{"id":4544,"date":"2016-05-16T11:49:00","date_gmt":"2016-05-16T06:19:00","guid":{"rendered":"http:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-1\/"},"modified":"2018-07-19T18:15:52","modified_gmt":"2018-07-19T12:45:52","slug":"ncert-solutions-class-12-maths-exercise-6-1","status":"publish","type":"post","link":"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-1\/","title":{"rendered":"NCERT Solutions class 12 Maths Exercise 6.1"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_76 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 eztoc-toggle-hide-by-default' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-1\/#NCERT_Solutions_for_Class_12_Maths_Application_of_Derivatives\" >NCERT Solutions for Class 12 Maths\u00a0Application of Derivatives<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-1\/#NCERT_Solutions_class_12_Maths_Exercise_61\" >NCERT Solutions class 12 Maths Exercise 6.1<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-1\/#2_The_volume_of_a_cube_is_increasing_at_the_rate_of_8_cm3sec_How_fast_is_the_surface_area_increasing_when_the_length_of_an_edge_is_12_cm\" >2. The volume of a cube is increasing at the rate of 8 cm3\/sec. How fast is the surface area increasing when the length of an edge is 12 cm?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-1\/#NCERT_Solutions_class_12_Maths_Exercise_61-2\" >NCERT Solutions class 12 Maths Exercise 6.1<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-1\/#3_The_radius_of_the_circle_is_increasing_uniformly_at_the_rate_of_3_cm_per_second_Find_the_rate_at_which_the_area_of_the_circle_is_increasing_when_the_radius_is_10_cm\" >3. The radius of the circle is increasing uniformly at the rate of 3 cm per second. Find the rate at which the area of the circle is increasing when the radius is 10 cm.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-1\/#NCERT_Solutions_class_12_Maths_Exercise_61-3\" >NCERT Solutions class 12 Maths Exercise 6.1<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-1\/#4_An_edge_of_a_variable_cube_is_increasing_at_the_rate_of_3_cm_per_second_How_fast_is_the_volume_of_the_cube_increasing_when_the_edge_if_10_cm_long\" >4. An edge of a variable cube is increasing at the rate of 3 cm per second. How fast is the volume of the cube increasing when the edge if 10 cm long?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-1\/#NCERT_Solutions_class_12_Maths_Exercise_61-4\" >NCERT Solutions class 12 Maths Exercise 6.1<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-1\/#5_A_stone_is_dropped_into_a_quite_lake_and_waves_move_in_circles_at_the_rate_of_5_cmsec_At_the_instant_when_radius_of_the_circular_wave_is_8_cm_how_fast_is_the_enclosed_area_increasing\" >5. A stone is dropped into a quite lake and waves move in circles at the rate of 5 cm\/sec. At the instant when radius of the circular wave is 8 cm, how fast is the enclosed area increasing?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-1\/#NCERT_Solutions_class_12_Maths_Exercise_61-5\" >NCERT Solutions class 12 Maths Exercise 6.1<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-11\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-1\/#6_The_radius_of_a_circle_is_increasing_at_the_rate_of_07_cms_What_is_the_rate_of_its_circumference\" >6. The radius of a circle is increasing at the rate of 0.7 cm\/s. What is the rate of its circumference?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-12\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-1\/#7_The_length_of_a_rectangle_is_decreasing_at_the_rate_of_5_cmminute_When_8_cm_and_6_cm_find_the_rates_of_change_of_a_the_perimeter_and_b_the_area_of_the_rectangle\" >7. The length \u00a0of a rectangle is decreasing at the rate of 5 cm\/minute. When \u00a0= 8 cm and \u00a0= 6 cm, find the rates of change of (a) the perimeter and (b) the area of the rectangle.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-13\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-1\/#8_A_balloon_which_always_remains_spherical_on_inflation_is_being_inflated_by_pumping_in_900_cubic_centimeters_of_gas_per_second_Find_the_rate_at_which_the_radius_of_the_balloon_increases_when_the_radius_is_15_cm\" >8. A balloon, which always remains spherical on inflation, is being inflated by pumping in 900 cubic centimeters of gas per second. Find the rate at which the radius of the balloon increases when the radius is 15 cm.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-14\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-1\/#9_A_balloon_which_always_remains_spherical_has_a_variables_radius_Find_the_rate_at_which_its_volume_is_increasing_with_the_radius_when_the_later_is_10_cm\" >9. A balloon, which always remains spherical has a variables radius. Find the rate at which its volume is increasing with the radius when the later is 10 cm.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-15\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-1\/#10_A_ladder_5_cm_long_is_leaning_against_a_wall_The_bottom_of_the_ladder_is_pulled_along_the_ground_away_from_the_wall_at_the_rate_of_2_cms_How_fast_is_its_height_on_the_wall_decreasing_when_the_foot_of_the_ladder_is_4_m_away_from_the_wall\" >10. A ladder 5 cm long is leaning against a wall. The bottom of the ladder is pulled along the ground, away from the wall, at the rate of 2 cm\/s. How fast is its height on the wall decreasing when the foot of the ladder is 4 m away from the wall?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-16\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-1\/#11_A_particle_moves_along_the_curve_Find_the_points_on_the_curve_at_which_the_coordinate_is_changing_8_times_as_fast_as_the_coordinate\" >11. A particle moves along the curve \u00a0Find the points on the curve at which the coordinate is changing 8 times as fast as the coordinate.