{"id":4508,"date":"2016-05-16T11:49:00","date_gmt":"2016-05-16T06:19:00","guid":{"rendered":"http:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-6\/"},"modified":"2018-05-08T09:29:39","modified_gmt":"2018-05-08T03:59:39","slug":"ncert-solutions-class-12-maths-miscellaneous-6","status":"publish","type":"post","link":"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-6\/","title":{"rendered":"NCERT Solutions class 12 Maths Miscellaneous"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_76 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 eztoc-toggle-hide-by-default' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-6\/#NCERT_Solutions_class_12_Maths_Linear_Programming\" >NCERT Solutions class 12 Maths Linear Programming<\/a><ul class='ez-toc-list-level-4' ><li class='ez-toc-heading-level-4'><ul class='ez-toc-list-level-4' ><li class='ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-6\/#NCERT_Solutions_class_12_Maths_Miscellaneous\" >NCERT Solutions class 12 Maths Miscellaneous<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-6\/#2_A_farmer_mixes_two_brands_P_and_Q_of_cattle_feed_Brand_P_costing_Rs_250_per_bag_contains_3_units_of_nutritional_element_A_25_units_of_element_B_and_2_units_of_element_C_Brand_Q_costing_Rs_200_per_bag_contains_15_units_of_nutritional_element_A_1125_units_of_element_B_and_3_units_of_element_C_The_minimum_requirements_of_nutrients_A_B_and_C_are_18_units_45_units_and_24_units_respectively_Determine_the_number_of_bags_of_each_brand_which_should_be_mixed_in_order_to_produce_a_mixture_having_a_minimum_cost_per_bag_What_is_the_minimum_cost_of_the_mixture_per_bag\" >2. A farmer mixes two brands P and Q of cattle feed. Brand P, costing Rs. 250 per bag, contains 3 units of nutritional element A, 2.5 units of element B and 2 units of element C. Brand Q costing Rs. 200 per bag contains 1.5 units of nutritional element A, 11.25 units of element B and 3 units of element C. The minimum requirements of nutrients A, B and C are 18 units, 45 units and 24 units respectively. Determine the number of bags of each brand which should be mixed in order to produce a mixture having a minimum cost per bag? What is the minimum cost of the mixture per bag?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-6\/#NCERT_Solutions_class_12_Maths_Miscellaneous-2\" >NCERT Solutions class 12 Maths Miscellaneous<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-6\/#3_A_dietician_wishes_to_mix_together_two_kinds_of_food_X_and_Y_in_such_a_way_that_the_mixture_contains_at_least_10_units_of_vitamin_A_12_units_of_vitamin_B_and_8_units_of_vitamin_C_The_vitamin_contents_of_one_kg_food_is_given_below\" >3. A dietician wishes to mix together two kinds of food X and Y in such a way that the mixture contains at least 10 units of vitamin A, 12 units of vitamin B and 8 units of vitamin C. The vitamin contents of one kg food is given below:<\/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-miscellaneous-6\/#NCERT_Solutions_class_12_Maths_Miscellaneous-3\" >NCERT Solutions class 12 Maths Miscellaneous<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-6\/#4_A_manufacturer_makes_two_types_of_toys_A_and_B_Three_machines_are_needed_for_this_purpose_and_the_time_in_minutes_requires_for_each_toy_on_the_machines_is_given_below\" >4. A manufacturer makes two types of toys A and B. Three machines are needed for this purpose and the time (in minutes) requires for each toy on the machines is given below:<\/a><ul class='ez-toc-list-level-5' ><li class='ez-toc-heading-level-5'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-6\/#now_the_profit_Z_earned_by_the_manufacturer_to_produce_units_of_type_A_and_units_of_type_B_is\" >now, the profit Z earned by the manufacturer to produce \u00a0units of type A and \u00a0units of type B is<\/a><\/li><\/ul><\/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-miscellaneous-6\/#NCERT_Solutions_class_12_Maths_Miscellaneous-4\" >NCERT Solutions class 12 Maths Miscellaneous<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-6\/#5_An_aeroplane_carries_a_maximum_of_200_passengers_A_profit_of_Rs_1000_is_made_on_each_executive_class_ticket_and_a_profit_of_Rs_600_is_made_on_each_economy_class_ticket_The_airline_reserves_at_least_20_seats_for_executive_class_than_by_the_executive_class_Determine_how_many_tickets_of_each_type_must_be_sold_in_order_to_maximize_the_profit_for_the_airline_What_is_the_maximum_profit\" >5. An aeroplane carries a maximum of 200 passengers. A profit of Rs. 1000 is made on each executive class ticket and a profit of Rs. 600 is made on each economy class ticket. The airline reserves at least 20 seats for executive class than by the executive class. Determine how many tickets of each type must be sold in order to maximize the profit for the airline. What is the maximum profit?<\/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-miscellaneous-6\/#_Points_A_200_0_and_B0_200_are_on_the_line_and_therefore_0_0_is_included_in_the_required_half_plane\" >\u00a0Points A (200, 0) and B(0, 200) are on the line and therefore (0, 0) is included in the required half plane.<\/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-miscellaneous-6\/#NCERT_Solutions_class_12_Maths_Miscellaneous-5\" >NCERT Solutions class 12 Maths Miscellaneous<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-13\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-6\/#6_Two_godowns_A_and_B_have_grain_capacity_of_100_quintals_and_50_quintals_respectively_They_supply_to_3_ration_shops_D_E_and_F_whose_requirements_are_60_50_and_40_quintals_respectively_The_cost_of_transportation_per_quintal_from_the_godowns_to_the_shops_are_given_in_the_following_table\" >6. Two godowns A and B have grain capacity of 100 quintals and 50 quintals respectively. They supply to 3 ration shops. D, E and F whose requirements are 60, 50 and 40 quintals respectively. The cost of transportation per quintal from the godowns to the shops are given in the following table:<\/a><ul class='ez-toc-list-level-5' ><li class='ez-toc-heading-level-5'><a class=\"ez-toc-link ez-toc-heading-14\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-6\/#Subject_to_and\" >Subject to , , \u00a0and ,<\/a><\/li><\/ul><\/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-miscellaneous-6\/#NCERT_Solutions_class_12_Maths_Miscellaneous-6\" >NCERT Solutions class 12 Maths Miscellaneous<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-16\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-6\/#7_An_oil_company_has_two_depots_A_and_B_with_capacities_of_7000_L_and_4000_L_respectively_The_company_is_to_supply_oil_to_three_petrol_pumps_D_E_and_F_whose_requirements_are_4500_L_3000_L_and_3500_L_respectively_The_distances_in_km_between_the_depots_and_the_petrol_pumps_is_given_in_the_following_table\" >7. An oil company has two depots A and B with capacities of 7000 L and 4000 L respectively. The company is to supply oil to three petrol pumps. D, E and F whose requirements are 4500 L, 3000 L and 3500 L respectively. The distances (in km) between the depots and the petrol pumps is given in the following table:<\/a><ul class='ez-toc-list-level-5' ><li class='ez-toc-heading-level-5'><a class=\"ez-toc-link ez-toc-heading-17\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-6\/#Assuming_that_the_transportation_cost_of_10_liters_of_oil_is_Re_1_per_km_how_should_the_delivery_be_scheduled_in_order_that_the_transportation_cost_is_minimum_What_is_the_minimum_cost\" >Assuming that the transportation cost of 10 liters of oil is Re. 1 per km, how should the delivery be scheduled in order that the transportation cost is minimum? What is the minimum cost?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-5'><a class=\"ez-toc-link ez-toc-heading-18\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-6\/#Since_requirements_of_oil_at_petrol_pump_D_E_and_F_are_and_liters_respectively\" >Since requirements of oil at petrol pump, D, E and F are \u00a0\u00a0and \u00a0liters respectively.<\/a><\/li><\/ul><\/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-miscellaneous-6\/#NCERT_Solutions_class_12_Maths_Miscellaneous-7\" >NCERT Solutions class 12 Maths Miscellaneous<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-20\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-6\/#8_A_fruit_grower_can_use_two_types_of_fertilizer_in_his_garden_brand_P_and_brand_Q_The_amounts_in_kg_of_nitrogen_phosphoric_acid_potash_and_chlorine_in_a_bag_of_each_brand_are_given_in_the_table_Tests_indicate_that_the_garden_needs_at_least_240_kg_of_phosphoric_acid_at_least_270_kg_of_potash_and_at_most_310_kg_of_chlorine\" >8. A fruit grower can use two types of fertilizer in his garden, brand P and brand Q. The amounts (in kg) of nitrogen, phosphoric acid, potash and chlorine in a bag of each brand are given in the table. Tests indicate that the garden needs at least 240 kg of phosphoric acid, at least 270 kg of potash and at most 310 kg of chlorine.