2<\/sup><\/strong><\/p>\n<p><strong>(v) (x<sup>2<\/sup>&nbsp;+ y<sup>2<\/sup>&nbsp;\u2212 z<sup>2<\/sup>)<sup>&nbsp;2<\/sup>&nbsp;\u2212 (x<sup>2<\/sup>&nbsp;\u2212 y<sup>2<\/sup>&nbsp;+ z<sup>2<\/sup>)<sup>&nbsp;2<\/sup><\/strong><\/p>\n<p><strong>Solution:<\/strong><\/p>\n<p><strong>(i)<\/strong>&nbsp;(a + b + c)<sup>2<\/sup>&nbsp;+ (a \u2212 b + c)<sup>&nbsp;2<\/sup><\/p>\n<p>= (a<sup>2&nbsp;<\/sup>+ b<sup>2&nbsp;<\/sup>+ c<sup>2&nbsp;<\/sup>+ 2ab+2bc+2ca) + (a<sup>2&nbsp;<\/sup>+ (\u2212b)<sup>&nbsp;2&nbsp;<\/sup>+ c<sup>2&nbsp;<\/sup>\u22122ab\u22122bc+2ca)<\/p>\n<p><em>= 2a<sup>2&nbsp;<\/sup>+ 2 b<sup>2&nbsp;<\/sup>+ 2c<sup>2&nbsp;<\/sup>+ 4ca<\/em><\/p>\n<p><strong>(ii)<\/strong>&nbsp;(a + b + c)<sup>2<\/sup>&nbsp;\u2212 (a \u2212 b + c)<sup>&nbsp;2<\/sup><\/p>\n<p>= (a<sup>2&nbsp;<\/sup>+ b<sup>2&nbsp;<\/sup>+ c<sup>2&nbsp;<\/sup>+ 2ab+2bc+2ca) \u2212 (a<sup>2&nbsp;<\/sup>+ (\u2212b)<sup>&nbsp;2&nbsp;<\/sup>+ c<sup>2&nbsp;<\/sup>\u22122ab\u22122bc+2ca)<\/p>\n<p>= a<sup>2&nbsp;<\/sup>+ b<sup>2&nbsp;<\/sup>+ c<sup>2&nbsp;<\/sup>+ 2ab + 2bc + 2ca \u2212 a<sup>2&nbsp;<\/sup>\u2212 b<sup>2&nbsp;<\/sup>\u2212 c<sup>2&nbsp;<\/sup>+ 2ab + 2bc \u2212 2ca<\/p>\n<p><em>= 4ab + 4bc<\/em><\/p>\n<p><strong>(iii)<\/strong>&nbsp;(a + b + c)<sup>2<\/sup>&nbsp;+ (a \u2013 b + c)<sup>&nbsp;2<\/sup>&nbsp;+ (a + b \u2212 c)<sup>&nbsp;2<\/sup><\/p>\n<p>= a<sup>2<\/sup>&nbsp;+ b<sup>2<\/sup>&nbsp;+ c<sup>2<\/sup>&nbsp;+ 2ab + 2bc + 2ca + (a<sup>2<\/sup>&nbsp;+ b<sup>2<\/sup>&nbsp;+ (c)<sup>&nbsp;2<\/sup>&nbsp;\u2212 2ab \u2212 2cb + 2ca) + (a<sup>2<\/sup>&nbsp;+ b<sup>2<\/sup>&nbsp;+ c<sup>2<\/sup>&nbsp;+ 2ab \u2212 2bc \u2013 2ca)<\/p>\n<p><em>= 3 a<sup>2<\/sup>&nbsp;+ 3b<sup>2<\/sup>&nbsp;+ 3c<sup>2<\/sup>&nbsp;+ 2ab \u2212 2bc + 2ca<\/em><\/p>\n<p><strong>(iv)<\/strong>&nbsp;(2x + p \u2212 c)<sup>2<\/sup>&nbsp;\u2212 (2x \u2212 p + c)<sup>&nbsp;2<\/sup><\/p>\n<p>= [4x<sup>2<\/sup>&nbsp;+ p<sup>2<\/sup>&nbsp;+ c<sup>2<\/sup>&nbsp;+ 4xp \u2212 2pc \u2212 4xc] \u2212 [4x<sup>2<\/sup>&nbsp;+ p<sup>2<\/sup>&nbsp;+ c<sup>2<\/sup>&nbsp;\u2212 4xp\u2212 2pc + 4xc]<\/p>\n<p>= 4x<sup>2<\/sup>&nbsp;+ p<sup>2<\/sup>&nbsp;+ c<sup>2<\/sup>&nbsp;+ 4xp \u2212 2pc \u2212 4cx \u2212 4x<sup>2<\/sup>&nbsp;\u2212 p<sup>2<\/sup>&nbsp;\u2212 c<sup>2<\/sup>&nbsp;+ 4xp + 2pc\u2212 4cx<\/p>\n<p>= 8xp \u2212 8xc<\/p>\n<p><em>= 8(xp \u2212 xc)<\/em><\/p>\n<p><strong>(v)<\/strong>&nbsp;(x<sup>2<\/sup>&nbsp;+ y<sup>2<\/sup>&nbsp;\u2212 z<sup>2<\/sup>)<sup>&nbsp;2<\/sup>&nbsp;\u2212 (x<sup>2<\/sup>&nbsp;\u2212 y<sup>2<\/sup>&nbsp;+ z<sup>2<\/sup>)<sup>&nbsp;2<\/sup><\/p>\n<p>= (x<sup>2<\/sup>&nbsp;+ y<sup>2<\/sup>&nbsp;+ (\u2212z)<sup>&nbsp;2<\/sup>)<sup>&nbsp;2<\/sup>&nbsp;\u2212 (x<sup>2<\/sup>&nbsp;\u2212 y<sup>2<\/sup>&nbsp;+ z<sup>2<\/sup>)<sup>&nbsp;2<\/sup><\/p>\n<p>= [x<sup>4<\/sup>&nbsp;+ y<sup>4<\/sup>&nbsp;+ z<sup>4<\/sup>&nbsp;+ 2x<sup>2<\/sup>y<sup>2<\/sup>&nbsp;\u2013&nbsp;2y<sup>2<\/sup>z<sup>&nbsp;2<\/sup>&nbsp;\u2013 2x<sup>2<\/sup>z<sup>2<\/sup>&nbsp;\u2212 [x<sup>4<\/sup>&nbsp;+ y<sup>4<\/sup>&nbsp;+ z<sup>4<\/sup>&nbsp;\u2212 2x<sup>2<\/sup>y<sup>2<\/sup>&nbsp;\u2212 2y<sup>2<\/sup>z<sup>2<\/sup>&nbsp;+ 2x<sup>2<\/sup>z<sup>2<\/sup>]<\/p>\n<p><em>= 4x<sup>2<\/sup>y<sup>2&nbsp;<\/sup>\u2013 4z<sup>2<\/sup>x<sup>2<\/sup><\/em><\/p>\n<p><strong>Question 3: If a + b + c = 0 and a<sup>2<\/sup>&nbsp;+ b<sup>2<\/sup>&nbsp;+ c<sup>2<\/sup>&nbsp;= 16, find the value of ab + bc + ca.<\/strong><\/p>\n<p><strong>Solution<\/strong>:<\/p>\n<p>a + b + c = 0 and a<sup>2<\/sup>&nbsp;+ b<sup>2<\/sup>&nbsp;+ c<sup>2<\/sup>&nbsp;= 16 (given)<\/p>\n<p>Choose a + b + c = 0<\/p>\n<p>Squaring both sides,<\/p>\n<p>(a + b + c)<sup>2<\/sup>&nbsp;= 0<\/p>\n<p>a<sup>2<\/sup>&nbsp;+ b<sup>2<\/sup>&nbsp;+ c<sup>2<\/sup>&nbsp;+ 2(ab + bc + ca) = 0<\/p>\n<p>16 + 2(ab + bc + c) = 0<\/p>\n<p>2(ab + bc + ca) = -16<\/p>\n<p>ab + bc + ca = -16\/2 = -8<\/p>\n<p>or&nbsp;<em>ab + bc + ca = -8<\/em><\/p>\n<hr>\n<h3><span class=\"ez-toc-section\" id=\"exercise-43-page-no-419\"><\/span>Exercise 4.3 Page No: 4.19<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Question 1: Find the cube of each of the following binomial expressions:<\/strong><\/p>\n<p><strong>(i) (1\/x + y\/3)<\/strong><\/p>\n<p><strong>(ii) (3\/x \u2013 2\/x<sup>2<\/sup>)<\/strong><\/p>\n<p><strong>(iii) (2x + 3\/x)<\/strong><\/p>\n<p><strong>(iv) (4 \u2013 1\/3x)<\/strong><\/p>\n<p><strong>Solution:<\/strong><\/p>\n<p><em>[Using identities: (a + b)<sup>3<\/sup>&nbsp;= a<sup>3<\/sup>&nbsp;+ b<sup>3<\/sup>&nbsp;+ 3ab(a + b) and (a \u2013 b)<sup>3<\/sup>&nbsp;= a<sup>3<\/sup>&nbsp;\u2013 b<sup>3<\/sup>&nbsp;\u2013 3ab(a \u2013 b) ]<\/em><\/p>\n<p><strong>(i)<\/strong><\/p>\n<p><img title=\"RD sharma class 9 maths chapter 4 ex 4.3 Q1 part 1 solution\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/10\/rd-sharma-class-9-maths-chapter-4-ex-4-3-q1-part-1.png\" alt=\"RD sharma class 9 maths chapter 4 ex 4.3 Q1 part 1 solution\"><\/p>\n<p><strong>(ii)<\/strong><\/p>\n<p><img title=\"RD sharma class 9 maths chapter 4 ex 4.3 Q1 part 2 