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-17\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-1\/#12_The_radius_of_an_air_bubble_is_increasing_at_the_rate_of_cms_At_what_rate_is_the_volume_of_the_bubble_increasing_when_the_radius_is_1_cm\" >12. The radius of an air bubble is increasing at the rate of \u00a0cm\/s. At what rate is the volume of the bubble increasing when the radius is 1 cm?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-18\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-1\/#13_A_balloon_which_always_remains_spherical_has_a_variable_diameter_Find_the_rate_of_change_of_its_volume_with_respect_to\" >13. A balloon which always remains spherical, has a variable diameter \u00a0Find the rate of change of its volume with respect to \u00a0<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-19\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-1\/#14_Sand_is_pouring_from_a_pipe_at_the_rate_of_12_cm3s_The_falling_sand_forms_a_cone_on_the_ground_in_such_a_way_that_the_height_of_the_cone_is_always_one-sixth_of_the_radius_of_the_base_How_fast_is_the_height_of_the_sand_cone_increasing_when_the_height_is_4cm\" >14. Sand is pouring from a pipe at the rate of 12 cm3\/s. The falling sand forms a cone on the ground in such a way that the height of the cone is always one-sixth of the radius of the base. How fast is the height of the sand cone increasing when the height is 4cm?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-20\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-1\/#15_The_total_cost_C_in_rupees_associated_with_the_production_of_units_of_an_item_given_by_C_Find_the_marginal_cost_when_17_units_are_produced\" >15. The total cost C\u00a0in rupees associated with the production of \u00a0units of an item given by \u00a0C\u00a0Find the marginal cost when 17 units are produced.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-21\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-1\/#16_The_total_revenue_in_rupees_received_from_the_sale_of_units_of_a_product_is_given_by\" >16. The total revenue in rupees received from the sale of \u00a0units of a product is given by<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-22\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-1\/#17_The_rate_of_change_of_the_area_of_a_circle_with_respect_to_its_radius_at_6_cm_is\" >17. The rate of change of the area of a circle with respect to its radius \u00a0at \u00a0=6 cm is:<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-23\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-1\/#NCERT_Solutions_class_12_Maths_Exercise_61-6\" >NCERT Solutions class 12 Maths Exercise 6.1<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-24\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-1\/#NCERT_Solutions_class_12_Maths_Exercise_61-7\" >NCERT Solutions class 12 Maths Exercise 6.1<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-25\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-1\/#NCERT_Solutions_for_Mathematics_Class_3rd_to_12th\" >NCERT Solutions for Mathematics Class 3rd to 12th<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-26\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-1\/#CBSE_App_for_Class_12\" >CBSE App for Class 12<\/a><\/li><\/ul><\/nav><\/div>\n<p>NCERT Solutions class 12 Maths Exercise 6.1 Class 12 Maths book solutions are available in PDF format for free download. These ncert book chapter wise questions and answers are very helpful for CBSE board exam. CBSE recommends NCERT books and most of the questions in CBSE exam are asked from NCERT text books. Class 12 Maths chapter wise NCERT solution for Maths part 1 and Maths part 2 for all the chapters can be downloaded from our website and myCBSEguide mobile app for free.<\/p>\n<p><strong>Download\u00a0<a href=\"https:\/\/mycbseguide.com\/downloads\/cbse-class-12-mathematics-application-of-derivatives\/1290\/ncert-solutions\/5\/\">NCERT solutions for Applications of Derivatives<\/a>\u00a0as PDF.<\/strong><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/media-mycbseguide.s3.ap-south-1.amazonaws.com\/images\/blog\/12%20Maths%20Book%20pic.jpg\" alt=\"NCERT Solutions class 12 Maths Exercise 6.1\" width=\"183\" height=\"238\" \/><\/p>\n<h2><span class=\"ez-toc-section\" id=\"NCERT_Solutions_for_Class_12_Maths_Application_of_Derivatives\"><\/span><strong>NCERT Solutions for Class 12 Maths\u00a0Application of Derivatives<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><strong>1. Find the rate of change of the area of a circle with respect to its radius <img decoding=\"async\" style=\"height: 13px; width: 12px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image005.png\" \/>\u00a0when<\/strong><\/p>\n<p><strong>(a) <img decoding=\"async\" style=\"height: 13px; width: 12px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image005.png\" \/>\u00a0= 3 cm<\/strong><\/p>\n<p><strong>(b) <img decoding=\"async\" style=\"height: 13px; width: 12px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image005.png\" \/>\u00a0= 4 cm<\/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\/Ex6.1\/image006.png\" \/>\u00a0denote the area of the circle of variable radius <img decoding=\"async\" style=\"height: 15px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image007.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image008.png\" \/>Area of circle <img decoding=\"async\" style=\"height: 27px; width: 65px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image009.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image010.png\" \/>Rate of change of area <img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image006.png\" \/>\u00a0w.r.t. <img decoding=\"async\" style=\"height: 13px; width: 12px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image005.