<\/a><ul class='ez-toc-list-level-5' ><li class='ez-toc-heading-level-5'><a class=\"ez-toc-link ez-toc-heading-21\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-6\/#And_0_0_does_not_lie_on_the_required_half-plane_of_this_in_equation\" >And (0, 0) does not lie on the required half-plane of this in equation.<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-22\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-6\/#NCERT_Solutions_class_12_Maths_Miscellaneous-8\" >NCERT Solutions class 12 Maths Miscellaneous<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-23\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-6\/#9_Refer_to_Question_8_If_the_grower_wants_to_maximize_the_amount_of_nitrogen_added_to_the_garden_how_many_bags_of_each_brand_should_be_added_What_is_the_maximum_amount_of_nitrogen_added\" >9. Refer to Question 8. If the grower wants to maximize the amount of nitrogen added to the garden, how many bags of each brand should be added? What is the maximum amount of nitrogen added?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-24\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-6\/#NCERT_Solutions_class_12_Maths_Miscellaneous-9\" >NCERT Solutions class 12 Maths Miscellaneous<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-25\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-6\/#10_A_toy_company_manufactures_two_types_of_dolls_A_and_B_Market_tests_and_available_resources_have_indicated_that_the_combined_production_level_should_not_exceed_1200_dolls_per_week_and_the_demand_for_dolls_of_type_B_is_at_most_half_of_that_for_dolls_of_type_A_Further_the_production_level_of_dolls_of_type_A_can_exceed_three_times_the_production_of_dolls_of_other_type_by_at_most_600_units_If_the_company_makes_profit_of_Rs12_and_Rs_16_per_doll_respectively_on_dolls_A_and_B_how_many_of_each_should_be_produced_weekly_in_order_to_maximize_the_profit\" >10. A toy company manufactures two types of dolls, A and B. Market tests and available resources have indicated that the combined production level should not exceed 1200 dolls per week and the demand for dolls of type B is at most half of that for dolls of type A. Further, the production level of dolls of type A can exceed three times the production of dolls of other type by at most 600 units. If the company makes profit of Rs.12 and Rs. 16 per doll respectively on dolls A and B, how many of each should be produced weekly in order to maximize the profit?<\/a><ul class='ez-toc-list-level-5' ><li class='ez-toc-heading-level-5'><a class=\"ez-toc-link ez-toc-heading-26\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-6\/#Here_100_0_satisfies_the_inequation_therefore_the_required_half_plane_includes_100_0\" >Here, (100, 0) satisfies the inequation , therefore the required half plane includes (100, 0).<\/a><\/li><\/ul><\/li><\/ul><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-27\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-6\/#NCERT_Solutions_class_12_Maths_Miscellaneous-10\" >NCERT Solutions class 12 Maths Miscellaneous<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-28\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-6\/#CBSE_App_for_Students\" >CBSE App for Students<\/a><\/li><\/ul><\/nav><\/div>\n<p>NCERT Solutions class 12 Maths Miscellaneous Class 12 Maths book solutions are available in PDF format for free download. These ncert book chapter wise questions and answers are very helpful for CBSE board exam. CBSE recommends NCERT books and most of the questions in CBSE exam are asked from NCERT text books. Class 12 Maths chapter wise NCERT solution for Maths part 1 and Maths part 2 for all the chapters can be downloaded from our website and myCBSEguide mobile app for free.<\/p>\n<p><strong>Download\u00a0<a href=\"https:\/\/mycbseguide.com\/downloads\/cbse-class-12-mathematics-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 Miscellaneous\" width=\"192\" height=\"246\" \/><\/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>. Refer to Example 9. How many packets of each food should be used to maximize the amount of vitamin A in the diet? What is the maximum amount of vitamin A in the diet?<\/strong><\/p>\n<p><strong>Ans. <\/strong>Let <img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image001.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image002.png\" \/>\u00a0be the number of packets of food P and Q respectively, <img decoding=\"async\" style=\"height: 21px; width: 79px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image003.png\" \/>\u00a0&#8211;<\/p>\n<p>We have to maximize Z = <img decoding=\"async\" style=\"height: 21px; width: 52px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image004.png\" \/>\u00a0(vitamin A) subject to the constraints <img decoding=\"async\" style=\"height: 21px; width: 99px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image005.png\" \/>\u00a0(constraints on Calcium), i.e., <img decoding=\"async\" style=\"height: 21px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image006.png\" \/>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u2026\u2026\u2026.(i)<\/p>\n<p>And <img decoding=\"async\" style=\"height: 21px; width: 100px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image007.png\" \/>\u00a0(constraints on Iron), i.e., <img decoding=\"async\" style=\"height: 21px; width: 81px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image008.png\" \/>\u00a0\u00a0\u00a0\u2026\u2026\u2026.(ii)<\/p>\n<p>Also <img decoding=\"async\" style=\"height: 21px; width: 92px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image009.png\" \/>\u00a0(constraints on Cholesterol), i.e., <img decoding=\"async\" style=\"height: 21px; width: 91px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image010.png\" \/>\u00a0\u00a0\u2026\u2026\u2026.(iii)<\/p>\n<p><img decoding=\"async\" style=\"height: 21px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image011.png\" \/>\u00a0\u00a0\u00a0\u2026\u2026\u2026.(iv)<\/p>\n<p><img decoding=\"async\" id=\"Picture 25\" style=\"height: 275px; width: 260px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image012.jpg\" \/><\/p>\n<p>Consider <img decoding=\"async\" style=\"height: 21px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image006.png\" \/><\/p>\n<p>Let\u00a0 <img decoding=\"async\" style=\"height: 21px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image013.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image014.png\" \/>\u00a0\u00a0 <img decoding=\"async\" style=\"height: 21px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image015.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\/12%20Misc\/image001.png\" \/><\/td>\n<td style=\"width: .5in;\">0<\/td>\n<td style=\"width: .5in;\">10<\/td>\n<td style=\"width: .5in;\">20<\/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\/12%20Misc\/image002.png\" \/><\/td>\n<td style=\"width: .5in;\">80<\/td>\n<td style=\"width: .5in;\">40<\/td>\n<td style=\"width: .5in;\">0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Here, (0, 0) does not satisfy this inequation, therefore the required half plane does not include the point (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\/12%20Misc\/image008.png\" \/><\/p>\n<p>Let <img decoding=\"async\" style=\"height: 21px; width: 83px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image016.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image014.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 81px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image017.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\/12%20Misc\/image001.png\" \/><\/td>\n<td style=\"width: .5in;\">115<\/td>\n<td style=\"width: .5in;\">65<\/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\/12%20Misc\/image002.png\" \/><\/td>\n<td style=\"width: .5in;\">0<\/td>\n<td style=\"width: .5in;\">10<\/td>\n<td style=\"width: .5in;\">23<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Here, also (0, 0) does not satisfy this inequation, therefore the required half plane does not include the point (0, 0)<\/p>\n<p>Again consider <img decoding=\"async\" style=\"height: 21px; width: 91px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image010.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\/12%20Misc\/image018.png\" \/>\u00a0\u00a0<img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image014.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image019.png\" \/><\/p>\n<p>Therefore, G (50, 0) and H (0, 75) satisfy the equation.<\/p>\n<p>As (0, 0) satisfies the inequation <img decoding=\"async\" style=\"height: 21px; width: 91px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image018.png\" \/>, therefore the required half plane contains (0, 0)<\/p>\n<p>The shaded region is the feasible solution and its corners are P (15, 20), Q (40, 15) and R (2, 72).