{"id":125249,"date":"2023-09-12T13:39:00","date_gmt":"2023-09-12T08:09:00","guid":{"rendered":"https:\/\/www.kopykitab.com\/blog\/?p=125249"},"modified":"2023-12-06T11:01:29","modified_gmt":"2023-12-06T05:31:29","slug":"rd-sharma-class-10-solutions-chapter-2-exercise-2-2","status":"publish","type":"post","link":"https:\/\/www.kopykitab.com\/blog\/rd-sharma-class-10-solutions-chapter-2-exercise-2-2\/","title":{"rendered":"RD Sharma Class 10 Solutions Chapter 2 Exercise 2.2 (Updated for 2024)"},"content":{"rendered":"\n<p><img class=\"alignnone size-full wp-image-125264\" src=\"https:\/\/www.kopykitab.com\/blog\/wp-content\/uploads\/2021\/09\/RD-Sharma-Class-10-Solutions-Chapter-2-Exercise-2.2.jpg\" alt=\"RD Sharma Class 10 Solutions Chapter 2 Exercise 2.2\" width=\"1200\" height=\"675\" srcset=\"https:\/\/www.kopykitab.com\/blog\/wp-content\/uploads\/2021\/09\/RD-Sharma-Class-10-Solutions-Chapter-2-Exercise-2.2.jpg 1200w, https:\/\/www.kopykitab.com\/blog\/wp-content\/uploads\/2021\/09\/RD-Sharma-Class-10-Solutions-Chapter-2-Exercise-2.2-768x432.jpg 768w\" sizes=\"(max-width: 1200px) 100vw, 1200px\" \/><\/p>\n<p><strong>RD Sharma Class 10 Solutions Chapter 2 Exercise 2.2:\u00a0<\/strong>This exercise focuses on verifying the zeros and the relationship between the zeros and the coefficients of a cubic polynomial. The experts at Kopykitab have prepared <a href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-class-10-solutions-for-maths\/\"><strong>RD Sharma Solutions Class 10<\/strong><\/a> to help students understand these concepts clearly while following the latest CBSE patterns. For a detailed explanation of the questions in this exercise, students can get <a href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-class-10-solutions-chapter-2-polynomials\/\"><strong>RD Sharma Solutions for Class 10 Maths Chapter 2 Polynomials<\/strong><\/a> Exercise 2.2 PDF from the link below.<\/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-69e191cc08af6\" 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\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/label><input  type=\"checkbox\" id=\"item-69e191cc08af6\"><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 eztoc-visibility-hide-by-default' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-class-10-solutions-chapter-2-exercise-2-2\/#download-rd-sharma-class-10-solutions-chapter-2-exercise-22-free-pdf\" title=\"Download RD Sharma Class 10 Solutions Chapter 2 Exercise 2.2 Free PDF\">Download RD Sharma Class 10 Solutions Chapter 2 Exercise 2.2 Free PDF<\/a><ul class='ez-toc-list-level-3'><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-class-10-solutions-chapter-2-exercise-2-2\/#access-answers-to-rd-sharma-solutions-class-10-maths-chapter-2-exercise-22-important-question-with-answers\" title=\"Access answers to RD Sharma Solutions Class 10 Maths Chapter 2 Exercise 2.2- Important Question with Answers\">Access answers to RD Sharma Solutions Class 10 Maths Chapter 2 Exercise 2.2- Important Question with Answers<\/a><\/li><\/ul><\/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-class-10-solutions-chapter-2-exercise-2-2\/#faqs-on-rd-sharma-class-10-solutions-chapter-2-exercise-22\" title=\"FAQs on RD Sharma Class 10 Solutions Chapter 2 Exercise 2.2\">FAQs on RD Sharma Class 10 Solutions Chapter 2 Exercise 2.2<\/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-class-10-solutions-chapter-2-exercise-2-2\/#where-can-i-download-rd-sharma-class-10-solutions-chapter-2-exercise-22-free-pdf\" title=\"Where can I download RD Sharma Class 10 Solutions Chapter 2 Exercise 2.2 free PDF?\">Where can I download RD Sharma Class 10 Solutions Chapter 2 Exercise 2.2 free PDF?