{"id":4495,"date":"2016-05-16T11:49:00","date_gmt":"2016-05-16T06:19:00","guid":{"rendered":"http:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-4\/"},"modified":"2018-05-04T17:18:06","modified_gmt":"2018-05-04T11:48:06","slug":"ncert-solutions-class-12-maths-miscellaneous-4","status":"publish","type":"post","link":"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-4\/","title":{"rendered":"NCERT Solutions class 12 Maths Miscellaneous"},"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-miscellaneous-4\/#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-miscellaneous-4\/#NCERT_Solutions_class_12_Maths_Miscellaneous\" >NCERT Solutions class 12 Maths Miscellaneous<\/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-miscellaneous-4\/#15_If_for_some_prove_that\" >15. If \u00a0for some \u00a0prove that \u00a0<\/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-miscellaneous-4\/#16_If_with_prove_that\" >16. If \u00a0with \u00a0prove that \u00a0<\/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-miscellaneous-4\/#NCERT_Solutions_class_12_Maths_Miscellaneous-2\" >NCERT Solutions class 12 Maths Miscellaneous<\/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-miscellaneous-4\/#17_If_and_find\" >17. If \u00a0and \u00a0find \u00a0<\/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-miscellaneous-4\/#NCERT_Solutions_class_12_Maths_Miscellaneous-3\" >NCERT Solutions class 12 Maths Miscellaneous<\/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-miscellaneous-4\/#18_If_show_that_exists_for_all_real_and_find_it\" >18. If \u00a0show that \u00a0exists for all real \u00a0and find it.<\/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-miscellaneous-4\/#NCERT_Solutions_class_12_Maths_Miscellaneous-4\" >NCERT Solutions class 12 Maths Miscellaneous<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-4\/#19_Using_mathematical_induction_prove_that_for_all_positive_integers\" >19. Using mathematical induction, prove that \u00a0for all positive integers \u00a0<\/a><\/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-miscellaneous-4\/#NCERT_Solutions_class_12_Maths_Miscellaneous-5\" >NCERT Solutions class 12 Maths Miscellaneous<\/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-miscellaneous-4\/#20_Using_the_fact_that_and_the_differentiation_obtain_the_sum_formula_for_cosines\" >20. Using the fact that \u00a0and the differentiation, obtain the sum formula for cosines.<\/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-miscellaneous-4\/#NCERT_Solutions_class_12_Maths_Miscellaneous-6\" >NCERT Solutions class 12 Maths Miscellaneous<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-14\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-4\/#21_Does_there_exist_a_function_which_is_continuous_everywhere_but_not_differentiable_at_exactly_two_points\" >21. Does there exist a function which is continuous everywhere but not differentiable at exactly two points?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-15\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-4\/#NCERT_Solutions_class_12_Maths_Miscellaneous-7\" >NCERT Solutions class 12 Maths Miscellaneous<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-16\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-4\/#23_If_show_that\" >.23. If \u00a0show that \u00a0<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-17\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-4\/#NCERT_Solutions_class_12_Maths_Miscellaneous-8\" >NCERT Solutions class 12 Maths Miscellaneous<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-18\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-4\/#CBSE_App_for_Class_12\" >CBSE App for Class 12<\/a><\/li><\/ul><\/nav><\/div>\n<p>NCERT Solutions class 12 Maths Miscellaneous 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 Miscellaneous\" width=\"177\" height=\"230\" \/><\/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><strong>Differentiate with respect to <\/strong><strong><img decoding=\"async\" id=\"Picture 45\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image001.png\" \/><\/strong><strong>\u00a0the functions in Exercises 1 to 5.<\/strong><\/p>\n<p><strong>1. <\/strong><strong><img decoding=\"async\" id=\"Picture 44\" style=\"height: 32px; width: 96px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image002.png\" \/><\/strong><strong>\u00a0<\/strong><\/p>\n<p><strong>Ans. <\/strong>Let <img decoding=\"async\" id=\"Picture 54\" style=\"height: 32px; width: 121px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image003.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image004.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 52\" style=\"height: 41px; width: 249px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image005.png\" \/><\/p>\n<p><img decoding=\"async\" id=\"Picture 51\" style=\"height: 45px; width: 249px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image006.png\" \/><\/p>\n<p><img decoding=\"async\" id=\"Picture 50\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image007.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 49\" style=\"height: 41px; width: 257px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image008.png\" \/>\u00a0= <img decoding=\"async\" id=\"Picture 48\" style=\"height: 32px; width: 156px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image009.png\" \/><\/p>\n<p><img decoding=\"async\" id=\"Picture 47\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image007.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 46\" style=\"height: 41px; width: 200px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image010.png\" \/><\/p>\n<hr \/>\n<p><strong>2. <\/strong><strong><img decoding=\"async\" id=\"Picture 43\" style=\"height: 21px; width: 93px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image011.png\" \/><\/strong><strong>\u00a0<\/strong><\/p>\n<p><strong>Ans. <\/strong>Let <img decoding=\"async\" id=\"Picture 61\" style=\"height: 24px; width: 119px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image012.png\" \/>\u00a0= <img decoding=\"async\" id=\"Picture 60\" style=\"height: 29px; width: 116px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image013.png\" \/><\/p>\n<p><img decoding=\"async\" id=\"Picture 59\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image004.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 58\" style=\"height: 41px; width: 276px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image014.png\" \/><\/p>\n<p><img decoding=\"async\" id=\"Picture 57\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image007.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 56\" style=\"height: 41px; width: 211px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image015.png\" \/>\u00a0= <img decoding=\"async\" id=\"Picture 55\" style=\"height: 29px; width: 183px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image016.png\" \/><\/p>\n<hr \/>\n<p><strong>3. <\/strong><strong><img decoding=\"async\" id=\"Picture 42\" style=\"height: 29px; width: 61px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image017.png\" \/><\/strong><strong>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Let <img decoding=\"async\" id=\"Picture 109\" style=\"height: 29px; width: 87px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image018.png\" \/>\u2026\u2026\u2026.(i)<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 108\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image007.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 107\" style=\"height: 29px; width: 132px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image019.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 106\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image007.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 105\" style=\"height: 27px; width: 153px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image020.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 104\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image004.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 103\" style=\"height: 41px; width: 207px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image021.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 102\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image007.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 101\" style=\"height: 45px; width: 315px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image022.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 100\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image007.