<\/p>\n<p>Now\u00a0\u00a0\u00a0 Z = <img decoding=\"async\" style=\"height: 21px; width: 52px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image004.png\" \/><\/p>\n<p>At P(15, 20)\u00a0 Z = 6 x 15 + 3 x 20 = 90 + 60 = 150<\/p>\n<p>At Q (40, 15)\u00a0 Z = 6 x 40 + 3 x 15 = 240 + 45 = 285<\/p>\n<p>At R (2, 72)\u00a0 Z = 6 x 2 + 3 x 72 = 12 + 216 = 228<\/p>\n<p>Hence, maximum Z = 285 units of vitamin A at <img decoding=\"async\" style=\"height: 21px; width: 93px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image020.png\" \/><\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4><span class=\"ez-toc-section\" id=\"2_A_farmer_mixes_two_brands_P_and_Q_of_cattle_feed_Brand_P_costing_Rs_250_per_bag_contains_3_units_of_nutritional_element_A_25_units_of_element_B_and_2_units_of_element_C_Brand_Q_costing_Rs_200_per_bag_contains_15_units_of_nutritional_element_A_1125_units_of_element_B_and_3_units_of_element_C_The_minimum_requirements_of_nutrients_A_B_and_C_are_18_units_45_units_and_24_units_respectively_Determine_the_number_of_bags_of_each_brand_which_should_be_mixed_in_order_to_produce_a_mixture_having_a_minimum_cost_per_bag_What_is_the_minimum_cost_of_the_mixture_per_bag\"><\/span><strong>2. A farmer mixes two brands P and Q of cattle feed. Brand P, costing Rs. 250 per bag, contains 3 units of nutritional element A, 2.5 units of element B and 2 units of element C. Brand Q costing Rs. 200 per bag contains 1.5 units of nutritional element A, 11.25 units of element B and 3 units of element C. The minimum requirements of nutrients A, B and C are 18 units, 45 units and 24 units respectively. Determine the number of bags of each brand which should be mixed in order to produce a mixture having a minimum cost per bag? What is the minimum cost of the mixture per bag?<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>Ans. <\/strong>Let number of bags of cattle feed brand P = <img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image001.png\" \/>\u00a0and number of bags of cattle feed brand Q = <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image002.png\" \/><\/p>\n<p><img decoding=\"async\" id=\"Picture 27\" style=\"height: 233px; width: 227px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image021.jpg\" \/><\/p>\n<p>We have to minimize Z = <img decoding=\"async\" style=\"height: 21px; width: 85px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image022.png\" \/>\u00a0subject to the constraints <img decoding=\"async\" style=\"height: 21px; width: 92px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image023.png\" \/>, <img decoding=\"async\" style=\"height: 21px; width: 123px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image024.png\" \/>, <img decoding=\"async\" style=\"height: 21px; width: 84px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image025.png\" \/>,<\/p>\n<p><img decoding=\"async\" style=\"height: 21px; width: 64px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image026.png\" \/>0<\/p>\n<p>Consider <img decoding=\"async\" style=\"height: 21px; width: 92px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image023.png\" \/><\/p>\n<p>Let\u00a0 <img decoding=\"async\" style=\"height: 21px; width: 92px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/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\/12%20Misc\/image014.png\" \/>\u00a0\u00a0 <img decoding=\"async\" style=\"height: 21px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image028.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image014.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 41px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image029.png\" \/><\/p>\n<p>Now the points are A (6, 0) and B (0, 12).<\/p>\n<p>Now clearly (0, 0) does not lie in the required half plane as (0, 0) does not satisfy the inequation <img decoding=\"async\" style=\"height: 21px; width: 92px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image023.png\" \/>.<\/p>\n<p>Again consider <img decoding=\"async\" style=\"height: 21px; width: 123px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image024.png\" \/><\/p>\n<p>Let <img decoding=\"async\" style=\"height: 21px; width: 123px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/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\/12%20Misc\/image014.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 84px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/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\/12%20Misc\/image014.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image032.png\" \/><\/p>\n<p>Now point C (18, 0) and D (0, 4) lie on the line.<\/p>\n<p>Now again (0, 0) does not lie in the required half plane as (0, 0) does not satisfy the inequation <img decoding=\"async\" style=\"height: 21px; width: 123px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image024.png\" \/>.<\/p>\n<p>Again consider <img decoding=\"async\" style=\"height: 21px; width: 84px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image025.png\" \/><\/p>\n<p>Let\u00a0 <img decoding=\"async\" style=\"height: 21px; width: 84px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image033.png\" \/>\u00a0\u00a0<img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image014.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image034.png\" \/><\/p>\n<p>Here, the points E (12, 0) and F (0, 8) lie on the line.<\/p>\n<p>Again also (0, 0) does not lie on the half plane as (0, 0) does not satisfy this inequation.<\/p>\n<p>The feasible region of XCPQEY and the co-ordinates of corners are C (18, 0), P (9, 2), Q (3, 6) and E (0, 12).<\/p>\n<p>Now\u00a0\u00a0\u00a0\u00a0 Z = <img decoding=\"async\" style=\"height: 21px; width: 85px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image022.png\" \/><\/p>\n<p>At C (18, 0)\u00a0\u00a0 Z = 250 x 18 + 200 x 0 = 4500<\/p>\n<p>At P (9, 2)\u00a0\u00a0 Z = 250 x 9 + 200 x 2 = 2450<\/p>\n<p>At Q (3, 6)\u00a0\u00a0 Z = 250 x 3 + 200 x 6 = 1950<\/p>\n<p>At E (0, 12)\u00a0\u00a0 Z = 250 x 0 + 200 x 12 = 2400<\/p>\n<p>&nbsp;<\/p>\n<p>Here, minimum cost Z = Rs. 1950 when <img decoding=\"async\" style=\"height: 21px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image035.png\" \/><\/p>\n<p>Hence, number of bags of brand P = 3 and number of bags of brand Q = 6 and minimum cost of the mixture per bag = Rs. <img decoding=\"async\" style=\"height: 41px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image036.png\" \/>\u00a0= Rs. 216.67 per bag.<\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-2\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4><span class=\"ez-toc-section\" id=\"3_A_dietician_wishes_to_mix_together_two_kinds_of_food_X_and_Y_in_such_a_way_that_the_mixture_contains_at_least_10_units_of_vitamin_A_12_units_of_vitamin_B_and_8_units_of_vitamin_C_The_vitamin_contents_of_one_kg_food_is_given_below\"><\/span><strong>3. A dietician wishes to mix together two kinds of food X and Y in such a way that the mixture contains at least 10 units of vitamin A, 12 units of vitamin B and 8 units of vitamin C. The vitamin contents of one kg food is given below:<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<table border=\"1\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td style=\"width: 78.3pt;\"><strong>Food<\/strong><\/td>\n<td style=\"width: 1.1in;\"><strong>Vitamin A<\/strong><\/td>\n<td style=\"width: 81.0pt;\"><strong>Vitamin B<\/strong><\/td>\n<td style=\"width: 81.0pt;\"><strong>Vitamin C<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 78.3pt;\"><strong>X<\/strong><\/td>\n<td style=\"width: 1.1in;\"><strong>1<\/strong><\/td>\n<td style=\"width: 81.0pt;\"><strong>2<\/strong><\/td>\n<td style=\"width: 81.0pt;\"><strong>3<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 78.3pt;\"><strong>Y<\/strong><\/td>\n<td style=\"width: 1.1in;\"><strong>2<\/strong><\/td>\n<td style=\"width: 81.0pt;\"><strong>2<\/strong><\/td>\n<td style=\"width: 81.0pt;\"><strong>1<\/strong><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><strong>One kg of food X costs Rs. 16 and one kg of food Y costs Rs. 20. Find the least cost of the mixture which will produce the required diet?<\/strong><\/p>\n<p><strong>Ans. <\/strong>Let in the mixture food X weighs = <img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image001.png\" \/>\u00a0kg and in the mixture food Y weighs = <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image002.png\" \/>\u00a0kg<\/p>\n<p>We have to minimize Z = <img decoding=\"async\" style=\"height: 21px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image037.png\" \/>\u00a0subject to constraints, <img decoding=\"async\" style=\"height: 21px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image038.png\" \/>, <img decoding=\"async\" style=\"height: 21px; width: 83px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image039.png\" \/>, <img decoding=\"async\" style=\"height: 21px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image040.png\" \/>, <img decoding=\"async\" style=\"height: 21px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image011.png\" \/><\/p>\n<p>Consider <img decoding=\"async\" style=\"height: 21px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image038.png\" \/><\/p>\n<p><img decoding=\"async\" id=\"Picture 28\" style=\"height: 224px; width: 227px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image041.jpg\" \/><\/p>\n<p>Let <img decoding=\"async\" style=\"height: 21px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image042.