<\/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-class-10-solutions-chapter-2-exercise-2-2\/#what-are-the-benefits-of-using-rd-sharma-class-10-solutions-chapter-2-exercise-22\" title=\"What are the benefits of using RD Sharma Class 10 Solutions Chapter 2 Exercise 2.2?\">What are the benefits of using RD Sharma Class 10 Solutions Chapter 2 Exercise 2.2?<\/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-class-10-solutions-chapter-2-exercise-2-2\/#is-it-required-to-remember-all-of-the-questions-in-chapter-2-exercise-22-of-rd-sharma-solutions-for-class-10-maths\" title=\"Is it required to remember all of the questions in Chapter 2 Exercise 2.2 of RD Sharma Solutions for Class 10 Maths?\">Is it required to remember all of the questions in Chapter 2 Exercise 2.2 of RD Sharma Solutions for Class 10 Maths?<\/a><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n<h2><span class=\"ez-toc-section\" id=\"download-rd-sharma-class-10-solutions-chapter-2-exercise-22-free-pdf\"><\/span>Download RD Sharma Class 10 Solutions Chapter 2 Exercise 2.2 Free PDF<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<div id=\"example1\" style=\"text-align: justify;\">\u00a0<\/div>\n<p style=\"text-align: justify;\"><style>\n.pdfobject-container { height: 800px;}<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-Sharma-Solution-Class-10-Chapter-2-Exercise-2.2.pdf\", \"#example1\");<\/script><\/p>\n<p><a href=\"https:\/\/www.kopykitab.com\/blog\/wp-content\/uploads\/2021\/09\/RD-Sharma-Solution-Class-10-Chapter-2-Exercise-2.2.pdf\">RD Sharma Class 10 Solutions Chapter 2 Exercise 2.2<\/a><\/p>\n<h3><span class=\"ez-toc-section\" id=\"access-answers-to-rd-sharma-solutions-class-10-maths-chapter-2-exercise-22-important-question-with-answers\"><\/span>Access answers to RD Sharma Solutions Class 10 Maths Chapter 2 Exercise 2.2- Important Question with Answers<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>1. Verify that the numbers given alongside of the cubic polynomials below are their zeroes. Also, verify the relationship between the zeros and coefficients in each of the following cases:<\/strong><\/p>\n<p><strong>(i) f(x) = 2x<sup>3\u00a0<\/sup>+ x<sup>2\u00a0<\/sup>\u2013 5x + 2; 1\/2, 1, -2<\/strong><\/p>\n<p><strong>Solution:<\/strong><\/p>\n<p>Given, f(x) = 2x<sup>3\u00a0<\/sup>+ x<sup>2\u00a0<\/sup>\u2013 5x + 2, where a= 2, b= 1, c= -5 and d= 2<\/p>\n<p>For x = 1\/2<\/p>\n<p>f(1\/2) = 2(1\/2)<sup>3<\/sup>\u00a0+ (1\/2)<sup>2<\/sup>\u00a0\u2013 5(1\/2) + 2<\/p>\n<p>= 1\/4 + 1\/4 \u2013 5\/2 + 2 = 0<\/p>\n<p>\u21d2 f(1\/2) = 0, hence x = 1\/2 is a root of the given polynomial.<\/p>\n<p>For x = 1<\/p>\n<p>f(1) = 2(1)<sup>3<\/sup>\u00a0+ (1)<sup>2<\/sup>\u00a0\u2013 5(1) + 2<\/p>\n<p>= 2 + 1 \u2013 5 + 2 = 0<\/p>\n<p>\u21d2 f(1) = 0, hence x = 1 is also a root of the given polynomial.<\/p>\n<p>For x = -2<\/p>\n<p>f(-2) = 2(-2)<sup>3<\/sup>\u00a0+ (-2)<sup>2<\/sup>\u00a0\u2013 5(-2) + 2<\/p>\n<p>= -16 + 4 + 10 + 2 = 0<\/p>\n<p>\u21d2 f(-2) = 0, hence x = -2 is also a root of the given polynomial.<\/p>\n<p>Now,<\/p>\n<p>Sum of zeros = -b\/a<\/p>\n<p>1\/2 + 1 \u2013 2 = \u2013 (1)\/2<\/p>\n<p>-1\/2 = -1\/2<\/p>\n<p>Some of the products of the zeros taken two at a time = c\/a<\/p>\n<p>(1\/2 x 1) + (1 x -2) + (1\/2 x -2) = -5\/ 2<\/p>\n<p>1\/2 \u2013 2 + (-1) = -5\/2<\/p>\n<p>-5\/2 = -5\/2<\/p>\n<p>Product of zeros = \u2013 d\/a<\/p>\n<p>1\/2 x 1 x (\u2013 2) = -(2)\/2<\/p>\n<p>-1 = -1<\/p>\n<p>Hence, the relationship between the zeros and coefficients is verified.<\/p>\n<p><strong>(ii) g(x) = x<sup>3\u00a0<\/sup>\u2013 4x<sup>2\u00a0<\/sup>+ 5x \u2013 2; 2, 1, 1<\/strong><\/p>\n<p><strong>Solution:<\/strong><\/p>\n<p>Given, g(x) = x<sup>3\u00a0<\/sup>\u2013 4x<sup>2\u00a0<\/sup>+ 5x \u2013 2, where a= 1, b= -4, c= 5 and d= -2<\/p>\n<p>For x = 2<\/p>\n<p>g(2) = (2)<sup>3<\/sup>\u00a0\u2013 4(2)<sup>2<\/sup>\u00a0+ 5(2) \u2013 2<\/p>\n<p>= 8 \u2013 16 + 10 \u2013 2 = 0<\/p>\n<p>\u21d2 f(2) = 0, hence x = 2 is a root of the given polynomial.