{"id":4506,"date":"2016-05-16T11:49:00","date_gmt":"2016-05-16T06:19:00","guid":{"rendered":"http:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-12-1\/"},"modified":"2018-05-07T18:11:47","modified_gmt":"2018-05-07T12:41:47","slug":"ncert-solutions-class-12-maths-exercise-12-1","status":"publish","type":"post","link":"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-12-1\/","title":{"rendered":"NCERT Solutions class 12 Maths Exercise 12.1"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_76 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 eztoc-toggle-hide-by-default' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-12-1\/#NCERT_Solutions_class_12_Maths_Linear_Programming\" >NCERT Solutions class 12 Maths Linear Programming<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-12-1\/#2_Minimize_Z_subject_to\" >2.\u00a0 Minimize Z = \u00a0subject to \u00a0<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-12-1\/#NCERT_Solutions_class_12_Maths_Exercise_121\" >NCERT Solutions class 12 Maths Exercise 12.1<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-12-1\/#3_Maximize_Z_subject_to\" >3.\u00a0 Maximize Z = \u00a0subject to \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-exercise-12-1\/#NCERT_Solutions_class_12_Maths_Exercise_121-2\" >NCERT Solutions class 12 Maths Exercise 12.1<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-12-1\/#4_Minimize_Z_such_that\" >4.\u00a0 Minimize Z = \u00a0such that \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-exercise-12-1\/#NCERT_Solutions_class_12_Maths_Exercise_121-3\" >NCERT Solutions class 12 Maths Exercise 12.1<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-12-1\/#5_Maximize_Z_subject_to\" >5.\u00a0 Maximize Z = \u00a0subject to \u00a0<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-12-1\/#NCERT_Solutions_class_12_Maths_Exercise_121-4\" >NCERT Solutions class 12 Maths Exercise 12.1<\/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-exercise-12-1\/#6_Minimize_Z_subject_to\" >6.\u00a0 Minimize Z = \u00a0subject to<\/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-exercise-12-1\/#NCERT_Solutions_class_12_Maths_Exercise_121-5\" >NCERT Solutions class 12 Maths Exercise 12.1<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-12\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-12-1\/#Show_that_the_minimum_of_Z_occurs_at_more_than_two_points\" >Show that the minimum of Z occurs at more than two points:<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-13\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-12-1\/#NCERT_Solutions_class_12_Maths_Exercise_121-6\" >NCERT Solutions class 12 Maths Exercise 12.1<\/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-exercise-12-1\/#8_Minimize_and_Maximize_Z_subject_to\" >8.\u00a0 Minimize and Maximize Z = subject to \u00a0<\/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-exercise-12-1\/#NCERT_Solutions_class_12_Maths_Exercise_121-7\" >NCERT Solutions class 12 Maths Exercise 12.1<\/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-exercise-12-1\/#9_Maximize_Z_subject_to_the_constraints\" >9.\u00a0 Maximize Z = \u00a0subject to the constraints: \u00a0<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-17\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-12-1\/#NCERT_Solutions_class_12_Maths_Exercise_121-8\" >NCERT Solutions class 12 Maths Exercise 12.1<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-18\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-12-1\/#10_Maximize_Z_subject_to\" >10.\u00a0 Maximize Z = \u00a0subject to \u00a0<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-19\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-12-1\/#NCERT_Solutions_class_12_Maths_Exercise_121-9\" >NCERT Solutions class 12 Maths Exercise 12.1<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-20\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-12-1\/#CBSE_App_for_Students\" >CBSE App for Students<\/a><\/li><\/ul><\/nav><\/div>\n<p>NCERT Solutions class 12 Maths Exercise 12.1 Class 12 Maths book solutions are available in PDF format for free download. These ncert book chapter wise questions and answers are very helpful for CBSE board exam. CBSE recommends NCERT books and most of the questions in CBSE exam are asked from NCERT text books. Class 12 Maths chapter wise NCERT solution for Maths part 1 and Maths part 2 for all the chapters can be downloaded from our website and myCBSEguide mobile app for free.<\/p>\n<p><strong>Download\u00a0<a href=\"https:\/\/mycbseguide.com\/downloads\/cbse-class-12-mathematics-linear-programming\/1296\/ncert-solutions\/5\/\">NCERT solutions for\u00a0Linear Programming\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%20Part-02%20.jpg\" alt=\"NCERT Solutions class 12 Maths Exercise 12.1\" width=\"195\" height=\"250\" \/><\/p>\n<h2><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Linear_Programming\"><\/span>NCERT Solutions class 12 Maths Linear Programming<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><strong>Solve the following Linear Programming Problems graphically:<\/strong><\/p>\n<p><strong>1.\u00a0 Maximize Z = <\/strong><strong><img decoding=\"async\" style=\"height: 21px; width: 52px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image001.png\" \/><\/strong><strong>\u00a0subject to the constraints: <\/strong><strong><img decoding=\"async\" style=\"height: 21px; width: 136px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image002.png\" \/><\/strong><strong>. <\/strong><\/p>\n<p><strong>Ans.