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image014.png\" \/>\u00a0\u00a0<img decoding=\"async\" style=\"height: 41px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image043.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image044.png\" \/>\u00a0Points A (10, 0) and B (0, 5) lies on the line.<\/p>\n<p>Here, (0, 0) does not satisfy the inequation <img decoding=\"async\" style=\"height: 21px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image038.png\" \/>, therefore the required half plane does not include (0, 0).<\/p>\n<p>Again consider <img decoding=\"async\" style=\"height: 21px; width: 83px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image039.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\/12%20Misc\/image045.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image014.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 60px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image046.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image014.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 63px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image047.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image044.png\" \/>\u00a0Points C (6, 0) and D (0, 6) lies on the line.<\/p>\n<p>Again consider <img decoding=\"async\" style=\"height: 21px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image040.png\" \/><\/p>\n<p>Let\u00a0 <img decoding=\"async\" style=\"height: 21px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image048.png\" \/>\u00a0\u00a0<img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image014.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image049.png\" \/><\/p>\n<p>Again in the inequation (0, 0) is not included in the required half plane.<\/p>\n<p>The shaded region is our feasible solution A (10, 0), P (2, 4), Q (1, 5), E (0, 8).<\/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;\">F<\/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\/12%20Misc\/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\/12%20Misc\/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>The corners of the feasible region are A (10, 0), P (2, 4), Q (1, 5), E (0, 8).<\/p>\n<p>Now\u00a0Z = <img decoding=\"async\" style=\"height: 21px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image037.png\" \/><\/p>\n<p>At A (10, 0)\u00a0\u00a0 Z = 16 x 10 + 20 x 0 = 160<\/p>\n<p>At P (2, 4)\u00a0\u00a0 Z = 16 x 2 + 20 x 4 = 112<\/p>\n<p>At Q (1, 5)\u00a0\u00a0 Z = 16 x 1 + 20 x 5 = 116<\/p>\n<p>At E (0, 8)\u00a0\u00a0 Z = 16 x 0 + 20 x 8 = 160<\/p>\n<p>Therefore minimum Z = Rs. 112 at <img decoding=\"async\" style=\"height: 21px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image050.png\" \/><\/p>\n<p>Hence, minimum cost of the mixture = Rs. 112 when he mixes 2 kg of food X and 4 kg of food Y.<\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-3\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4><span class=\"ez-toc-section\" id=\"4_A_manufacturer_makes_two_types_of_toys_A_and_B_Three_machines_are_needed_for_this_purpose_and_the_time_in_minutes_requires_for_each_toy_on_the_machines_is_given_below\"><\/span><strong>4. A manufacturer makes two types of toys A and B. Three machines are needed for this purpose and the time (in minutes) requires for each toy on the machines is given below:<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<table border=\"1\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td style=\"width: 112.5pt;\" rowspan=\"2\"><strong>Types of Toys<\/strong><\/td>\n<td style=\"width: 135.0pt;\" colspan=\"3\"><strong>Machines<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 45.0pt;\"><strong>I<\/strong><\/td>\n<td style=\"width: 45.0pt;\"><strong>II<\/strong><\/td>\n<td style=\"width: 45.0pt;\"><strong>III<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 112.5pt;\"><strong>A<\/strong><\/td>\n<td style=\"width: 45.0pt;\"><strong>12<\/strong><\/td>\n<td style=\"width: 45.0pt;\"><strong>18<\/strong><\/td>\n<td style=\"width: 45.0pt;\"><strong>6<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 112.5pt;\"><strong>B<\/strong><\/td>\n<td style=\"width: 45.0pt;\"><strong>6<\/strong><\/td>\n<td style=\"width: 45.0pt;\"><strong>0<\/strong><\/td>\n<td style=\"width: 45.0pt;\"><strong>9<\/strong><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><strong>Each machine is available for a maximum of 6 hours per day. If the profit on each type of type A is Rs. 7.50 and that on each toy of type B is Rs. 5, shows that 15 toys of type A and 30 of type B should be manufactures in a day to get maximum profit.<\/strong><\/p>\n<p><strong>Ans. <\/strong>Let <img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image001.png\" \/>\u00a0units of toys A and <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image002.png\" \/>\u00a0units of toys B are produced by the manufacturer. Time spent on machine I to produce <img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image001.png\" \/>\u00a0units of toys A and <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image002.png\" \/>\u00a0units of toys B = <img decoding=\"async\" style=\"height: 27px; width: 72px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image051.png\" \/>\u00a0minutes.<\/p>\n<p><img decoding=\"async\" id=\"Picture 29\" style=\"height: 180px; width: 185px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image052.jpg\" \/><\/p>\n<p>Since each machine is available for a maximum of 6 x 60 = 360 minutes.<\/p>\n<p>Therefore, we have <img decoding=\"async\" style=\"height: 21px; width: 99px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image053.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image014.png\" \/>\u00a0\u00a0<img decoding=\"async\" style=\"height: 21px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image054.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 19px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image055.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 21px; width: 91px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image056.png\" \/><\/p>\n<h5><span class=\"ez-toc-section\" id=\"now_the_profit_Z_earned_by_the_manufacturer_to_produce_units_of_type_A_and_units_of_type_B_is\"><\/span>now, the profit Z earned by the manufacturer to produce <img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image001.png\" \/>\u00a0units of type A and <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image002.png\" \/>\u00a0units of type B is <img decoding=\"async\" style=\"height: 21px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image057.png\" \/><span class=\"ez-toc-section-end\"><\/span><\/h5>\n<p>we have to maximize Z = <img decoding=\"async\" style=\"height: 21px; width: 72px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image058.png\" \/>\u00a0i.e., 4Z = <img decoding=\"async\" style=\"height: 21px; width: 52px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image059.png\" \/>\u00a0subject to constraints <img decoding=\"async\" style=\"height: 21px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image060.png\" \/>, <img decoding=\"async\" style=\"height: 19px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image061.png\" \/>, <img decoding=\"async\" style=\"height: 21px; width: 91px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image056.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 21px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image062.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image063.png\" \/><\/p>\n<p>Consider <img decoding=\"async\" style=\"height: 21px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image060.png\" \/><\/p>\n<p>Let <img decoding=\"async\" style=\"height: 21px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image064.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image014.png\" \/>\u00a0\u00a0<img decoding=\"async\" style=\"height: 41px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image065.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image044.png\" \/>\u00a0Points A (30, 0) and B (0, 60) lies on the line. Also (0, 0) lies in the required half plane.<\/p>\n<p>Again consider <img decoding=\"async\" style=\"height: 19px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image061.png\" \/><\/p>\n<p>Let <img decoding=\"async\" style=\"height: 19px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image066.png\" \/><\/p>\n<p>It represent the half plane to the left of <img decoding=\"async\" style=\"height: 19px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image066.png\" \/>.<\/p>\n<p>Again consider <img decoding=\"async\" style=\"height: 21px; width: 91px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/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\/12%20Misc\/image067.png\" \/>\u00a0\u00a0<img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image014.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 77px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image068.