{"id":4507,"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-12-2\/"},"modified":"2018-05-07T18:25:19","modified_gmt":"2018-05-07T12:55:19","slug":"ncert-solutions-class-12-maths-exercise-12-2","status":"publish","type":"post","link":"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-12-2\/","title":{"rendered":"NCERT Solutions class 12 Maths Exercise 12.2"},"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-12-2\/#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-12-2\/#NCERT_Solutions_class_12_Maths_Exercise_122\" >NCERT Solutions class 12 Maths Exercise 12.2<\/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-12-2\/#2_One_kind_of_cake_requires_200g_of_flour_and_25_g_of_fat_and_another_kind_of_cake_requires_100_g_of_flour_and_50_g_of_fat_Find_the_maximum_number_of_cake_which_can_be_made_from_5_kg_of_flour_and_1_kg_of_fat_assuming_that_there_is_no_shortage_of_the_other_ingredients_used_in_making_the_cakes\" >2. One kind of cake requires 200g of flour and 25 g of fat and another kind of cake requires 100 g of flour and 50 g of fat. Find the maximum number of cake which can be made from 5 kg of flour and 1 kg of fat assuming that there is no shortage of the other ingredients used in making the cakes.<\/a><ul class='ez-toc-list-level-4' ><li class='ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-12-2\/#NCERT_Solutions_class_12_Maths_Exercise_122-2\" >NCERT Solutions class 12 Maths Exercise 12.2<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-12-2\/#3_A_factory_makes_tennis_rackets_and_cricket_bats_A_tennis_racket_takes_15_hours_of_machine_time_and_3_hours_of_craftmans_time_in_its_making_while_a_cricket_bat_takes_3_hours_of_machine_time_and_1_hour_of_craftmans_time_In_a_day_the_factory_has_the_availability_of_not_more_than_42_hours_of_machine_time_and_24_hours_of_craftmans_time\" >3. A factory makes tennis rackets and cricket bats. A tennis racket takes 1.5 hours of machine time and 3 hours of craftman\u2019s time in its making while a cricket bat takes 3 hours of machine time and 1 hour of craftman\u2019s time. In a day, the factory has the availability of not more than 42 hours of machine time and 24 hours of craftman\u2019s time.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-12-2\/#NCERT_Solutions_class_12_Maths_Exercise_122-3\" >NCERT Solutions class 12 Maths Exercise 12.2<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-12-2\/#4_A_manufacturer_produces_nuts_and_bolts_It_takes_1_hour_of_work_on_machine_A_and_3_hours_on_machine_B_to_produce_a_package_of_nuts_It_takes_3_hours_on_machine_A_and_1_hour_on_machine_B_to_produce_a_package_of_bolts_He_earns_a_profit_of_Rs_1750_per_package_on_nuts_and_Rs_700_per_package_on_bolts_How_many_packages_of_each_should_be_produced_each_day_so_as_to_maximize_his_profit_if_he_operates_his_machines_for_at_the_most_12_hours_a_day\" >4. A manufacturer produces nuts and bolts. It takes 1 hour of work on machine A and 3 hours on machine B to produce a package of nuts. It takes 3 hours on machine A and 1 hour on machine B to produce a package of bolts. He earns a profit of Rs. 17.50 per package on nuts and Rs. 7.00 per package on bolts. How many packages of each should be produced each day so as to maximize his profit, if he operates his machines for at the most 12 hours a day?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-12-2\/#NCERT_Solutions_class_12_Maths_Exercise_122-4\" >NCERT Solutions class 12 Maths Exercise 12.2<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-12-2\/#5_A_factory_manufacturers_two_types_of_screws_A_and_B_Each_type_of_screw_requires_the_use_of_two_machines_an_automatic_and_a_hand_operated_It_takes_4_minutes_on_the_automatic_and_6_minutes_on_hand_operated_machines_to_manufacture_a_package_of_screw_A_while_it_takes_6_minutes_on_automatic_and_3_minutes_on_hand_operated_machines_to_manufacture_a_package_of_screws_B_Each_machine_is_available_for_at_the_most_4_hours_on_any_day_The_manufacturer_can_sell_a_package_of_screws_A_at_a_profit_of_Rs_7_and_screws_B_at_a_profit_of_Rs_10_Assuming_that_he_can_sell_all_the_screws_he_manufactures_how_many_packages_of_each_type_should_the_factory_owner_produce_a_day_in_order_to_maximize_his_profit_Determine_the_maximum_profit\" >5. A factory manufacturers two types of screws, A and B. Each type of screw requires the use of two machines, an automatic and a hand operated. It takes 4 minutes on the automatic and 6 minutes on hand operated machines to manufacture a package of screw A, while it takes 6 minutes on automatic and 3 minutes on hand operated machines to manufacture a package of screws B. Each machine is available for at the most 4 hours on any day. The manufacturer can sell a package of screws A at a profit of Rs. 7 and screws B at a profit of Rs. 10. Assuming that he can sell all the screws he manufactures, how many packages of each type should the factory owner produce a day in order to maximize his profit? Determine the maximum profit.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-12-2\/#NCERT_Solutions_class_12_Maths_Exercise_122-5\" >NCERT Solutions class 12 Maths Exercise 12.2<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-11\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-12-2\/#6_A_cottage_industry_manufactures_pedestal_lamps_and_wooden_shades_each_requiring_the_use_of_a_grindingcutting_machine_and_a_sprayer_It_takes_2_hours_on_grindingcutting_machine_and_3_hours_on_the_sprayer_to_manufacture_a_pedestal_lamp_It_takes_1_hour_on_the_grindingcutting_machine_and_2_hours_on_the_sprayer_to_manufacture_a_shade_On_any_day_the_sprayer_is_available_for_at_the_most_20_hours_and_the_grindingcutting_machine_for_at_the_most_12_hours_The_profit_from_the_sale_of_a_lamp_is_Rs5_and_that_from_a_shade_is_Rs3_Assuming_that_the_manufacturer_can_sell_all_the_lamps_and_shades_that_he_produces_how_should_he_schedule_his_daily_production_in_order_to_maximize_his_profit\" >6. A cottage industry manufactures pedestal lamps and wooden shades, each requiring the use of a grinding\/cutting machine and a sprayer. It takes 2 hours on grinding\/cutting machine and 3 hours on the sprayer to manufacture a pedestal lamp. It takes 1 hour on the grinding\/cutting machine and 2 hours on the sprayer to manufacture a shade. On any day, the sprayer is available for at the most 20 hours and the grinding\/cutting machine for at the most 12 hours. The profit from the sale of a lamp is Rs.5 and that from a shade is Rs.3. Assuming that the manufacturer can sell all the lamps and shades that he produces, how should he schedule his daily production in order to maximize his profit?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-12\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-12-2\/#NCERT_Solutions_class_12_Maths_Exercise_122-6\" >NCERT Solutions class 12 Maths Exercise 12.2<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-13\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-12-2\/#7_A_company_manufactures_two_types_of_novelty_souvenirs_made_of_plywood_Souvenirs_of_type_A_requires_5_minutes_each_for_cutting_and_10_minutes_each_for_assembling_Souvenirs_of_type_B_require_8_minutes_each_for_cutting_and_8_minutes_each_for_assembling_There_are_3_hours_20_minutes_available_for_cutting_and_4_hours_for_assembling_The_profit_is_Rs5_each_for_type_A_and_Rs6_each_for_type_B_souvenirs_How_many_souvenirs_of_each_type_should_the_company_manufacture_in_order_to_maximize_the_profit\" >7. A company manufactures two types of novelty souvenirs made of plywood. Souvenirs of type A requires 5 minutes each for cutting and 10 minutes each for assembling. Souvenirs of type B require 8 minutes each for cutting and 8 minutes each for assembling. There are 3 hours 20 minutes available for cutting and 4 hours for assembling. The profit is Rs.5 each for type A and Rs.6 each for type B souvenirs. How many souvenirs of each type should the company manufacture in order to maximize the profit?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-14\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-12-2\/#NCERT_Solutions_class_12_Maths_Exercise_122-7\" >NCERT Solutions class 12 Maths Exercise 12.2<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-15\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-12-2\/#8_A_merchant_plans_to_sell_two_types_of_personal_computers_%E2%80%93_a_desktop_model_and_a_portable_model_that_will_cost_Rs_25000_and_Rs_40000_respectively_He_estimates_that_the_total_monthly_demand_of_computers_will_not_exceed_250_units_Determine_the_number_of_units_of_each_type_of_computers_which_the_merchant_should_stock_to_get_maximum_profit_if_he_does_not_want_to_invest_more_than_Rs_70_lakhs_and_if_his_profit_on_the_desktop_model_is_Rs_4500_and_on_portable_model_is_Rs_5000\" >8. A merchant plans to sell two types of personal computers \u2013 a desktop model and a portable model that will cost Rs. 25000 and Rs. 40000 respectively. He estimates that the total monthly demand of computers will not exceed 250 units. Determine the number of units of each type of computers which the merchant should stock to get maximum profit if he does not want to invest more than Rs, 70 lakhs and if his profit on the desktop model is Rs. 4500 and on portable model is Rs. 5000.