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 99\" style=\"height: 45px; width: 340px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image023.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 98\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image007.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 97\" style=\"height: 45px; width: 263px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image024.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 96\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image007.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 63\" style=\"height: 45px; width: 228px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image025.png\" \/><\/p>\n<p><img decoding=\"async\" id=\"Picture 62\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image007.png\" alt=\"*\" \/>\u00a0<img decoding=\"async\" id=\"Picture 53\" style=\"height: 45px; width: 279px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image026.png\" \/><\/p>\n<hr \/>\n<p><strong>4. <\/strong><strong><img decoding=\"async\" id=\"Picture 41\" style=\"height: 32px; width: 139px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image027.png\" \/><\/strong><strong>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Let <img decoding=\"async\" id=\"Picture 117\" style=\"height: 32px; width: 104px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image028.png\" \/>\u00a0= <img decoding=\"async\" id=\"Picture 115\" style=\"height: 51px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image029.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 114\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image004.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 113\" style=\"height: 81px; width: 149px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image030.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 112\" style=\"height: 48px; width: 87px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image031.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 111\" style=\"height: 49px; width: 61px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image032.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 110\" style=\"height: 47px; width: 65px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image033.png\" \/><\/p>\n<hr \/>\n<p><strong>5. <\/strong><strong><img decoding=\"async\" id=\"Picture 40\" style=\"height: 63px; width: 131px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image034.png\" \/><\/strong><strong>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Let <img decoding=\"async\" id=\"Picture 128\" style=\"height: 63px; width: 84px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image035.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 127\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image004.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 126\" style=\"height: 75px; width: 292px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image036.png\" \/>\u00a0\u00a0\u00a0\u00a0[By Quotient Rule]\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 125\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image007.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 124\" style=\"height: 139px; width: 436px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image037.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 123\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image007.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 122\" style=\"height: 88px; width: 259px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image038.png\" \/>\u00a0= <img decoding=\"async\" id=\"Picture 121\" style=\"height: 105px; width: 153px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image039.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 120\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image007.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 119\" style=\"height: 83px; width: 220px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image040.png\" \/>\u00a0= <img decoding=\"async\" id=\"Picture 118\" style=\"height: 83px; width: 183px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image041.png\" \/><\/p>\n<hr \/>\n<p><strong>Differentiate with respect to <\/strong><strong><img decoding=\"async\" id=\"Picture 39\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image001.png\" \/><\/strong><strong>\u00a0the functions in Exercises 6 to 11.<\/strong><\/p>\n<p><strong>6. <\/strong><strong><img decoding=\"async\" id=\"Picture 38\" style=\"height: 53px; width: 257px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image042.png\" \/><\/strong><strong>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Let <img decoding=\"async\" id=\"Picture 143\" style=\"height: 53px; width: 284px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image043.png\" \/>\u00a0\u00a0\u2026\u2026\u2026.(i)<\/p>\n<p style=\"text-align: justify;\">Now, <img decoding=\"async\" id=\"Picture 142\" style=\"height: 47px; width: 279px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image044.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 141\" style=\"height: 53px; width: 120px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image045.png\" \/>\u00a0= <img decoding=\"async\" id=\"Picture 140\" style=\"height: 41px; width: 85px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image046.png\" \/><\/p>\n<p style=\"text-align: justify;\">And <img decoding=\"async\" id=\"Picture 139\" style=\"height: 47px; width: 279px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image047.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 138\" style=\"height: 53px; width: 119px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image048.png\" \/>\u00a0= <img decoding=\"async\" id=\"Picture 137\" style=\"height: 41px; width: 85px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image049.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 136\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image004.png\" \/>\u00a0From eq. (i),\u00a0 <img decoding=\"async\" id=\"Picture 135\" style=\"height: 83px; width: 256px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image050.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 134\" style=\"height: 83px; width: 100px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image051.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 133\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image007.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 132\" style=\"height: 45px; width: 112px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image052.png\" \/>\u00a0= <img decoding=\"async\" id=\"Picture 131\" style=\"height: 41px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image053.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 130\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image004.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 129\" style=\"height: 41px; width: 96px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image054.png\" \/><\/p>\n<hr \/>\n<p><strong>7. <\/strong><strong><img decoding=\"async\" id=\"Picture 37\" style=\"height: 29px; width: 104px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image055.png\" \/><\/strong><strong>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Let <img decoding=\"async\" id=\"Picture 157\" style=\"height: 29px; width: 129px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image056.png\" \/>\u00a0\u2026\u2026\u2026.(i)<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 156\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image007.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 155\" style=\"height: 29px; width: 136px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image057.png\" \/>\u00a0= <img decoding=\"async\" id=\"Picture 154\" style=\"height: 27px; width: 103px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image058.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 153\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image004.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 152\" style=\"height: 41px; width: 216px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image059.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 151\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image007.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 150\" style=\"height: 44px; width: 304px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image060.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 149\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image007.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 148\" style=\"height: 44px; width: 276px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image061.