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image044.png\" \/>\u00a0Points C (60, 0) and D (0, 40) lies on the line. Therefore, (0, 0) lies in the required half plane.<\/p>\n<p>The shaded portion is our feasible region. Its corners are O (0, 0), P (20, 0), Q (20, 20), R (15, 30), D (0, 40).<\/p>\n<p>Now\u00a0\u00a0\u00a0\u00a0 Z = <img decoding=\"async\" style=\"height: 21px; width: 72px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image058.png\" \/><\/p>\n<p>At O (0, 0)\u00a0\u00a0 Z = 7.5 x 0 + 5 x 0 = 0<\/p>\n<p>At P (20, 0)\u00a0\u00a0 Z = 7.5 x 00 + 5 x 0 = 150<\/p>\n<p>At Q (20, 20)\u00a0\u00a0 Z = 7.5 x 20 + 5 x 20 = 250<\/p>\n<p>At R (15, 30)\u00a0\u00a0 Z = 7.5 x 15 + 5 x 30 = 262.50<\/p>\n<p>At D (0, 40)\u00a0\u00a0 Z = 7.5 x 0 + 5 x 40 = 200<\/p>\n<p>Now maximum profit = Z = Rs. 262.50, when he manufacturers 15 toys of types A and 30 of type B in a day.<\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-4\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4><span class=\"ez-toc-section\" id=\"5_An_aeroplane_carries_a_maximum_of_200_passengers_A_profit_of_Rs_1000_is_made_on_each_executive_class_ticket_and_a_profit_of_Rs_600_is_made_on_each_economy_class_ticket_The_airline_reserves_at_least_20_seats_for_executive_class_than_by_the_executive_class_Determine_how_many_tickets_of_each_type_must_be_sold_in_order_to_maximize_the_profit_for_the_airline_What_is_the_maximum_profit\"><\/span><strong>5. An aeroplane carries a maximum of 200 passengers. A profit of Rs. 1000 is made on each executive class ticket and a profit of Rs. 600 is made on each economy class ticket. The airline reserves at least 20 seats for executive class than by the executive class. Determine how many tickets of each type must be sold in order to maximize the profit for the airline. What is the maximum profit?<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>Ans. <\/strong>Let number of tickets of executive class sold = <img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image001.png\" \/>\u00a0and number of tickets of economy class sold = <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image002.png\" \/><\/p>\n<p>We have to maximize = Z = <img decoding=\"async\" style=\"height: 21px; width: 91px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image069.png\" \/>\u00a0subject to <img decoding=\"async\" style=\"height: 21px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image070.png\" \/>, <img decoding=\"async\" style=\"height: 19px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image071.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 21px; width: 47px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image072.png\" \/><\/p>\n<p><img decoding=\"async\" id=\"Picture 30\" style=\"height: 215px; width: 192px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image073.jpg\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 21px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image011.png\" \/><\/p>\n<p>Consider <img decoding=\"async\" style=\"height: 21px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image070.png\" \/><\/p>\n<p>Let\u00a0 <img decoding=\"async\" style=\"height: 21px; width: 77px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image074.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image014.png\" \/>\u00a0\u00a0<img decoding=\"async\" style=\"height: 41px; width: 93px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image075.png\" \/><\/p>\n<h4><span class=\"ez-toc-section\" id=\"_Points_A_200_0_and_B0_200_are_on_the_line_and_therefore_0_0_is_included_in_the_required_half_plane\"><\/span><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image044.png\" \/>\u00a0Points A (200, 0) and B(0, 200) are on the line and therefore (0, 0) is included in the required half plane.<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p>Again consider <img decoding=\"async\" style=\"height: 19px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image071.png\" \/><\/p>\n<p>Let <img decoding=\"async\" style=\"height: 19px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image066.png\" \/><\/p>\n<p>It is the line parallel to <img decoding=\"async\" style=\"height: 17px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image076.png\" \/>axis at a positive distance 20 and the half plane lies towards right of it.<\/p>\n<p>Again consider <img decoding=\"async\" style=\"height: 21px; width: 47px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image072.png\" \/><\/p>\n<p>Let <img decoding=\"async\" style=\"height: 21px; width: 47px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image077.png\" \/><\/p>\n<table border=\"1\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td style=\"width: 42.3pt;\"><\/td>\n<td style=\"width: 47.7pt;\">O<\/td>\n<td style=\"width: 47.7pt;\">C<\/td>\n<td style=\"width: 47.7pt;\">D<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 42.3pt;\"><img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image001.png\" \/><\/td>\n<td style=\"width: 47.7pt;\">0<\/td>\n<td style=\"width: 47.7pt;\">20<\/td>\n<td style=\"width: 47.7pt;\">40<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 42.3pt;\"><img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image002.png\" \/><\/td>\n<td style=\"width: 47.7pt;\">0<\/td>\n<td style=\"width: 47.7pt;\">80<\/td>\n<td style=\"width: 47.7pt;\">160<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Here, (40, 0) does not satisfy <img decoding=\"async\" style=\"height: 21px; width: 47px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image072.png\" \/>, therefore plane does not include (40, 0).<\/p>\n<p>The shaded portion is the feasible region. Its corners are C (20, 80), D (40, 160) and P (20, 180)<\/p>\n<p>Now\u00a0\u00a0\u00a0 Z = <img decoding=\"async\" style=\"height: 21px; width: 91px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image069.png\" \/><\/p>\n<p>At C (20, 80) Z = 1000 x 20 + 600 x 80 = 20000 + 48000 = 68,000<\/p>\n<p>At D (40, 60) Z = 1000 x 40 + 600 x 60 = 40000 + 96000 = 1,36,000<\/p>\n<p>At P (20, 180) Z = 1000 x 20 + 600 x 180 = 20000 + 108000 = 1,28,000<\/p>\n<p>Hence Maximum profit Z = Rs. 1,36,000 at <img decoding=\"async\" style=\"height: 21px; width: 101px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image078.png\" \/><\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-5\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4><span class=\"ez-toc-section\" id=\"6_Two_godowns_A_and_B_have_grain_capacity_of_100_quintals_and_50_quintals_respectively_They_supply_to_3_ration_shops_D_E_and_F_whose_requirements_are_60_50_and_40_quintals_respectively_The_cost_of_transportation_per_quintal_from_the_godowns_to_the_shops_are_given_in_the_following_table\"><\/span><strong>6. Two godowns A and B have grain capacity of 100 quintals and 50 quintals respectively. They supply to 3 ration shops. D, E and F whose requirements are 60, 50 and 40 quintals respectively. The cost of transportation per quintal from the godowns to the shops are given in the following table:<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<table border=\"1\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td style=\"width: 3.25in;\" colspan=\"3\"><strong>Transportation cost per quintal (in Rs.)<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 83.7pt;\"><strong>From \/ To<\/strong><\/td>\n<td style=\"width: 73.8pt;\"><strong>A<\/strong><\/td>\n<td style=\"width: 76.5pt;\"><strong>B<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 83.7pt;\"><strong>D<\/strong><\/p>\n<p><strong>E<\/strong><\/p>\n<p><strong>F<\/strong><\/td>\n<td style=\"width: 73.8pt;\"><strong>6<\/strong><\/p>\n<p><strong>3<\/strong><\/p>\n<p><strong>2.50<\/strong><\/td>\n<td style=\"width: 76.5pt;\"><strong>4<\/strong><\/p>\n<p><strong>2<\/strong><\/p>\n<p><strong>3<\/strong><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><strong>How should the supplies be transported in order that the transportation cost is minimum? What is the minimum cost?<\/strong><\/p>\n<p><strong>Ans. <\/strong>Let godown A supplies <img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image001.png\" \/>\u00a0quintals of grain to the ration shop D and <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image002.png\" \/>\u00a0quintals to ration shop E.<\/p>\n<p><img decoding=\"async\" id=\"Picture 32\" style=\"height: 240px; width: 244px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image079.jpg\" \/><\/p>\n<p>We have to minimize Z = <img decoding=\"async\" style=\"height: 41px; width: 77px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image080.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 27px; width: 160px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image081.png\" \/>+<img decoding=\"async\" style=\"height: 27px; width: 65px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image082.png\" \/><\/p>\n<p>+<img decoding=\"async\" style=\"height: 27px; width: 87px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image083.