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-16\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-12-2\/#NCERT_Solutions_class_12_Maths_Exercise_122-8\" >NCERT Solutions class 12 Maths Exercise 12.2<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-17\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-12-2\/#9_A_diet_is_to_contain_at_least_80_units_of_vitamin_A_and_100_units_of_minerals_Two_foods_F1_and_F2_are_available_Food_F1_costs_Rs_4_per_unit_food_and_F2_costs_Rs_6_per_unit_One_unit_of_food_F1_contains_3_units_of_vitamin_A_and_4_units_of_minerals_One_unit_of_food_F2_contains_6_units_of_vitamin_A_and_3_units_of_minerals_Formulate_this_as_a_linear_programming_problem_Find_the_minimum_cost_for_diet_that_consists_of_mixture_of_these_two_foods_and_also_meets_the_minimal_nutritional_requirements\" >9. A diet is to contain at least 80 units of vitamin A and 100 units of minerals. Two foods F1 and F2 are available. Food F1 costs Rs. 4 per unit food and F2 costs Rs. 6 per unit. One unit of food F1 contains 3 units of vitamin A and 4 units of minerals. One unit of food F2 contains 6 units of vitamin A and 3 units of minerals. Formulate this as a linear programming problem. Find the minimum cost for diet that consists of mixture of these two foods and also meets the minimal nutritional requirements.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-18\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-12-2\/#NCERT_Solutions_class_12_Maths_Exercise_122-9\" >NCERT Solutions class 12 Maths Exercise 12.2<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-19\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-12-2\/#10_There_are_two_types_of_fertilizers_F1_and_F2_F1_consists_of_10_nitrogen_and_6_phosphoric_acid_and_F2_consists_of_5_nitrogen_and_10_phosphoric_acid_After_testing_the_soil_conditions_a_farmer_finds_that_she_needs_at_least_14_kg_of_nitrogen_and_14_kg_of_phosphoric_acid_for_her_crop_If_F1_costs_Rs_6kg_and_F2_costs_Rs_5kg_determine_how_much_of_each_type_of_fertilizer_should_be_used_so_that_nutrient_requirements_are_met_at_a_minimum_cost_What_is_the_minimum_cost\" >10. There are two types of fertilizers F1 and F2. F1 consists of 10% nitrogen and 6% phosphoric acid and F2 consists of 5% nitrogen and 10% phosphoric acid. After testing the soil conditions, a farmer finds that she needs at least 14 kg of nitrogen and 14 kg of phosphoric acid for her crop. If F1 costs Rs. 6\/kg and F2 costs Rs. 5\/kg, determine how much of each type of fertilizer should be used so that nutrient requirements are met at a minimum cost. What is the minimum cost?<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-20\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-12-2\/#11_The_corner_points_of_the_feasible_region_determined_by_the_following_system_of_linear_inequalities\" >11. The corner points of the feasible region determined by the following system of linear inequalities:<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-21\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-12-2\/#NCERT_Solutions_class_12_Maths_Exercise_122-10\" >NCERT Solutions class 12 Maths Exercise 12.2<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-22\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-12-2\/#CBSE_App_for_Students\" >CBSE App for Students<\/a><\/li><\/ul><\/nav><\/div>\n<p>NCERT Solutions class 12 Maths Exercise 12.2 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 12.2\" width=\"200\" height=\"256\" \/><\/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>\u00a0<\/strong><strong>1. Reshma wishes to mix two types of food P and Q in such a way that the vitamin contents of the mixture contain at least 8 units of vitamin A and 11 units of vitamin B. Food P costs Rs. 60\/kg and Food Q costs Rs. 80\/kg. Food P contains 3 units\/kg of vitamin A and 5 units\/kg of vitamin B while Food Q contains 4 units\/kg of vitamin A and 2 units\/kg of vitamin B. Determine the minimum cost of the mixture.<\/strong><\/p>\n<p><strong>Ans. <\/strong>Let food P consists of <img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image001.png\" \/>\u00a0kg and food Q consists of <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image002.png\" \/>\u00a0kg.<\/p>\n<p>Minimum Z = <img decoding=\"async\" style=\"height: 21px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image003.png\" \/><\/p>\n<p>According to question \u00a0<img decoding=\"async\" style=\"height: 21px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image004.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 21px; width: 81px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image005.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image006.png\" \/>\u00a0and\u00a0\u00a0 <img decoding=\"async\" style=\"height: 20px; width: 27px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image007.png\" \/>\u00a00<\/p>\n<table border=\"1\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td style=\"width: .5in;\"><\/td>\n<td style=\"width: .5in;\">A<\/td>\n<td style=\"width: .5in;\">B<\/td>\n<td style=\"width: .5in;\">C<\/td>\n<\/tr>\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\/Ex12.2\/image001.png\" \/><\/td>\n<td style=\"width: .5in;\">0<\/td>\n<td style=\"width: .5in;\">4<\/td>\n<td style=\"width: .5in;\"><img decoding=\"async\" style=\"height: 17px; width: 21px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image008.png\" \/><\/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\/Ex12.2\/image002.png\" \/><\/td>\n<td style=\"width: .5in;\">2<\/td>\n<td style=\"width: .5in;\"><img decoding=\"async\" style=\"height: 17px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image009.png\" \/><\/td>\n<td style=\"width: .5in;\">5<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Consider <img decoding=\"async\" style=\"height: 21px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image004.png\" \/><\/p>\n<p>Let\u00a0 <img decoding=\"async\" style=\"height: 21px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image010.png\" \/><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image011.png\" \/><img decoding=\"async\" style=\"height: 41px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image012.png\" \/><\/p>\n<p>But (0, 0) does not satisfy this inequation, therefore the required half-plane does not contain (0, 0).<\/p>\n<p>Again consider <img decoding=\"async\" style=\"height: 21px; width: 81px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image005.png\" \/><\/p>\n<p>Let\u00a0 <img decoding=\"async\" style=\"height: 21px; width: 81px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image013.png\" \/><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image011.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image014.png\" \/><\/p>\n<table border=\"1\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td style=\"width: .5in;\">\n<p style=\"text-align: justify;\">\n<\/td>\n<td style=\"width: .5in;\">D<\/td>\n<td style=\"width: .5in;\">E<\/td>\n<td style=\"width: .5in;\">F<\/td>\n<\/tr>\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\/Ex12.2\/image001.png\" \/><\/td>\n<td style=\"width: .5in;\">0<\/td>\n<td style=\"width: .5in;\">4<\/td>\n<td style=\"width: .5in;\"><img decoding=\"async\" style=\"height: 17px; width: 21px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image008.png\" \/><\/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\/Ex12.2\/image002.png\" \/><\/td>\n<td style=\"width: .5in;\">2<\/td>\n<td style=\"width: .5in;\"><img decoding=\"async\" style=\"height: 17px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image009.png\" \/><\/td>\n<td style=\"width: .5in;\">5<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Again (0, 0) does not satisfy this inequation, therefore the required half-plane does not contain (0, 0).<\/p>\n<p><img decoding=\"async\" id=\"Picture 14\" style=\"height: 231px; width: 231px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image015.jpg\" \/><\/p>\n<p>The double shaded region is our solution set. The corners of this region are D (0, 5.5), Q (2, 1.5) and P<img decoding=\"async\" style=\"height: 45px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image016.png\" \/><\/p>\n<p>Now Z = <img decoding=\"async\" style=\"height: 21px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image003.png\" \/><\/p>\n<p>AtD(0,5.5)<\/p>\n<p>Z = 60 x 0 + 80 x 5.5 = 440<\/p>\n<p>AtQ (2, 1.5)<\/p>\n<p>Z = 60 x 2 + 80 x 1.5 = 160<\/p>\n<p>AtP<img decoding=\"async\" style=\"height: 45px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image017.png\" \/><\/p>\n<p>Z = 60 x <img decoding=\"async\" style=\"height: 41px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image018.png\" \/>\u00a0\u00a080 x 0 = 160<\/p>\n<p>Hence, minimum Z = 160 at <img decoding=\"async\" style=\"height: 21px; width: 87px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image019.png\" \/>\u00a0means minimum cost Z = Rs. 160 when Reshma mixes food P = 2 kg and food Q = 1.5 kg.<\/p>\n<hr \/>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_122\"><\/span><strong>NCERT Solutions class 12 Maths Exercise 12.2<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"2_One_kind_of_cake_requires_200g_of_flour_and_25_g_of_fat_and_another_kind_of_cake_requires_100_g_of_flour_and_50_g_of_fat_Find_the_maximum_number_of_cake_which_can_be_made_from_5_kg_of_flour_and_1_kg_of_fat_assuming_that_there_is_no_shortage_of_the_other_ingredients_used_in_making_the_cakes\"><\/span><strong>2. One kind of cake requires 200g of flour and 25 g of fat and another kind of cake requires 100 g of flour and 50 g of fat. Find the maximum number of cake which can be made from 5 kg of flour and 1 kg of fat assuming that there is no shortage of the other ingredients used in making the cakes.