\u00a0 <\/strong>As <img decoding=\"async\" style=\"height: 21px; width: 79px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image003.png\" \/>\u00a0therefore we shall shade the other inequalities in the first quadrant only.<\/p>\n<p>Now\u00a0 <img decoding=\"async\" style=\"height: 21px; width: 60px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image004.png\" \/><\/p>\n<p><img decoding=\"async\" id=\"Picture 2\" style=\"height: 154px; width: 149px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image005.jpg\" \/><\/p>\n<p>Let\u00a0 <img decoding=\"async\" style=\"height: 21px; width: 61px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image006.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image007.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 63px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image008.png\" \/><\/p>\n<p>Thus the line has 4 and 4 as intercepts along the axes. Now, (0, 0) satisfies the inequation, i.e., <img decoding=\"async\" style=\"height: 19px; width: 61px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image009.png\" \/>\u00a0\u00a0Therefore, shaded region OAB is the feasible solution.<\/p>\n<p>Its corners are O (0, 0), A (4, 0), B (0, 4)<\/p>\n<p>At O (0, 0)\u00a0\u00a0\u00a0 Z = 0<\/p>\n<p>At A (4, 0)\u00a0\u00a0\u00a0 Z = 3 x 4 = 12<\/p>\n<p>At B (0, 4)\u00a0\u00a0\u00a0 Z = 4 x 4 = 16<\/p>\n<p>Hence, max Z = 16 at <img decoding=\"async\" style=\"height: 21px; width: 79px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image010.png\" \/><\/p>\n<hr \/>\n<h3><span class=\"ez-toc-section\" id=\"2_Minimize_Z_subject_to\"><\/span><strong>2.\u00a0 Minimize Z = <\/strong><strong><img decoding=\"async\" style=\"height: 21px; width: 61px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image011.png\" \/><\/strong><strong>\u00a0subject to <\/strong><strong><img decoding=\"async\" style=\"height: 21px; width: 229px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image012.png\" \/><\/strong><strong>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Ans.\u00a0 <\/strong>Consider\u00a0 <img decoding=\"async\" style=\"height: 21px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image013.png\" \/><\/p>\n<p>Let <img decoding=\"async\" style=\"height: 21px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image014.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image007.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 63px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image015.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image016.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 73px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image017.png\" \/><\/p>\n<p><img decoding=\"async\" id=\"Picture 4\" style=\"height: 225px; width: 231px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image018.png\" \/><\/p>\n<p>Since, (0, 0) satisfies the inequaitons <img decoding=\"async\" style=\"height: 21px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image013.png\" \/><\/p>\n<p>Therefore, its solution contains (0, 0)<\/p>\n<p>Again\u00a0 <img decoding=\"async\" style=\"height: 21px; width: 83px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image019.png\" \/><\/p>\n<p>Let\u00a0 <img decoding=\"async\" style=\"height: 21px; width: 83px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image020.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image007.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 41px; width: 63px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image021.png\" \/><\/p>\n<p>Again, (0, 0) satisfies <img decoding=\"async\" style=\"height: 21px; width: 83px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image019.png\" \/><\/p>\n<p>Therefore its solution contains (0, 0).<\/p>\n<p>The feasible region is the solution set which is double shaded and is OABCO.<\/p>\n<p>At\u00a0 O (0, 0)\u00a0 Z = 0<\/p>\n<p>At\u00a0 A (4, 0)\u00a0 Z = \u20133 x 4 = \u201312<\/p>\n<p>At\u00a0 B (2, 3)\u00a0 Z = \u20133 x 2 + 4 x 3 = 6<\/p>\n<p>At\u00a0 C (0, 4)\u00a0 Z = 4 x 4 = 16<\/p>\n<p>Hence, minimum Z = \u201312 at <img decoding=\"async\" style=\"height: 21px; width: 79px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image022.png\" \/><\/p>\n<hr \/>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_121\"><\/span>NCERT Solutions class 12 Maths Exercise 12.1<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"3_Maximize_Z_subject_to\"><\/span><strong>3.\u00a0 Maximize Z = <\/strong><strong><img decoding=\"async\" style=\"height: 21px; width: 52px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image023.png\" \/><\/strong><strong>\u00a0subject to <\/strong><strong><img decoding=\"async\" style=\"height: 21px; width: 244px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image024.png\" \/><\/strong><strong>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Ans.\u00a0 <\/strong>We first draw the graph of equation <img decoding=\"async\" style=\"height: 21px; width: 81px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image025.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image007.