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image014.png\" \/>\u00a0\u00a0Z = <img decoding=\"async\" style=\"height: 41px; width: 95px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image084.png\" \/><\/p>\n<h5><span class=\"ez-toc-section\" id=\"Subject_to_and\"><\/span>Subject to <img decoding=\"async\" style=\"height: 21px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image085.png\" \/>, <img decoding=\"async\" style=\"height: 19px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image086.png\" \/>, <img decoding=\"async\" style=\"height: 21px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image087.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 21px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image088.png\" \/>, <img decoding=\"async\" style=\"height: 21px; width: 79px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image003.png\" \/><span class=\"ez-toc-section-end\"><\/span><\/h5>\n<p>Consider <img decoding=\"async\" style=\"height: 21px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image085.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\/12%20Misc\/image089.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image014.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 89px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image090.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image044.png\" \/>\u00a0Points P (100, 0) and Q (0, 100) lie on the line and it represents the half-plane containing (0, 0).<\/p>\n<p>Again we consider <img decoding=\"async\" style=\"height: 19px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image086.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 21px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image087.png\" \/><\/p>\n<p>We draw <img decoding=\"async\" style=\"height: 19px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image091.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 21px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image092.png\" \/><\/p>\n<p>Again consider <img decoding=\"async\" style=\"height: 21px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image088.png\" \/><\/p>\n<p>Let <img decoding=\"async\" style=\"height: 21px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image093.png\" \/>\u00a0\u00a0<img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image014.png\" \/>\u00a0\u00a0<img decoding=\"async\" style=\"height: 41px; width: 77px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image094.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image044.png\" \/>\u00a0Points A (60, 0) and R (0, 60) lie on the line and it represents the half-plane containing (0, 0)<\/p>\n<p>The shaded region is the feasible solution. Its corners are A (60, 0), B (60, 40), C (50, 50) and D (10, 50).<\/p>\n<p>Now\u00a0\u00a0\u00a0 Z = <img decoding=\"async\" style=\"height: 41px; width: 95px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image084.png\" \/><\/p>\n<p>At A (60, 0) Z = <img decoding=\"async\" style=\"height: 41px; width: 243px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image095.png\" \/><\/p>\n<p>At B (60, 40) Z = <img decoding=\"async\" style=\"height: 41px; width: 281px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image096.png\" \/><\/p>\n<p>At C (50, 50) Z = <img decoding=\"async\" style=\"height: 41px; width: 280px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image097.png\" \/><\/p>\n<p>At D (10, 50) Z = <img decoding=\"async\" style=\"height: 41px; width: 272px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image098.png\" \/><\/p>\n<p>Hence minimum value is Z = 510 at <img decoding=\"async\" style=\"height: 21px; width: 93px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image099.png\" \/><\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-6\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4><span class=\"ez-toc-section\" id=\"7_An_oil_company_has_two_depots_A_and_B_with_capacities_of_7000_L_and_4000_L_respectively_The_company_is_to_supply_oil_to_three_petrol_pumps_D_E_and_F_whose_requirements_are_4500_L_3000_L_and_3500_L_respectively_The_distances_in_km_between_the_depots_and_the_petrol_pumps_is_given_in_the_following_table\"><\/span><strong>7. An oil company has two depots A and B with capacities of 7000 L and 4000 L respectively. The company is to supply oil to three petrol pumps. D, E and F whose requirements are 4500 L, 3000 L and 3500 L respectively. The distances (in km) between the depots and the petrol pumps is given in the following table:<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<table border=\"1\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td style=\"width: 3.25in;\" colspan=\"3\"><strong>Distance in km.<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 83.7pt;\"><strong>From \/ To<\/strong><\/td>\n<td style=\"width: 73.8pt;\"><strong>A<\/strong><\/td>\n<td style=\"width: 76.5pt;\"><strong>B<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 83.7pt;\"><strong>D<\/strong><\/p>\n<p><strong>E<\/strong><\/p>\n<p><strong>F<\/strong><\/td>\n<td style=\"width: 73.8pt;\"><strong>7<\/strong><\/p>\n<p><strong>6<\/strong><\/p>\n<p><strong>3<\/strong><\/td>\n<td style=\"width: 76.5pt;\"><strong>3<\/strong><\/p>\n<p><strong>4<\/strong><\/p>\n<p><strong>2<\/strong><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h5><span class=\"ez-toc-section\" id=\"Assuming_that_the_transportation_cost_of_10_liters_of_oil_is_Re_1_per_km_how_should_the_delivery_be_scheduled_in_order_that_the_transportation_cost_is_minimum_What_is_the_minimum_cost\"><\/span><strong>Assuming that the transportation cost of 10 liters of oil is Re. 1 per km, how should the delivery be scheduled in order that the transportation cost is minimum? What is the minimum cost?<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h5>\n<p><strong>Ans. <\/strong>Let <img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image001.png\" \/>\u00a0liters of oil is supplied from depot A to petrol pump D and <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image002.png\" \/>\u00a0liters of oil supplied from depot A to petrol pump E then <img decoding=\"async\" style=\"height: 27px; width: 95px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image100.png\" \/>\u00a0liters of oil will be supplied from depot A to petrol pump F.<\/p>\n<p><img decoding=\"async\" id=\"Picture 34\" style=\"height: 192px; width: 214px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image101.jpg\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image044.png\" \/>\u00a0We have <img decoding=\"async\" style=\"height: 21px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image011.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 27px; width: 119px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/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\/12%20Misc\/image014.png\" \/>\u00a0\u00a0<img decoding=\"async\" style=\"height: 21px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image103.png\" \/><\/p>\n<h5><span class=\"ez-toc-section\" id=\"Since_requirements_of_oil_at_petrol_pump_D_E_and_F_are_and_liters_respectively\"><\/span>Since requirements of oil at petrol pump, D, E and F are <img decoding=\"async\" style=\"height: 27px; width: 77px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image104.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 71px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image105.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 29px; width: 109px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image106.png\" \/>\u00a0liters respectively.<span class=\"ez-toc-section-end\"><\/span><\/h5>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image044.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 19px; width: 83px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image107.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image014.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 19px; width: 61px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image108.png\" \/><\/p>\n<p>And <img decoding=\"async\" style=\"height: 21px; width: 83px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image109.png\" \/>\u00a0\u00a0\u00a0\u00a0\u00a0<img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image014.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 61px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image110.png\" \/><\/p>\n<p>And <img decoding=\"async\" style=\"height: 29px; width: 179px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image111.png\" \/>\u00a0\u00a0\u00a0<img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image014.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 84px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image112.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image113.png\" \/>\u00a0\u00a0The cost of transportation per km for 10 liters oil is Re 1<\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image044.png\" \/>\u00a0\u00a0The cost of transportation per km per liter = Rs. <img decoding=\"async\" style=\"height: 41px; width: 21px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image114.