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Ans. <\/strong>Let number of cakes made of first kind are <img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image001.png\" \/>\u00a0and that of second kind is <img decoding=\"async\" style=\"height: 17px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image020.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image021.png\" \/>Let to maximize Z = <img decoding=\"async\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image022.png\" \/><\/p>\n<p>According to question\u00a0 <img decoding=\"async\" style=\"height: 21px; width: 131px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image023.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 21px; width: 115px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image024.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 21px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image025.png\" \/><\/p>\n<p>Consider <img decoding=\"async\" style=\"height: 21px; width: 131px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image023.png\" \/><\/p>\n<p>Let <img decoding=\"async\" style=\"height: 21px; width: 131px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image026.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image011.png\" \/><img decoding=\"async\" style=\"height: 21px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image027.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image011.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image028.png\" \/><\/p>\n<p>Here, (0, 0) satisfies this inequation, therefore the required half plane contains (0, 0).<\/p>\n<p><img decoding=\"async\" id=\"Picture 15\" style=\"height: 182px; width: 199px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image029.jpg\" \/><\/p>\n<p>Again consider <img decoding=\"async\" style=\"height: 21px; width: 115px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image024.png\" \/><\/p>\n<p>Let <img decoding=\"async\" style=\"height: 21px; width: 115px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image030.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image011.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 77px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image031.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image011.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 77px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image032.png\" \/><\/p>\n<p>Here, again (0, 0) satisfies this inequation, therefore the required half plane contains (0, 0).<\/p>\n<p>The double shaded region is the feasible region which is solution set.<\/p>\n<p>The corner points of this region are O (0, 0), A (25, 0), B (20, 10) and C (0, 20).<\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image021.png\" \/>Z = <img decoding=\"async\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image022.png\" \/><\/p>\n<p>At O (0, 0) Z = 0 + 0 = 0<\/p>\n<p>At A (25, 0) Z = 25 + 0 = 25<\/p>\n<p>At B (20, 10) Z = 20 + 10 = 30<\/p>\n<p>AtC (0, 20) Z = 0 + 20 = 20<\/p>\n<p>Hence, maximum number of cakes Z = 30 when <img decoding=\"async\" style=\"height: 21px; width: 93px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image033.png\" \/><\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_122-2\"><\/span><strong>NCERT Solutions class 12 Maths Exercise 12.2<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4><span class=\"ez-toc-section\" id=\"3_A_factory_makes_tennis_rackets_and_cricket_bats_A_tennis_racket_takes_15_hours_of_machine_time_and_3_hours_of_craftmans_time_in_its_making_while_a_cricket_bat_takes_3_hours_of_machine_time_and_1_hour_of_craftmans_time_In_a_day_the_factory_has_the_availability_of_not_more_than_42_hours_of_machine_time_and_24_hours_of_craftmans_time\"><\/span><strong>3. A factory makes tennis rackets and cricket bats. A tennis racket takes 1.5 hours of machine time and 3 hours of craftman\u2019s time in its making while a cricket bat takes 3 hours of machine time and 1 hour of craftman\u2019s time. In a day, the factory has the availability of not more than 42 hours of machine time and 24 hours of craftman\u2019s time.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>(i) What number of rackets and bats must be made if the factory is to work at full capacity?<\/strong><\/p>\n<p><strong>(ii) If the profit on a racket and on a bat is Rs. 20 and Rs. 10 respectively, find maximum profit of the factory when it works at full capacity.<\/strong><\/p>\n<p><strong>Ans. <\/strong>Let number of rackets = <img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image001.png\" \/>\u00a0and number of bats = <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image002.png\" \/><\/p>\n<table border=\"1\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td style=\"width: .5in;\"><\/td>\n<td style=\"width: .5in;\">A<\/td>\n<td style=\"width: .5in;\">B<\/td>\n<td style=\"width: .5in;\">C<\/td>\n<\/tr>\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\/Ex12.2\/image001.png\" \/><\/td>\n<td style=\"width: .5in;\">28<\/td>\n<td style=\"width: .5in;\">20<\/td>\n<td style=\"width: .5in;\">0<\/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\/Ex12.2\/image002.png\" \/><\/td>\n<td style=\"width: .5in;\">0<\/td>\n<td style=\"width: .5in;\">4<\/td>\n<td style=\"width: .5in;\">14<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><strong>(i)<\/strong> To maximize Z = <img decoding=\"async\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image022.png\" \/><\/p>\n<p>According to question \u00a0\u00a0<img decoding=\"async\" style=\"height: 21px; width: 93px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image034.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 21px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image035.png\" \/><\/p>\n<p><img decoding=\"async\" id=\"Picture 16\" style=\"height: 245px; width: 241px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image036.jpg\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 21px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image025.png\" \/><\/p>\n<p>Consider <img decoding=\"async\" style=\"height: 21px; width: 93px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image034.png\" \/><\/p>\n<p>Let <img decoding=\"async\" style=\"height: 21px; width: 95px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image037.png\" \/><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image011.png\" \/><img decoding=\"async\" style=\"height: 21px; width: 77px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image038.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image011.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image039.png\" \/><\/p>\n<p>Here (0, 0) satisfies this inequaiton, therefore the required half-plane contains (0, 0).<\/p>\n<p>Again consider<\/p>\n<p><img decoding=\"async\" style=\"height: 21px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image035.png\" \/><\/p>\n<p>Let <img decoding=\"async\" style=\"height: 21px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image040.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image011.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image041.png\" \/><\/p>\n<p>Here (0, 0) satisfies this inequaiton, therefore the required half-plane contains (0, 0).<\/p>\n<p>Now the feasible region is the double-shaded region i.e., the solution set. The corner points of this region are O (0, 0), P (8, 0), Q (4, 12) and R (0, 14).<\/p>\n<p>Now Z = <img decoding=\"async\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image022.png\" \/><\/p>\n<p>At O (0, 0) Z = 0 + 0 = 0<\/p>\n<p>At P (8, 0) Z = 8 + 0 = 0<\/p>\n<p>At Q (4, 12) Z = 4 + 12 = 16<\/p>\n<p>At R (0, 14) Z = 0 + 14 = 14<\/p>\n<p>Now maximum Z = 16 at <img decoding=\"async\" style=\"height: 21px; width: 85px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image042.png\" \/><\/p>\n<p><strong>(ii)<\/strong> The profit on a racket = Rs. 20 and the profit on a bat = Rs. 10<\/p>\n<p>Let maximum profit P = <img decoding=\"async\" style=\"height: 21px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image043.png\" \/><\/p>\n<p>Hence maximum profit P = 20 x 4 + 10 x 12 = Rs.200<\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_122-3\"><\/span><strong>NCERT Solutions class 12 Maths Exercise 12.2<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4><span class=\"ez-toc-section\" id=\"4_A_manufacturer_produces_nuts_and_bolts_It_takes_1_hour_of_work_on_machine_A_and_3_hours_on_machine_B_to_produce_a_package_of_nuts_It_takes_3_hours_on_machine_A_and_1_hour_on_machine_B_to_produce_a_package_of_bolts_He_earns_a_profit_of_Rs_1750_per_package_on_nuts_and_Rs_700_per_package_on_bolts_How_many_packages_of_each_should_be_produced_each_day_so_as_to_maximize_his_profit_if_he_operates_his_machines_for_at_the_most_12_hours_a_day\"><\/span><strong>4. A manufacturer produces nuts and bolts. It takes 1 hour of work on machine A and 3 hours on machine B to produce a package of nuts. It takes 3 hours on machine A and 1 hour on machine B to produce a package of bolts. He earns a profit of Rs. 17.50 per package on nuts and Rs. 7.00 per package on bolts. How many packages of each should be produced each day so as to maximize his profit, if he operates his machines for at the most 12 hours a day?