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image026.png\" \/><\/p>\n<p>For\u00a0 <img decoding=\"async\" style=\"height: 21px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image027.png\" \/>\u00a0\u00a0\u00a0<img decoding=\"async\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image028.png\" \/><\/p>\n<p>And for <img decoding=\"async\" style=\"height: 21px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image029.png\" \/>\u00a0\u00a0\u00a0<img decoding=\"async\" style=\"height: 19px; width: 36px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image030.png\" \/><\/p>\n<table border=\"1\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td style=\"width: .4in;\"><img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image031.png\" \/><\/td>\n<td style=\"width: .4in;\">0<\/td>\n<td style=\"width: .4in;\">5<\/td>\n<\/tr>\n<tr>\n<td style=\"width: .4in;\"><img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image032.png\" \/><\/td>\n<td style=\"width: .4in;\">3<\/td>\n<td style=\"width: .4in;\">0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<table border=\"1\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td style=\"width: .4in;\"><img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image031.png\" \/><\/td>\n<td style=\"width: .4in;\">2<\/td>\n<td style=\"width: .4in;\">0<\/td>\n<\/tr>\n<tr>\n<td style=\"width: .4in;\"><img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image032.png\" \/><\/td>\n<td style=\"width: .4in;\">0<\/td>\n<td style=\"width: .4in;\">5<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Similarly, for equation <img decoding=\"async\" style=\"height: 21px; width: 83px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image033.png\" \/>, the points are (2, 0) and (0, 5).<\/p>\n<p>As (0, 0) satisfies both the inequations and also <img decoding=\"async\" style=\"height: 21px; width: 79px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image003.png\" \/>\u00a0then the feasible require contains the half-plane containing (0, 0).<\/p>\n<p>Therefore, the feasible portion is OABC which is shown as shaded in the graph.<\/p>\n<p><img decoding=\"async\" id=\"Picture 7\" style=\"height: 218px; width: 214px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image034.jpg\" \/><\/p>\n<p>Co-ordinates of point B can be obtained by solving <img decoding=\"async\" style=\"height: 21px; width: 81px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image025.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 21px; width: 83px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image033.png\" \/>\u00a0and it is B<img decoding=\"async\" style=\"height: 45px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image035.png\" \/><\/p>\n<p>Thus, co-ordinates of O, A, B and C are (0, 0), (2, 0), <img decoding=\"async\" style=\"height: 45px; width: 65px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image036.png\" \/>\u00a0and (0, 3).<\/p>\n<p>Z = <img decoding=\"async\" style=\"height: 21px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image037.png\" \/>\u00a0(if <img decoding=\"async\" style=\"height: 21px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image038.png\" \/>)<\/p>\n<p>Z = 5 x 2 + 3 x 0 = 10 (if <img decoding=\"async\" style=\"height: 21px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image039.png\" \/>)<\/p>\n<p>Z = 5 x <img decoding=\"async\" style=\"height: 41px; width: 24px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image040.png\" \/>\u00a0+ 3 x <img decoding=\"async\" style=\"height: 41px; width: 23px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image041.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 41px; width: 30px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image042.png\" \/><\/p>\n<p>(if <img decoding=\"async\" style=\"height: 41px; width: 99px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image043.png\" \/>)<\/p>\n<p>Z = 5 x 0 + 3 x 3 = 9 (if <img decoding=\"async\" style=\"height: 21px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image044.png\" \/>)<\/p>\n<p>Hence, Z = <img decoding=\"async\" style=\"height: 41px; width: 30px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image042.png\" \/>\u00a0is maximum when <img decoding=\"async\" style=\"height: 41px; width: 99px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image043.png\" \/>.<\/p>\n<hr \/>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_121-2\"><\/span>NCERT Solutions class 12 Maths Exercise 12.1<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"4_Minimize_Z_such_that\"><\/span><strong>4.\u00a0 Minimize Z = <\/strong><strong><img decoding=\"async\" style=\"height: 21px; width: 52px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image045.png\" \/><\/strong><strong>\u00a0such that <\/strong><strong><img decoding=\"async\" style=\"height: 21px; width: 184px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image046.png\" \/><\/strong><strong>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Ans.\u00a0 <\/strong>For plotting the graphs of <img decoding=\"async\" style=\"height: 21px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image047.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 21px; width: 61px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image048.png\" \/>, we have the following tables:<\/p>\n<table border=\"1\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td style=\"width: .4in;\"><img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image031.png\" \/><\/td>\n<td style=\"width: .4in;\">1<\/td>\n<td style=\"width: .4in;\">0<\/td>\n<\/tr>\n<tr>\n<td style=\"width: .4in;\"><img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image032.png\" \/><\/td>\n<td style=\"width: .4in;\">1<\/td>\n<td style=\"width: .4in;\">2<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<table border=\"1\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td style=\"width: .4in;\"><img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image031.png\" \/><\/td>\n<td style=\"width: .4in;\">0<\/td>\n<td style=\"width: .4in;\">3<\/td>\n<\/tr>\n<tr>\n<td style=\"width: .4in;\"><img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image032.png\" \/><\/td>\n<td style=\"width: .4in;\">1<\/td>\n<td style=\"width: .4in;\">0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The feasible portion represented by the inequalities<\/p>\n<p><img decoding=\"async\" style=\"height: 21px; width: 128px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image049.