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 41px; width: 120px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image011.png\" \/><\/p>\n<p><strong>(a)<\/strong> When <img decoding=\"async\" style=\"height: 19px; width: 35px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image012.png\" \/>\u00a0cm, then<\/p>\n<p><img decoding=\"async\" style=\"height: 41px; width: 113px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image013.png\" \/>\u00a0sq. cm<\/p>\n<p><strong>(b)<\/strong> When <img decoding=\"async\" style=\"height: 17px; width: 36px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image014.png\" \/>\u00a0cm, then<\/p>\n<p><img decoding=\"async\" style=\"height: 41px; width: 113px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image015.png\" \/>\u00a0sq. cm<\/p>\n<hr \/>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_61\"><\/span>NCERT Solutions class 12 Maths Exercise 6.1<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"2_The_volume_of_a_cube_is_increasing_at_the_rate_of_8_cm3sec_How_fast_is_the_surface_area_increasing_when_the_length_of_an_edge_is_12_cm\"><\/span><strong>2. The volume of a cube is increasing at the rate of 8 cm<sup>3<\/sup>\/sec. How fast is the surface area increasing when the length of an edge is 12 cm?<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\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\/Ex6.1\/image006.png\" \/>\u00a0cm be the edge of the cube.<\/p>\n<p>Given: Rate of increase of volume of cube = 8 cm<sup>3<\/sup>\/sec<\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image016.png\" \/><img decoding=\"async\" style=\"height: 41px; width: 126px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image017.png\" \/>\u00a0is positive = 8<\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image016.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 79px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image018.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image016.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 65px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image019.png\" \/> \u2026\u2026\u2026.(i)<\/p>\n<p>Let <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image020.png\" \/>\u00a0be the surface area of the cube, i.e., \u00a0<img decoding=\"async\" style=\"height: 24px; width: 52px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image021.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image010.png\" \/>\u00a0Rate of change of surface area of the cube = <img decoding=\"async\" style=\"height: 41px; width: 80px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image022.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 45px; width: 66px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image023.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 45px; width: 71px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image024.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 45px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image025.png\" \/>\u00a0cm<sup>2<\/sup>\/sec<\/p>\n<p>Putting <img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image006.png\" \/>\u00a0= 12 cm (given),<\/p>\n<p><img decoding=\"async\" style=\"height: 41px; width: 83px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image026.png\" \/>\u00a0cm<sup>2<\/sup>\/sec<\/p>\n<p>Since <img decoding=\"async\" style=\"height: 41px; width: 23px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image027.png\" \/>\u00a0is positive, therefore surface area is increasing at the rate of <img decoding=\"async\" style=\"height: 41px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image028.png\" \/>\u00a0cm<sup>2<\/sup>\/sec.<\/p>\n<hr \/>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_61-2\"><\/span>NCERT Solutions class 12 Maths Exercise 6.1<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"3_The_radius_of_the_circle_is_increasing_uniformly_at_the_rate_of_3_cm_per_second_Find_the_rate_at_which_the_area_of_the_circle_is_increasing_when_the_radius_is_10_cm\"><\/span><strong>3. The radius of the circle is increasing uniformly at the rate of 3 cm per second. Find the rate at which the area of the circle is increasing when the radius is 10 cm.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\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\/Ex6.1\/image006.png\" \/>\u00a0cm be the radius of the circle at time <img decoding=\"async\" style=\"height: 16px; width: 12px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image029.png\" \/><\/p>\n<p>Rate of increase of radius of circle = 3 cm\/sec<\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image016.png\" \/><img decoding=\"async\" style=\"height: 41px; width: 23px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image030.png\" \/>\u00a0is positive and = 3 cm\/sec<\/p>\n<p>Let <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image020.png\" \/>\u00a0be the area of the circle.<\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image016.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 24px; width: 54px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image031.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image010.png\" \/>Rate of change of area of circle = <img decoding=\"async\" style=\"height: 41px; width: 23px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image027.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 41px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image032.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 41px; width: 119px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image033.