solution\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/10\/rd-sharma-class-9-maths-chapter-4-ex-4-3-q1-part-2.png\" alt=\"RD sharma class 9 maths chapter 4 ex 4.3 Q1 part 2 solution\"><\/p>\n<p><strong>(iii)<\/strong><\/p>\n<p><img title=\"RD sharma class 9 maths chapter 4 ex 4.3 Q1 part 3 solution\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/10\/rd-sharma-class-9-maths-chapter-4-ex-4-3-q1-part-3.png\" alt=\"RD sharma class 9 maths chapter 4 ex 4.3 Q1 part 3 solution\"><\/p>\n<p><strong>(iv)<\/strong><\/p>\n<p><img title=\"RD sharma class 9 maths chapter 4 ex 4.3 Q1 part 4 solution\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/10\/rd-sharma-class-9-maths-chapter-4-ex-4-3-q1-part-4.png\" alt=\"RD sharma class 9 maths chapter 4 ex 4.3 Q1 part 4 solution\"><\/p>\n<p><strong>Question 2: Simplify each of the following:<\/strong><\/p>\n<p><strong>(i) (x + 3)<sup>3<\/sup>&nbsp;+ (x \u2013 3)<sup>&nbsp;3<\/sup><\/strong><\/p>\n<p><strong>(ii) (x\/2 + y\/3)<sup>&nbsp;3&nbsp;<\/sup>\u2013 (x\/2 \u2013 y\/3)<sup>&nbsp;3<\/sup><\/strong><\/p>\n<p><strong>(iii) (x + 2\/x)<sup>&nbsp;3<\/sup>&nbsp;+ (x \u2013 2\/x)<sup>&nbsp;3<\/sup><\/strong><\/p>\n<p><strong>(iv) (2x \u2013 5y)<sup>&nbsp;3&nbsp;<\/sup>\u2013 (2x + 5y)<sup>&nbsp;3<\/sup><\/strong><\/p>\n<p><strong>Solution:<\/strong><\/p>\n<p>[Using identities:<\/p>\n<p>a<sup>3<\/sup>&nbsp;+ b<sup>3<\/sup>&nbsp;= (a + b)(a<sup>2<\/sup>&nbsp;+ b<sup>2<\/sup>&nbsp;\u2013 ab)<\/p>\n<p>a<sup>3<\/sup>&nbsp;\u2013 b<sup>3<\/sup>&nbsp;= (a \u2013 b)(a<sup>2<\/sup>&nbsp;+ b<sup>2<\/sup>&nbsp;+ ab)<\/p>\n<p>(a + b)(a-b) = a<sup>2<\/sup>&nbsp;\u2013 b<sup>2<\/sup><\/p>\n<p>(a + b)<sup>2&nbsp;<\/sup>= a<sup>2<\/sup>&nbsp;+ b<sup>2<\/sup>&nbsp;+ 2ab and<\/p>\n<p>(a \u2013 b)<sup>2&nbsp;<\/sup>= a<sup>2<\/sup>&nbsp;+ b<sup>2<\/sup>&nbsp;\u2013 2ab]<\/p>\n<p><strong>(i)<\/strong>&nbsp;(x + 3)<sup>3<\/sup>&nbsp;+ (x \u2013 3)<sup>&nbsp;3<\/sup><\/p>\n<p>Here a = (x + 3), b = (x \u2013 3)<\/p>\n<p><img title=\"RD sharma class 9 maths chapter 4 ex 4.3 Q2 solution part 1\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2020\/10\/rd-sharma-class-9-maths-chapter-4-ex-4-3-q2-soluti.png\" alt=\"RD sharma class 9 maths chapter 4 ex 4.3 Q2 solution part 1\"><\/p>\n<p><strong>(ii)<\/strong>&nbsp;(x\/2 + y\/3)<sup>&nbsp;3&nbsp;<\/sup>\u2013 (x\/2 \u2013 y\/3)<sup>&nbsp;3<\/sup><\/p>\n<p>Here a = (x\/2 + y\/3) and b = (x\/2 \u2013 y\/3)<\/p>\n<p><img title=\"RD sharma class 9 maths chapter 4 ex 4.3 Q2 solution part 2\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/10\/rd-sharma-class-9-maths-chapter-4-ex-4-3-q2-soluti-1.png\" alt=\"RD sharma class 9 maths chapter 4 ex 4.3 Q2 solution part 2\"><\/p>\n<p><strong>(iii)<\/strong>&nbsp;(x + 2\/x)<sup>&nbsp;3<\/sup>&nbsp;+ (x \u2013 2\/x)<sup>&nbsp;3<\/sup><\/p>\n<p>Here a = (x + 2\/x) and b = (x \u2013 2\/x)<\/p>\n<p><sup><img title=\"RD sharma class 9 maths chapter 4 ex 4.3 Q2 solution part 3\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/10\/rd-sharma-class-9-maths-chapter-4-ex-4-3-q2-soluti-2.png\" alt=\"RD sharma class 9 maths chapter 4 ex 4.3 Q2 solution part 3\"><\/sup><\/p>\n<p><strong>(iv)<\/strong>&nbsp;(2x \u2013 5y)<sup>&nbsp;3&nbsp;<\/sup>\u2013 (2x + 5y)<sup>&nbsp;3<\/sup><\/p>\n<p>Here a = (2x \u2013 5y) and b = 2x + 5y<\/p>\n<p><img title=\"RD sharma class 9 maths chapter 4 ex 4.3 Q2 solution part 4\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/10\/rd-sharma-class-9-maths-chapter-4-ex-4-3-q2-soluti-3.png\" alt=\"RD sharma class 9 maths chapter 4 ex 4.3 Q2 solution part 4\"><\/p>\n<p><strong>Question 3: If a + b = 10 and ab = 21, find the value of a<sup>3<\/sup>&nbsp;+ b<sup>3<\/sup>.<\/strong><\/p>\n<p><strong>Solution<\/strong>:<\/p>\n<p>a + b = 10, ab = 21 (given)<\/p>\n<p>Choose a + b = 10<\/p>\n<p>Cubing both sides,<\/p>\n<p>(a + b)<sup>3<\/sup>&nbsp;= (10)<sup>3<\/sup><\/p>\n<p>a<sup>3<\/sup>&nbsp;+ b<sup>3<\/sup>&nbsp;+ 3ab(a + b) = 1000<\/p>\n<p>a<sup>3<\/sup>&nbsp;+ b<sup>3<\/sup>&nbsp;+ 3 x 21 x 10 = 1000 (using given values)<\/p>\n<p>a<sup>3<\/sup>&nbsp;+ b<sup>3<\/sup>&nbsp;+ 630 = 1000<\/p>\n<p>a<sup>3<\/sup>&nbsp;+ b<sup>3<\/sup>&nbsp;= 1000 \u2013 630 = 370<\/p>\n<p>or&nbsp;<em>a<sup>3<\/sup>&nbsp;+ b<sup>3<\/sup>&nbsp;= 370<\/em><\/p>\n<p><strong>Question 4: If a \u2013 b = 4 and ab = 21, find the value of a<sup>3<\/sup>&nbsp;\u2013 b<sup>3<\/sup>.<\/strong><\/p>\n<p><strong>Solution<\/strong>:<\/p>\n<p>a \u2013 b = 4, ab= 21 (given)<\/p>\n<p>Choose a \u2013 b = 4<\/p>\n<p>Cubing both sides,<\/p>\n<p>(a \u2013 b)<sup>3<\/sup>&nbsp;= (4)<sup>3<\/sup><\/p>\n<p>a<sup>3<\/sup>&nbsp;\u2013 b<sup>3<\/sup>&nbsp;\u2013 3ab (a \u2013 b) = 64<\/p>\n<p>a<sup>3<\/sup>&nbsp;\u2013 b<sup>3<\/sup>&nbsp;\u2013 3 \u00d7 21 x 4 = 64 (using given values)<\/p>\n<p>a<sup>3<\/sup>&nbsp;\u2013 b<sup>3<\/sup>&nbsp;\u2013 252 = 64<\/p>\n<p>a<sup>3<\/sup>&nbsp;\u2013 b<sup>3<\/sup>&nbsp;= 64 + 252<\/p>\n<p>= 316<\/p>\n<p>Or&nbsp;<em>a<sup>3<\/sup>&nbsp;\u2013 b<sup>3<\/sup>&nbsp;= 316<\/em><\/p>\n<p><strong>Question 5: If x + 1\/x = 5, find the value of x<sup>3<\/sup>&nbsp;+ 1\/x<sup>3&nbsp;<\/sup>.