{"id":4486,"date":"2016-05-16T11:49:00","date_gmt":"2016-05-16T06:19:00","guid":{"rendered":"http:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-5-1-part-1\/"},"modified":"2018-05-04T15:14:42","modified_gmt":"2018-05-04T09:44:42","slug":"ncert-solutions-class-12-maths-exercise-5-1-part-1","status":"publish","type":"post","link":"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-5-1-part-1\/","title":{"rendered":"NCERT Solutions class 12 Maths Exercise 5.1 Part-1"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_76 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\" style=\"cursor:inherit\">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=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" 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: #999;color:#999\" 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><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 eztoc-toggle-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:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-5-1-part-1\/#NCERT_Solutions_class_12_Continuity_Differentiability\" >NCERT Solutions class 12\u00a0Continuity &amp; Differentiability<\/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:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-5-1-part-1\/#NCERT_Solutions_class_12_Maths_Exercise_51_Part-1\" >NCERT Solutions class 12 Maths Exercise 5.1 Part-1<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-5-1-part-1\/#2_Examine_the_continuity_of_the_function_at\" >2. Examine the continuity of the function \u00a0at<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-5-1-part-1\/#NCERT_Solutions_class_12_Maths_Exercise_51_Part-1-2\" >NCERT Solutions class 12 Maths Exercise 5.1 Part-1<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-5-1-part-1\/#3_Examine_the_following_functions_for_continuity\" >3. Examine the following functions for continuity:<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-5-1-part-1\/#NCERT_Solutions_class_12_Maths_Exercise_51_Part-1-3\" >NCERT Solutions class 12 Maths Exercise 5.1 Part-1<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-5-1-part-1\/#4_Prove_that_the_function_is_continuous_at_where_is_a_positive_integer\" >4.\u00a0 Prove that the function \u00a0is continuous at \u00a0where \u00a0is a positive integer.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-5-1-part-1\/#NCERT_Solutions_class_12_Maths_Exercise_51_Part-1-4\" >NCERT Solutions class 12 Maths Exercise 5.1 Part-1<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-5-1-part-1\/#Find_all_points_of_discontinuity_of_where_is_defined_by_Exercise_6_to_12\" >Find all points of discontinuity of \u00a0where \u00a0is defined by: (Exercise 6 to 12)<\/a><ul class='ez-toc-list-level-4' ><li class='ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-5-1-part-1\/#11\" >11.<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-11\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-5-1-part-1\/#NCERT_Solutions_class_12_Maths_Exercise_51_Part-1-5\" >NCERT Solutions class 12 Maths Exercise 5.1 Part-1<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-12\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-5-1-part-1\/#13_Is_the_function_defined_by_a_continuous_function\" >13. Is the function defined by \u00a0a continuous function?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-13\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-5-1-part-1\/#Discuss_the_continuity_of_the_function_where_is_defined_by\" >Discuss the continuity of the function \u00a0where \u00a0is defined by:<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-14\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-5-1-part-1\/#NCERT_Solutions_class_12_Maths_Exercise_51_Part-1-6\" >NCERT Solutions class 12 Maths Exercise 5.1 Part-1<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-15\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-5-1-part-1\/#CBSE_App_for_Class_12\" >CBSE App for Class 12<\/a><\/li><\/ul><\/nav><\/div>\n<p>NCERT Solutions class 12 Maths Exercise 5.1 Part-1 Class 12 Maths book solutions are available in PDF format for free download. These ncert book chapter wise questions and answers are very helpful for CBSE board exam. CBSE recommends NCERT books and most of the questions in CBSE exam are asked from NCERT text books. Class 12 Maths chapter wise NCERT solution for Maths part 1 and Maths part 2 for all the chapters can be downloaded from our website and myCBSEguide mobile app for free.<\/p>\n<p><strong>Download\u00a0<a href=\"https:\/\/mycbseguide.com\/downloads\/cbse-class-12-mathematics-continuity-and-differentiability\/1289\/ncert-solutions\/5\/\">NCERT solutions for\u00a0Continuity and Differentiability\u00a0<\/a>as PDF.<\/strong><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/media-mycbseguide.s3.ap-south-1.amazonaws.com\/images\/blog\/12%20Maths%20Book%20pic.jpg\" alt=\"NCERT Solutions class 12 Maths Exercise 5.1 Part-1\" width=\"162\" height=\"211\" \/><\/p>\n<h2><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Continuity_Differentiability\"><\/span>NCERT Solutions class 12\u00a0Continuity &amp; Differentiability<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p style=\"text-align: justify;\"><strong>1. Prove that the function <\/strong><strong><img decoding=\"async\" style=\"height: 27px; width: 92px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image001.png\" \/><\/strong><strong>\u00a0is continuous at <\/strong><strong><img decoding=\"async\" style=\"height: 21px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image002.png\" \/><\/strong><strong>\u00a0at <\/strong><strong><img decoding=\"async\" style=\"height: 19px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image003.png\" \/><\/strong><strong>\u00a0and at <\/strong><strong><img decoding=\"async\" style=\"height: 19px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image004.png\" \/><\/strong><strong>\u00a0<\/strong><\/p>\n<p><strong>Ans. <\/strong>Given: <img decoding=\"async\" style=\"height: 27px; width: 92px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image001.png\" \/><\/p>\n<p>Continuity at <img decoding=\"async\" style=\"height: 21px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image002.png\" \/>\u00a0\u00a0<img decoding=\"async\" style=\"height: 31px; width: 151px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image005.png\" \/>\u00a0= 5 (0) \u2013 3 = 0 \u2013 3 = \u2013 3<\/p>\n<p>And <img decoding=\"async\" style=\"height: 27px; width: 51px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image006.png\" \/>\u00a05 (0) \u2013 3 = 0 \u2013 3 = \u2013 3<\/p>\n<p>Since <img decoding=\"async\" style=\"height: 31px; width: 112px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image007.png\" \/>, therefore, <img decoding=\"async\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image008.png\" \/>\u00a0is continuous at <img decoding=\"async\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image009.png\" \/>.