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 147\" style=\"height: 44px; width: 97px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image062.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 146\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image007.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 145\" style=\"height: 51px; width: 153px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image063.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 144\" style=\"height: 51px; width: 172px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image064.png\" \/><\/p>\n<hr \/>\n<p><strong>8. <\/strong><strong><img decoding=\"async\" id=\"Picture 36\" style=\"height: 27px; width: 136px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image065.png\" \/><\/strong><strong>\u00a0for some constants <\/strong><strong><img decoding=\"async\" id=\"Picture 35\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image066.png\" \/><\/strong><strong>\u00a0and <\/strong><strong><img decoding=\"async\" id=\"Picture 34\" style=\"height: 19px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image067.png\" \/><\/strong><strong>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Let <img decoding=\"async\" id=\"Picture 168\" style=\"height: 27px; width: 161px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image068.png\" \/>\u00a0for some constants <img decoding=\"async\" id=\"Picture 167\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image066.png\" \/>\u00a0and <img decoding=\"async\" id=\"Picture 166\" style=\"height: 19px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image069.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 165\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image004.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 164\" style=\"height: 41px; width: 308px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image070.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 163\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image007.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 162\" style=\"height: 41px; width: 297px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image071.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 161\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image007.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 160\" style=\"height: 41px; width: 297px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image072.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 159\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image007.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 158\" style=\"height: 41px; width: 287px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image073.png\" \/><\/p>\n<hr \/>\n<p><strong>9. <\/strong><strong><img decoding=\"async\" id=\"Picture 33\" style=\"height: 41px; width: 215px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image074.png\" \/><\/strong><strong>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Let <img decoding=\"async\" id=\"Picture 184\" style=\"height: 29px; width: 159px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image075.png\" \/>\u00a0\u2026\u2026\u2026.(i)<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 183\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image007.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 182\" style=\"height: 29px; width: 204px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image076.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 181\" style=\"height: 27px; width: 204px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image077.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 180\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image004.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 179\" style=\"height: 41px; width: 501px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image078.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 178\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image007.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 177\" style=\"height: 47px; width: 557px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image079.png\" \/><\/p>\n<p style=\"text-align: justify;\">\u00a0<img decoding=\"async\" id=\"Picture 176\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image007.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 175\" style=\"height: 44px; width: 353px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image080.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 174\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image007.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 173\" style=\"height: 44px; width: 285px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image081.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 172\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image007.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 171\" style=\"height: 41px; width: 283px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image082.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 170\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image007.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 169\" style=\"height: 41px; width: 404px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image083.png\" \/><\/p>\n<hr \/>\n<p><strong>10. <\/strong><strong><img decoding=\"async\" id=\"Picture 32\" style=\"height: 24px; width: 113px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image084.png\" \/><\/strong><strong>\u00a0for some fixed <\/strong><strong><img decoding=\"async\" id=\"Picture 31\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image085.png\" \/><\/strong><strong>\u00a0and <\/strong><strong><img decoding=\"async\" id=\"Picture 30\" style=\"height: 19px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image086.png\" \/><\/strong><strong>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Let <img decoding=\"async\" id=\"Picture 206\" style=\"height: 24px; width: 133px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image087.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 205\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image004.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 204\" style=\"height: 41px; width: 221px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image088.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 203\" style=\"height: 41px; width: 171px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image089.png\" \/>\u00a0 \u2026\u2026.(i)<\/p>\n<p style=\"text-align: justify;\">Now taking <img decoding=\"async\" id=\"Picture 202\" style=\"height: 41px; width: 52px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image090.png\" \/>,\u00a0 let <img decoding=\"async\" id=\"Picture 201\" style=\"height: 21px; width: 43px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image091.png\" \/>\u00a0\u2026\u2026\u2026.(ii)<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 200\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image007.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 199\" style=\"height: 24px; width: 88px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image092.png\" \/>\u00a0= <img decoding=\"async\" id=\"Picture 198\" style=\"height: 21px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image093.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 197\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image007.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 196\" style=\"height: 41px; width: 145px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image094.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 195\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image007.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 194\" style=\"height: 41px; width: 189px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image095.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 193\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image007.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 192\" style=\"height: 41px; width: 129px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image096.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 191\" style=\"height: 21px; width: 55px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image097.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 190\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image007.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 189\" style=\"height: 41px; width: 113px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image098.