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 21px; width: 52px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image050.png\" \/>\u00a0is ABC which is shaded<\/p>\n<p>&nbsp;<\/p>\n<p><img decoding=\"async\" id=\"Picture 6\" style=\"height: 178px; width: 194px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image051.jpg\" \/><\/p>\n<p>in the figure. The coordinates of point B are <img decoding=\"async\" style=\"height: 45px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image052.png\" \/><\/p>\n<p>Which can be obtained by solving <img decoding=\"async\" style=\"height: 21px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image047.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 21px; width: 61px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image048.png\" \/>.<\/p>\n<p>At A (0, 2)<\/p>\n<p>Z = 3 x 0 + 5 x 2 = 10<\/p>\n<p>At B<img decoding=\"async\" style=\"height: 45px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image052.png\" \/><\/p>\n<p>Z = <img decoding=\"async\" style=\"height: 41px; width: 188px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image053.png\" \/><\/p>\n<p>At C (3, 0)<\/p>\n<p>Z = 3 x 3 + 5 x 0 = 9<\/p>\n<p>Hence, Z is minimum is 7 when <img decoding=\"async\" style=\"height: 41px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image054.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 41px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image055.png\" \/><\/p>\n<hr \/>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_121-3\"><\/span>NCERT Solutions class 12 Maths Exercise 12.1<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"5_Maximize_Z_subject_to\"><\/span><strong>5.\u00a0 Maximize Z = <\/strong><strong><img decoding=\"async\" style=\"height: 21px; width: 52px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image056.png\" \/><\/strong><strong>\u00a0subject to <\/strong><strong><img decoding=\"async\" style=\"height: 21px; width: 205px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image057.png\" \/><\/strong><strong>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Ans.\u00a0 <\/strong>Consider\u00a0 <img decoding=\"async\" style=\"height: 21px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image058.png\" \/><\/p>\n<p>Let\u00a0\u00a0\u00a0 <img decoding=\"async\" style=\"height: 21px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image059.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image007.png\" \/>\u00a0\u00a0 <img decoding=\"async\" style=\"height: 41px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image060.png\" \/><\/p>\n<p><img decoding=\"async\" id=\"Picture 8\" style=\"height: 168px; width: 188px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image061.jpg\" \/><\/p>\n<p>Since, (0, 0) satisfies the inequation, therefore the half plane containing (0, 0) is the required plane.<\/p>\n<p>Again\u00a0 <img decoding=\"async\" style=\"height: 21px; width: 81px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image062.png\" \/><\/p>\n<p>Let\u00a0 <img decoding=\"async\" style=\"height: 21px; width: 73px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image063.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image007.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 41px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image064.png\" \/><\/p>\n<p>It also satisfies by (0, 0) and its required half plane contains (0, 0).<\/p>\n<p>Now double shaded region in the first quadrant contains the solution.<\/p>\n<p>Now OABC represents the feasible region.<\/p>\n<p>Z = <img decoding=\"async\" style=\"height: 21px; width: 52px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image056.png\" \/><\/p>\n<p>At\u00a0 O (0, 0)\u00a0\u00a0\u00a0 Z = 3 x 0 + 2 x 0 = 0<\/p>\n<p>At\u00a0 A (5, 0)\u00a0\u00a0\u00a0 Z = 3 x 5 + 2 x 0 = 15<\/p>\n<p>At\u00a0 B (4, 3)\u00a0\u00a0\u00a0 Z = 3 x 4 + 2 x 3 = 18<\/p>\n<p>At\u00a0 C (0, 5)\u00a0\u00a0\u00a0 Z = 3 x 0 + 2 x 5 = 10<\/p>\n<p><img decoding=\"async\" id=\"Picture 9\" style=\"height: 225px; width: 229px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image065.jpg\" \/><\/p>\n<p>Hence, Z is maximum i.e., 18 at <img decoding=\"async\" style=\"height: 21px; width: 79px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image066.png\" \/><\/p>\n<hr \/>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_121-4\"><\/span>NCERT Solutions class 12 Maths Exercise 12.1<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"6_Minimize_Z_subject_to\"><\/span><strong>6.\u00a0 Minimize Z = <\/strong><strong><img decoding=\"async\" style=\"height: 21px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image067.png\" \/><\/strong><strong>\u00a0subject to <\/strong><strong><img decoding=\"async\" style=\"height: 21px; width: 193px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image068.png\" \/><\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Ans.