<\/p>\n<p>Continuity at <img decoding=\"async\" style=\"height: 21px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image010.png\" \/>\u00a0\u00a0<img decoding=\"async\" style=\"height: 31px; width: 156px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image011.png\" \/>\u00a0= 5 (\u2013 3) \u2013 3 = \u2013 15 \u2013 3 = \u2013 18<\/p>\n<p>And <img decoding=\"async\" style=\"height: 27px; width: 59px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image012.png\" \/>\u00a05 (\u2013 3) \u2013 3 = \u2013 15 \u2013 3 = \u2013 18<\/p>\n<p>Since <img decoding=\"async\" style=\"height: 31px; width: 115px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image013.png\" \/>, therefore, <img decoding=\"async\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image008.png\" \/>\u00a0is continuous at <img decoding=\"async\" style=\"height: 19px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image003.png\" \/><\/p>\n<p>Continuity at <img decoding=\"async\" style=\"height: 21px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image014.png\" \/>\u00a0\u00a0<img decoding=\"async\" style=\"height: 31px; width: 151px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image015.png\" \/>\u00a0= 5 (5) \u2013 3 = 25 \u2013 3 = 2<\/p>\n<p>And <img decoding=\"async\" style=\"height: 27px; width: 51px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image016.png\" \/>\u00a05 (5) \u2013 3 = 25 \u2013 3 = 22<\/p>\n<p>Since <img decoding=\"async\" style=\"height: 31px; width: 112px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image017.png\" \/>, therefore, <img decoding=\"async\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image008.png\" \/>\u00a0is continuous at <img decoding=\"async\" style=\"height: 19px; width: 36px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image018.png\" \/>.<\/p>\n<hr \/>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_51_Part-1\"><\/span>NCERT Solutions class 12 Maths Exercise 5.1 Part-1<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"2_Examine_the_continuity_of_the_function_at\"><\/span><strong>2. Examine the continuity of the function <\/strong><strong><img decoding=\"async\" style=\"height: 27px; width: 97px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image019.png\" \/><\/strong><strong>\u00a0at <\/strong><strong><img decoding=\"async\" style=\"height: 19px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image020.png\" \/><\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Ans. <\/strong>Given: <img decoding=\"async\" style=\"height: 27px; width: 97px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image019.png\" \/><\/p>\n<p>Continuity at <img decoding=\"async\" style=\"height: 19px; width: 36px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image021.png\" \/>, \u00a0 <img decoding=\"async\" style=\"height: 32px; width: 156px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image022.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 29px; width: 140px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image023.png\" \/><\/p>\n<p>And <img decoding=\"async\" style=\"height: 27px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image024.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 29px; width: 140px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image023.png\" \/><\/p>\n<p>Since <img decoding=\"async\" style=\"height: 31px; width: 112px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image025.png\" \/>, therefore, <img decoding=\"async\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image008.png\" \/>\u00a0is continuous at <img decoding=\"async\" style=\"height: 19px; width: 36px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image021.png\" \/>.<\/p>\n<hr \/>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_51_Part-1-2\"><\/span>NCERT Solutions class 12 Maths Exercise 5.1 Part-1<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"3_Examine_the_following_functions_for_continuity\"><\/span><strong>3. Examine the following functions for continuity:<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>(a) <\/strong><strong><img decoding=\"async\" style=\"height: 27px; width: 84px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image026.png\" \/><\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(b) <\/strong><strong><img decoding=\"async\" style=\"height: 41px; width: 125px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image027.png\" \/><\/strong><strong>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(c) <\/strong><strong><img decoding=\"async\" style=\"height: 44px; width: 151px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image028.png\" \/><\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(d) <\/strong><strong><img decoding=\"async\" style=\"height: 27px; width: 89px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image029.png\" \/><\/strong><strong>\u00a0<\/strong><\/p>\n<p><strong>Ans. (a)<\/strong> Given: <img decoding=\"async\" style=\"height: 27px; width: 84px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image026.png\" \/><\/p>\n<p>It is evident that <img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image030.png\" \/>\u00a0is defined at every real number <img decoding=\"async\" style=\"height: 19px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image031.png\" \/>\u00a0and its value at <img decoding=\"async\" style=\"height: 19px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image031.png\" \/>\u00a0is <img decoding=\"async\" style=\"height: 19px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image032.png\" \/><\/p>\n<p>It is also observed that <img decoding=\"async\" style=\"height: 31px; width: 241px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image033.png\" \/><\/p>\n<p style=\"text-align: justify;\">Since <img decoding=\"async\" style=\"height: 31px; width: 112px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image034.png\" \/>, therefore, <img decoding=\"async\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image008.png\" \/>\u00a0is continuous at every real number and it is a continuous function.<\/p>\n<p><strong>(b)<\/strong> Given: <img decoding=\"async\" style=\"height: 41px; width: 125px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image027.png\" \/><\/p>\n<p>For any real number <img decoding=\"async\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image035.png\" \/>, we get\u00a0 <img decoding=\"async\" style=\"height: 41px; width: 181px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image036.png\" \/><\/p>\n<p>And <img decoding=\"async\" style=\"height: 41px; width: 87px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image037.png\" \/><\/p>\n<p style=\"text-align: justify;\">Since <img decoding=\"async\" style=\"height: 31px; width: 112px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image034.png\" \/>, therefore, <img decoding=\"async\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image008.png\" \/>\u00a0is continuous at every point of domain of <img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image030.png\" \/>\u00a0and it is a continuous function.