<\/strong><\/p>\n<p><strong>Solution<\/strong>:<\/p>\n<p>Given: x + 1\/x = 5<\/p>\n<p>Apply Cube on x + 1\/x<\/p>\n<p><img title=\"RD sharma class 9 maths chapter 4 ex 4.3 Q5 solution\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/10\/rd-sharma-class-9-maths-chapter-4-ex-4-3-q5-soluti.png\" alt=\"RD sharma class 9 maths chapter 4 ex 4.3 Q5 solution\"><\/p>\n<p><strong>Question 6: If x \u2013 1\/x = 7, find the value of x<sup>3<\/sup>&nbsp;\u2013 1\/x<sup>3&nbsp;<\/sup>.<\/strong><\/p>\n<p><strong>Solution<\/strong>:<\/p>\n<p>Given: x \u2013 1\/x = 7<\/p>\n<p>Apply Cube on x \u2013 1\/x<\/p>\n<p><img title=\"RD sharma class 9 maths chapter 4 ex 4.3 Q6 solution\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/10\/rd-sharma-class-9-maths-chapter-4-ex-4-3-q6-soluti.png\" alt=\"RD sharma class 9 maths chapter 4 ex 4.3 Q6 solution\"><\/p>\n<p><strong>Question 7: If x \u2013 1\/x = 5, find the value of x<sup>3<\/sup>&nbsp;\u2013 1\/x<sup>3&nbsp;<\/sup>.<\/strong><\/p>\n<p><strong>Solution:<\/strong><\/p>\n<p>Given: x \u2013 1\/x = 5<\/p>\n<p>Apply Cube on x \u2013 1\/x<\/p>\n<p><strong><img title=\"RD sharma class 9 maths chapter 4 ex 4.3 Ques 7 solution\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/10\/rd-sharma-class-9-maths-chapter-4-ex-4-3-ques-7-so.png\" alt=\"RD sharma class 9 maths chapter 4 ex 4.3 Ques 7 solution\"><\/strong><\/p>\n<p><strong>Question 8: If (x<sup>2<\/sup>&nbsp;+ 1\/x<sup>2<\/sup>) = 51, find the value of x<sup>3<\/sup>&nbsp;\u2013 1\/x<sup>3<\/sup>.<\/strong><\/p>\n<p><strong>Solution:<\/strong><\/p>\n<p>We know that: (x \u2013 y)<sup>2<\/sup>&nbsp;= x<sup>2<\/sup>&nbsp;+ y<sup>2<\/sup>&nbsp;\u2013 2xy<\/p>\n<p>Replace y with 1\/x, we get<\/p>\n<p>(x \u2013 1\/x)<sup>2<\/sup>&nbsp;= x<sup>2<\/sup>&nbsp;+ 1\/x<sup>2<\/sup>&nbsp;\u2013 2<\/p>\n<p>Since (x<sup>2<\/sup>&nbsp;+ 1\/x<sup>2<\/sup>) = 51 (given)<\/p>\n<p>(x \u2013 1\/x)<sup>2<\/sup>&nbsp;= 51 \u2013 2 = 49<\/p>\n<p>or (x \u2013 1\/x) = \u00b17<\/p>\n<p>Now, Find x<sup>3<\/sup>&nbsp;\u2013 1\/x<sup>3<\/sup><\/p>\n<p>We know that, x<sup>3<\/sup>&nbsp;\u2013 y<sup>3<\/sup>&nbsp;= (x \u2013 y)(x<sup>2<\/sup>&nbsp;+ y<sup>2<\/sup>&nbsp;+ xy)<\/p>\n<p>Replace y with 1\/x, we get<\/p>\n<p>x<sup>3<\/sup>&nbsp;\u2013 1\/x<sup>3<\/sup>&nbsp;= (x \u2013 1\/x)(x<sup>2<\/sup>&nbsp;+ 1\/x<sup>2<\/sup>&nbsp;+ 1)<\/p>\n<p>Use (x \u2013 1\/x) = 7 and (x<sup>2<\/sup>&nbsp;+ 1\/x<sup>2<\/sup>) = 51<\/p>\n<p>x<sup>3<\/sup>&nbsp;\u2013 1\/x<sup>3<\/sup>&nbsp;= 7 x 52 = 364<\/p>\n<p><em>x<sup>3<\/sup>&nbsp;\u2013 1\/x<sup>3<\/sup>&nbsp;= 364<\/em><\/p>\n<p><strong>Question 9: If (x<sup>2<\/sup>&nbsp;+ 1\/x<sup>2<\/sup>) = 98, find the value of x<sup>3<\/sup>&nbsp;+ 1\/x<sup>3<\/sup>.<\/strong><\/p>\n<p><strong>Solution:<\/strong><\/p>\n<p>We know that: (x + y)<sup>2<\/sup>&nbsp;= x<sup>2<\/sup>&nbsp;+ y<sup>2<\/sup>&nbsp;+ 2xy<\/p>\n<p>Replace y with 1\/x, we get<\/p>\n<p>(x + 1\/x)<sup>2<\/sup>&nbsp;= x<sup>2<\/sup>&nbsp;+ 1\/x<sup>2<\/sup>&nbsp;+ 2<\/p>\n<p>Since (x<sup>2<\/sup>&nbsp;+ 1\/x<sup>2<\/sup>) = 98 (given)<\/p>\n<p>(x + 1\/x)<sup>2<\/sup>&nbsp;= 98 + 2 = 100<\/p>\n<p>or (x + 1\/x) = \u00b110<\/p>\n<p>Now, Find x<sup>3<\/sup>&nbsp;+ 1\/x<sup>3<\/sup><\/p>\n<p>We know that, x<sup>3<\/sup>&nbsp;+ y<sup>3<\/sup>&nbsp;= (x + y)(x<sup>2<\/sup>&nbsp;+ y<sup>2<\/sup>&nbsp;\u2013 xy)<\/p>\n<p>Replace y with 1\/x, we get<\/p>\n<p>x<sup>3<\/sup>&nbsp;+ 1\/x<sup>3<\/sup>&nbsp;= (x + 1\/x)(x<sup>2<\/sup>&nbsp;+ 1\/x<sup>2<\/sup>&nbsp;\u2013 1)<\/p>\n<p>Use (x + 1\/x) = 10 and (x<sup>2<\/sup>&nbsp;+ 1\/x<sup>2<\/sup>) = 98<\/p>\n<p>x<sup>3<\/sup>&nbsp;+ 1\/x<sup>3<\/sup>&nbsp;= 10 x 97 = 970<\/p>\n<p><em>x<sup>3<\/sup>&nbsp;+ 1\/x<sup>3<\/sup>&nbsp;= 970<\/em><\/p>\n<p><strong>Question 10: If 2x + 3y = 13 and xy = 6, find the value of 8x<sup>3<\/sup>&nbsp;+ 27y<sup>3<\/sup>.<\/strong><\/p>\n<p><strong>Solution:<\/strong><\/p>\n<p>Given: 2x + 3y = 13, xy = 6<\/p>\n<p>Cubing 2x + 3y = 13 both sides, we get<\/p>\n<p>(2x + 3y)<sup>3<\/sup>&nbsp;= (13)<sup>3<\/sup><\/p>\n<p>(2x)<sup>3<\/sup>&nbsp;+ (3y)&nbsp;<sup>3<\/sup>&nbsp;+ 3( 2x )(3y) (2x + 3y) = 2197<\/p>\n<p>8x<sup>3<\/sup>&nbsp;+ 27y<sup>3<\/sup>&nbsp;+ 18xy(2x + 3y) = 2197<\/p>\n<p>8x<sup>3<\/sup>&nbsp;+ 27y<sup>3<\/sup>&nbsp;+ 18 x 6 x 13 = 2197<\/p>\n<p>8x<sup>3<\/sup>&nbsp;+ 27y<sup>3<\/sup>&nbsp;+ 1404 = 2197<\/p>\n<p>8x<sup>3<\/sup>&nbsp;+ 27y<sup>3<\/sup>&nbsp;= 2197 \u2013 1404 = 793<\/p>\n<p><em>8x<sup>3<\/sup>&nbsp;+ 27y<sup>3<\/sup>&nbsp;= 793<\/em><\/p>\n<p><strong>Question 11: If 3x \u2013 2y= 11 and xy = 12, find the value of 27x<sup>3<\/sup>&nbsp;\u2013 8y<sup>3<\/sup>.