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 188\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image004.png\" \/>\u00a0From eq. (ii), <img decoding=\"async\" id=\"Picture 187\" style=\"height: 41px; width: 136px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image099.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 186\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image004.png\" \/>\u00a0From eq. (i), <img decoding=\"async\" id=\"Picture 185\" style=\"height: 41px; width: 215px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image100.png\" \/><\/p>\n<hr \/>\n<p><strong>11. <\/strong><strong><img decoding=\"async\" id=\"Picture 29\" style=\"height: 32px; width: 100px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image101.png\" \/><\/strong><strong>\u00a0for <\/strong><strong><img decoding=\"async\" id=\"Picture 28\" style=\"height: 19px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image102.png\" \/><\/strong><strong>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Let <img decoding=\"async\" id=\"Picture 239\" style=\"height: 32px; width: 125px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image103.png\" \/>\u00a0for <img decoding=\"async\" id=\"Picture 238\" style=\"height: 19px; width: 36px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image104.png\" \/><\/p>\n<p style=\"text-align: justify;\">Putting <img decoding=\"async\" id=\"Picture 237\" style=\"height: 24px; width: 57px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image105.png\" \/>\u00a0and <img decoding=\"async\" id=\"Picture 236\" style=\"height: 32px; width: 80px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image106.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 235\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image004.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 234\" style=\"height: 41px; width: 91px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image107.png\" \/>\u00a0\u2026\u2026\u2026.(i)<\/p>\n<p style=\"text-align: justify;\">Now <img decoding=\"async\" id=\"Picture 233\" style=\"height: 24px; width: 57px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image105.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 232\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image007.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 231\" style=\"height: 27px; width: 103px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image108.png\" \/>\u00a0= <img decoding=\"async\" id=\"Picture 230\" style=\"height: 29px; width: 85px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image109.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 229\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image004.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 228\" style=\"height: 41px; width: 269px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image110.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 227\" style=\"height: 41px; width: 165px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image111.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 226\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image007.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 225\" style=\"height: 44px; width: 155px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image112.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 224\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image007.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 223\" style=\"height: 51px; width: 168px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image113.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 222\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image007.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 221\" style=\"height: 51px; width: 189px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image114.png\" \/>\u00a0\u2026\u2026\u2026.(ii)<\/p>\n<p style=\"text-align: justify;\">Again\u00a0 <img decoding=\"async\" id=\"Picture 220\" style=\"height: 32px; width: 80px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image106.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 219\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image007.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 218\" style=\"height: 32px; width: 125px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image115.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 217\" style=\"height: 27px; width: 87px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image116.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 216\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image004.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 215\" style=\"height: 41px; width: 265px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image117.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 214\" style=\"height: 41px; width: 215px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image118.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 213\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image007.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 212\" style=\"height: 44px; width: 179px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image119.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 211\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image007.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 210\" style=\"height: 51px; width: 191px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image120.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 209\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image007.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 208\" style=\"height: 51px; width: 237px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image121.png\" \/>\u00a0\u2026\u2026\u2026.(iii)<\/p>\n<p style=\"text-align: justify;\">Putting the values from eq. (ii) and (iii) in eq. (i),<\/p>\n<p><img decoding=\"async\" id=\"Picture 207\" style=\"height: 51px; width: 401px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image122.png\" \/><\/p>\n<hr \/>\n<p><strong>12. Find <\/strong><strong><img decoding=\"async\" id=\"Picture 27\" style=\"height: 41px; width: 23px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image123.png\" \/><\/strong><strong>\u00a0if <\/strong><strong><img decoding=\"async\" id=\"Picture 26\" style=\"height: 27px; width: 105px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image124.png\" \/><\/strong><strong>\u00a0and <\/strong><strong><img decoding=\"async\" id=\"Picture 25\" style=\"height: 41px; width: 188px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image125.png\" \/><\/strong><strong>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Given:\u00a0\u00a0\u00a0\u00a0 <img decoding=\"async\" id=\"Picture 249\" style=\"height: 27px; width: 105px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image124.png\" \/>\u00a0and <img decoding=\"async\" id=\"Picture 248\" style=\"height: 27px; width: 103px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image126.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 247\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image004.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 246\" style=\"height: 41px; width: 284px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image127.png\" \/>\u00a0and <img decoding=\"async\" id=\"Picture 245\" style=\"height: 41px; width: 133px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image128.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 244\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image004.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 243\" style=\"height: 47px; width: 176px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image129.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 242\" style=\"height: 80px; width: 101px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image130.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 241\" style=\"height: 80px; width: 57px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image131.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 240\" style=\"height: 41px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image132.png\" \/><\/p>\n<hr \/>\n<p><strong>13. Find <\/strong><strong><img decoding=\"async\" id=\"Picture 24\" style=\"height: 41px; width: 23px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image123.png\" \/><\/strong><strong>\u00a0if <\/strong><strong><img decoding=\"async\" id=\"Picture 23\" style=\"height: 28px; width: 233px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image133.png\" \/><\/strong><strong>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Given:\u00a0\u00a0\u00a0\u00a0 <img decoding=\"async\" id=\"Picture 260\" style=\"height: 28px; width: 163px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image134.