<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>Ans. <\/strong>Let number of package of nuts = <img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image001.png\" \/>\u00a0and number of packages of bolts = <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image002.png\" \/><\/p>\n<p>To maximum profit Z = <img decoding=\"async\" style=\"height: 21px; width: 80px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image044.png\" \/><\/p>\n<p>According to question <img decoding=\"async\" style=\"height: 21px; width: 73px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image045.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 21px; width: 73px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image046.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image025.png\" \/><\/p>\n<p><img decoding=\"async\" id=\"Picture 17\" style=\"height: 191px; width: 219px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image047.jpg\" \/><\/p>\n<p>Consider <img decoding=\"async\" style=\"height: 21px; width: 73px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image045.png\" \/><\/p>\n<p>Let <img decoding=\"async\" style=\"height: 21px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image048.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image011.png\" \/><img decoding=\"async\" style=\"height: 41px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image049.png\" \/><\/p>\n<p>Therefore, points are A (12, 0), B (0, 4). Also, (0, 0) satisfies this inequation, therefore the required half-plane contains (0, 0).<\/p>\n<p>Again consider <img decoding=\"async\" style=\"height: 21px; width: 73px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image046.png\" \/><\/p>\n<p>Let <img decoding=\"async\" style=\"height: 21px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image050.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image011.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image051.png\" \/><\/p>\n<p>Therefore, points are C (4, 0), D (0, 12).<\/p>\n<p>Again (0, 0) satisfies this inequation, therefore the required half-plane contains (0, 0).<\/p>\n<p>The double shaded region is OBPCO and its corners are O (0, 0), B (0, 4), D (3, 3), C (4, 0).<\/p>\n<p>Now Z = <img decoding=\"async\" style=\"height: 21px; width: 80px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image044.png\" \/><\/p>\n<p>At O (0, 0) Z = 17.5 x 0 + 7 x 0 = 0<\/p>\n<p>At B (0, 4) Z = 17.5 x 0 + 7 x 4 = 28<\/p>\n<p>At P (3, 3) Z = 17.5 x 3 + 7 x 3 = 73.50<\/p>\n<p>At C (4, 0) Z = 17.5 x 4 + 7 x 0 = 70<\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image021.png\" \/>maximum profit = Rs. 73.50 at <img decoding=\"async\" style=\"height: 21px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image052.png\" \/><\/p>\n<p>Hence maximum profit Z = Rs. 73.50 when he produces number of packets of nuts = 3 and number of packets of bolts = 3.<\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_122-4\"><\/span><strong>NCERT Solutions class 12 Maths Exercise 12.2<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4><span class=\"ez-toc-section\" id=\"5_A_factory_manufacturers_two_types_of_screws_A_and_B_Each_type_of_screw_requires_the_use_of_two_machines_an_automatic_and_a_hand_operated_It_takes_4_minutes_on_the_automatic_and_6_minutes_on_hand_operated_machines_to_manufacture_a_package_of_screw_A_while_it_takes_6_minutes_on_automatic_and_3_minutes_on_hand_operated_machines_to_manufacture_a_package_of_screws_B_Each_machine_is_available_for_at_the_most_4_hours_on_any_day_The_manufacturer_can_sell_a_package_of_screws_A_at_a_profit_of_Rs_7_and_screws_B_at_a_profit_of_Rs_10_Assuming_that_he_can_sell_all_the_screws_he_manufactures_how_many_packages_of_each_type_should_the_factory_owner_produce_a_day_in_order_to_maximize_his_profit_Determine_the_maximum_profit\"><\/span><strong>5. A factory manufacturers two types of screws, A and B. Each type of screw requires the use of two machines, an automatic and a hand operated. It takes 4 minutes on the automatic and 6 minutes on hand operated machines to manufacture a package of screw A, while it takes 6 minutes on automatic and 3 minutes on hand operated machines to manufacture a package of screws B. Each machine is available for at the most 4 hours on any day. The manufacturer can sell a package of screws A at a profit of Rs. 7 and screws B at a profit of Rs. 10. Assuming that he can sell all the screws he manufactures, how many packages of each type should the factory owner produce a day in order to maximize his profit? Determine the maximum profit.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>Ans. <\/strong>Let the manufacturers produces <img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image001.png\" \/>\u00a0packages of screws A and <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image002.png\" \/>\u00a0packages of screws B, then time taken by <img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image001.png\" \/>\u00a0packages of screws A and <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image002.png\" \/>\u00a0packages of screws B on automatic machine = <img decoding=\"async\" style=\"height: 27px; width: 65px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image053.png\" \/>\u00a0minutes. And time taken by <img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image001.png\" \/>\u00a0packages of screws A and <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image002.png\" \/>\u00a0packages of screws B on hand operated machine = <img decoding=\"async\" style=\"height: 27px; width: 64px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image054.png\" \/>\u00a0minutes.<\/p>\n<p>Since, each machine is available for at the most 4 hours, i.e., 4 x 60 = 240 minutes.<\/p>\n<p>Therefore, we have <img decoding=\"async\" style=\"height: 21px; width: 93px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image055.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image011.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 91px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image056.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 21px; width: 92px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image057.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image011.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 79px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image058.png\" \/><\/p>\n<p>Profit on selling <img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image001.png\" \/>\u00a0packages of screws A and <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image002.png\" \/>\u00a0packages of screws B = Z = <img decoding=\"async\" style=\"height: 21px; width: 60px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image059.png\" \/><\/p>\n<p><img decoding=\"async\" id=\"Picture 18\" style=\"height: 259px; width: 257px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image060.jpg\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image021.png\" \/>\u00a0To find: <img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image001.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image002.png\" \/>\u00a0such that Z = <img decoding=\"async\" style=\"height: 21px; width: 60px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image059.png\" \/>\u00a0is maximum subject to <img decoding=\"async\" style=\"height: 21px; width: 91px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image056.png\" \/>, <img decoding=\"async\" style=\"height: 21px; width: 80px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image061.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image006.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 21px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image062.png\" \/><\/p>\n<p>Consider <img decoding=\"async\" style=\"height: 21px; width: 91px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image056.png\" \/><\/p>\n<p>Let <img decoding=\"async\" style=\"height: 21px; width: 91px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/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\/Ex12.2\/image011.png\" \/><img decoding=\"async\" style=\"height: 41px; width: 77px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image064.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image021.png\" \/>\u00a0\u00a0 A (60, 0) and B (0, 4)<\/p>\n<p>Here, (0, 0) satisfies this inequation, therefore the required hal-plane contains (0, 0).<\/p>\n<p>Again <img decoding=\"async\" style=\"height: 21px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image065.png\" \/><\/p>\n<p>Let <img decoding=\"async\" style=\"height: 21px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image066.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image011.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image067.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image021.png\" \/>\u00a0\u00a0 C (40, 0) and D (0, 80)<\/p>\n<p>Here also (0, 0) satisfies this inequation, therefore the required hal-plane contains (0, 0).<\/p>\n<p>The feasible portion of the graph satisfying the inequalities <img decoding=\"async\" style=\"height: 21px; width: 91px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image056.