<\/p>\n<p style=\"text-align: justify;\"><strong>(c)<\/strong> Given: <img decoding=\"async\" style=\"height: 44px; width: 151px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image028.png\" \/><\/p>\n<p>For any real number <img decoding=\"async\" style=\"height: 19px; width: 47px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image038.png\" \/>, we get<\/p>\n<p><img decoding=\"async\" style=\"height: 44px; width: 403px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image039.png\" \/><\/p>\n<p>And <img decoding=\"async\" style=\"height: 44px; width: 189px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image040.png\" \/><\/p>\n<p style=\"text-align: justify;\">Since <img decoding=\"async\" style=\"height: 31px; width: 112px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image034.png\" \/>, therefore, <img decoding=\"async\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image008.png\" \/>\u00a0is continuous at every point of domain of <img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image030.png\" \/>\u00a0and it is a continuous function.<\/p>\n<p style=\"text-align: justify;\"><strong>(d)<\/strong> Given: <img decoding=\"async\" style=\"height: 27px; width: 89px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image029.png\" \/><\/p>\n<p style=\"text-align: justify;\">Domain of <img decoding=\"async\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image008.png\" \/>\u00a0is real and infinite for all real <img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image041.png\" \/><\/p>\n<p style=\"text-align: justify;\">Here <img decoding=\"async\" style=\"height: 27px; width: 89px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image029.png\" \/>\u00a0is a modulus function.<\/p>\n<p style=\"text-align: justify;\">Since, every modulus function is continuous, therefore, <img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image030.png\" \/>\u00a0is continuous in its domain R.<\/p>\n<hr \/>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_51_Part-1-3\"><\/span>NCERT Solutions class 12 Maths Exercise 5.1 Part-1<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"4_Prove_that_the_function_is_continuous_at_where_is_a_positive_integer\"><\/span><strong>4.\u00a0 Prove that the function <\/strong><strong><img decoding=\"async\" style=\"height: 27px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image042.png\" \/><\/strong><strong>\u00a0is continuous at <\/strong><strong><img decoding=\"async\" style=\"height: 15px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image043.png\" \/><\/strong><strong>\u00a0where <\/strong><strong><img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image044.png\" \/><\/strong><strong>\u00a0is a positive integer.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Ans. <\/strong>Given: <img decoding=\"async\" style=\"height: 27px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image042.png\" \/>\u00a0where <img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image044.png\" \/>\u00a0is a positive integer.<\/p>\n<p>Continuity at <img decoding=\"async\" style=\"height: 15px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image043.png\" \/>, \u00a0 <img decoding=\"async\" style=\"height: 32px; width: 159px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image045.png\" \/><\/p>\n<p>And <img decoding=\"async\" style=\"height: 27px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image046.png\" \/><\/p>\n<p>Since <img decoding=\"async\" style=\"height: 31px; width: 112px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image047.png\" \/>, therefore, <img decoding=\"async\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image008.png\" \/>\u00a0is continuous at <img decoding=\"async\" style=\"height: 15px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image043.png\" \/>.<\/p>\n<hr \/>\n<p style=\"text-align: justify;\"><strong>5. Is the function <\/strong><strong><img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image030.png\" \/><\/strong><strong>\u00a0defined by <\/strong><strong><img decoding=\"async\" style=\"height: 48px; width: 131px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image048.png\" \/><\/strong><strong>\u00a0continuous at <\/strong><strong><img decoding=\"async\" style=\"height: 21px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image002.png\" \/><\/strong><strong>\u00a0at <\/strong><strong><img decoding=\"async\" style=\"height: 21px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image049.png\" \/><\/strong><strong>\u00a0at <\/strong><strong><img decoding=\"async\" style=\"height: 19px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image050.png\" \/><\/strong><strong>\u00a0<\/strong><\/p>\n<p><strong>Ans. <\/strong>Given: <img decoding=\"async\" style=\"height: 48px; width: 155px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image051.png\" \/><\/p>\n<p>At <img decoding=\"async\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image009.png\" \/>, It is evident that <img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image030.png\" \/>\u00a0is defined at 0 and its value at 0 is 0.<\/p>\n<p>Then <img decoding=\"async\" style=\"height: 31px; width: 133px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image052.png\" \/>\u00a0and\u00a0<img decoding=\"async\" style=\"height: 27px; width: 61px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image053.png\" \/><\/p>\n<p>Therefore, <img decoding=\"async\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image008.png\" \/>\u00a0is continuous at <img decoding=\"async\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image009.png\" \/>.<\/p>\n<p style=\"text-align: justify;\">At <img decoding=\"async\" style=\"height: 19px; width: 35px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image054.png\" \/>, Left Hand limit of <img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image030.png\" \/><img decoding=\"async\" style=\"height: 31px; width: 147px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image055.png\" \/><\/p>\n<p style=\"text-align: justify;\">Right Hand limit of <img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image030.png\" \/><img decoding=\"async\" style=\"height: 31px; width: 148px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image056.png\" \/><\/p>\n<p style=\"text-align: justify;\">Here <img decoding=\"async\" style=\"height: 31px; width: 139px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image057.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, <img decoding=\"async\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image008.png\" \/>\u00a0is not continuous at <img decoding=\"async\" style=\"height: 19px; width: 35px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image054.png\" \/>.<\/p>\n<p style=\"text-align: justify;\">At <img decoding=\"async\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image058.png\" \/>, <img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image030.png\" \/>\u00a0is defined at 2 and its value at 2 is 5.<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 31px; width: 144px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image059.png\" \/>, therefore, <img decoding=\"async\" style=\"height: 31px; width: 112px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image060.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, <img decoding=\"async\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image008.png\" \/>\u00a0is not continuous at <img decoding=\"async\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image058.png\" \/>.