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 259\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image004.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 258\" style=\"height: 64px; width: 269px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image135.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 257\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image007.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 256\" style=\"height: 67px; width: 343px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image136.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 255\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image007.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 254\" style=\"height: 45px; width: 268px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image137.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 253\" style=\"height: 51px; width: 164px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image138.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 252\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image007.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 251\" style=\"height: 45px; width: 159px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image139.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 250\" style=\"height: 45px; width: 116px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image140.png\" \/>\u00a0= 0<\/p>\n<hr \/>\n<p><strong>14. If <\/strong><strong><img decoding=\"async\" id=\"Picture 22\" style=\"height: 27px; width: 145px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image141.png\" \/><\/strong><strong>\u00a0for <\/strong><strong><img decoding=\"async\" id=\"Picture 21\" style=\"height: 21px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image142.png\" \/><\/strong><strong>\u00a0prove that <\/strong><strong><img decoding=\"async\" id=\"Picture 20\" style=\"height: 49px; width: 92px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image143.png\" \/><\/strong><strong>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Given: <img decoding=\"async\" id=\"Picture 283\" style=\"height: 27px; width: 141px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image144.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 282\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image007.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 281\" style=\"height: 27px; width: 128px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image145.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 280\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image007.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 279\" style=\"height: 27px; width: 136px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image146.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 278\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image007.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 277\" style=\"height: 24px; width: 127px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image147.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 276\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image007.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 275\" style=\"height: 24px; width: 136px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image148.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 274\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image007.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 273\" style=\"height: 27px; width: 181px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image149.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 272\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image007.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 271\" style=\"height: 19px; width: 77px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image150.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 270\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image007.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 269\" style=\"height: 19px; width: 77px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image151.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 268\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image007.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 267\" style=\"height: 27px; width: 89px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image152.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 266\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image007.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 265\" style=\"height: 41px; width: 60px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image153.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 264\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image004.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 263\" style=\"height: 68px; width: 207px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image154.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 262\" style=\"height: 52px; width: 103px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image155.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 261\" style=\"height: 49px; width: 65px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image156.png\" \/>\u00a0\u00a0\u00a0\u00a0 Proved.<\/p>\n<hr \/>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"15_If_for_some_prove_that\"><\/span><strong>15. If <\/strong><strong><img decoding=\"async\" id=\"Picture 19\" style=\"height: 29px; width: 153px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image157.png\" \/><\/strong><strong>\u00a0for some <\/strong><strong><img decoding=\"async\" id=\"Picture 18\" style=\"height: 21px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image158.png\" \/><\/strong><strong>\u00a0prove that <\/strong><strong><img decoding=\"async\" id=\"Picture 17\" style=\"height: 103px; width: 87px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image159.png\" \/><\/strong><strong>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Given: <img decoding=\"async\" id=\"Picture 306\" style=\"height: 29px; width: 141px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image160.png\" \/>\u2026\u2026\u2026.(i)<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 305\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image004.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 304\" style=\"height: 41px; width: 169px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image161.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 303\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image007.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 302\" style=\"height: 41px; width: 156px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image162.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 301\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image007.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 300\" style=\"height: 48px; width: 100px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image163.png\" \/>\u00a0\u2026\u2026\u2026.(ii)<\/p>\n<p style=\"text-align: justify;\">Again <img decoding=\"async\" id=\"Picture 299\" style=\"height: 72px; width: 208px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image164.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 298\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image007.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 297\" style=\"height: 80px; width: 263px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image165.png\" \/> \u00a0[From eq. (ii)<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 296\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image007.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 295\" style=\"height: 85px; width: 201px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image166.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 294\" style=\"height: 61px; width: 147px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image167.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 293\" style=\"height: 52px; width: 57px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image168.png\" \/>\u00a0\u2026\u2026\u2026.