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 21px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image065.png\" \/>\u00a0is OABC which is shown shaded in the figure.<\/p>\n<p>Co-ordinates of O, A, B and C are (0, 0), (0, 40), (30, 20) and (40, 0) respectively.<\/p>\n<p>Now Z = <img decoding=\"async\" style=\"height: 21px; width: 60px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image059.png\" \/><\/p>\n<p>At O (0, 0) Z = 7 x 0 + 10 x 0 = 0<\/p>\n<p>At A (0, 40) Z = 7 x 0 + 10 x 40 = 400<\/p>\n<p>At B (30, 20) Z = 7 x 30 + 10 x 20 = 410<\/p>\n<p>At C (40, 0) Z = 7 x 40 + 10 x 0 = 280<\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image021.png\" \/>maximum profit Z = Rs. 410 at <img decoding=\"async\" style=\"height: 21px; width: 92px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image068.png\" \/><\/p>\n<p>Hence, if the manufacturer produces 30 screws of type A and 20 screws of type B, he earn a maximum profit of Rs. 410.<\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_122-5\"><\/span><strong>NCERT Solutions class 12 Maths Exercise 12.2<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4><span class=\"ez-toc-section\" id=\"6_A_cottage_industry_manufactures_pedestal_lamps_and_wooden_shades_each_requiring_the_use_of_a_grindingcutting_machine_and_a_sprayer_It_takes_2_hours_on_grindingcutting_machine_and_3_hours_on_the_sprayer_to_manufacture_a_pedestal_lamp_It_takes_1_hour_on_the_grindingcutting_machine_and_2_hours_on_the_sprayer_to_manufacture_a_shade_On_any_day_the_sprayer_is_available_for_at_the_most_20_hours_and_the_grindingcutting_machine_for_at_the_most_12_hours_The_profit_from_the_sale_of_a_lamp_is_Rs5_and_that_from_a_shade_is_Rs3_Assuming_that_the_manufacturer_can_sell_all_the_lamps_and_shades_that_he_produces_how_should_he_schedule_his_daily_production_in_order_to_maximize_his_profit\"><\/span><strong>6. A cottage industry manufactures pedestal lamps and wooden shades, each requiring the use of a grinding\/cutting machine and a sprayer. It takes 2 hours on grinding\/cutting machine and 3 hours on the sprayer to manufacture a pedestal lamp. It takes 1 hour on the grinding\/cutting machine and 2 hours on the sprayer to manufacture a shade. On any day, the sprayer is available for at the most 20 hours and the grinding\/cutting machine for at the most 12 hours. The profit from the sale of a lamp is Rs.5 and that from a shade is Rs.3. Assuming that the manufacturer can sell all the lamps and shades that he produces, how should he schedule his daily production in order to maximize his profit?<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>Ans. <\/strong>Let the manufacturer produces <img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image001.png\" \/>\u00a0pedestal lamps and <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image002.png\" \/>\u00a0wooden shades, then the time taken by <img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image001.png\" \/>\u00a0pedestal lamps and <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image002.png\" \/>\u00a0wooden shades on grinding\/cutting machines = <img decoding=\"async\" style=\"height: 27px; width: 57px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image069.png\" \/>\u00a0hours and the time taken by <img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image001.png\" \/>\u00a0pedestal lamps and <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image002.png\" \/>\u00a0wooden shades on the sprayer = <img decoding=\"async\" style=\"height: 27px; width: 64px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image070.png\" \/>\u00a0hours.<\/p>\n<p><img decoding=\"async\" id=\"Picture 19\" style=\"height: 255px; width: 225px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image071.jpg\" \/><\/p>\n<p>Since grinding\/cutting machine is available for at the most 12 hours, i.e., <img decoding=\"async\" style=\"height: 21px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image072.png\" \/>\u00a0and sprayer is available for at the most 20 hours, i.e., <img decoding=\"async\" style=\"height: 21px; width: 84px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image073.png\" \/><\/p>\n<p>Profit from the sale of <img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image001.png\" \/>\u00a0lamps and <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image002.png\" \/>\u00a0shades Z = <img decoding=\"async\" style=\"height: 21px; width: 52px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image074.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image021.png\" \/>\u00a0To find: <img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image001.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image002.png\" \/>\u00a0such that Z = <img decoding=\"async\" style=\"height: 21px; width: 52px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image074.png\" \/>\u00a0is maximum subject to constraints <img decoding=\"async\" style=\"height: 21px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image072.png\" \/>, <img decoding=\"async\" style=\"height: 21px; width: 84px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image073.png\" \/>, <img decoding=\"async\" style=\"height: 21px; width: 79px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image075.png\" \/><\/p>\n<p>Consider <img decoding=\"async\" style=\"height: 21px; width: 84px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image073.png\" \/><\/p>\n<p>Let <img decoding=\"async\" style=\"height: 21px; width: 84px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image076.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image011.png\" \/><img decoding=\"async\" style=\"height: 41px; width: 79px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image077.png\" \/><\/p>\n<table border=\"1\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td style=\"width: .5in;\"><\/td>\n<td style=\"width: .5in;\">A<\/td>\n<td style=\"width: .5in;\">B<\/td>\n<td style=\"width: 35.1pt;\">C<\/td>\n<\/tr>\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\/Ex12.2\/image001.png\" \/><\/td>\n<td style=\"width: .5in;\">0<\/td>\n<td style=\"width: .5in;\">2<\/td>\n<td style=\"width: 35.1pt;\">4<\/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\/Ex12.2\/image002.png\" \/><\/td>\n<td style=\"width: .5in;\">10<\/td>\n<td style=\"width: .5in;\">7<\/td>\n<td style=\"width: 35.1pt;\">4<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Now the area represented by <img decoding=\"async\" style=\"height: 21px; width: 84px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image073.png\" \/>\u00a0is the half-plane containing (0, 0) as (0, 0) satisfies the inequaiton.<\/p>\n<p>Consider <img decoding=\"async\" style=\"height: 21px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image072.png\" \/><\/p>\n<p>Let <img decoding=\"async\" style=\"height: 21px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/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\/Ex12.2\/image011.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image079.png\" \/><\/p>\n<table border=\"1\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td style=\"width: .5in;\"><\/td>\n<td style=\"width: .5in;\">D<\/td>\n<td style=\"width: .5in;\">E<\/td>\n<td style=\"width: .5in;\">C<\/td>\n<\/tr>\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\/Ex12.2\/image001.png\" \/><\/td>\n<td style=\"width: .5in;\">0<\/td>\n<td style=\"width: .5in;\">2<\/td>\n<td style=\"width: .5in;\">4<\/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\/Ex12.2\/image002.png\" \/><\/td>\n<td style=\"width: .5in;\">12<\/td>\n<td style=\"width: .5in;\">8<\/td>\n<td style=\"width: .5in;\">4<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The inequation consists of the half-plane containing (0, ) as (0, 0) satisfies this inequation.<\/p>\n<p>The double shaded region OPCAO is our solution where O (0, 0), P (6, 0), C (4, 4), A (0, 10).<\/p>\n<p>Now Z = <img decoding=\"async\" style=\"height: 21px; width: 52px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image074.png\" \/><\/p>\n<p>At O (0, 0) Z = 5 x 0 + 3 x 0 = 0<\/p>\n<p>At P (6, 0) Z = 5 x 6 + 3 x 0 = 30<\/p>\n<p>At C (4, 4) Z = 5 x 4 + 3 x 4 = 32<\/p>\n<p>At A (0, 10) Z = 5 x 0 + 3 x 10 = 30<\/p>\n<p>Now maximum Z = 32 at <img decoding=\"async\" style=\"height: 21px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image080.png\" \/><\/p>\n<p>Hence, maximum profit Z = Rs. 32 when he manufactures 4 pedestal lamps and 4 wooden shades.<\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_122-6\"><\/span><strong>NCERT Solutions class 12 Maths Exercise 12.2<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4><span class=\"ez-toc-section\" id=\"7_A_company_manufactures_two_types_of_novelty_souvenirs_made_of_plywood_Souvenirs_of_type_A_requires_5_minutes_each_for_cutting_and_10_minutes_each_for_assembling_Souvenirs_of_type_B_require_8_minutes_each_for_cutting_and_8_minutes_each_for_assembling_There_are_3_hours_20_minutes_available_for_cutting_and_4_hours_for_assembling_The_profit_is_Rs5_each_for_type_A_and_Rs6_each_for_type_B_souvenirs_How_many_souvenirs_of_each_type_should_the_company_manufacture_in_order_to_maximize_the_profit\"><\/span><strong>7. A company manufactures two types of novelty souvenirs made of plywood. Souvenirs of type A requires 5 minutes each for cutting and 10 minutes each for assembling. Souvenirs of type B require 8 minutes each for cutting and 8 minutes each for assembling. There are 3 hours 20 minutes available for cutting and 4 hours for assembling. The profit is Rs.5 each for type A and Rs.6 each for type B souvenirs. How many souvenirs of each type should the company manufacture in order to maximize the profit?