<\/p>\n<hr \/>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_51_Part-1-4\"><\/span>NCERT Solutions class 12 Maths Exercise 5.1 Part-1<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"Find_all_points_of_discontinuity_of_where_is_defined_by_Exercise_6_to_12\"><\/span><strong>Find all points of discontinuity of <\/strong><strong><img decoding=\"async\" style=\"height: 21px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image061.png\" \/><\/strong><strong>\u00a0where <\/strong><strong><img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image030.png\" \/><\/strong><strong>\u00a0is defined by: (Exercise 6 to 12)<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>6. <\/strong><strong><img decoding=\"async\" style=\"height: 48px; width: 151px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image062.png\" \/><\/strong><strong>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Given: <img decoding=\"async\" style=\"height: 48px; width: 188px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image063.png\" \/><\/p>\n<p style=\"text-align: justify;\">Here <img decoding=\"async\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image008.png\" \/>\u00a0is defined for <img decoding=\"async\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image064.png\" \/>\u00a0i.e., on <img decoding=\"async\" style=\"height: 27px; width: 52px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image065.png\" \/>\u00a0and also for <img decoding=\"async\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image066.png\" \/>\u00a0i.e., on <img decoding=\"async\" style=\"height: 27px; width: 47px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image067.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image068.png\" \/>\u00a0Domain of <img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image030.png\" \/>\u00a0is <img decoding=\"async\" style=\"height: 27px; width: 172px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image069.png\" \/>\u00a0= R<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image068.png\" \/>\u00a0For all <img decoding=\"async\" style=\"height: 27px; width: 131px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image070.png\" \/>\u00a0is a polynomial and hence continuous and for all<img decoding=\"async\" style=\"height: 27px; width: 131px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image071.png\" \/>\u00a0is a continuous and hence it is also continuous on R \u2013 {2}.<\/p>\n<p style=\"text-align: justify;\">Now Left Hand limit = <img decoding=\"async\" style=\"height: 31px; width: 159px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image072.png\" \/>\u00a0= 2 x 2 + 3 = 4 + 3 = 7<\/p>\n<p style=\"text-align: justify;\">Right Hand limit = <img decoding=\"async\" style=\"height: 31px; width: 159px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image073.png\" \/>\u00a0= 2 x 2 \u2013 3 = 4 \u2013 3 = 1<\/p>\n<p style=\"text-align: justify;\">Since\u00a0\u00a0 <img decoding=\"async\" style=\"height: 31px; width: 143px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image074.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, <img decoding=\"async\" style=\"height: 31px; width: 61px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image075.png\" \/>\u00a0does not exist and hence <img decoding=\"async\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image008.png\" \/>\u00a0is discontinuous at <img decoding=\"async\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image058.png\" \/>\u00a0(only)<\/p>\n<hr \/>\n<p style=\"text-align: justify;\"><strong>7. <\/strong><strong><img decoding=\"async\" style=\"height: 80px; width: 196px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image076.png\" \/><\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Given: <img decoding=\"async\" style=\"height: 80px; width: 185px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image077.png\" \/><\/p>\n<p style=\"text-align: justify;\">Here <img decoding=\"async\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image008.png\" \/>\u00a0is defined for <img decoding=\"async\" style=\"height: 19px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image078.png\" \/>\u00a0i.e., on <img decoding=\"async\" style=\"height: 27px; width: 60px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image079.png\" \/>\u00a0and for <img decoding=\"async\" style=\"height: 19px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image080.png\" \/>\u00a0and also for <img decoding=\"async\" style=\"height: 19px; width: 36px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image081.png\" \/>\u00a0i.e., on <img decoding=\"async\" style=\"height: 27px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image082.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image068.png\" \/>\u00a0Domain of <img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image030.png\" \/>\u00a0is <img decoding=\"async\" style=\"height: 27px; width: 237px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image083.png\" \/>\u00a0= R<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image068.png\" \/>\u00a0For all <img decoding=\"async\" style=\"height: 27px; width: 191px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image084.png\" \/>\u00a0is a polynomial and hence continuous and for all <img decoding=\"async\" style=\"height: 27px; width: 173px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image085.png\" \/>\u00a0is a continuous and hence it is also continuous and also for all <img decoding=\"async\" style=\"height: 27px; width: 131px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image086.png\" \/>. Therefore, <img decoding=\"async\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image008.png\" \/>\u00a0is continuous on R \u2013 <img decoding=\"async\" style=\"height: 27px; width: 51px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image087.png\" \/><\/p>\n<p style=\"text-align: justify;\">It is observed that <img decoding=\"async\" style=\"height: 19px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image003.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 19px; width: 36px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image021.png\" \/>\u00a0are partitioning points of domain R.<\/p>\n<p style=\"text-align: justify;\">Now Left Hand limit = <img decoding=\"async\" style=\"height: 32px; width: 315px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image088.png\" \/><\/p>\n<p style=\"text-align: justify;\">Right Hand limit = <img decoding=\"async\" style=\"height: 31px; width: 244px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image089.png\" \/><\/p>\n<p style=\"text-align: justify;\">And <img decoding=\"async\" style=\"height: 27px; width: 173px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image090.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, <img decoding=\"async\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image008.png\" \/>\u00a0is continuous at <img decoding=\"async\" style=\"height: 19px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image003.png\" \/>.<\/p>\n<p style=\"text-align: justify;\">Again\u00a0 Left Hand limit = <img decoding=\"async\" style=\"height: 31px; width: 231px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image091.png\" \/><\/p>\n<p style=\"text-align: justify;\">Right Hand limit = <img decoding=\"async\" style=\"height: 31px; width: 309px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image092.png\" \/><\/p>\n<p style=\"text-align: justify;\">Since <img decoding=\"async\" style=\"height: 31px; width: 141px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image093.