\u00a0 <\/strong>Consider\u00a0 <img decoding=\"async\" style=\"height: 21px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image069.png\" \/><\/p>\n<p>Let\u00a0\u00a0\u00a0 <img decoding=\"async\" style=\"height: 21px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image070.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image007.png\" \/>\u00a0\u00a0<img decoding=\"async\" style=\"height: 21px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image071.png\" \/><\/p>\n<table border=\"1\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td style=\"width: .4in;\"><img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image031.png\" \/><\/td>\n<td style=\"width: .4in;\">0<\/td>\n<td style=\"width: .4in;\">3<\/td>\n<td style=\"width: .4in;\">\u20131<\/td>\n<\/tr>\n<tr>\n<td style=\"width: .4in;\"><img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image032.png\" \/><\/td>\n<td style=\"width: .4in;\">3<\/td>\n<td style=\"width: .4in;\">\u20131<\/td>\n<td style=\"width: .4in;\">5<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>(0, 0) is not contained in the required half plane as (0, 0) does not satisfy the inequation <img decoding=\"async\" style=\"height: 21px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image069.png\" \/>.<\/p>\n<p>Again\u00a0 <img decoding=\"async\" style=\"height: 21px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image072.png\" \/><\/p>\n<p>Let\u00a0 <img decoding=\"async\" style=\"height: 21px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image073.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image007.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 41px; width: 63px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image074.png\" \/><\/p>\n<p>Here also (0, 0) does not contain the required half plane. The double shaded region XABY is the solution set. Its corners are A (6, 0) and B (0, 3).<\/p>\n<p>At A (6, 0)\u00a0 Z = 6 + 0 = 6<\/p>\n<p>At B (0, 3)\u00a0 Z = 0 + 2 x 3 = 6<\/p>\n<p>Therefore, at both points the value of Z = 6 which is minimum. In fact at every point on the line AB makes Z = 6 which is also minimum.<\/p>\n<hr \/>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_121-5\"><\/span>NCERT Solutions class 12 Maths Exercise 12.1<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"Show_that_the_minimum_of_Z_occurs_at_more_than_two_points\"><\/span><strong>Show that the minimum of Z occurs at more than two points:<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>7.\u00a0 Minimize and Maximize Z = <\/strong><strong><img decoding=\"async\" style=\"height: 21px; width: 59px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image075.png\" \/><\/strong><strong>\u00a0subject to <\/strong><strong><img decoding=\"async\" style=\"height: 21px; width: 277px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image076.png\" \/><\/strong><\/p>\n<p><strong>Ans.\u00a0 <\/strong>Consider\u00a0 <img decoding=\"async\" style=\"height: 21px; width: 83px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image077.png\" \/><\/p>\n<p>Let\u00a0\u00a0\u00a0 <img decoding=\"async\" style=\"height: 21px; width: 83px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image078.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image007.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 83px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image079.png\" \/><\/p>\n<p>The half plane containing(0, 0) is the required half plane as (0, 0) makes <img decoding=\"async\" style=\"height: 21px; width: 83px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image077.png\" \/>, true.<\/p>\n<table border=\"1\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td style=\"width: .5in;\"><img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image031.png\" \/><\/td>\n<td style=\"width: .5in;\">0<\/td>\n<td style=\"width: .5in;\">30<\/td>\n<td style=\"width: 36.25pt;\">60<\/td>\n<\/tr>\n<tr>\n<td style=\"width: .5in;\"><img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image032.png\" \/><\/td>\n<td style=\"width: .5in;\">60<\/td>\n<td style=\"width: .5in;\">30<\/td>\n<td style=\"width: 36.25pt;\">0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><img decoding=\"async\" id=\"Picture 10\" style=\"height: 204px; width: 218px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image080.jpg\" \/><\/p>\n<p>Again\u00a0\u00a0 <img decoding=\"async\" style=\"height: 21px; width: 60px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image081.png\" \/><\/p>\n<p>Let\u00a0\u00a0\u00a0 <img decoding=\"async\" style=\"height: 21px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image082.png\" \/><\/p>\n<p>Also the half plane containing (0, 0) does not make <img decoding=\"async\" style=\"height: 21px; width: 60px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image081.png\" \/>\u00a0true.<\/p>\n<p>Therefore, the required half plane does not contain (0, 0).<\/p>\n<p>Again\u00a0\u00a0\u00a0 <img decoding=\"async\" style=\"height: 21px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image083.png\" \/><\/p>\n<p>Let\u00a0\u00a0\u00a0 <img decoding=\"async\" style=\"height: 21px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image084.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image007.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 47px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image085.png\" \/><\/p>\n<p>Let test point be (30, 0).