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image044.png\" \/>\u00a0The cost of transportation<\/p>\n<p>Z = <img decoding=\"async\" style=\"height: 29px; width: 531px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image115.png\" \/><\/p>\n<p>Z = <img decoding=\"async\" style=\"height: 21px; width: 121px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image116.png\" \/><\/p>\n<p>Therefore, the feasible region is ABECD.<\/p>\n<p>Its corners are A (500, 3000), B (35, 0), E (4500, 0), C (4500, 2500), D (4000, 3000).<\/p>\n<p>Now\u00a0\u00a0\u00a0 Z = <img decoding=\"async\" style=\"height: 21px; width: 121px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image116.png\" \/><\/p>\n<p>At A (500, 3000) Z = 0.3 x 500 + 0.1 x 3000 + 3950 = 4400<\/p>\n<p>At B (3500, 0) Z = 0.3 x 3500 + 0.1 x 0 + 3950 = 5000<\/p>\n<p>At E (4500, 0) Z = 0.3 x 4500 + 0.1 x 0 + 3950 = 5300<\/p>\n<p>At C (4500, 2500) Z = 0.3 x 4500 + 0.1 x 2500 + 3950 = 5550<\/p>\n<p>At D (4000, 3000) Z = 0.3 x 4000 + 0.1 x 3000 + 3950 = 5450<\/p>\n<p>Minimum transportation charges are Rs. 4400 at <img decoding=\"async\" style=\"height: 21px; width: 115px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image117.png\" \/><\/p>\n<p>Hence, 500 liters, 3000 liters and 3500 liters of oil should be transported from depot A to petrol pumps D, E, F and 4000 liters, 0 liter and 0 liter of oil be transported from depot B to petrol pumps D, E and F with minimum cost of transportation of Rs. 4400.<\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-7\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4><span class=\"ez-toc-section\" id=\"8_A_fruit_grower_can_use_two_types_of_fertilizer_in_his_garden_brand_P_and_brand_Q_The_amounts_in_kg_of_nitrogen_phosphoric_acid_potash_and_chlorine_in_a_bag_of_each_brand_are_given_in_the_table_Tests_indicate_that_the_garden_needs_at_least_240_kg_of_phosphoric_acid_at_least_270_kg_of_potash_and_at_most_310_kg_of_chlorine\"><\/span><strong>8. A fruit grower can use two types of fertilizer in his garden, brand P and brand Q. The amounts (in kg) of nitrogen, phosphoric acid, potash and chlorine in a bag of each brand are given in the table. Tests indicate that the garden needs at least 240 kg of phosphoric acid, at least 270 kg of potash and at most 310 kg of chlorine.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>If the grower wants to minimize the amount of nitrogen added to the garden, how many bags of each brand should be used? What is the minimum amount of nitrogen added in the garden?<\/strong><\/p>\n<table border=\"1\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td style=\"width: 247.5pt;\" colspan=\"3\"><strong>kg per bag<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 1.5in;\"><strong>\u00a0<\/strong><\/td>\n<td style=\"width: 1.0in;\"><strong>Brand P<\/strong><\/td>\n<td style=\"width: 67.5pt;\"><strong>Brand Q<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 1.5in;\"><strong>Nitrogen<\/strong><\/p>\n<p><strong>Phosphoric acid<\/strong><\/p>\n<p><strong>Potash<\/strong><\/p>\n<p><strong>Chlorine<\/strong><\/td>\n<td style=\"width: 1.0in;\"><strong>3<\/strong><\/p>\n<p><strong>1<\/strong><\/p>\n<p><strong>3<\/strong><\/p>\n<p><strong>1.5<\/strong><\/td>\n<td style=\"width: 67.5pt;\"><strong>3.5<\/strong><\/p>\n<p><strong>2<\/strong><\/p>\n<p><strong>1.5<\/strong><\/p>\n<p><strong>2<\/strong><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><strong>Ans. <\/strong>Let amount of Brand P of fertilizer = <img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image001.png\" \/>\u00a0bags and amount of Brand Q of fertilizer = <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image002.png\" \/>\u00a0bags<\/p>\n<p>We have to minimize Z = <img decoding=\"async\" style=\"height: 21px; width: 64px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image118.png\" \/>\u00a0subject to <img decoding=\"async\" style=\"height: 21px; width: 85px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image119.png\" \/>, <img decoding=\"async\" style=\"height: 21px; width: 101px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image120.png\" \/>, <img decoding=\"async\" style=\"height: 21px; width: 103px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image121.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 21px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image011.png\" \/><\/p>\n<p><img decoding=\"async\" id=\"Picture 35\" style=\"height: 219px; width: 231px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image122.jpg\" \/><\/p>\n<p>Consider <img decoding=\"async\" style=\"height: 21px; width: 85px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image119.png\" \/><\/p>\n<p>Let <img decoding=\"async\" style=\"height: 21px; width: 85px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image123.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image014.png\" \/>\u00a0\u00a0<img decoding=\"async\" style=\"height: 41px; width: 92px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image124.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image044.png\" \/>\u00a0Points A (240, 0) and B (0, 120) lie on the line.<\/p>\n<h5><span class=\"ez-toc-section\" id=\"And_0_0_does_not_lie_on_the_required_half-plane_of_this_in_equation\"><\/span>And (0, 0) does not lie on the required half-plane of this in equation.<span class=\"ez-toc-section-end\"><\/span><\/h5>\n<p>Again consider<\/p>\n<p><img decoding=\"async\" style=\"height: 21px; width: 101px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image120.png\" \/><\/p>\n<p>Let <img decoding=\"async\" style=\"height: 21px; width: 103px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image125.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image014.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 83px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image126.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image014.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 83px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image127.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image044.png\" \/>\u00a0Points C (90, 0) and D (0, 180) lie on the line.<\/p>\n<p>Here also (0, 0) does not lie on the required half-plane of this inequation.<\/p>\n<table border=\"1\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td style=\"width: 42.3pt;\"><\/td>\n<td style=\"width: 47.7pt;\">E<\/td>\n<td style=\"width: 47.7pt;\">F<\/td>\n<td style=\"width: 47.7pt;\">G<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 42.3pt;\"><img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image001.png\" \/><\/td>\n<td style=\"width: 47.7pt;\">0<\/td>\n<td style=\"width: 47.7pt;\">100<\/td>\n<td style=\"width: 47.7pt;\">200<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 42.3pt;\"><img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image002.png\" \/><\/td>\n<td style=\"width: 47.7pt;\">155<\/td>\n<td style=\"width: 47.7pt;\">80<\/td>\n<td style=\"width: 47.7pt;\">5<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Again consider <img decoding=\"async\" style=\"height: 21px; width: 103px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image121.png\" \/><\/p>\n<p>Let <img decoding=\"async\" style=\"height: 21px; width: 103px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image128.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image014.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 92px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/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\/12%20Misc\/image014.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 85px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image130.png\" \/><\/p>\n<p>Here also (0, 0) does not lie on the required half-plane of this inequation.<\/p>\n<p>The shaded portion is the feasible region. Its corners are P (140, 50), Q (20, 140) and R (40, 100).<\/p>\n<p>Now\u00a0\u00a0\u00a0 Z = <img decoding=\"async\" style=\"height: 21px; width: 64px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image118.png\" \/><\/p>\n<p>At P (140, 50) Z = 3 x 140 + 3.5 x 50 = 420 + 175 = 595<\/p>\n<p>At Q (20, 140) Z = 3 x 20 + 3.5 x 140 = 60 + 490 = 550<\/p>\n<p>At R (40, 100) Z = 3 x 40 + 3.5 x 100 = 120 + 350 = 470<\/p>\n<p>Hence minimum Z = 470 at <img decoding=\"async\" style=\"height: 21px; width: 101px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image131.png\" \/><\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-8\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4><span class=\"ez-toc-section\" id=\"9_Refer_to_Question_8_If_the_grower_wants_to_maximize_the_amount_of_nitrogen_added_to_the_garden_how_many_bags_of_each_brand_should_be_added_What_is_the_maximum_amount_of_nitrogen_added\"><\/span><strong>9. Refer to Question 8. If the grower wants to maximize the amount of nitrogen added to the garden, how many bags of each brand should be added? What is the maximum amount of nitrogen added?<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>Ans. <\/strong>We have Z = <img decoding=\"async\" style=\"height: 21px; width: 64px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image118.png\" \/>\u00a0and Z is maximum at P (140, 50).