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, <img decoding=\"async\" style=\"height: 31px; width: 61px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image094.png\" \/>\u00a0does not exist and hence <img decoding=\"async\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image008.png\" \/>\u00a0is discontinuous at <img decoding=\"async\" style=\"height: 19px; width: 36px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image021.png\" \/>\u00a0(only).<\/p>\n<hr \/>\n<p style=\"text-align: justify;\"><strong>8. <\/strong><strong><img decoding=\"async\" style=\"height: 72px; width: 144px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image095.png\" \/><\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Given: <img decoding=\"async\" style=\"height: 72px; width: 168px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image096.png\" \/>\u00a0i.e., <img decoding=\"async\" style=\"height: 41px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image097.png\" \/>\u00a0if <img decoding=\"async\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image098.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 41px; width: 56px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image099.png\" \/>\u00a0if <img decoding=\"async\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image100.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image101.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 27px; width: 60px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image102.png\" \/>\u00a0if <img decoding=\"async\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image098.png\" \/>, <img decoding=\"async\" style=\"height: 27px; width: 71px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image103.png\" \/>\u00a0if <img decoding=\"async\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image100.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 27px; width: 63px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image104.png\" \/>\u00a0if <img decoding=\"async\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image009.png\" \/><\/p>\n<p style=\"text-align: justify;\">It is clear that domain of <img decoding=\"async\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image008.png\" \/>\u00a0is R as <img decoding=\"async\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image008.png\" \/>\u00a0is defined for <img decoding=\"async\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image098.png\" \/>, <img decoding=\"async\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image100.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image009.png\" \/>.<\/p>\n<p style=\"text-align: justify;\">For all <img decoding=\"async\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image098.png\" \/>, <img decoding=\"async\" style=\"height: 27px; width: 60px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image102.png\" \/>\u00a0is a constant function and hence continuous.<\/p>\n<p style=\"text-align: justify;\">For all <img decoding=\"async\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image100.png\" \/>, <img decoding=\"async\" style=\"height: 27px; width: 71px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image103.png\" \/>\u00a0is a constant function and hence continuous.<\/p>\n<p style=\"text-align: justify;\">Therefore <img decoding=\"async\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image008.png\" \/>\u00a0is continuous on R \u2013 {0}.<\/p>\n<p style=\"text-align: justify;\">Now Left Hand limit = <img decoding=\"async\" style=\"height: 31px; width: 167px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image105.png\" \/><\/p>\n<p style=\"text-align: justify;\">Right Hand limit = <img decoding=\"async\" style=\"height: 31px; width: 147px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image106.png\" \/><\/p>\n<p style=\"text-align: justify;\">Since <img decoding=\"async\" style=\"height: 31px; width: 141px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image107.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, <img decoding=\"async\" style=\"height: 31px; width: 61px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image108.png\" \/>\u00a0does not exist and hence <img decoding=\"async\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image008.png\" \/>\u00a0is discontinuous at <img decoding=\"async\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image009.png\" \/>\u00a0(only).<\/p>\n<hr \/>\n<p style=\"text-align: justify;\"><strong>9. <\/strong><strong><img decoding=\"async\" style=\"height: 75px; width: 145px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image109.png\" \/><\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Given: <img decoding=\"async\" style=\"height: 75px; width: 169px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image110.png\" \/><\/p>\n<p style=\"text-align: justify;\">At <img decoding=\"async\" style=\"height: 21px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image002.png\" \/>\u00a0L.H.L. = <img decoding=\"async\" style=\"height: 47px; width: 79px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image111.png\" \/> And\u00a0<img decoding=\"async\" style=\"height: 27px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image112.png\" \/><\/p>\n<p style=\"text-align: justify;\">R.H.L. = <img decoding=\"async\" style=\"height: 31px; width: 97px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image113.png\" \/><\/p>\n<p style=\"text-align: justify;\">Since\u00a0\u00a0 L.H.L. = R.H.L. = <img decoding=\"async\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image114.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, <img decoding=\"async\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image008.png\" \/>\u00a0is a continuous function.<\/p>\n<p style=\"text-align: justify;\">Now, for <img decoding=\"async\" style=\"height: 19px; width: 60px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image115.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 47px; width: 128px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image116.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image068.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 31px; width: 79px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image117.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, <img decoding=\"async\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image008.png\" \/>\u00a0is a continuous at <img decoding=\"async\" style=\"height: 19px; width: 60px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image115.png\" \/><\/p>\n<p style=\"text-align: justify;\">Now, for <img decoding=\"async\" style=\"height: 19px; width: 60px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image118.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 31px; width: 135px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image119.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, <img decoding=\"async\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image008.png\" \/>\u00a0is a continuous at <img decoding=\"async\" style=\"height: 19px; width: 60px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image118.png\" \/><\/p>\n<p style=\"text-align: justify;\">Hence, the function is continuous at all points of its domain.