(iii)<\/p>\n<p style=\"text-align: justify;\">Putting values of <img decoding=\"async\" id=\"Picture 292\" style=\"height: 41px; width: 23px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image123.png\" \/>\u00a0and <img decoding=\"async\" id=\"Picture 291\" style=\"height: 44px; width: 32px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image169.png\" \/>\u00a0in the given expression,<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 290\" style=\"height: 103px; width: 91px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image170.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 289\" style=\"height: 113px; width: 101px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image171.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 288\" style=\"height: 69px; width: 208px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image172.png\" \/>\u00a0= <img decoding=\"async\" id=\"Picture 287\" style=\"height: 57px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image173.png\" \/>\u00a0= <img decoding=\"async\" id=\"Picture 286\" style=\"height: 15px; width: 21px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image174.png\" \/><\/p>\n<p>which is a constant and is independent of <img decoding=\"async\" id=\"Picture 285\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image066.png\" \/>\u00a0and <img decoding=\"async\" id=\"Picture 284\" style=\"height: 19px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image067.png\" \/><\/p>\n<hr \/>\n<h3><span class=\"ez-toc-section\" id=\"16_If_with_prove_that\"><\/span><strong>16. If <\/strong><strong><img decoding=\"async\" id=\"Picture 16\" style=\"height: 27px; width: 131px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image175.png\" \/><\/strong><strong>\u00a0with <\/strong><strong><img decoding=\"async\" id=\"Picture 15\" style=\"height: 21px; width: 72px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image176.png\" \/><\/strong><strong>\u00a0prove that <\/strong><strong><img decoding=\"async\" id=\"Picture 14\" style=\"height: 45px; width: 120px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image177.png\" \/><\/strong><strong>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Given: <img decoding=\"async\" id=\"Picture 321\" style=\"height: 27px; width: 131px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image175.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 320\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image007.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 319\" style=\"height: 47px; width: 99px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image178.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 318\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image004.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 317\" style=\"height: 53px; width: 145px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image179.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 316\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image007.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 315\" style=\"height: 68px; width: 300px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image180.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 314\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image007.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 313\" style=\"height: 52px; width: 303px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image181.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 312\" style=\"height: 49px; width: 232px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image182.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 311\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image007.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 310\" style=\"height: 49px; width: 131px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image183.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 309\" style=\"height: 47px; width: 81px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image184.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 308\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image007.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 307\" style=\"height: 45px; width: 116px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image185.png\" \/>\u00a0 [Taking reciprocal]\n<hr \/>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-2\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"17_If_and_find\"><\/span><strong>17. If <\/strong><strong><img decoding=\"async\" id=\"Picture 13\" style=\"height: 27px; width: 128px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image186.png\" \/><\/strong><strong>\u00a0and <\/strong><strong><img decoding=\"async\" id=\"Picture 12\" style=\"height: 27px; width: 133px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image187.png\" \/><\/strong><strong>\u00a0find <\/strong><strong><img decoding=\"async\" id=\"Picture 11\" style=\"height: 44px; width: 36px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image188.png\" \/><\/strong><strong>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Given: <img decoding=\"async\" id=\"Picture 341\" style=\"height: 27px; width: 128px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image186.png\" \/>\u00a0and <img decoding=\"async\" id=\"Picture 340\" style=\"height: 27px; width: 128px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image189.png\" \/><\/p>\n<p style=\"text-align: justify;\">Differentiating both sides with respect to <img decoding=\"async\" id=\"Picture 339\" style=\"height: 19px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image190.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 338\" style=\"height: 45px; width: 168px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image191.png\" \/>\u00a0and <img decoding=\"async\" id=\"Picture 337\" style=\"height: 45px; width: 161px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image192.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 336\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image007.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 335\" style=\"height: 45px; width: 232px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image193.png\" \/>\u00a0and <img decoding=\"async\" id=\"Picture 334\" style=\"height: 48px; width: 244px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image194.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 333\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image007.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 332\" style=\"height: 41px; width: 188px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image195.png\" \/>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 and <img decoding=\"async\" id=\"Picture 331\" style=\"height: 41px; width: 201px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image196.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 330\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image007.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 329\" style=\"height: 41px; width: 83px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image197.png\" \/>\u00a0and\u00a0 <img decoding=\"async\" id=\"Picture 328\" style=\"height: 41px; width: 81px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image198.png\" \/><\/p>\n<p style=\"text-align: justify;\">Now\u00a0\u00a0\u00a0 <img decoding=\"async\" id=\"Picture 327\" style=\"height: 41px; width: 233px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image199.png\" \/><\/p>\n<p style=\"text-align: justify;\">Again\u00a0 <img decoding=\"async\" id=\"Picture 326\" style=\"height: 44px; width: 96px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image200.png\" \/>\u00a0= <img decoding=\"async\" id=\"Picture 325\" style=\"height: 41px; width: 65px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image201.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 324\" style=\"height: 41px; width: 59px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image202.png\" \/>\u00a0= <img decoding=\"async\" id=\"Picture 323\" style=\"height: 41px; width: 87px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image203.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 322\" style=\"height: 44px; width: 121px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image204.png\" \/><\/p>\n<hr \/>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-3\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"18_If_show_that_exists_for_all_real_and_find_it\"><\/span><strong>18. If <\/strong><strong><img decoding=\"async\" id=\"Picture 10\" style=\"height: 29px; width: 80px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image205.png\" \/><\/strong><strong>\u00a0show that <\/strong><strong><img decoding=\"async\" id=\"Picture 9\" style=\"height: 27px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image206.png\" \/><\/strong><strong>\u00a0exists for all real <\/strong><strong><img decoding=\"async\" id=\"Picture 8\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image001.png\" \/><\/strong><strong>\u00a0and find it.