<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>Ans. <\/strong>Let the company manufactures <img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image001.png\" \/>\u00a0souvenirs of type A and <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image002.png\" \/>\u00a0souvenirs of type B, then time taken for cutting <img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image001.png\" \/>\u00a0souvenirs of type A and <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image002.png\" \/>\u00a0souvenirs of type B = <img decoding=\"async\" style=\"height: 27px; width: 64px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image081.png\" \/>\u00a0minutes and time taken for assembling <img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image001.png\" \/>\u00a0souvenirs of type A and <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image002.png\" \/>\u00a0souvenirs of type B = <img decoding=\"async\" style=\"height: 27px; width: 71px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image082.png\" \/>\u00a0minutes.<\/p>\n<p><img decoding=\"async\" id=\"Picture 20\" style=\"height: 247px; width: 271px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image083.jpg\" \/><\/p>\n<p>Since 3 hours 29 minutes i.e., 200 minutes are available for cutting, therefore we have <img decoding=\"async\" style=\"height: 21px; width: 95px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image084.png\" \/><\/p>\n<p>Also since 4 hours i.e., 240 minutes are available for assembling, therefore we have <img decoding=\"async\" style=\"height: 21px; width: 99px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image085.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image011.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 91px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image086.png\" \/><\/p>\n<p>Thus, our L.P.P. is to maximize profit Z = <img decoding=\"async\" style=\"height: 21px; width: 52px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image087.png\" \/>\u00a0subject to constraints <img decoding=\"async\" style=\"height: 21px; width: 91px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image088.png\" \/>, <img decoding=\"async\" style=\"height: 21px; width: 91px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image086.png\" \/>, <img decoding=\"async\" style=\"height: 21px; width: 79px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image075.png\" \/><\/p>\n<p>Now <img decoding=\"async\" style=\"height: 21px; width: 91px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image088.png\" \/><\/p>\n<p>Let <img decoding=\"async\" style=\"height: 21px; width: 92px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image089.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image011.png\" \/><img decoding=\"async\" style=\"height: 41px; width: 77px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image090.png\" \/><\/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\/Ex12.2\/image001.png\" \/><\/td>\n<td style=\"width: .5in;\">40<\/td>\n<td style=\"width: .5in;\">0<\/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\/Ex12.2\/image002.png\" \/><\/td>\n<td style=\"width: .5in;\">0<\/td>\n<td style=\"width: .5in;\">25<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Again <img decoding=\"async\" style=\"height: 21px; width: 91px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image086.png\" \/><\/p>\n<p>Let <img decoding=\"async\" style=\"height: 21px; width: 91px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image091.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image011.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 77px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image092.png\" \/><\/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\/Ex12.2\/image001.png\" \/><\/td>\n<td style=\"width: .5in;\">24<\/td>\n<td style=\"width: .5in;\">20<\/td>\n<td style=\"width: .5in;\">0<\/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\/Ex12.2\/image002.png\" \/><\/td>\n<td style=\"width: .5in;\">0<\/td>\n<td style=\"width: .5in;\">5<\/td>\n<td style=\"width: .5in;\">30<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>In both the equations origin (0, 0) satisfies them and therefore the required half planes are there which contains (0, 0).<\/p>\n<p>The portion of graph satisfying the inequalities <img decoding=\"async\" style=\"height: 21px; width: 91px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image088.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 21px; width: 91px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image086.png\" \/>\u00a0is OABC and is shown in shaded in the figure. Coordinates of the points O, A, B and C are (0, 0), (24, 0), (8, 20) and (0, 25) respectively.<\/p>\n<p>Now Z = <img decoding=\"async\" style=\"height: 21px; width: 52px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image087.png\" \/><\/p>\n<p>At O (0, 0) Z = 5 x 0 + 6 x 0 = 0<\/p>\n<p>At A (24, 0) Z = 5 x 24 + 6 x 0 = 120<\/p>\n<p>At B (8, 20) Z = 5 x 8 + 6 x 20 = 160<\/p>\n<p>At C (0, 25) Z = 5 x 0 + 6 x 25 = 150<\/p>\n<p>Now maximum Z = 160 at <img decoding=\"async\" style=\"height: 21px; width: 83px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image093.png\" \/><\/p>\n<p>Hence, maximum profit = Rs. 160 when he manufactures 8 souvenirs of type A and 20 souvenirs of type B.<\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_122-7\"><\/span><strong>NCERT Solutions class 12 Maths Exercise 12.2<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4><span class=\"ez-toc-section\" id=\"8_A_merchant_plans_to_sell_two_types_of_personal_computers_%E2%80%93_a_desktop_model_and_a_portable_model_that_will_cost_Rs_25000_and_Rs_40000_respectively_He_estimates_that_the_total_monthly_demand_of_computers_will_not_exceed_250_units_Determine_the_number_of_units_of_each_type_of_computers_which_the_merchant_should_stock_to_get_maximum_profit_if_he_does_not_want_to_invest_more_than_Rs_70_lakhs_and_if_his_profit_on_the_desktop_model_is_Rs_4500_and_on_portable_model_is_Rs_5000\"><\/span><strong>8. A merchant plans to sell two types of personal computers \u2013 a desktop model and a portable model that will cost Rs. 25000 and Rs. 40000 respectively. He estimates that the total monthly demand of computers will not exceed 250 units. Determine the number of units of each type of computers which the merchant should stock to get maximum profit if he does not want to invest more than Rs, 70 lakhs and if his profit on the desktop model is Rs. 4500 and on portable model is Rs. 5000.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>Ans. <\/strong>Let number of desktop model computer = <img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image001.png\" \/>\u00a0and number of a portable model computer = <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image002.png\" \/><\/p>\n<p>To maximum profit Z = <img decoding=\"async\" style=\"height: 21px; width: 101px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image094.png\" \/>\u00a0subject to <img decoding=\"async\" style=\"height: 21px; width: 200px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image095.png\" \/>, <img decoding=\"async\" style=\"height: 21px; width: 80px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image096.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image025.png\" \/><\/p>\n<p><img decoding=\"async\" id=\"Picture 21\" style=\"height: 236px; width: 240px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image097.jpg\" \/><\/p>\n<p>Consider <img decoding=\"async\" style=\"height: 21px; width: 200px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image095.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image011.png\" \/><img decoding=\"async\" style=\"height: 21px; width: 97px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image098.png\" \/><\/p>\n<p>Let <img decoding=\"async\" style=\"height: 21px; width: 97px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image099.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image011.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 91px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image100.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image021.png\" \/>\u00a0Points are A (280, 0), B (0, 175)<\/p>\n<p>Now (0, 0) satisfies the inequation, therefore the required half-plane contains (0, 0).<\/p>\n<p>Again consider <img decoding=\"async\" style=\"height: 21px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image101.png\" \/><\/p>\n<p>Let <img decoding=\"async\" style=\"height: 21px; width: 77px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image102.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image011.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 93px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image103.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image021.png\" \/>\u00a0Points are C (250, 0), D (0, 250).<\/p>\n<p>Again (0, 0) satisfies the inequation, therefore the required half-plane contains (0, 0).<\/p>\n<p>The double shaded region OCPBO is the solution set. Its corners are O (0, 0), C (280, 0), P (200, 50), B (0, 200).