<\/strong><\/p>\n<p><strong>Solution:<\/strong><\/p>\n<p>Given: 3x \u2013 2y = 11 and xy = 12<\/p>\n<p>Cubing 3x \u2013 2y = 11 both sides, we get<\/p>\n<p>(3x \u2013 2y)<sup>3<\/sup>&nbsp;= (11)<sup>3<\/sup><\/p>\n<p>(3x)<sup>3<\/sup>&nbsp;\u2013 (2y)<sup>3<\/sup>&nbsp;\u2013 3 ( 3x)( 2y) (3x \u2013 2y) =1331<\/p>\n<p>27x<sup>3<\/sup>&nbsp;\u2013 8y<sup>3<\/sup>&nbsp;\u2013 18xy(3x -2y) =1331<\/p>\n<p>27x<sup>3<\/sup>&nbsp;\u2013 8y<sup>3<\/sup>&nbsp;\u2013 18 x 12 x 11 = 1331<\/p>\n<p>27x<sup>3<\/sup>&nbsp;\u2013 8y<sup>3<\/sup>&nbsp;\u2013 2376 = 1331<\/p>\n<p>27x<sup>3<\/sup>&nbsp;\u2013 8y<sup>3<\/sup>&nbsp;= 1331 + 2376 = 3707<\/p>\n<p><em>27x<sup>3<\/sup>&nbsp;\u2013 8y<sup>3<\/sup>&nbsp;= 3707<\/em><\/p>\n<hr>\n<h3><span class=\"ez-toc-section\" id=\"exercise-44-page-no-423\"><\/span>Exercise 4.4 Page No: 4.23<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Question 1: Find the following products:<\/strong><\/p>\n<p><strong>(i) (3x + 2y)(9x<sup>2<\/sup>&nbsp;\u2013 6xy + 4y<sup>2<\/sup>)<\/strong><\/p>\n<p><strong>(ii) (4x \u2013 5y)(16x<sup>2<\/sup>&nbsp;+ 20xy + 25y<sup>2<\/sup>)<\/strong><\/p>\n<p><strong>(iii) (7p<sup>4<\/sup>&nbsp;+ q)(49p<sup>8<\/sup>&nbsp;\u2013 7p<sup>4<\/sup>q + q<sup>2<\/sup>)<\/strong><\/p>\n<p><strong>(iv) (x\/2 + 2y)(x<sup>2<\/sup>\/4 \u2013 xy + 4y<sup>2<\/sup>)<\/strong><\/p>\n<p><strong>(v) (3\/x \u2013 5\/y)(9\/x<sup>2<\/sup>&nbsp;+ 25\/y<sup>2<\/sup>&nbsp;+ 15\/xy)<\/strong><\/p>\n<p><strong>(vi) (3 + 5\/x)(9 \u2013 15\/x + 25\/x<sup>2<\/sup>)<\/strong><\/p>\n<p><strong>(vii) (2\/x + 3x)(4\/x<sup>2&nbsp;<\/sup>+ 9x<sup>2&nbsp;<\/sup>\u2013 6)<\/strong><\/p>\n<p><strong>(viii) (3\/x \u2013 2x<sup>2<\/sup>)(9\/x<sup>2&nbsp;<\/sup>+ 4x<sup>4<\/sup>&nbsp;\u2013 6x)<\/strong><\/p>\n<p><strong>(ix) (1 \u2013 x)(1 + x + x<sup>2<\/sup>)<\/strong><\/p>\n<p><strong>(x) (1 + x)(1 \u2013 x + x<sup>2<\/sup>)<\/strong><\/p>\n<p><strong>(xi) (x<sup>2&nbsp;<\/sup>\u2013 1)(x<sup>4<\/sup>&nbsp;+ x<sup>2&nbsp;<\/sup>+1)<\/strong><\/p>\n<p><strong>(xii) (x<sup>3&nbsp;<\/sup>+ 1)(x<sup>6<\/sup>&nbsp;\u2013 x<sup>3<\/sup>&nbsp;+ 1)<\/strong><\/p>\n<p><strong>Solution:<\/strong><\/p>\n<p><strong>(i)<\/strong>&nbsp;(3x + 2y)(9x<sup>2<\/sup>&nbsp;\u2013 6xy + 4y<sup>2<\/sup>)<\/p>\n<p>= (3x + 2y)[(3x)<sup>2<\/sup>&nbsp;\u2013 (3x)(2y) + (2y)<sup>2<\/sup>)]<\/p>\n<p>We know, a<sup>3<\/sup>&nbsp;+ b<sup>3<\/sup>&nbsp;= (a + b)(a<sup>2<\/sup>&nbsp;+ b<sup>2<\/sup>&nbsp;\u2013 ab)<\/p>\n<p>= (3x)<sup>3<\/sup>&nbsp;+ (2y)<sup>&nbsp;3<\/sup><\/p>\n<p><em>= 27x<sup>3<\/sup>&nbsp;+ 8y<sup>3<\/sup><\/em><\/p>\n<p><strong>(ii)<\/strong>&nbsp;(4x \u2013 5y)(16x<sup>2<\/sup>&nbsp;+ 20xy + 25y<sup>2<\/sup>)<\/p>\n<p>= (4x \u2013 5y)[(4x)<sup>2<\/sup>&nbsp;+ (4x)(5y) + (5y)<sup>2<\/sup>)]<\/p>\n<p>We know, a<sup>3<\/sup>&nbsp;\u2013 b<sup>3<\/sup>&nbsp;= (a \u2013 b)(a<sup>2<\/sup>&nbsp;+ b<sup>2<\/sup>&nbsp;+ ab)<\/p>\n<p>= (4x)<sup>3<\/sup>&nbsp;\u2013 (5y)<sup>&nbsp;3<\/sup><\/p>\n<p><em>= 64x<sup>3<\/sup>&nbsp;\u2013 125y<sup>3<\/sup><\/em><\/p>\n<p><strong>(iii)<\/strong>&nbsp;(7p<sup>4<\/sup>&nbsp;+ q)(49p<sup>8<\/sup>&nbsp;\u2013 7p<sup>4<\/sup>q + q<sup>2<\/sup>)<\/p>\n<p>= (7p<sup>4<\/sup>&nbsp;+ q)[(7p<sup>4<\/sup>)<sup>2<\/sup>&nbsp;\u2013 (7p<sup>4<\/sup>)(q) + (q)<sup>2<\/sup>)]<\/p>\n<p>We know, a<sup>3<\/sup>&nbsp;+ b<sup>3<\/sup>&nbsp;= (a + b)(a<sup>2<\/sup>&nbsp;+ b<sup>2<\/sup>&nbsp;\u2013 ab)<\/p>\n<p>= (7p<sup>4<\/sup>)<sup>3<\/sup>&nbsp;+ (q)<sup>&nbsp;3<\/sup><\/p>\n<p><em>= 343 p<sup>12<\/sup>&nbsp;+ q<sup>3<\/sup><\/em><\/p>\n<p><strong>(iv)<\/strong>&nbsp;(x\/2 + 2y)(x<sup>2<\/sup>\/4 \u2013 xy + 4y<sup>2<\/sup>)<\/p>\n<p>We know,&nbsp;<em>a<sup>3<\/sup>&nbsp;\u2013 b<sup>3<\/sup>&nbsp;= (a \u2013 b)(a<sup>2<\/sup>&nbsp;+ b<sup>2<\/sup>&nbsp;+ ab)<\/em><\/p>\n<p>(x\/2 + 2y)(x<sup>2<\/sup>\/4 \u2013 xy + 4y<sup>2<\/sup>)<\/p>\n<p><img title=\"RD sharma class 9 maths chapter 4 ex 4.4 question 2 Solution\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/10\/rd-sharma-class-9-maths-chapter-4-image-1.png\" alt=\"RD sharma class 9 maths chapter 4 ex 4.4 question 2 Solution\"><\/p>\n<p><strong>(v)<\/strong>&nbsp;(3\/x \u2013 5\/y)(9\/x<sup>2<\/sup>&nbsp;+ 25\/y<sup>2<\/sup>&nbsp;+ 15\/xy)<\/p>\n<p><img title=\"RD sharma class 9 maths chapter 4 ex 4.4\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/10\/rd-sharma-class-9-maths-chapter-4-ex-4-4.png\" alt=\"RD sharma class 9 maths chapter 4 ex 4.4\"><\/p>\n<p><em>[Using a<sup>3<\/sup>&nbsp;\u2013 b<sup>3<\/sup>&nbsp;= (a \u2013 b)(a<sup>2<\/sup>&nbsp;+ b<sup>2<\/sup>&nbsp;+ ab) ]<\/em><\/p>\n<p><strong>(vi)<\/strong>&nbsp;(3 + 5\/x)(9 \u2013 15\/x + 25\/x<sup>2<\/sup>)<\/p>\n<p><img title=\"RD sharma class 9 maths chapter 4 ex 4.4 solution\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/10\/rd-sharma-class-9-maths-chapter-4-ex-4-4-solution.png\" alt=\"RD sharma class 9 maths chapter 4 ex 4.4 solution\"><\/p>\n<p><em>[Using: a<sup>3<\/sup>&nbsp;+ b<sup>3<\/sup>&nbsp;= (a + b)(a<sup>2<\/sup>&nbsp;+ b<sup>2<\/sup>&nbsp;\u2013 ab)]<\/em><\/p>\n<p><strong>(vii)<\/strong>&nbsp;(2\/x + 3x)(4\/x<sup>2&nbsp;<\/sup>+ 9x<sup>2&nbsp;<\/sup>\u2013 6)<\/p>\n<p><img title=\"RD sharma class 9 maths chapter 4 ex 4.4 question 1\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2020\/10\/rd-sharma-class-9-maths-chapter-4-ex-4-4-question.png\" alt=\"RD sharma class 9 maths chapter 4 ex 4.4 question 1\"><\/p>\n<p><em>[Using: a<sup>3<\/sup>&nbsp;+ b<sup>3<\/sup>&nbsp;= (a + b)(a<sup>2<\/sup>&nbsp;+ b<sup>2<\/sup>&nbsp;\u2013 ab)]<\/em><\/p>\n<p><strong>(viii)<\/strong>&nbsp;(3\/x \u2013 2x<sup>2<\/sup>)(9\/x<sup>2&nbsp;<\/sup>+ 4x<sup>4<\/sup>&nbsp;\u2013 6x)<\/p>\n<p><img