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Given: <img decoding=\"async\" id=\"Picture 369\" style=\"height: 29px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image207.png\" \/>= <img decoding=\"async\" id=\"Picture 368\" style=\"height: 59px; width: 139px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image208.png\" \/><\/p>\n<p style=\"text-align: justify;\">Now, L.H.D. at <img decoding=\"async\" id=\"Picture 367\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image209.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 366\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image007.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 365\" style=\"height: 44px; width: 115px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image210.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 364\" style=\"height: 44px; width: 80px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image211.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 363\" style=\"height: 31px; width: 80px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image212.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 362\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image004.png\" \/>\u00a0L.H.D. at <img decoding=\"async\" id=\"Picture 361\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image209.png\" \/>\u00a0= R.H.D. at <img decoding=\"async\" id=\"Picture 360\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image209.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, <img decoding=\"async\" id=\"Picture 359\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image213.png\" \/>\u00a0is differentiable at <img decoding=\"async\" id=\"Picture 358\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image209.png\" \/>.<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 357\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image004.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 356\" style=\"height: 27px; width: 56px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image214.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 355\" style=\"height: 53px; width: 135px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image215.png\" \/><\/p>\n<p style=\"text-align: justify;\">Now, L.H.D. at <img decoding=\"async\" id=\"Picture 354\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image209.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 353\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image007.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 352\" style=\"height: 44px; width: 124px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image216.png\" \/>\u00a0= <img decoding=\"async\" id=\"Picture 351\" style=\"height: 44px; width: 87px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image217.png\" \/>\u00a0= <img decoding=\"async\" id=\"Picture 350\" style=\"height: 31px; width: 92px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image218.png\" \/><\/p>\n<p style=\"text-align: justify;\">And R.H.D. at <img decoding=\"async\" id=\"Picture 349\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image209.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 348\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image007.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 347\" style=\"height: 44px; width: 124px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image219.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 346\" style=\"height: 44px; width: 87px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image217.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 345\" style=\"height: 29px; width: 72px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image220.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 344\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image004.png\" \/>\u00a0Again L.H.D. at <img decoding=\"async\" id=\"Picture 343\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image209.png\" \/>\u00a0= R.H.D. at <img decoding=\"async\" id=\"Picture 342\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image209.png\" \/><\/p>\n<hr \/>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-4\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"19_Using_mathematical_induction_prove_that_for_all_positive_integers\"><\/span><strong>19. Using mathematical induction, prove that <\/strong><strong><img decoding=\"async\" id=\"Picture 7\" style=\"height: 41px; width: 99px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image221.png\" \/><\/strong><strong>\u00a0for all positive integers <\/strong><strong><img decoding=\"async\" id=\"Picture 6\" style=\"height: 15px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image222.png\" \/><\/strong><strong>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Let <img decoding=\"async\" id=\"Picture 399\" style=\"height: 27px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image223.png\" \/>\u00a0be the given statement in the problem.<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 398\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image007.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 397\" style=\"height: 41px; width: 147px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image224.png\" \/>\u00a0\u2026..(i)<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 396\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image004.png\" \/><img decoding=\"async\" id=\"Picture 395\" style=\"height: 41px; width: 152px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image225.png\" \/>\u00a0= <img decoding=\"async\" id=\"Picture 394\" style=\"height: 27px; width: 60px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image226.png\" \/>, which is true as <img decoding=\"async\" id=\"Picture 393\" style=\"height: 41px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image227.png\" \/><\/p>\n<p style=\"text-align: justify;\">Now we suppose <img decoding=\"async\" id=\"Picture 392\" style=\"height: 27px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image228.png\" \/>\u00a0is true.<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 391\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image004.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 390\" style=\"height: 41px; width: 159px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image229.png\" \/>\u00a0\u2026\u2026.(ii)<\/p>\n<p style=\"text-align: justify;\">To establish the truth of <img decoding=\"async\" id=\"Picture 389\" style=\"height: 27px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image230.png\" \/>\u00a0we prove,<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 388\" style=\"height: 41px; width: 231px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image231.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 387\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image007.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 386\" style=\"height: 21px; width: 91px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image232.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 385\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image007.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 384\" style=\"height: 41px; width: 128px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image233.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 383\" style=\"height: 41px; width: 116px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image234.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 382\" style=\"height: 27px; width: 104px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image235.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 381\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image004.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 380\" style=\"height: 27px; width: 143px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image236.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 379\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image007.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 378\" style=\"height: 27px; width: 144px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image237.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 377\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image007.