<\/p>\n<p>For x = 1<\/p>\n<p>g(1) = (1)<sup>3<\/sup>\u00a0\u2013 4(1)<sup>2<\/sup>\u00a0+ 5(1) \u2013 2<\/p>\n<p>= 1 \u2013 4 + 5 \u2013 2 = 0<\/p>\n<p>\u21d2 g(1) = 0, hence x = 1 is also a root of the given polynomial.<\/p>\n<p>Now,<\/p>\n<p>Sum of zeros = -b\/a<\/p>\n<p>1 + 1 + 2 = \u2013 (-4)\/1<\/p>\n<p>4 = 4<\/p>\n<p>Some of the products of the zeros taken two at a time = c\/a<\/p>\n<p>(1 x 1) + (1 x 2) + (2 x 1) = 5\/ 1<\/p>\n<p>1 + 2 + 2 = 5<\/p>\n<p>5 = 5<\/p>\n<p>Product of zeros = \u2013 d\/a<\/p>\n<p>1 x 1 x 2 = -(-2)\/1<\/p>\n<p>2 = 2<\/p>\n<p>Hence, the relationship between the zeros and coefficients is verified.<\/p>\n<p><strong>2.<\/strong>\u00a0<strong>Find a cubic polynomial with the sum, the sum of the product of its zeroes taken two at a time, and the product of its zeros as 3, -1, and -3 respectively.<\/strong><\/p>\n<p><strong>Solution:<\/strong><\/p>\n<p><strong>Generally,<\/strong><\/p>\n<p>A cubic polynomial say f(x) is of the form ax<sup>3\u00a0<\/sup>+ bx<sup>2\u00a0<\/sup>+ cx + d.<\/p>\n<p>And, can be shown w.r.t its relationship between roots as.<\/p>\n<p>\u21d2 f(x) = k [x<sup>3<\/sup>\u00a0\u2013 (sum of roots)x<sup>2<\/sup>\u00a0+ (sum of products of roots taken two at a time)x \u2013 (product of roots)]<\/p>\n<p>Where k is any non-zero real number.<\/p>\n<p>Here,<\/p>\n<p>f(x) = k [x<sup>3<\/sup>\u00a0\u2013 (3)x<sup>2<\/sup>\u00a0+ (-1)x \u2013 (-3)]<\/p>\n<p>\u2234 f(x) = k [x<sup>3<\/sup>\u00a0\u2013 3x<sup>2<\/sup>\u00a0\u2013 x + 3)]<\/p>\n<p>where k is any non-zero real number.<\/p>\n<p><strong>3.<\/strong>\u00a0<strong>If the zeros of the polynomial f(x) = 2x<sup>3<\/sup>\u00a0\u2013 15x<sup>2<\/sup>\u00a0+ 37x \u2013 30 are in A.P., find them.<\/strong><\/p>\n<p><strong>Solution:<\/strong><\/p>\n<p>Let the zeros of the given polynomial be \u03b1, \u03b2 and \u03b3. (3 zeros as it\u2019s a cubic polynomial)<\/p>\n<p><strong>And given, the zeros are in A.P.<\/strong><\/p>\n<p><strong>So, let\u2019s consider the roots as<\/strong><\/p>\n<p>\u03b1 = a \u2013 d, \u03b2 = a and \u03b3 = a +d<\/p>\n<p>Where a is the first term and d is the common difference.<\/p>\n<p>From given f(x), a= 2, b= -15, c= 37 and d= 30<\/p>\n<p>\u21d2 Sum of roots = \u03b1 + \u03b2 + \u03b3 = (a \u2013 d) + a + (a + d) = 3a = (-b\/a) = -(-15\/2) = 15\/2<\/p>\n<p>So, calculating for a, we get 3a = 15\/2 \u21d2 a = 5\/2<\/p>\n<p>\u21d2 Product of roots = (a \u2013 d) x (a) x (a + d) = a(a<sup>2<\/sup>\u00a0\u2013d<sup>2<\/sup>) = -d\/a = -(30)\/2 = 15<\/p>\n<p>\u21d2 a(a<sup>2<\/sup>\u00a0\u2013d<sup>2<\/sup>) = 15<\/p>\n<p>Substituting the value of a, we get<\/p>\n<p>\u21d2 (5\/2)[(5\/2)<sup>2<\/sup>\u00a0\u2013d<sup>2<\/sup>] = 15<\/p>\n<p>\u21d2 5[(25\/4) \u2013d<sup>2<\/sup>] = 30<\/p>\n<p>\u21d2 (25\/4) \u2013 d<sup>2<\/sup>\u00a0= 6<\/p>\n<p>\u21d2 25 \u2013 4d<sup>2<\/sup>\u00a0= 24<\/p>\n<p>\u21d2 1 = 4d<sup>2<\/sup><\/p>\n<p>\u2234 d = 1\/2 or -1\/2<\/p>\n<p>Taking d = 1\/2 and a = 5\/2<\/p>\n<p>We get,<\/p>\n<p>the zeros as 2, 5\/2 and 3<\/p>\n<p>Taking d = -1\/2 and a = 5\/2<\/p>\n<p>We get,<\/p>\n<p>the zeros as 3, 5\/2 and 2<\/p>\n<p>We have provided complete details of RD Sharma Class 10 Solutions Chapter 2 Exercise 2.2. If you have any queries related to <a href=\"https:\/\/www.cbse.gov.in\/\" target=\"_blank\" rel=\"noopener\"><strong>CBSE<\/strong><\/a>\u00a0Class 10, feel free to ask us in the comment section below.