title=\"RD sharma class 9 maths chapter 4 ex 4.4 question 1 solution\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/10\/rd-sharma-class-9-maths-chapter-4-ex-4-4-question-1.png\" alt=\"RD sharma class 9 maths chapter 4 ex 4.4 question 1 solution\"><\/p>\n<p><em>[Using : a<sup>3<\/sup>&nbsp;\u2013 b<sup>3<\/sup>&nbsp;= (a \u2013 b)(a<sup>2<\/sup>&nbsp;+ b<sup>2<\/sup>&nbsp;+ ab)]<\/em><\/p>\n<p><strong>(ix)<\/strong>&nbsp;(1 \u2013 x)(1 + x + x<sup>2<\/sup>)<\/p>\n<p>And we know, a<sup>3<\/sup>&nbsp;\u2013 b<sup>3<\/sup>&nbsp;= (a \u2013 b)(a<sup>2<\/sup>&nbsp;+ b<sup>2<\/sup>&nbsp;+ ab)<\/p>\n<p>(1 \u2013 x)(1 + x + x<sup>2<\/sup>) can be written as<\/p>\n<p>(1 \u2013 x)[(1<sup>2<\/sup>&nbsp;+ (1)(x)+ x<sup>2<\/sup>)]<\/p>\n<p>= (1)<sup>3<\/sup>&nbsp;\u2013 (x)<sup>3<\/sup><\/p>\n<p><em>= 1 \u2013 x<sup>3<\/sup><\/em><\/p>\n<p><strong>(x)<\/strong>&nbsp;(1 + x)(1 \u2013 x + x<sup>2<\/sup>)<\/p>\n<p>And we know,&nbsp;<em>a<sup>3<\/sup>&nbsp;+ b<sup>3<\/sup>&nbsp;= (a + b)(a<sup>2<\/sup>&nbsp;+ b<sup>2<\/sup>&nbsp;\u2013 ab)]<\/em><\/p>\n<p>(1 + x)(1 \u2013 x + x<sup>2<\/sup>) can be written as,<\/p>\n<p>(1 + x)[(1<sup>2<\/sup>&nbsp;\u2013 (1)(x) + x<sup>2<\/sup>)]<\/p>\n<p>= (1)<sup>3<\/sup>&nbsp;+ (x)<sup>&nbsp;3<\/sup><\/p>\n<p><em>= 1 + x<sup>3<\/sup><\/em><\/p>\n<p><strong>(xi)<\/strong>&nbsp;(x<sup>2&nbsp;<\/sup>\u2013 1)(x<sup>4<\/sup>&nbsp;+ x<sup>2&nbsp;<\/sup>+1) can be written as,<\/p>\n<p>(x<sup>2<\/sup>&nbsp;\u2013 1)[(x<sup>2<\/sup>)<sup>2<\/sup>&nbsp;\u2013 1<sup>2<\/sup>&nbsp;+ (x<sup>2<\/sup>)(1)]<\/p>\n<p>= (x<sup>2<\/sup>)<sup>3<\/sup>&nbsp;\u2013 1<sup>3<\/sup><\/p>\n<p><em>= x<sup>6<\/sup>&nbsp;\u2013 1<\/em><\/p>\n<p><em>[using a<sup>3<\/sup>&nbsp;\u2013 b<sup>3<\/sup>&nbsp;= (a \u2013 b)(a<sup>2<\/sup>&nbsp;+ b<sup>2<\/sup>&nbsp;+ ab) ]<\/em><\/p>\n<p><strong>(xii)<\/strong>&nbsp;(x<sup>3&nbsp;<\/sup>+ 1)(x<sup>6<\/sup>&nbsp;\u2013 x<sup>3<\/sup>&nbsp;+ 1) can be written as,<\/p>\n<p>(x<sup>3<\/sup>&nbsp;+ 1)[(x<sup>3<\/sup>)<sup>2<\/sup>&nbsp;\u2013 (x<sup>3<\/sup>)(1) + 1<sup>2<\/sup>]<\/p>\n<p>= (x<sup>3<\/sup>)<sup>&nbsp;3<\/sup>&nbsp;+ 1<sup>3<\/sup><\/p>\n<p><em>= x<sup>9<\/sup>&nbsp;+ 1<\/em><\/p>\n<p><em>[using a<sup>3<\/sup>&nbsp;+ b<sup>3<\/sup>&nbsp;= (a + b)(a<sup>2<\/sup>&nbsp;+ b<sup>2<\/sup>&nbsp;\u2013 ab) ]<\/em><\/p>\n<p><strong>Question 2: If x = 3 and y = -1, find the values of each of the following using in identity:<\/strong><\/p>\n<p><strong>(i) (9y<sup>2<\/sup>&nbsp;\u2013 4x<sup>2<\/sup>)(81y<sup>4<\/sup>&nbsp;+ 36x<sup>2<\/sup>y<sup>2<\/sup>&nbsp;+ 16x<sup>4<\/sup>)<\/strong><\/p>\n<p><strong>(ii) (3\/x \u2013 x\/3)(x<sup>2<\/sup>&nbsp;\/9 + 9\/x<sup>2<\/sup>&nbsp;+ 1)<\/strong><\/p>\n<p><strong>(iii) (x\/7 + y\/3)(x<sup>2<\/sup>\/49 + y<sup>2<\/sup>\/9 \u2013 xy\/21)<\/strong><\/p>\n<p><strong>(iv) (x\/4 \u2013 y\/3)(x<sup>2<\/sup>\/16 + xy\/12 + y<sup>2<\/sup>\/9)<br>(v) (5\/x + 5x)(25\/x<sup>2<\/sup>&nbsp;\u2013 25 + 25x<sup>2<\/sup>)<\/strong><\/p>\n<p><strong>Solution:<\/strong><\/p>\n<p><strong>(i)<\/strong>&nbsp;(9y<sup>2<\/sup>&nbsp;\u2013 4x<sup>2<\/sup>)(81y<sup>4<\/sup>&nbsp;+ 36x<sup>2<\/sup>y<sup>2<\/sup>&nbsp;+ 16x<sup>4<\/sup>)<\/p>\n<p>= (9y<sup>2<\/sup>&nbsp;\u2013 4x<sup>2<\/sup>) [(9y<sup>2<\/sup>&nbsp;)<sup>&nbsp;2&nbsp;<\/sup>+ 9y<sup>2<\/sup>&nbsp;x 4x<sup>2&nbsp;<\/sup>+ (4x<sup>2<\/sup>)<sup>&nbsp;2<\/sup>&nbsp;]<\/p>\n<p>= (9y<sup>2<\/sup>&nbsp;)<sup>&nbsp;3&nbsp;<\/sup>\u2013 (4x<sup>2<\/sup>)<sup>3<\/sup><\/p>\n<p>= 729 y<sup>6<\/sup>&nbsp;\u2013 64 x<sup>6<\/sup><\/p>\n<p>Put x = 3 and y = -1<\/p>\n<p>= 729 \u2013 46656<\/p>\n<p><em>= \u2013 45927<\/em><\/p>\n<p><strong>(ii)<\/strong>&nbsp;Put x = 3 and y = -1<\/p>\n<p>(3\/x \u2013 x\/3)(x<sup>2<\/sup>&nbsp;\/9 + 9\/x<sup>2<\/sup>&nbsp;+ 1)<\/p>\n<p><img title=\"RD sharma class 9 maths chapter 4 ex 4.4 question 2 solution part 2\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/10\/rd-sharma-class-9-maths-chapter-4-ex-4-4-question-2.png\" alt=\"RD sharma class 9 maths chapter 4 ex 4.4 question 2 solution part 2\"><\/p>\n<p><strong>(iii)<\/strong>&nbsp;Put x = 3 and y = -1<\/p>\n<p>(x\/7 + y\/3)(x<sup>2<\/sup>\/49 + y<sup>2<\/sup>\/9 \u2013 xy\/21)<\/p>\n<p><img title=\"RD sharma class 9 maths chapter 4 ex 4.4 question 2 solution part 3\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2020\/10\/rd-sharma-class-9-maths-chapter-4-ex-4-4-question-3.png\" alt=\"RD sharma class 9 maths chapter 4 ex 4.4 question 2 solution part 3\"><\/p>\n<p><strong>(iv)<\/strong>&nbsp;Put x = 3 and y = -1<\/p>\n<p>(x\/4 \u2013 y\/3)(x<sup>2<\/sup>\/16 + xy\/12 + y<sup>2<\/sup>\/9)<\/p>\n<p><img title=\"RD sharma class 9 maths chapter 4 ex 4.4 question 2 solution part 4\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/10\/rd-sharma-class-9-maths-chapter-4-ex-4-4-question-4.png\" alt=\"RD sharma class 9 maths chapter 4 ex 4.4 question 2 solution part 4\"><\/p>\n<p><strong>(v)<\/strong>&nbsp;Put x = 3 and y = -1<\/p>\n<p>(5\/x + 5x)(25\/x<sup>2<\/sup>&nbsp;\u2013 25 + 25x<sup>2<\/sup>)<\/p>\n<p><img title=\"RD sharma class 9 maths chapter 4 ex 4.4 question 2 solution part 5\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2020\/10\/rd-sharma-class-9-maths-chapter-4-ex-4-4-question-5.png\" alt=\"RD sharma class 9 maths chapter 4 ex 4.4 question 2 solution part 5\"><\/p>\n<p><strong>Question 3: If a + b = 10 and ab = 16, find the value of a<sup>2<\/sup>&nbsp;\u2013 ab + b<sup>2<\/sup>&nbsp;and a<sup>2<\/sup>&nbsp;+ ab + b<sup>2<\/sup>.