<\/p>\n<table border=\"1\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td style=\"width: .5in;\"><img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image031.png\" \/><\/td>\n<td style=\"width: .5in;\">0<\/td>\n<td style=\"width: .5in;\">30<\/td>\n<td style=\"width: .5in;\">60<\/td>\n<\/tr>\n<tr>\n<td style=\"width: .5in;\"><img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image032.png\" \/><\/td>\n<td style=\"width: .5in;\">0<\/td>\n<td style=\"width: .5in;\">15<\/td>\n<td style=\"width: .5in;\">30<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image007.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 21px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image083.png\" \/>\u00a0\u00a0<img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image007.png\" \/>\u00a0\u00a030 \u2013 2 x 0 <img decoding=\"async\" style=\"height: 16px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image086.png\" \/>\u00a00\u00a0\u00a0\u00a0 It is true.<\/p>\n<p>Therefore, the half plane contains (30, 0).<\/p>\n<p>The region CFEKC represents the feasible region.<\/p>\n<p>At\u00a0 C (60, 0)\u00a0\u00a0\u00a0 Z = 5 x 60 = 300<\/p>\n<p>At\u00a0 F (120, 0)\u00a0\u00a0\u00a0 Z = 5 x 120 = 600<\/p>\n<p>At\u00a0 E (60, 30)\u00a0\u00a0\u00a0 Z = 5 x 60 + 10 x 30 = 600<\/p>\n<p>At\u00a0 K (40, 20)\u00a0\u00a0\u00a0 Z = 5 x 40 + 10 x 20 = 400<\/p>\n<p>Hence, minimum Z = 300 at <img decoding=\"async\" style=\"height: 21px; width: 84px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image087.png\" \/>\u00a0and maximum Z = 600 at <img decoding=\"async\" style=\"height: 21px; width: 92px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image088.png\" \/>\u00a0or <img decoding=\"async\" style=\"height: 21px; width: 95px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image089.png\" \/><\/p>\n<hr \/>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_121-6\"><\/span>NCERT Solutions class 12 Maths Exercise 12.1<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"8_Minimize_and_Maximize_Z_subject_to\"><\/span><strong>8.\u00a0 Minimize and Maximize Z = <\/strong><strong><img decoding=\"async\" style=\"height: 21px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image067.png\" \/><\/strong><strong>subject to <\/strong><strong><img decoding=\"async\" style=\"height: 21px; width: 293px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image090.png\" \/><\/strong><strong>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Ans.\u00a0 <\/strong>Consider\u00a0 <img decoding=\"async\" style=\"height: 21px; width: 83px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image091.png\" \/><\/p>\n<p>Let\u00a0\u00a0\u00a0 <img decoding=\"async\" style=\"height: 21px; width: 83px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image092.png\" \/>\u00a0\u00a0\u00a0<img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image007.png\" \/>\u00a0\u00a0\u00a0<img decoding=\"async\" style=\"height: 41px; width: 83px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image093.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 21px; width: 83px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image091.png\" \/>\u00a0represents which does not include (0, 0) as it does not made it true.<\/p>\n<table border=\"1\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td style=\"width: .5in;\"><img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image031.png\" \/><\/td>\n<td style=\"width: .5in;\">0<\/td>\n<td style=\"width: .5in;\">25<\/td>\n<td style=\"width: .5in;\">50<\/td>\n<td style=\"width: .5in;\">100<\/td>\n<\/tr>\n<tr>\n<td style=\"width: .5in;\"><img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image032.png\" \/><\/td>\n<td style=\"width: .5in;\">0<\/td>\n<td style=\"width: .5in;\">50<\/td>\n<td style=\"width: .5in;\">100<\/td>\n<td style=\"width: .5in;\">200<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Again consider\u00a0 <img decoding=\"async\" style=\"height: 21px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image094.png\" \/><\/p>\n<p>Let\u00a0\u00a0\u00a0 <img decoding=\"async\" style=\"height: 21px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image095.png\" \/>\u00a0\u00a0\u00a0<img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image007.png\" \/>\u00a0\u00a0<img decoding=\"async\" style=\"height: 21px; width: 47px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image096.png\" \/><\/p>\n<p>Let the test point be (10, 0).<\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image016.png\" \/>\u00a02 x 10 \u2013 0 <img decoding=\"async\" style=\"height: 16px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image097.png\" \/>\u00a00 which is false.<\/p>\n<p><img decoding=\"async\" id=\"Picture 11\" style=\"height: 225px; width: 223px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image098.jpg\" \/><\/p>\n<p>Therefore, the required half does not contain (10, 0).<\/p>\n<p>Again consider <img decoding=\"async\" style=\"height: 21px; width: 85px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image099.png\" \/><\/p>\n<p>Let\u00a0 <img decoding=\"async\" style=\"height: 21px; width: 85px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image100.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image007.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 91px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image101.png\" \/><\/p>\n<p>Now (0, 0) satisfies <img decoding=\"async\" style=\"height: 21px; width: 85px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image099.png\" \/><\/p>\n<p>Therefore, the required half place contains (0, 0).