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 376\" style=\"height: 28px; width: 91px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image238.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, <img decoding=\"async\" id=\"Picture 375\" style=\"height: 27px; width: 61px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image239.png\" \/>\u00a0is true if <img decoding=\"async\" id=\"Picture 374\" style=\"height: 27px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image228.png\" \/>\u00a0is true but <img decoding=\"async\" id=\"Picture 373\" style=\"height: 27px; width: 35px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image240.png\" \/>\u00a0is true.<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 372\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image004.png\" \/>\u00a0By Principal of Induction <img decoding=\"async\" id=\"Picture 371\" style=\"height: 27px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image223.png\" \/>\u00a0is true for all <img decoding=\"async\" id=\"Picture 370\" style=\"height: 15px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image241.png\" \/>\u00a0N.<\/p>\n<hr \/>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-5\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"20_Using_the_fact_that_and_the_differentiation_obtain_the_sum_formula_for_cosines\"><\/span><strong>20. Using the fact that <\/strong><strong><img decoding=\"async\" id=\"Picture 5\" style=\"height: 27px; width: 251px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image242.png\" \/><\/strong><strong>\u00a0and the differentiation, obtain the sum formula for cosines.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Given:\u00a0\u00a0\u00a0\u00a0 <img decoding=\"async\" id=\"Picture 407\" style=\"height: 27px; width: 259px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image243.png\" \/><\/p>\n<p style=\"text-align: justify;\">Consider A and B as function of <img decoding=\"async\" id=\"Picture 406\" style=\"height: 16px; width: 9px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image244.png\" \/>\u00a0and differentiating both sides w.r.t. <img decoding=\"async\" id=\"Picture 405\" style=\"height: 17px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image245.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 404\" style=\"height: 45px; width: 644px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image246.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 403\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image007.png\" \/>\u00a0\u00a0<img decoding=\"async\" id=\"Picture 402\" style=\"height: 45px; width: 409px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image247.png\" \/><\/p>\n<p><img decoding=\"async\" id=\"Picture 401\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image007.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 400\" style=\"height: 27px; width: 259px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image248.png\" \/><\/p>\n<hr \/>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-6\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"21_Does_there_exist_a_function_which_is_continuous_everywhere_but_not_differentiable_at_exactly_two_points\"><\/span><strong>21. Does there exist a function which is continuous everywhere but not differentiable at exactly two points?<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Let us consider the function\u00a0\u00a0\u00a0 <img decoding=\"async\" id=\"Picture 411\" style=\"height: 27px; width: 115px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image249.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 410\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image250.png\" \/>\u00a0is continuous everywhere but it is not differentiable at <img decoding=\"async\" id=\"Picture 409\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image209.png\" \/>\u00a0and <img decoding=\"async\" id=\"Picture 408\" style=\"height: 19px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image251.png\" \/><\/p>\n<hr \/>\n<p><strong>22. If <\/strong><strong><img decoding=\"async\" id=\"Picture 4\" style=\"height: 80px; width: 181px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image252.png\" \/><\/strong><strong>\u00a0prove that <\/strong><strong><img decoding=\"async\" id=\"Picture 3\" style=\"height: 80px; width: 203px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image253.png\" \/><\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Given: <img decoding=\"async\" id=\"Picture 415\" style=\"height: 80px; width: 176px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image254.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 414\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image004.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 413\" style=\"height: 112px; width: 597px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image255.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 412\" style=\"height: 101px; width: 151px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image256.png\" \/><\/p>\n<hr \/>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-7\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"23_If_show_that\"><\/span><strong>.23. If <\/strong><strong><img decoding=\"async\" id=\"Picture 2\" style=\"height: 27px; width: 141px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image257.png\" \/><\/strong><strong>\u00a0show that <\/strong><strong><img decoding=\"async\" id=\"Picture 1\" style=\"height: 44px; width: 188px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image258.png\" \/><\/strong><strong>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Given:\u00a0\u00a0\u00a0\u00a0 <img decoding=\"async\" id=\"Picture 430\" style=\"height: 27px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image259.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 429\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image004.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 428\" style=\"height: 41px; width: 157px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image260.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 427\" style=\"height: 51px; width: 123px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image261.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 426\" style=\"height: 45px; width: 53px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image262.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 425\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image007.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 424\" style=\"height: 49px; width: 99px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image263.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 423\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image007.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 422\" style=\"height: 49px; width: 139px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image264.png\" \/><\/p>\n<p style=\"text-align: justify;\">Differentiating both sides with respect to <img decoding=\"async\" id=\"Picture 421\" style=\"height: 17px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image245.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 420\" style=\"height: 49px; width: 281px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image265.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 419\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image007.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 418\" style=\"height: 44px; width: 171px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image266.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 417\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image007.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 416\" style=\"height: 44px; width: 192px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/05%20Misc\/image267.png\" \/>\u00a0\u00a0\u00a0\u00a0\u00a0 Proved.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-8\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<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 Miscellaneous 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 &#8230; <a title=\"NCERT Solutions class 12 Maths Miscellaneous\" class=\"read-more\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-4\/\" aria-label=\"More on NCERT Solutions class 12 Maths Miscellaneous\">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-4495","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 Miscellaneous | myCBSEguide<\/title>\n<meta name=\"description\" content=\"NCERT Solutions class 12 Maths Miscellaneous in PDF format for free download. 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