<\/strong><\/p>\n<p><strong>Solution:<\/strong><\/p>\n<p>a + b = 10, ab = 16<\/p>\n<p>Squaring, a + b = 10, both sides<\/p>\n<p>(a + b)<sup>2<\/sup>&nbsp;= (10)<sup>2<\/sup><\/p>\n<p>a<sup>2<\/sup>&nbsp;+ b<sup>2<\/sup>&nbsp;+ 2ab = 100<\/p>\n<p>a<sup>2<\/sup>&nbsp;+ b<sup>2<\/sup>&nbsp;+ 2 x 16 = 100<\/p>\n<p>a<sup>2<\/sup>&nbsp;+ b<sup>2<\/sup>&nbsp;+ 32 = 100<\/p>\n<p>a<sup>2<\/sup>&nbsp;+ b<sup>2<\/sup>&nbsp;= 100 \u2013 32 = 68<\/p>\n<p>a<sup>2<\/sup>&nbsp;+ b<sup>2<\/sup>&nbsp;= 68<\/p>\n<p>Again, a<sup>2<\/sup>&nbsp;\u2013 ab + b<sup>2<\/sup>&nbsp;= a<sup>2<\/sup>&nbsp;+ b<sup>2<\/sup>&nbsp;\u2013 ab = 68 \u2013 16 = 52 and<\/p>\n<p><em>a<sup>2<\/sup>&nbsp;+ ab + b<sup>2<\/sup>&nbsp;= a<sup>2<\/sup>&nbsp;+ b<sup>2<\/sup>&nbsp;+ ab = 68 + 16 = 84<\/em><\/p>\n<p><strong>Question 4: If a + b = 8 and ab = 6, find the value of a<sup>3<\/sup>&nbsp;+ b<sup>3<\/sup>.<\/strong><\/p>\n<p><strong>Solution:<\/strong><\/p>\n<p>a + b = 8, ab = 6<\/p>\n<p>Cubing, a + b = 8, both sides, we get<\/p>\n<p>(a + b)<sup>3<\/sup>&nbsp;= (8)<sup>3<\/sup><\/p>\n<p>a<sup>3<\/sup>&nbsp;+ b<sup>3<\/sup>&nbsp;+ 3ab(a + b) = 512<\/p>\n<p>a<sup>3<\/sup>&nbsp;+ b<sup>3<\/sup>&nbsp;+ 3 x 6 x 8 = 512<\/p>\n<p>a<sup>3<\/sup>&nbsp;+ b<sup>3<\/sup>&nbsp;+ 144 = 512<\/p>\n<p>a<sup>3<\/sup>&nbsp;+ b<sup>3<\/sup>&nbsp;= 512 \u2013 144 = 368<\/p>\n<p><em>a<sup>3<\/sup>&nbsp;+ b<sup>3<\/sup>&nbsp;= 368<\/em><\/p>\n<hr>\n<h3><span class=\"ez-toc-section\" id=\"exercise-45-page-no-428\"><\/span>Exercise 4.5 Page No: 4.28<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Question 1: Find the following products:<\/strong><\/p>\n<p><strong>(i) (3x + 2y + 2z) (9x<sup>2<\/sup>&nbsp;+ 4y<sup>2<\/sup>&nbsp;+ 4z<sup>2<\/sup>&nbsp;\u2013 6xy \u2013 4yz \u2013 6zx)<\/strong><\/p>\n<p><strong>(ii) (4x \u2013 3y + 2z) (16x<sup>2<\/sup>&nbsp;+ 9y<sup>2&nbsp;<\/sup>+ 4z<sup>2&nbsp;<\/sup>+ 12xy + 6yz \u2013 8zx)<\/strong><\/p>\n<p><strong>(iii) (2a \u2013 3b \u2013 2c) (4a<sup>2<\/sup>&nbsp;+ 9b<sup>2<\/sup>&nbsp;+ 4c<sup>2<\/sup>&nbsp;+ 6ab \u2013 6bc + 4ca)<\/strong><\/p>\n<p><strong>(iv) (3x -4y + 5z) (9x<sup>2<\/sup>&nbsp;+ 16y<sup>2<\/sup>&nbsp;+ 25z<sup>2<\/sup>&nbsp;+ 12xy- 15zx + 20yz)<\/strong><\/p>\n<p><strong>Solution:<\/strong><\/p>\n<p><strong>(i)<\/strong>&nbsp;(3x + 2y + 2z) (9x<sup>2<\/sup>&nbsp;+ 4y<sup>2<\/sup>&nbsp;+ 4z<sup>2<\/sup>&nbsp;\u2013 6xy \u2013 4yz \u2013 6zx)<\/p>\n<p>= (3x + 2y + 2z) [(3x)<sup>2<\/sup>&nbsp;+ (2y)<sup>&nbsp;2<\/sup>&nbsp;+ (2z)<sup>&nbsp;2<\/sup>&nbsp;\u2013 3x x 2y \u2013 2y x 2z \u2013 2z x 3x]<\/p>\n<p>= (3x)<sup>3<\/sup>&nbsp;+ (2y)<sup>3<\/sup>&nbsp;+ (2z)<sup>3<\/sup>&nbsp;\u2013 3 x 3x x 2y x 2z<\/p>\n<p><em>= 27x<sup>3<\/sup>&nbsp;+ 8y<sup>3<\/sup>&nbsp;+ 8Z<sup>3<\/sup>&nbsp;\u2013 36xyz<\/em><\/p>\n<p><strong>(ii)<\/strong>&nbsp;(4x \u2013 3y + 2z) (16x<sup>2<\/sup>&nbsp;+ 9y<sup>2<\/sup>&nbsp;+ 4z<sup>2<\/sup>&nbsp;+ 12xy + 6yz \u2013 8zx)<\/p>\n<p>= (4x -3y + 2z) [(4x)<sup>2<\/sup>&nbsp;+ (-3y)<sup>&nbsp;2<\/sup>&nbsp;+ (2z)<sup>&nbsp;2<\/sup>&nbsp;\u2013 4x x (-3y) \u2013 (-3y) x (2z) \u2013 (2z x 4x)]<\/p>\n<p>= (4x)<sup>&nbsp;3<\/sup>&nbsp;+ (-3y)<sup>&nbsp;3<\/sup>&nbsp;+ (2z)<sup>&nbsp;3<\/sup>&nbsp;\u2013 3 x 4x x (-3y) x (2z)<\/p>\n<p><em>= 64x<sup>3<\/sup>&nbsp;\u2013 27y<sup>3<\/sup>&nbsp;+ 8z<sup>3<\/sup>&nbsp;+ 72xyz<\/em><\/p>\n<p><strong>(iii)<\/strong>&nbsp;(2a -3b- 2c) (4a<sup>2<\/sup>&nbsp;+ 9b<sup>2<\/sup>&nbsp;+ 4c<sup>2<\/sup>&nbsp;+ 6ab \u2013 6bc + 4ca)<\/p>\n<p>= (2a -3b- 2c) [(2a)<sup>&nbsp;2<\/sup>&nbsp;+ (-3b)<sup>&nbsp;2<\/sup>&nbsp;+ (-2c)<sup>&nbsp;2<\/sup>&nbsp;\u2013 2a x (-3b) \u2013 (-3b) x (-2c) \u2013 (-2c) x 2a]<\/p>\n<p>= (2a)<sup>3<\/sup>&nbsp;+ (-3b)<sup>&nbsp;3<\/sup>&nbsp;+ (-2c)<sup>&nbsp;3<\/sup>&nbsp;-3x 2a x (-3 b) (-2c)<\/p>\n<p><em>= 8a<sup>3<\/sup>&nbsp;\u2013 21b<sup>3<\/sup>&nbsp;\u2013 8c<sup>3<\/sup>&nbsp;\u2013 36abc<\/em><\/p>\n<p><strong>(iv)<\/strong>&nbsp;(3x \u2013 4y + 5z) (9x<sup>2<\/sup>&nbsp;+ 16y<sup>2<\/sup>&nbsp;+ 25z<sup>2<\/sup>&nbsp;+ 12xy \u2013 15zx + 20yz)<\/p>\n<p>= [3x + (-4y) + 5z] [(3x)<sup>&nbsp;2<\/sup>&nbsp;+ (-4y)<sup>&nbsp;2<\/sup>&nbsp;+ (5z)<sup>&nbsp;2<\/sup>&nbsp;\u2013 3x x (-4y) -(-4y) (5z) \u2013 5z x 3x]<\/p>\n<p>= (3x)<sup>&nbsp;3<\/sup>&nbsp;+ (-4y)<sup>&nbsp;3<\/sup>&nbsp;+ (5z)<sup>&nbsp;3<\/sup>&nbsp;\u2013 3 x 3x x (-4y) (5z)<\/p>\n<p><em>= 27x<sup>3<\/sup>&nbsp;\u2013 64y<sup>3<\/sup>&nbsp;+ 125z<sup>3<\/sup>&nbsp;+ 180xyz<\/em><\/p>\n<p><strong>Question 2: If x + y + z = 8 and xy + yz+ zx = 20, find the value of x<sup>3<\/sup>&nbsp;+ y<sup>3<\/sup>&nbsp;+ z<sup>3<\/sup>&nbsp;\u2013 3xyz.