<\/p>\n<p>Now triple shaded region is ABCDA which is the required feasible region.<\/p>\n<p>At\u00a0 A (0, 50)<\/p>\n<p>Z = <img decoding=\"async\" style=\"height: 21px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image067.png\" \/>\u00a0= 0 + 2 x 50 = 100<\/p>\n<p>At\u00a0 B (20, 40)\u00a0 Z = 20 + 2 x 40 = 100<\/p>\n<p>At\u00a0 C (50, 100)\u00a0 Z = 50 + 2 x 100 = 250<\/p>\n<p>At\u00a0 D (0, 200)\u00a0 Z = 0 + 2 x 200 = 400<\/p>\n<p>Hence maximum Z = 400 at <img decoding=\"async\" style=\"height: 21px; width: 91px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image102.png\" \/>\u00a0and minimum Z = 100 at <img decoding=\"async\" style=\"height: 21px; width: 84px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image103.png\" \/>\u00a0or <img decoding=\"async\" style=\"height: 21px; width: 96px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image104.png\" \/><\/p>\n<hr \/>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_121-7\"><\/span>NCERT Solutions class 12 Maths Exercise 12.1<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"9_Maximize_Z_subject_to_the_constraints\"><\/span><strong>9.\u00a0 Maximize Z = <\/strong><strong><img decoding=\"async\" style=\"height: 21px; width: 55px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image105.png\" \/><\/strong><strong>\u00a0subject to the constraints: <\/strong><strong><img decoding=\"async\" style=\"height: 21px; width: 208px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image106.png\" \/><\/strong><strong>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Ans.\u00a0 <\/strong>Consider <img decoding=\"async\" style=\"height: 19px; width: 36px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image107.png\" \/><\/p>\n<p>Let <img decoding=\"async\" style=\"height: 19px; width: 36px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image108.png\" \/>\u00a0which is a line parallel to <img decoding=\"async\" style=\"height: 17px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image109.png\" \/>axis at a positive distance of 3 from it.<\/p>\n<p><img decoding=\"async\" id=\"Picture 12\" style=\"height: 230px; width: 209px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image110.jpg\" \/><\/p>\n<p>Since <img decoding=\"async\" style=\"height: 19px; width: 36px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image107.png\" \/>, therefore the required half-plane does not contain (0, 0).<\/p>\n<p>Now consider <img decoding=\"async\" style=\"height: 20px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image111.png\" \/>\u00a05<\/p>\n<p>Let\u00a0 <img decoding=\"async\" style=\"height: 21px; width: 60px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image112.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image007.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 63px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image113.png\" \/><\/p>\n<p>Now (0, 0) does not satisfy <img decoding=\"async\" style=\"height: 20px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image111.png\" \/>\u00a05, therefore the required half plane does not contain (0, 0).<\/p>\n<p>Again consider <img decoding=\"async\" style=\"height: 21px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image072.png\" \/><\/p>\n<p>Let\u00a0 <img decoding=\"async\" style=\"height: 21px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image073.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image007.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 41px; width: 63px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image074.png\" \/><\/p>\n<p>Here also (0, 0) does not satisfy <img decoding=\"async\" style=\"height: 21px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image072.png\" \/>, therefore the required half plane does not contain (0, 0).<\/p>\n<p>The corners of the feasible region are A (6, 0), B (4, 1) and C (3, 2).<\/p>\n<p>At\u00a0 A (6, 0)\u00a0 Z = \u20136\u00a0 + 2 x 0 = \u20136<\/p>\n<p>At\u00a0 B (4, 1)\u00a0 Z = \u20134 + 2 x 1 = \u20132<\/p>\n<p>At\u00a0 C (3, 2)\u00a0 Z = \u20133 + 2 x 2 = 1<\/p>\n<p>Hence, maximum Z = 1 at <img decoding=\"async\" style=\"height: 21px; width: 79px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image114.png\" \/><\/p>\n<hr \/>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_121-8\"><\/span>NCERT Solutions class 12 Maths Exercise 12.1<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"10_Maximize_Z_subject_to\"><\/span><strong>10.\u00a0 Maximize Z = <\/strong><strong><img decoding=\"async\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image115.png\" \/><\/strong><strong>\u00a0subject to <\/strong><strong><img decoding=\"async\" style=\"height: 21px; width: 193px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image116.png\" \/><\/strong><strong>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Ans.\u00a0 <\/strong>Consider\u00a0 <img decoding=\"async\" style=\"height: 21px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image117.png\" \/><\/p>\n<p>Let\u00a0 <img decoding=\"async\" style=\"height: 21px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image118.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image007.