<\/p>\n<hr \/>\n<p style=\"text-align: justify;\"><strong>10. <\/strong><strong><img decoding=\"async\" style=\"height: 51px; width: 159px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image120.png\" \/><\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Given: <img decoding=\"async\" style=\"height: 51px; width: 185px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image121.png\" \/><\/p>\n<p style=\"text-align: justify;\">It is observed that <img decoding=\"async\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image008.png\" \/>\u00a0being polynomial is continuous for <img decoding=\"async\" style=\"height: 19px; width: 33px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image122.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 19px; width: 33px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image123.png\" \/>\u00a0for all <img decoding=\"async\" style=\"height: 15px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image124.png\" \/>\u00a0R.<\/p>\n<p style=\"text-align: justify;\">Continuity at <img decoding=\"async\" style=\"height: 21px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image049.png\" \/>\u00a0R.H.L. = <img decoding=\"async\" style=\"height: 31px; width: 267px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image125.png\" \/><\/p>\n<p style=\"text-align: justify;\">L.H.L. = <img decoding=\"async\" style=\"height: 35px; width: 292px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image126.png\" \/><\/p>\n<p style=\"text-align: justify;\">And <img decoding=\"async\" style=\"height: 27px; width: 60px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image127.png\" \/><\/p>\n<p style=\"text-align: justify;\">Since\u00a0\u00a0 L.H.L. = R.H.L. = <img decoding=\"async\" style=\"height: 27px; width: 36px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image128.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, <img decoding=\"async\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image008.png\" \/>\u00a0is a continuous at <img decoding=\"async\" style=\"height: 19px; width: 35px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image054.png\" \/>\u00a0for all <img decoding=\"async\" style=\"height: 15px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image124.png\" \/>\u00a0R.<\/p>\n<p style=\"text-align: justify;\">Hence, <img decoding=\"async\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image008.png\" \/>\u00a0has no point of discontinuity.<\/p>\n<hr \/>\n<h4 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"11\"><\/span><strong>11. <\/strong><strong><img decoding=\"async\" style=\"height: 53px; width: 169px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image129.png\" \/><\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Given: <img decoding=\"async\" style=\"height: 53px; width: 188px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image130.png\" \/><\/p>\n<p style=\"text-align: justify;\">At <img decoding=\"async\" style=\"height: 21px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image131.png\" \/>\u00a0L.H.L. = <img decoding=\"async\" style=\"height: 32px; width: 147px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image132.png\" \/><\/p>\n<p style=\"text-align: justify;\">R.H.L. = <img decoding=\"async\" style=\"height: 32px; width: 145px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image133.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 27px; width: 157px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image134.png\" \/><\/p>\n<p style=\"text-align: justify;\">Since\u00a0\u00a0 L.H.L. = R.H.L. = <img decoding=\"async\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image135.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, <img decoding=\"async\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image008.png\" \/>\u00a0is a continuous at <img decoding=\"async\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image058.png\" \/><\/p>\n<p style=\"text-align: justify;\">\u00a0Now, for <img decoding=\"async\" style=\"height: 19px; width: 60px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image115.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 32px; width: 175px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image136.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 32px; width: 173px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image137.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image068.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 31px; width: 79px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image117.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, <img decoding=\"async\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image008.png\" \/>\u00a0is a continuous for all <img decoding=\"async\" style=\"height: 15px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image124.png\" \/>\u00a0R.<\/p>\n<p style=\"text-align: justify;\">Hence the function has no point of discontinuity.<\/p>\n<hr \/>\n<p style=\"text-align: justify;\"><strong>12. <\/strong><strong><img decoding=\"async\" style=\"height: 53px; width: 171px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image138.png\" \/><\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Given: <img decoding=\"async\" style=\"height: 53px; width: 197px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image139.png\" \/><\/p>\n<p style=\"text-align: justify;\">At <img decoding=\"async\" style=\"height: 21px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image049.png\" \/>\u00a0L.H.L. = <img decoding=\"async\" style=\"height: 32px; width: 179px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image140.png\" \/><\/p>\n<p style=\"text-align: justify;\">R.H.L. = <img decoding=\"async\" style=\"height: 32px; width: 153px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image141.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 27px; width: 109px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image142.png\" \/><\/p>\n<p style=\"text-align: justify;\">Since\u00a0\u00a0 L.H.L. <img decoding=\"async\" style=\"height: 15px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image143.png\" \/>\u00a0R.H.L.<\/p>\n<p style=\"text-align: justify;\">Therefore, <img decoding=\"async\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image008.png\" \/>\u00a0is discontinuous at <img decoding=\"async\" style=\"height: 19px; width: 35px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image054.png\" \/><\/p>\n<p style=\"text-align: justify;\">\u00a0Now, \u00a0 for <img decoding=\"async\" style=\"height: 19px; width: 57px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image144.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 32px; width: 180px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image145.png\" \/>\u00a0and for <img decoding=\"async\" style=\"height: 19px; width: 57px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image146.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 32px; width: 132px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image147.