<\/strong><\/p>\n<p><strong>Solution:<\/strong><\/p>\n<p>We know, x<sup>3<\/sup>&nbsp;+ y<sup>3<\/sup>&nbsp;+ z<sup>3<\/sup>&nbsp;\u2013 3xyz = (x + y + z) (x<sup>2<\/sup>&nbsp;+ y<sup>2<\/sup>&nbsp;+ z<sup>2<\/sup>&nbsp;\u2013 xy \u2013 yz \u2013 zx)<\/p>\n<p>Squaring, x + y + z = 8 both sides, we get<\/p>\n<p>(x + y + z)<sup>2<\/sup>&nbsp;= (8)<sup>&nbsp;2<\/sup><\/p>\n<p>x<sup>2<\/sup>&nbsp;+ y<sup>2<\/sup>&nbsp;+ z<sup>2<\/sup>&nbsp;+ 2(xy + yz + zx) = 64<\/p>\n<p>x<sup>2<\/sup>&nbsp;+ y<sup>2<\/sup>&nbsp;+ z<sup>2<\/sup>&nbsp;+ 2 x 20 = 64<\/p>\n<p>x<sup>2<\/sup>&nbsp;+ y<sup>2<\/sup>&nbsp;+ z<sup>2<\/sup>&nbsp;+ 40 = 64<\/p>\n<p>x<sup>2<\/sup>&nbsp;+ y<sup>2<\/sup>&nbsp;+ z<sup>2<\/sup>&nbsp;= 24<\/p>\n<p>Now,<\/p>\n<p>x<sup>3<\/sup>&nbsp;+ y<sup>3<\/sup>&nbsp;+ z<sup>3<\/sup>&nbsp;\u2013 3xyz = (x + y + z) [x<sup>2<\/sup>&nbsp;+ y<sup>2<\/sup>&nbsp;+ z<sup>2<\/sup>&nbsp;\u2013 (xy + yz + zx)]<\/p>\n<p>= 8(24 \u2013 20)<\/p>\n<p>= 8 x 4<\/p>\n<p>= 32<\/p>\n<p><em>\u21d2 x<sup>3<\/sup>&nbsp;+ y<sup>3<\/sup>&nbsp;+ z<sup>3<\/sup>&nbsp;\u2013 3xyz = 32<\/em><\/p>\n<p><strong>Question 3: If a +b + c = 9 and ab + bc + ca = 26, find the value of a<sup>3<\/sup>&nbsp;+ b<sup>3<\/sup>&nbsp;+ c<sup>3<\/sup>&nbsp;\u2013 3abc.<\/strong><\/p>\n<p><strong>Solution:<\/strong><\/p>\n<p>a + b + c = 9, ab + bc + ca = 26<\/p>\n<p>Squaring, a + b + c = 9 both sides, we get<\/p>\n<p>(a + b + c)<sup>2<\/sup>&nbsp;= (9)<sup>2<\/sup><\/p>\n<p>a<sup>2<\/sup>&nbsp;+ b<sup>2<\/sup>&nbsp;+ c<sup>2<\/sup>&nbsp;+ 2 (ab + bc + ca) = 81<\/p>\n<p>a<sup>2<\/sup>&nbsp;+ b<sup>2<\/sup>&nbsp;+ c<sup>2<\/sup>&nbsp;+ 2 x 26 = 81<\/p>\n<p>a<sup>2<\/sup>&nbsp;+ b<sup>2<\/sup>&nbsp;+ c<sup>2<\/sup>&nbsp;+ 52 = 81<\/p>\n<p>a<sup>2<\/sup>&nbsp;+ b<sup>2<\/sup>&nbsp;+ c<sup>2<\/sup>&nbsp;= 29<\/p>\n<p>Now, a<sup>3<\/sup>&nbsp;+ b<sup>3<\/sup>&nbsp;+ c<sup>3<\/sup>&nbsp;\u2013 3abc = (a + b + c) [(a<sup>2<\/sup>&nbsp;+ b<sup>2<\/sup>&nbsp;+ c<sup>2<\/sup>&nbsp;\u2013 (ab + bc + ca)]<\/p>\n<p>= 9[29 \u2013 26]<\/p>\n<p>= 9 x 3<\/p>\n<p>= 27<\/p>\n<p><em>\u21d2 a<sup>3<\/sup>&nbsp;+ b<sup>3<\/sup>&nbsp;+ c<sup>3<\/sup>&nbsp;\u2013 3abc = 27<\/em><\/p>\n<hr>\n<h3><span class=\"ez-toc-section\" id=\"exercise-vsaqs-page-no-428\"><\/span>Exercise VSAQs Page No: 4.28<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Question 1: If x + 1\/x = 3, then find the value of x<sup>2<\/sup>&nbsp;+ 1\/x<sup>2<\/sup>.<\/strong><\/p>\n<p><strong>Solution:<\/strong><\/p>\n<p>x + 1\/x = 3<\/p>\n<p>Squaring both sides, we have<\/p>\n<p>(x + 1\/x)<sup>2<\/sup>&nbsp;= 3<sup>2<\/sup><\/p>\n<p>x<sup>2<\/sup>&nbsp;+ 1\/x<sup>2<\/sup>&nbsp;+ 2 = 9<\/p>\n<p><em>x<sup>2<\/sup>&nbsp;+ 1\/x<sup>2<\/sup>&nbsp;= 9 \u2013 2 = 7<\/em><\/p>\n<p><strong>Question 2: If x + 1\/x = 3, then find the value of x^6 + 1\/x^6.<\/strong><\/p>\n<p><strong>Solution<\/strong>:<\/p>\n<p>x + 1\/x = 3<\/p>\n<p>Squaring both sides, we have<\/p>\n<p>(x + 1\/x)<sup>2<\/sup>&nbsp;= 3<sup>2<\/sup><\/p>\n<p>x<sup>2<\/sup>&nbsp;+ 1\/x<sup>2<\/sup>&nbsp;+ 2 = 9<\/p>\n<p>x<sup>2<\/sup>&nbsp;+ 1\/x<sup>2<\/sup>&nbsp;= 9 \u2013 2 = 7<\/p>\n<p>x<sup>2<\/sup>&nbsp;+ 1\/x<sup>2<\/sup>&nbsp;= 7 \u2026(1)<\/p>\n<p>Cubing equation (1) both sides,<\/p>\n<p><img title=\"RD sharma class 9 maths chapter 4 ex vsaqs solutions\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/10\/rd-sharma-class-9-maths-chapter-4-ex-vsaqs-solutio.png\" alt=\"RD sharma class 9 maths chapter 4 ex vsaqs solutions\"><\/p>\n<p><strong>Question 3: If a + b = 7 and ab = 12, find the value of a<sup>2<\/sup>&nbsp;+ b<sup>2<\/sup>.<\/strong><\/p>\n<p><strong>Solution:<\/strong><\/p>\n<p>a + b = 7, ab = 12<\/p>\n<p>Squaring, a + b = 7, both sides,<\/p>\n<p>(a + b)<sup>&nbsp;2<\/sup>&nbsp;= (7)<sup>&nbsp;2<\/sup><\/p>\n<p>a<sup>2<\/sup>&nbsp;+ b<sup>2<\/sup>&nbsp;+ 2ab = 49<\/p>\n<p>a<sup>2<\/sup>&nbsp;+ b<sup>2<\/sup>&nbsp;+ 2 x 12 = 49<\/p>\n<p>a<sup>2<\/sup>&nbsp;+ b<sup>2<\/sup>&nbsp;+ 24 = 49<\/p>\n<p><em>a<sup>2<\/sup>&nbsp;+ b<sup>2<\/sup>&nbsp;= 25<\/em><\/p>\n<p><strong>Question 4: If a \u2013 b = 5 and ab = 12, find the value of a<sup>2<\/sup>&nbsp;+ b<sup>2<\/sup>.<\/strong><\/p>\n<p><strong>Solution:<\/strong><\/p>\n<p>a \u2013 b = 5, ab = 12<\/p>\n<p>Squaring, a \u2013 b = 5, both sides,<\/p>\n<p>(a \u2013 b)<sup>2<\/sup>&nbsp;= (5)<sup>2<\/sup><\/p>\n<p>a<sup>2<\/sup>&nbsp;+ b<sup>2<\/sup>&nbsp;\u2013 2ab = 25<\/p>\n<p>a<sup>2<\/sup>&nbsp;+ b<sup>2<\/sup>&nbsp;\u2013 2 x 12 = 25<\/p>\n<p>a<sup>2<\/sup>&nbsp;+ b<sup>2<\/sup>&nbsp;\u2013 24 = 25<\/p>\n<p><em>a<sup>2<\/sup>&nbsp;+ b<sup>2<\/sup>&nbsp;= 49<\/em><\/p>\n<h3><span class=\"ez-toc-section\" id=\"rd-sharma-solutions-for-class-9-maths-chapter-4-algebraic-identities\"><\/span>RD Sharma Solutions for Class 9 Maths Chapter 4 Algebraic Identities<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>In the 4<sup>th<\/sup>&nbsp;Chapter of Class 9&nbsp;<strong>RD Sharma Solutions<\/strong>, students will study important identities, as listed below.<\/p>\n<ul>\n<li>Algebraic identities introduction<\/li>\n<li>Identity for the square of a trinomial<\/li>\n<li>Sum and difference of cubes identity<\/li>\n<\/ul>\n<p>These books are widely used by students to score high in the final exam. For RD Sharma Class 9 Maths Solutions, students can visit BYJU\u2019S website and access step-by-step answers to all the questions provided in the RD Sharma textbook.