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"faqs-on-rd-sharma-class-10-solutions-chapter-2-exercise-22\"><\/span>FAQs on RD Sharma Class 10 Solutions Chapter 2 Exercise 2.2<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-1631087873373\" class=\"rank-math-list-item\">\n<h3 class=\"rank-math-question \"><span class=\"ez-toc-section\" id=\"where-can-i-download-rd-sharma-class-10-solutions-chapter-2-exercise-22-free-pdf\"><\/span>Where can I download RD Sharma Class 10 Solutions Chapter 2 Exercise 2.2 free PDF?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<div class=\"rank-math-answer \">\n\n<p>You can download RD Sharma Class 10 Solutions Chapter 2 Exercise 2.2 free PDF from the above article.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"faq-question-1631087980843\" class=\"rank-math-list-item\">\n<h3 class=\"rank-math-question \"><span class=\"ez-toc-section\" id=\"what-are-the-benefits-of-using-rd-sharma-class-10-solutions-chapter-2-exercise-22\"><\/span>What are the benefits of using RD Sharma Class 10 Solutions Chapter 2 Exercise 2.2?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<div class=\"rank-math-answer \">\n\n<p>1. Correct answers according to the last CBSE guidelines and syllabus.<br \/>2. The solutions are written in simple language to assist students in their board examination, &amp; competitive examination preparation.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"faq-question-1631088042662\" class=\"rank-math-list-item\">\n<h3 class=\"rank-math-question \"><span class=\"ez-toc-section\" id=\"is-it-required-to-remember-all-of-the-questions-in-chapter-2-exercise-22-of-rd-sharma-solutions-for-class-10-maths\"><\/span>Is it required to remember all of the questions in Chapter 2 Exercise 2.2 of RD Sharma Solutions for Class 10 Maths?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<div class=\"rank-math-answer \">\n\n<p>Yes, all of the questions in RD Sharma Solutions for Class 10 Maths Chapter 2 Exercise 2.2 must be learned. These questions may appear on both board exams and class tests. Students will be prepared for their board exams if they learn these questions.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>","protected":false},"excerpt":{"rendered":"<p>RD Sharma Class 10 Solutions Chapter 2 Exercise 2.2:\u00a0This exercise focuses on verifying the zeros and the relationship between the zeros and the coefficients of a cubic polynomial. The experts at Kopykitab have prepared RD Sharma Solutions Class 10 to help students understand these concepts clearly while following the latest CBSE patterns. For a detailed &#8230; <a title=\"RD Sharma Class 10 Solutions Chapter 2 Exercise 2.2 (Updated for 2024)\" class=\"read-more\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-class-10-solutions-chapter-2-exercise-2-2\/\" aria-label=\"More on RD Sharma Class 10 Solutions Chapter 2 Exercise 2.2 (Updated for 2024)\">Read more<\/a><\/p>\n","protected":false},"author":238,"featured_media":125264,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"fifu_image_url":"","fifu_image_alt":""},"categories":[73411,2985,73410],"tags":[3243,9206,73520,4388],"amp_enabled":true,"_links":{"self":[{"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/posts\/125249"}],"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\/238"}],"replies":[{"embeddable":true,"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/comments?post=125249"}],"version-history":[{"count":5,"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/posts\/125249\/revisions"}],"predecessor-version":[{"id":517696,"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/posts\/125249\/revisions\/517696"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/media\/125264"}],"wp:attachment":[{"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/media?parent=125249"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/categories?post=125249"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/tags?post=125249"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}