<\/p>\n<p>To maximize the amount of nitrogen, 140 bags of brand P and 50 bags of brand Q are required.<\/p>\n<p>Therefore maximum amount of nitrogen required = 595 kg<\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-9\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4><span class=\"ez-toc-section\" id=\"10_A_toy_company_manufactures_two_types_of_dolls_A_and_B_Market_tests_and_available_resources_have_indicated_that_the_combined_production_level_should_not_exceed_1200_dolls_per_week_and_the_demand_for_dolls_of_type_B_is_at_most_half_of_that_for_dolls_of_type_A_Further_the_production_level_of_dolls_of_type_A_can_exceed_three_times_the_production_of_dolls_of_other_type_by_at_most_600_units_If_the_company_makes_profit_of_Rs12_and_Rs_16_per_doll_respectively_on_dolls_A_and_B_how_many_of_each_should_be_produced_weekly_in_order_to_maximize_the_profit\"><\/span><strong>10. A toy company manufactures two types of dolls, A and B. Market tests and available resources have indicated that the combined production level should not exceed 1200 dolls per week and the demand for dolls of type B is at most half of that for dolls of type A. Further, the production level of dolls of type A can exceed three times the production of dolls of other type by at most 600 units. If the company makes profit of Rs.12 and Rs. 16 per doll respectively on dolls A and B, how many of each should be produced weekly in order to maximize the profit?<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>Ans.<\/strong> Let the number of dolls of type A = <img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image001.png\" \/>\u00a0and number of dolls of type B = <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image002.png\" \/><\/p>\n<p>We have to maximize Z = <img decoding=\"async\" style=\"height: 21px; width: 65px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image132.png\" \/>\u00a0subject to <img decoding=\"async\" style=\"height: 21px; width: 83px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image133.png\" \/>, <img decoding=\"async\" style=\"height: 41px; width: 51px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image134.png\" \/>, <img decoding=\"async\" style=\"height: 21px; width: 83px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image135.png\" \/>, <img decoding=\"async\" style=\"height: 21px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image011.png\" \/><\/p>\n<table border=\"1\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td style=\"width: 42.3pt;\"><\/td>\n<td style=\"width: 47.7pt;\">A<\/td>\n<td style=\"width: 47.7pt;\">B<\/td>\n<td style=\"width: 47.7pt;\">C<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 42.3pt;\"><img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image001.png\" \/><\/td>\n<td style=\"width: 47.7pt;\">0<\/td>\n<td style=\"width: 47.7pt;\">600<\/td>\n<td style=\"width: 47.7pt;\">1200<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 42.3pt;\"><img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image002.png\" \/><\/td>\n<td style=\"width: 47.7pt;\">1200<\/td>\n<td style=\"width: 47.7pt;\">600<\/td>\n<td style=\"width: 47.7pt;\">0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Consider <img decoding=\"async\" style=\"height: 21px; width: 83px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image133.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\/12%20Misc\/image136.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image014.png\" \/>\u00a0\u00a0<img decoding=\"async\" style=\"height: 21px; width: 83px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image137.png\" \/><\/p>\n<p>Here, (0, 0) is included in the required half plane and satisfies this inequation.<\/p>\n<p>Again consider <img decoding=\"async\" style=\"height: 41px; width: 51px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image134.png\" \/><\/p>\n<p>Let <img decoding=\"async\" style=\"height: 41px; width: 51px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image138.png\" \/><\/p>\n<table border=\"1\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td style=\"width: 40.5pt;\"><\/td>\n<td style=\"width: 40.5pt;\">O<\/td>\n<td style=\"width: 43.6pt;\">E<\/td>\n<td style=\"width: 37.4pt;\">F<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 40.5pt;\"><img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image001.png\" \/><\/td>\n<td style=\"width: 40.5pt;\">0<\/td>\n<td style=\"width: 43.6pt;\">600<\/td>\n<td style=\"width: 37.4pt;\">1200<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 40.5pt;\"><img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image002.png\" \/><\/td>\n<td style=\"width: 40.5pt;\">1200<\/td>\n<td style=\"width: 43.6pt;\">600<\/td>\n<td style=\"width: 37.4pt;\">0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h5><span class=\"ez-toc-section\" id=\"Here_100_0_satisfies_the_inequation_therefore_the_required_half_plane_includes_100_0\"><\/span>Here, (100, 0) satisfies the inequation <img decoding=\"async\" style=\"height: 41px; width: 51px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image134.png\" \/>, therefore the required half plane includes (100, 0).<span class=\"ez-toc-section-end\"><\/span><\/h5>\n<p>Again consider <img decoding=\"async\" style=\"height: 21px; width: 83px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image135.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\/12%20Misc\/image139.png\" \/><\/p>\n<table border=\"1\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td style=\"width: 40.5pt;\"><\/td>\n<td style=\"width: 40.5pt;\">D<\/td>\n<td style=\"width: 43.6pt;\">P<\/td>\n<td style=\"width: 37.4pt;\">Q<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 40.5pt;\"><img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image001.png\" \/><\/td>\n<td style=\"width: 40.5pt;\">600<\/td>\n<td style=\"width: 43.6pt;\">900<\/td>\n<td style=\"width: 37.4pt;\">1200<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 40.5pt;\"><img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image002.png\" \/><\/td>\n<td style=\"width: 40.5pt;\">0<\/td>\n<td style=\"width: 43.6pt;\">100<\/td>\n<td style=\"width: 37.4pt;\">200<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image014.png\" \/>\u00a0\u00a0<img decoding=\"async\" style=\"height: 21px; width: 84px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image140.png\" \/><\/p>\n<p>Here, also (0, 0) is included in the required half-plane.<\/p>\n<p>The shaded region DRSOD is the feasible region whose corners are D(600,0), R(1050,150),<\/p>\n<p><img decoding=\"async\" id=\"Picture 37\" style=\"height: 199px; width: 246px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image141.jpg\" \/><\/p>\n<p>S(800,400) and O(0,0).<\/p>\n<p>Now\u00a0\u00a0\u00a0\u00a0 Z = <img decoding=\"async\" style=\"height: 21px; width: 65px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image132.png\" \/><\/p>\n<p>At D (600, 0)\u00a0\u00a0 Z = 12 x 600 + 16 x 0 = 7200<\/p>\n<p>At R (1050, 150)\u00a0\u00a0 Z = 12 x 105 + 16 x 150 = 12600 + 2400 = 15,000<\/p>\n<p>At S (800, 400)\u00a0\u00a0 Z = 12 x 800 + 16 x 400 = 9600 + 6400 = 16,000<\/p>\n<p>At O (0, 0)\u00a0\u00a0 Z = 12 x 0 + 16 x 0 = 0<\/p>\n<p>Hence maximum profit Z = Rs. 16,000 at <img decoding=\"async\" style=\"height: 21px; width: 111px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/12%20Misc\/image142.png\" \/><\/p>\n<h2><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-10\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>NCERT Solutions Class 12 Maths PDF (Download) Free from myCBSEguide app and myCBSEguide website. Ncert solution class 12 Maths includes text book solutions from both part 1 and part 2. NCERT Solutions for CBSE Class 12 Maths have total 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 Miscellaneous Class 12 Maths book solutions are available in PDF format for free download. These ncert book chapter wise questions and answers are very helpful for CBSE board exam. CBSE recommends NCERT books and most of the questions in CBSE exam are asked from NCERT text books. Class 12 Maths &#8230; <a title=\"NCERT Solutions class 12 Maths Miscellaneous\" class=\"read-more\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-6\/\" aria-label=\"More on NCERT Solutions class 12 Maths Miscellaneous\">Read more<\/a><\/p>\n","protected":false},"author":1,"featured_media":-1,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1432,281],"tags":[283,1342,321,216],"class_list":["post-4508","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mathematics-cbse-class-12","category-ncert-solutions","tag-cbse-study-material","tag-class-12","tag-mathematics","tag-ncert-solutions"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.0 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>NCERT Solutions class 12 Maths Miscellaneous | myCBSEguide<\/title>\n<meta name=\"description\" content=\"NCERT Solutions class 12 Maths Miscellaneous in PDF format for free download. 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