<\/p>\n<p>Now Z = <img decoding=\"async\" style=\"height: 21px; width: 101px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image094.png\" \/><\/p>\n<p>AtO (0, 0) Z = 4500 x 0 + 5000 x 0 = 0<\/p>\n<p>At C (280, 0) Z = 4500 x 280 + 5000 x 0 = 12,60,000<\/p>\n<p>(Rejected as <img decoding=\"async\" style=\"height: 19px; width: 53px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image104.png\" \/>)<\/p>\n<p>At P (200, 50) Z = 4500 x 200 + 5000 x 50 = 11,50,000<\/p>\n<p>At B (0, 200) Z = 4500 x 0 + 5000 x 200 = 10,00,000<\/p>\n<p>Now maximum Z = Rs. 11,50,000 at <img decoding=\"async\" style=\"height: 21px; width: 100px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image105.png\" \/><\/p>\n<p>Hence, maximum profit = Rs. 11,50,000 when he sells 200 desktop models and 50 portable models.<\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_122-8\"><\/span><strong>NCERT Solutions class 12 Maths Exercise 12.2<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4><span class=\"ez-toc-section\" id=\"9_A_diet_is_to_contain_at_least_80_units_of_vitamin_A_and_100_units_of_minerals_Two_foods_F1_and_F2_are_available_Food_F1_costs_Rs_4_per_unit_food_and_F2_costs_Rs_6_per_unit_One_unit_of_food_F1_contains_3_units_of_vitamin_A_and_4_units_of_minerals_One_unit_of_food_F2_contains_6_units_of_vitamin_A_and_3_units_of_minerals_Formulate_this_as_a_linear_programming_problem_Find_the_minimum_cost_for_diet_that_consists_of_mixture_of_these_two_foods_and_also_meets_the_minimal_nutritional_requirements\"><\/span><strong>9. A diet is to contain at least 80 units of vitamin A and 100 units of minerals. Two foods F<sub>1<\/sub> and F<sub>2<\/sub> are available. Food F<sub>1<\/sub> costs Rs. 4 per unit food and F<sub>2<\/sub> costs Rs. 6 per unit. One unit of food F<sub>1<\/sub> contains 3 units of vitamin A and 4 units of minerals. One unit of food F<sub>2<\/sub> contains 6 units of vitamin A and 3 units of minerals. Formulate this as a linear programming problem. Find the<\/strong> <strong>minimum cost for diet that consists of mixture of these two foods and also meets the minimal nutritional requirements.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>Ans. <\/strong>Let the number of units of food F<sub>1<\/sub> = <img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image001.png\" \/>\u00a0and number of units of food F<sub>2<\/sub> = <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image002.png\" \/><\/p>\n<p>To minimize Z = <img decoding=\"async\" style=\"height: 21px; width: 53px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image106.png\" \/>\u00a0subject to <img decoding=\"async\" style=\"height: 21px; width: 83px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image107.png\" \/>, <img decoding=\"async\" style=\"height: 21px; width: 91px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image108.png\" \/>\u00a0, <img decoding=\"async\" style=\"height: 21px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image025.png\" \/><\/p>\n<p>Consider <img decoding=\"async\" style=\"height: 21px; width: 83px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image107.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\/Ex12.2\/image109.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image011.png\" \/><img decoding=\"async\" style=\"height: 41px; width: 77px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image110.png\" \/><\/p>\n<table border=\"1\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td style=\"width: .5in;\"><\/td>\n<td style=\"width: .5in;\">A<\/td>\n<td style=\"width: .5in;\">B<\/td>\n<td style=\"width: .5in;\">C<\/td>\n<\/tr>\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\/Ex12.2\/image001.png\" \/><\/td>\n<td style=\"width: .5in;\">0<\/td>\n<td style=\"width: .5in;\">5<\/td>\n<td style=\"width: .5in;\">10<\/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\/Ex12.2\/image002.png\" \/><\/td>\n<td style=\"width: .5in;\"><img decoding=\"async\" style=\"height: 41px; width: 24px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image111.png\" \/><\/td>\n<td style=\"width: .5in;\"><img decoding=\"async\" style=\"height: 41px; width: 23px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image112.png\" \/><\/td>\n<td style=\"width: .5in;\"><img decoding=\"async\" style=\"height: 41px; width: 23px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image113.png\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Clearly, (0, 0) is not included in the half-plane.<\/p>\n<p>Now again consider <img decoding=\"async\" style=\"height: 21px; width: 91px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image108.png\" \/><\/p>\n<p>Let\u00a0 <img decoding=\"async\" style=\"height: 21px; width: 91px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image114.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image011.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 84px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image115.png\" \/><\/p>\n<table border=\"1\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td style=\"width: .5in;\"><\/td>\n<td style=\"width: .5in;\">D<\/td>\n<td style=\"width: .5in;\">E<\/td>\n<td style=\"width: .5in;\">F<\/td>\n<\/tr>\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\/Ex12.2\/image001.png\" \/><\/td>\n<td style=\"width: .5in;\">25<\/td>\n<td style=\"width: .5in;\">22<\/td>\n<td style=\"width: .5in;\">10<\/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\/Ex12.2\/image002.png\" \/><\/td>\n<td style=\"width: .5in;\">0<\/td>\n<td style=\"width: .5in;\">4<\/td>\n<td style=\"width: .5in;\">20<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Here, also (0, 0) is not included in the required half-plane.<\/p>\n<p>The double shaded region is our feasible region and its corners are<\/p>\n<p>NCERT Solutions class 12 Maths Exercise 12.2<\/p>\n<p><img decoding=\"async\" id=\"Picture 22\" style=\"height: 250px; width: 259px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image116.jpg\" \/><\/p>\n<p>A<img decoding=\"async\" style=\"height: 45px; width: 59px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image117.png\" \/>\u00a0P<img decoding=\"async\" style=\"height: 45px; width: 53px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image118.png\" \/>\u00a0and Q<img decoding=\"async\" style=\"height: 45px; width: 59px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image119.png\" \/><\/p>\n<p>Now Z = <img decoding=\"async\" style=\"height: 21px; width: 53px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image106.png\" \/><\/p>\n<p>At A<img decoding=\"async\" style=\"height: 45px; width: 53px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image120.png\" \/>\u00a0Z = 4 x <img decoding=\"async\" style=\"height: 41px; width: 23px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image121.png\" \/>\u00a0+ 6 x 0 = 106.67<\/p>\n<p>At P<img decoding=\"async\" style=\"height: 45px; width: 53px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image118.png\" \/>\u00a0Z = 4 x 24 + 6 x <img decoding=\"async\" style=\"height: 41px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image122.png\" \/>\u00a0= 104<\/p>\n<p>At Q<img decoding=\"async\" style=\"height: 45px; width: 59px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image119.png\" \/>\u00a0Z = 4 x 0 + 6 x <img decoding=\"async\" style=\"height: 41px; width: 29px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image123.png\" \/>\u00a0= 200<\/p>\n<p>Hence Z = Rs. 104 is minimum at <img decoding=\"async\" style=\"height: 41px; width: 87px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image124.png\" \/>.<\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_122-9\"><\/span><strong>NCERT Solutions class 12 Maths Exercise 12.2<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4><span class=\"ez-toc-section\" id=\"10_There_are_two_types_of_fertilizers_F1_and_F2_F1_consists_of_10_nitrogen_and_6_phosphoric_acid_and_F2_consists_of_5_nitrogen_and_10_phosphoric_acid_After_testing_the_soil_conditions_a_farmer_finds_that_she_needs_at_least_14_kg_of_nitrogen_and_14_kg_of_phosphoric_acid_for_her_crop_If_F1_costs_Rs_6kg_and_F2_costs_Rs_5kg_determine_how_much_of_each_type_of_fertilizer_should_be_used_so_that_nutrient_requirements_are_met_at_a_minimum_cost_What_is_the_minimum_cost\"><\/span><strong>10. There are two types of fertilizers F<sub>1<\/sub> and F<sub>2<\/sub>. F<sub>1<\/sub> consists of 10% nitrogen and 6% phosphoric acid and F<sub>2<\/sub> consists of 5% nitrogen and 10% phosphoric acid. After testing the soil conditions, a farmer finds that she needs at least 14 kg of nitrogen and 14 kg of phosphoric acid for her crop. If F<sub>1<\/sub> costs Rs. 6\/kg and F<sub>2<\/sub> costs Rs. 5\/kg, determine how much of each type of fertilizer should be used so that nutrient requirements are met at a minimum cost. What is the minimum cost?