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 57px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image119.png\" \/><\/p>\n<table border=\"1\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td style=\"width: .6in;\"><\/td>\n<td style=\"width: .6in;\">A<\/td>\n<td style=\"width: .6in;\">B<\/td>\n<td style=\"width: .6in;\">C<\/td>\n<td style=\"width: .6in;\">D<\/td>\n<\/tr>\n<tr>\n<td style=\"width: .6in;\">\u00a0<img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image031.png\" \/><\/td>\n<td style=\"width: .6in;\"><img decoding=\"async\" style=\"height: 17px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image120.png\" \/><\/td>\n<td style=\"width: .6in;\">0<\/td>\n<td style=\"width: .6in;\">0<\/td>\n<td style=\"width: .6in;\">3<\/td>\n<\/tr>\n<tr>\n<td style=\"width: .6in;\"><img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image032.png\" \/><\/td>\n<td style=\"width: .6in;\">0<\/td>\n<td style=\"width: .6in;\">1<\/td>\n<td style=\"width: .6in;\">3<\/td>\n<td style=\"width: .6in;\">4<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>If (0, 0) is the test point then <img decoding=\"async\" style=\"height: 21px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image117.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image007.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 19px; width: 44px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image121.png\" \/>\u00a0which is false and thus the required plane does not include (0, 0).<\/p>\n<p>Again\u00a0 <img decoding=\"async\" style=\"height: 21px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image122.png\" \/><\/p>\n<p>Let\u00a0 <img decoding=\"async\" style=\"height: 21px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image123.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image007.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 17px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image124.png\" \/><\/p>\n<table border=\"1\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td style=\"width: .6in;\"><\/td>\n<td style=\"width: .6in;\">O<\/td>\n<td style=\"width: .6in;\">E<\/td>\n<td style=\"width: .6in;\">F<\/td>\n<\/tr>\n<tr>\n<td style=\"width: .6in;\"><img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image031.png\" \/><\/td>\n<td style=\"width: .6in;\">0<\/td>\n<td style=\"width: .6in;\">1<\/td>\n<td style=\"width: .6in;\">2<\/td>\n<\/tr>\n<tr>\n<td style=\"width: .6in;\"><img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image032.png\" \/><\/td>\n<td style=\"width: .6in;\">0<\/td>\n<td style=\"width: .6in;\">1<\/td>\n<td style=\"width: .6in;\">2<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>For (1, 0) \u2013 1 <img decoding=\"async\" style=\"height: 16px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image097.png\" \/>\u00a00 which is true, therefore the required half-plane include (1, 0).<\/p>\n<p><img decoding=\"async\" id=\"Picture 13\" style=\"height: 179px; width: 220px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.1\/image125.jpg\" \/><\/p>\n<p>It is clear that the two required half planes do not intersect at all, i.e., they do not have a common region.<\/p>\n<p>Hence there is no maximum Z.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_121-9\"><\/span>NCERT Solutions class 12 Maths Exercise 12.1<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>NCERT Solutions Class 12 Maths PDF (Download) Free from myCBSEguide app and myCBSEguide website. Ncert solution class 12 Maths includes text book solutions from both part 1 and part 2. NCERT Solutions for CBSE Class 12 Maths have total 13 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_Students\"><\/span>CBSE App for Students<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>To download\u00a0NCERT Solutions for class 12 Physics, Chemistry, Biology, History, Political Science, Economics, Geography, Computer Science, Home Science, Accountancy, Business Studies and Home Science; do check myCBSEguide app or website. myCBSEguide provides sample papers with solution, test papers for chapter-wise practice, NCERT solutions, NCERT Exemplar solutions, quick revision notes for ready reference, CBSE guess papers and CBSE important question papers. Sample Paper all are made available through\u00a0<a href=\"https:\/\/play.google.com\/store\/apps\/details?id=in.techchefs.MyCBSEGuide&amp;referrer=utm_source%3Dmycbse_bottom%26utm_medium%3Dtext%26utm_campaign%3Dmycbseads\"><strong>the best app for CBSE students<\/strong><\/a>\u00a0and myCBSEguide website.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>NCERT Solutions class 12 Maths Exercise 12.1 Class 12 Maths book solutions are available in PDF format for free download. These ncert book chapter wise questions and answers are very helpful for CBSE board exam. CBSE recommends NCERT books and most of the questions in CBSE exam are asked from NCERT text books. Class 12 &#8230; <a title=\"NCERT Solutions class 12 Maths Exercise 12.1\" class=\"read-more\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-12-1\/\" aria-label=\"More on NCERT Solutions class 12 Maths Exercise 12.1\">Read more<\/a><\/p>\n","protected":false},"author":1,"featured_media":-1,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[281],"tags":[283,1342,321,216],"class_list":["post-4506","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-ncert-solutions","tag-cbse-study-material","tag-class-12","tag-mathematics","tag-ncert-solutions"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.0 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>NCERT Solutions class 12 Maths Exercise 12.1 | myCBSEguide<\/title>\n<meta name=\"description\" content=\"NCERT Solutions class 12 Maths Exercise 12.1 in PDF format for free download. 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