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, <img decoding=\"async\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image008.png\" \/>\u00a0is a continuous for all <img decoding=\"async\" style=\"height: 15px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image124.png\" \/>\u00a0R \u2013 {1}<\/p>\n<p style=\"text-align: justify;\">Hence for all given function <img decoding=\"async\" style=\"height: 19px; width: 35px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image054.png\" \/>\u00a0is a point of discontinuity.<\/p>\n<hr \/>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_51_Part-1-5\"><\/span>NCERT Solutions class 12 Maths Exercise 5.1 Part-1<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"13_Is_the_function_defined_by_a_continuous_function\"><\/span><strong>13. Is the function defined by <\/strong><strong><img decoding=\"async\" style=\"height: 48px; width: 169px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image148.png\" \/><\/strong><strong>\u00a0a continuous function?<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Given: <img decoding=\"async\" style=\"height: 48px; width: 185px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image149.png\" \/><\/p>\n<p style=\"text-align: justify;\">At <img decoding=\"async\" style=\"height: 21px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image049.png\" \/>\u00a0L.H.L. = <img decoding=\"async\" style=\"height: 31px; width: 171px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image150.png\" \/><\/p>\n<p style=\"text-align: justify;\">R.H.L. = <img decoding=\"async\" style=\"height: 31px; width: 179px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image151.png\" \/><\/p>\n<p style=\"text-align: justify;\">Since\u00a0\u00a0 L.H.L. <img decoding=\"async\" style=\"height: 15px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image143.png\" \/>\u00a0R.H.L.<\/p>\n<p style=\"text-align: justify;\">Therefore, <img decoding=\"async\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image008.png\" \/>\u00a0is discontinuous at <img decoding=\"async\" style=\"height: 19px; width: 35px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image054.png\" \/><\/p>\n<p style=\"text-align: justify;\">\u00a0Now, \u00a0 for <img decoding=\"async\" style=\"height: 19px; width: 57px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image144.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 31px; width: 164px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image152.png\" \/>\u00a0and for <img decoding=\"async\" style=\"height: 19px; width: 57px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image146.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 31px; width: 163px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image153.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, <img decoding=\"async\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image008.png\" \/>\u00a0is a continuous for all <img decoding=\"async\" style=\"height: 15px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image124.png\" \/>\u00a0R \u2013 {1}<\/p>\n<p style=\"text-align: justify;\">Hence <img decoding=\"async\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image008.png\" \/>\u00a0is not a continuous function.<\/p>\n<hr \/>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"Discuss_the_continuity_of_the_function_where_is_defined_by\"><\/span><strong>Discuss the continuity of the function <\/strong><strong><img decoding=\"async\" style=\"height: 21px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image061.png\" \/><\/strong><strong>\u00a0where <\/strong><strong><img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image030.png\" \/><\/strong><strong>\u00a0is defined by:<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>14. <\/strong><strong><img decoding=\"async\" style=\"height: 75px; width: 173px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image154.png\" \/><\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Given: <img decoding=\"async\" style=\"height: 75px; width: 196px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image155.png\" \/><\/p>\n<p style=\"text-align: justify;\">In the interval <img decoding=\"async\" style=\"height: 21px; width: 61px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image156.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 61px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image157.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image068.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image030.png\" \/>\u00a0is continuous in this interval.<\/p>\n<p style=\"text-align: justify;\">At <img decoding=\"async\" style=\"height: 21px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image049.png\" \/>\u00a0L.H.L. = <img decoding=\"async\" style=\"height: 31px; width: 87px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image158.png\" \/>\u00a0and R.H.L. = <img decoding=\"async\" style=\"height: 31px; width: 88px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image159.png\" \/><\/p>\n<p style=\"text-align: justify;\">Since\u00a0\u00a0 L.H.L. <img decoding=\"async\" style=\"height: 15px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image143.png\" \/>\u00a0R.H.L.<\/p>\n<p style=\"text-align: justify;\">Therefore, <img decoding=\"async\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image008.png\" \/>\u00a0is discontinuous at <img decoding=\"async\" style=\"height: 19px; width: 35px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image054.png\" \/><\/p>\n<p style=\"text-align: justify;\">At <img decoding=\"async\" style=\"height: 21px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image160.png\" \/>\u00a0L.H.L. = <img decoding=\"async\" style=\"height: 31px; width: 89px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image161.png\" \/>\u00a0and R.H.L. = <img decoding=\"async\" style=\"height: 31px; width: 88px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image162.png\" \/><\/p>\n<p style=\"text-align: justify;\">Since\u00a0\u00a0 L.H.L. <img decoding=\"async\" style=\"height: 15px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image143.png\" \/>\u00a0R.H.L.<\/p>\n<p style=\"text-align: justify;\">Therefore, <img decoding=\"async\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image008.png\" \/>\u00a0is discontinuous at <img decoding=\"async\" style=\"height: 19px; width: 36px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image021.png\" \/><\/p>\n<p style=\"text-align: justify;\">Hence, <img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image030.png\" \/>\u00a0is discontinuous at <img decoding=\"async\" style=\"height: 19px; width: 35px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image054.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 19px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image020.png\" \/><\/p>\n<hr \/>\n<p style=\"text-align: justify;\"><strong>15. <\/strong><strong><img decoding=\"async\" style=\"height: 75px; width: 164px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image163.png\" \/><\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Given: <img decoding=\"async\" style=\"height: 75px; width: 195px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image164.png\" \/><\/p>\n<p style=\"text-align: justify;\">At <img decoding=\"async\" style=\"height: 21px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image002.png\" \/>\u00a0L.H.L. = <img decoding=\"async\" style=\"height: 29px; width: 72px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image165.png\" \/>\u00a0and R.H.L. = <img decoding=\"async\" style=\"height: 31px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image166.png\" \/><\/p>\n<p style=\"text-align: justify;\">Since\u00a0\u00a0 L.H.L. = R.H.L.