<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>Ans. <\/strong>Let the quantity of F<sub>1<\/sub> = <img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image001.png\" \/>\u00a0kg and the quantity of F<sub>2<\/sub> = <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image002.png\" \/>\u00a0kg<\/p>\n<p>We have to minimize Z = <img decoding=\"async\" style=\"height: 21px; width: 52px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image125.png\" \/>\u00a0\u00a0subject to <img decoding=\"async\" style=\"height: 41px; width: 99px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image126.png\" \/>, <img decoding=\"async\" style=\"height: 41px; width: 100px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image127.png\" \/><img decoding=\"async\" style=\"height: 21px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image025.png\" \/><\/p>\n<p>Consider <img decoding=\"async\" style=\"height: 41px; width: 99px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image126.png\" \/><\/p>\n<p><img decoding=\"async\" id=\"Picture 23\" style=\"height: 228px; width: 226px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image128.jpg\" \/><\/p>\n<p>Let\u00a0 <img decoding=\"async\" style=\"height: 41px; width: 100px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image129.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image011.png\" \/><img decoding=\"async\" style=\"height: 21px; width: 105px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image130.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image011.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 85px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image131.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image011.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 85px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image132.png\" \/><\/p>\n<table border=\"1\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td style=\"width: .5in;\"><\/td>\n<td style=\"width: .5in;\">A<\/td>\n<td style=\"width: .5in;\">B<\/td>\n<td style=\"width: .5in;\">C<\/td>\n<td style=\"width: .5in;\">D<\/td>\n<\/tr>\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\/Ex12.2\/image001.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;\">140<\/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\/Ex12.2\/image002.png\" \/><\/td>\n<td style=\"width: .5in;\">280<\/td>\n<td style=\"width: .5in;\">180<\/td>\n<td style=\"width: .5in;\">80<\/td>\n<td style=\"width: .5in;\">0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Here, (0, 0) is not included in the required half plane and (0, 0) does not satisfy this inequation.<\/p>\n<p>NCERT Solutions class 12 Maths Exercise 12.2<\/p>\n<p>Again consider <img decoding=\"async\" style=\"height: 41px; width: 100px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image127.png\" \/><\/p>\n<p>Let\u00a0 <img decoding=\"async\" style=\"height: 41px; width: 100px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image133.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image011.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 105px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image134.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image011.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 91px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image135.png\" \/>\u00a0\u00a0<img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image011.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 41px; width: 85px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image136.png\" \/><\/p>\n<table border=\"1\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td style=\"width: .5in;\"><\/td>\n<td style=\"width: .5in;\">E<\/td>\n<td style=\"width: .5in;\">C<\/td>\n<td style=\"width: .5in;\">G<\/td>\n<\/tr>\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\/Ex12.2\/image001.png\" \/><\/td>\n<td style=\"width: .5in;\">0<\/td>\n<td style=\"width: .5in;\">100<\/td>\n<td style=\"width: .5in;\">200<\/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\/Ex12.2\/image002.png\" \/><\/td>\n<td style=\"width: .5in;\">140<\/td>\n<td style=\"width: .5in;\">80<\/td>\n<td style=\"width: .5in;\">20<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Again Here, (0, 0) is not included in the required half plane and (0, 0) does not satisfy this inequation.<\/p>\n<p>NCERT Solutions class 12 Maths Exercise 12.2<\/p>\n<p>The shaded region XPCAY is our feasible region. Its corners are P<img decoding=\"async\" style=\"height: 45px; width: 63px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image137.png\" \/>, C (100, 80) and A (0, 280).<\/p>\n<p>Now Z = <img decoding=\"async\" style=\"height: 21px; width: 52px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image125.png\" \/><\/p>\n<p>At P<img decoding=\"async\" style=\"height: 45px; width: 63px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image137.png\" \/>\u00a0 Z = 6 x <img decoding=\"async\" style=\"height: 41px; width: 31px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image138.png\" \/>\u00a0+ 5 x 0 = 1400<\/p>\n<p>At C (100, 80) Z = 6 x 100 + 5 x 80 = 1000<\/p>\n<p>At A (0, 280) Z = 6 x 0 + 5 x 280 = 1400<\/p>\n<p>Now minimum cost Z = Rs. 1000 at <img decoding=\"async\" style=\"height: 21px; width: 101px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image139.png\" \/><\/p>\n<hr \/>\n<p style=\"text-align: center;\"><strong>NCERT Solutions class 12 Maths Exercise 12.2<\/strong><\/p>\n<h3><span class=\"ez-toc-section\" id=\"11_The_corner_points_of_the_feasible_region_determined_by_the_following_system_of_linear_inequalities\"><\/span><strong>11. The corner points of the feasible region determined by the following system of linear inequalities:<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong><img decoding=\"async\" style=\"height: 21px; width: 213px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image140.png\" \/>\u00a0are (0, 0), (5, 0), (3, 4) and (0, 5). Let <img decoding=\"async\" style=\"height: 21px; width: 81px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image141.png\" \/>\u00a0where <img decoding=\"async\" style=\"height: 21px; width: 57px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image142.png\" \/>\u00a0Condition on <img decoding=\"async\" style=\"height: 17px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image143.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 17px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image144.png\" \/>\u00a0so that the maximum of Z occurs at both (3, 4) and (0, 5) is:<\/strong><\/p>\n<p><strong>(A) <img decoding=\"async\" style=\"height: 17px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image145.png\" \/>\u00a0<\/strong><\/p>\n<p><strong>(B) <img decoding=\"async\" style=\"height: 21px; width: 48px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image146.png\" \/>\u00a0<\/strong><\/p>\n<p><strong>(C) <img decoding=\"async\" style=\"height: 21px; width: 47px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image147.png\" \/>\u00a0<\/strong><\/p>\n<p><strong>(D) <img decoding=\"async\" style=\"height: 21px; width: 47px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image148.png\" \/>\u00a0<\/strong><\/p>\n<p><strong>Ans. <\/strong>To maximize Z = <img decoding=\"async\" style=\"height: 21px; width: 125px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image149.png\" \/>\u00a0subject to <img decoding=\"async\" style=\"height: 21px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image150.png\" \/>, <img decoding=\"async\" style=\"height: 21px; width: 149px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image151.png\" \/><\/p>\n<p>The corner points are (0, 0), (5, 0), (3, 4) and (0, 5).<\/p>\n<p>Maximum of Z occurs at both (3, 4) and (0, 5)<\/p>\n<p>Z = <img decoding=\"async\" style=\"height: 19px; width: 52px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image152.png\" \/><\/p>\n<p>At (3, 4) Z = <img decoding=\"async\" style=\"height: 21px; width: 53px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image153.png\" \/><\/p>\n<p>At (0, 5) Z = <img decoding=\"async\" style=\"height: 21px; width: 21px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image154.png\" \/><\/p>\n<p>Now maxima occurs at both points,<\/p>\n<p><img decoding=\"async\" style=\"height: 21px; width: 85px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image155.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image011.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 47px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image156.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image011.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 47px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex12.2\/image148.png\" \/><\/p>\n<p>Hence option (D) is correct.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_122-10\"><\/span>NCERT Solutions class 12 Maths Exercise 12.2<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>NCERT Solutions class 12 Maths Exercise 12.2 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 12.2 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 12.2\" class=\"read-more\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-12-2\/\" aria-label=\"More on NCERT Solutions class 12 Maths Exercise 12.2\">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-4507","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mathematics-cbse-class-12","category-ncert-solutions","tag-cbse-study-material","tag-class-12","tag-mathematics","tag-ncert-solutions"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.0 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>NCERT Solutions class 12 Maths Exercise 12.2 | myCBSEguide<\/title>\n<meta name=\"description\" content=\"NCERT Solutions class 12 Maths Exercise 12.2 in PDF format for free download. 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