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"important-topics-rd-sharma-solutions-class-9-maths-chapter-4\"><\/span><strong>Important Topics: RD Sharma Solutions Class 9 Maths Chapter 4&nbsp;<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><span style=\"font-weight: 400;\">Important topics always let you run a thorough check on the titles that are there in the chapters. So, the important topics that the chapter includes are-<\/span><\/p>\n<ul>\n<li style=\"font-weight: 400;\"><span style=\"font-weight: 400;\">Algebraic Identities Introduction<\/span><\/li>\n<li style=\"font-weight: 400;\"><span style=\"font-weight: 400;\">Identity for the square of a trinomial<\/span><\/li>\n<li style=\"font-weight: 400;\"><span style=\"font-weight: 400;\">Sum and difference of cubes Identity<\/span><\/li>\n<\/ul>\n<p><span style=\"font-weight: 400;\">Here is everything that students need to smoothly finish their CBSE Class 9 Mathematics syllabus. <\/span><span style=\"font-weight: 400;\">Start with <a href=\"https:\/\/cbse.nic.in\/\" target=\"_blank\" rel=\"noopener noreferrer\">CBSE<\/a> Class 9 to build a solid future ahead. For any doubts, ask in the comments.&nbsp;<\/span><\/p>\n<h2><span class=\"ez-toc-section\" id=\"faqs-on-rd-sharma-solutions-class-9-maths-chapter-4-%e2%80%93-algebraic-identities\"><\/span><strong>FAQs on RD Sharma Solutions Class 9 Maths Chapter 4 &#8211; Algebraic Identities<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n<div id=\"rank-math-faq\" class=\"rank-math-block\">\n<div class=\"rank-math-list \">\n<div id=\"faq-question-1630663002536\" class=\"rank-math-list-item\">\n<h3 class=\"rank-math-question \"><span class=\"ez-toc-section\" id=\"can-i-access-the-rd-sharma-solutions-for-class-9-maths-chapter-4-pdf-offline\"><\/span>Can I access the RD Sharma Solutions for Class 9 Maths Chapter 4\u00a0PDF offline?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<div class=\"rank-math-answer \">\n\n<p>Once you have downloaded the PDF online, you can access it offline as well.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"faq-question-1630663030507\" class=\"rank-math-list-item\">\n<h3 class=\"rank-math-question \"><span class=\"ez-toc-section\" id=\"from-where-can-i-download-the-pdf-of-rd-sharma-solutions-class-9-maths-chapter-4\"><\/span>From where can I download the PDF of RD Sharma Solutions Class 9 Maths Chapter 4?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<div class=\"rank-math-answer \">\n\n<p>You can find the download link from the above blog.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"faq-question-1630663050137\" class=\"rank-math-list-item\">\n<h3 class=\"rank-math-question \"><span class=\"ez-toc-section\" id=\"how-much-does-it-cost-to-download-the-pdf-of-rd-sharma-solutions-for-class-9-maths-chapter-4\"><\/span>How much does it cost to download the PDF of RD Sharma Solutions for Class 9 Maths Chapter 4?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<div class=\"rank-math-answer \">\n\n<p>You can download it for free.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>","protected":false},"excerpt":{"rendered":"<p>RD Sharma Solutions Class 9 Maths Chapter 4 &#8211; Algebraic Identities: Have doubts with Mathematics exam preparation? Have you started looking for some good study material yet? If not, don&#8217;t worry, we bring you RD Sharma Solutions Class 9 Maths Chapter 4 Algebraic Identities for solving the confusion with Algebraic identities. To know more, read &#8230; <a title=\"RD Sharma Solutions Class 9 Maths Chapter 4 &#8211; Algebraic Identities (Updated for 2024)\" class=\"read-more\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-9-maths-chapter-4-algebraic-identities\/\" aria-label=\"More on RD Sharma Solutions Class 9 Maths Chapter 4 &#8211; Algebraic Identities (Updated for 2024)\">Read more<\/a><\/p>\n","protected":false},"author":243,"featured_media":124413,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"fifu_image_url":"","fifu_image_alt":""},"categories":[73411,2985,73410],"tags":[3081,3037,3086],"amp_enabled":true,"_links":{"self":[{"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/posts\/61551"}],"collection":[{"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/users\/243"}],"replies":[{"embeddable":true,"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/comments?post=61551"}],"version-history":[{"count":5,"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/posts\/61551\/revisions"}],"predecessor-version":[{"id":501531,"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/posts\/61551\/revisions\/501531"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/media\/124413"}],"wp:attachment":[{"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/media?parent=61551"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/categories?post=61551"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/tags?post=61551"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}