<\/p>\n<p style=\"text-align: justify;\">Therefore, <img decoding=\"async\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image008.png\" \/>\u00a0is continuous at <img decoding=\"async\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image009.png\" \/><\/p>\n<p style=\"text-align: justify;\">At <img decoding=\"async\" style=\"height: 21px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image049.png\" \/>\u00a0L.H.L. = <img decoding=\"async\" style=\"height: 31px; width: 73px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image167.png\" \/>\u00a0 and R.H.L. = <img decoding=\"async\" style=\"height: 31px; width: 83px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image168.png\" \/><\/p>\n<p style=\"text-align: justify;\">Since\u00a0\u00a0 L.H.L. <img decoding=\"async\" style=\"height: 15px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image143.png\" \/>\u00a0R.H.L.<\/p>\n<p style=\"text-align: justify;\">Therefore, <img decoding=\"async\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image008.png\" \/>\u00a0is discontinuous at <img decoding=\"async\" style=\"height: 19px; width: 35px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image054.png\" \/><\/p>\n<p style=\"text-align: justify;\">When <img decoding=\"async\" style=\"height: 27px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image169.png\" \/>\u00a0being a polynomial function is continuous for all <img decoding=\"async\" style=\"height: 19px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image170.png\" \/><\/p>\n<p style=\"text-align: justify;\">When <img decoding=\"async\" style=\"height: 27px; width: 107px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image171.png\" \/>. It is being a polynomial function is continuous for all <img decoding=\"async\" style=\"height: 19px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image172.png\" \/><\/p>\n<p style=\"text-align: justify;\">Hence <img decoding=\"async\" style=\"height: 19px; width: 35px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image054.png\" \/>\u00a0is a point of discontinuity.<\/p>\n<hr \/>\n<p style=\"text-align: justify;\"><strong>16. <\/strong><strong><img decoding=\"async\" style=\"height: 75px; width: 184px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image173.png\" \/><\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Given: <img decoding=\"async\" style=\"height: 75px; width: 200px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image174.png\" \/><\/p>\n<p style=\"text-align: justify;\">At <img decoding=\"async\" style=\"height: 21px; width: 48px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image175.png\" \/>\u00a0L.H.L. = <img decoding=\"async\" style=\"height: 31px; width: 103px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image176.png\" \/>\u00a0and\u00a0R.H.L. = <img decoding=\"async\" style=\"height: 31px; width: 99px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image177.png\" \/><\/p>\n<p style=\"text-align: justify;\">Since\u00a0\u00a0 L.H.L. = R.H.L.<\/p>\n<p style=\"text-align: justify;\">Therefore, <img decoding=\"async\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image008.png\" \/>\u00a0is continuous at <img decoding=\"async\" style=\"height: 19px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image178.png\" \/><\/p>\n<p style=\"text-align: justify;\">At <img decoding=\"async\" style=\"height: 21px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image049.png\" \/>\u00a0L.H.L. = <img decoding=\"async\" style=\"height: 31px; width: 88px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image179.png\" \/>\u00a0and R.H.L. = <img decoding=\"async\" style=\"height: 31px; width: 88px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image180.png\" \/><\/p>\n<p style=\"text-align: justify;\">Since\u00a0\u00a0 L.H.L. = R.H.L.<\/p>\n<p style=\"text-align: justify;\">Therefore, <img decoding=\"async\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image008.png\" \/>\u00a0is continuous at <img decoding=\"async\" style=\"height: 19px; width: 35px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex5.1\/image054.png\" \/><\/p>\n<h2><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_51_Part-1-6\"><\/span>NCERT Solutions class 12 Maths Exercise 5.1 Part-1<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>NCERT Solutions Class 12 Maths PDF (Download) Free from myCBSEguide app and myCBSEguide website. Ncert solution class 12 Maths includes text book solutions from both part 1 and part 2. NCERT Solutions for CBSE Class 12 Maths have total 20 chapters. 12 Maths NCERT Solutions in PDF for free Download on our website. Ncert Maths class 12 solutions PDF and Maths ncert class 12 PDF solutions with latest modifications and as per the latest CBSE syllabus are only available in myCBSEguide<\/p>\n<h2><span class=\"ez-toc-section\" id=\"CBSE_App_for_Class_12\"><\/span>CBSE App for Class 12<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>To download\u00a0NCERT Solutions for class 12 Physics, Chemistry, Biology, History, Political Science, Economics, Geography, Computer Science, Home Science, Accountancy, Business Studies and Home Science; do check myCBSEguide app or website. myCBSEguide provides sample papers with solution, test papers for chapter-wise practice, NCERT solutions, NCERT Exemplar solutions, quick revision notes for ready reference, CBSE guess papers and CBSE important question papers. Sample Paper all are made available through\u00a0<a href=\"https:\/\/play.google.com\/store\/apps\/details?id=in.techchefs.MyCBSEGuide&amp;referrer=utm_source%3Dmycbse_bottom%26utm_medium%3Dtext%26utm_campaign%3Dmycbseads\"><strong>the best app for CBSE students<\/strong><\/a>\u00a0and myCBSEguide website.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>NCERT Solutions class 12 Maths Exercise 5.1 Part-1 Class 12 Maths book solutions are available in PDF format for free download. These ncert book chapter wise questions and answers are very helpful for CBSE board exam. CBSE recommends NCERT books and most of the questions in CBSE exam are asked from NCERT text books. Class &#8230; <a title=\"NCERT Solutions class 12 Maths Exercise 5.1 Part-1\" class=\"read-more\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-5-1-part-1\/\" aria-label=\"More on NCERT Solutions class 12 Maths Exercise 5.1 Part-1\">Read more<\/a><\/p>\n","protected":false},"author":1,"featured_media":-1,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1432,281],"tags":[283,1342,321,216],"class_list":["post-4486","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mathematics-cbse-class-12","category-ncert-solutions","tag-cbse-study-material","tag-class-12","tag-mathematics","tag-ncert-solutions"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.0 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>NCERT Solutions class 12 Maths Exercise 5.1 Part-1 math<\/title>\n<meta name=\"description\" content=\"NCERT Solutions class 12 Maths Exercise 5.1 Part-1 in PDF format for free download. 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