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 19px; width: 30px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image034.png\" \/><\/p>\n<p>Putting <img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image006.png\" \/>\u00a0= 10 cm (given), <img decoding=\"async\" style=\"height: 41px; width: 23px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image027.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 27px; width: 92px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image035.png\" \/>\u00a0cm<sup>2<\/sup>\/sec<\/p>\n<p>Since <img decoding=\"async\" style=\"height: 41px; width: 23px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image027.png\" \/>\u00a0is positive, therefore surface area is increasing at the rate of <img decoding=\"async\" style=\"height: 19px; width: 30px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image036.png\" \/>\u00a0cm<sup>2<\/sup>\/sec.<\/p>\n<hr \/>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_61-3\"><\/span>NCERT Solutions class 12 Maths Exercise 6.1<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"4_An_edge_of_a_variable_cube_is_increasing_at_the_rate_of_3_cm_per_second_How_fast_is_the_volume_of_the_cube_increasing_when_the_edge_if_10_cm_long\"><\/span><strong>4. An edge of a variable cube is increasing at the rate of 3 cm per second. How fast is the volume of the cube increasing when the edge if 10 cm long?<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\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\/Ex6.1\/image006.png\" \/>\u00a0cm be the edge of variable cube at time <img decoding=\"async\" style=\"height: 16px; width: 12px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image029.png\" \/><\/p>\n<p>Rate of increase of edge = 3 cm\/sec<\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image016.png\" \/><img decoding=\"async\" style=\"height: 41px; width: 23px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image030.png\" \/>\u00a0is positive and = 3 cm\/sec<\/p>\n<p>Let <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image020.png\" \/>\u00a0be the volume of the cube.<\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image016.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 24px; width: 42px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image037.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image010.png\" \/>Rate of change of volume of cube = <img decoding=\"async\" style=\"height: 41px; width: 23px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image027.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 41px; width: 36px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image038.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 41px; width: 107px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image039.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 21px; width: 27px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image040.png\" \/>\u00a0cm<sup>3<\/sup>\/sec<\/p>\n<p>Putting <img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image006.png\" \/>\u00a0= 10 cm (given),<\/p>\n<p><img decoding=\"async\" style=\"height: 41px; width: 23px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image027.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 30px; width: 87px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image041.png\" \/>\u00a0cm<sup>3<\/sup>\/sec<\/p>\n<p>Since <img decoding=\"async\" style=\"height: 41px; width: 23px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image027.png\" \/>\u00a0is positive, therefore volume of cube is increasing at the rate of 900 cm<sup>3<\/sup>\/sec.<\/p>\n<hr \/>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_61-4\"><\/span>NCERT Solutions class 12 Maths Exercise 6.1<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"5_A_stone_is_dropped_into_a_quite_lake_and_waves_move_in_circles_at_the_rate_of_5_cmsec_At_the_instant_when_radius_of_the_circular_wave_is_8_cm_how_fast_is_the_enclosed_area_increasing\"><\/span><strong>5. A stone is dropped into a quite lake and waves move in circles at the rate of 5 cm\/sec. At the instant when radius of the circular wave is 8 cm, how fast is the enclosed area increasing?<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\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\/Ex6.1\/image006.png\" \/>\u00a0cm be the radius of the circular wave at time <img decoding=\"async\" style=\"height: 16px; width: 12px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image029.png\" \/><\/p>\n<p>Rate of increase of radius of circular wave = 5 cm\/sec<\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image016.png\" \/><img decoding=\"async\" style=\"height: 41px; width: 23px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image030.png\" \/>\u00a0is positive and = 5 cm\/sec<\/p>\n<p>Let <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image020.png\" \/>\u00a0be the enclosed area of the circular wave.<\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image016.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 24px; width: 54px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image042.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image010.png\" \/>Rate of change of area = <img decoding=\"async\" style=\"height: 41px; width: 23px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image027.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 41px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image032.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 41px; width: 119px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image043.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image044.png\" \/><\/p>\n<p>Putting <img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image006.png\" \/>\u00a0= 8 cm (given),<\/p>\n<p><img decoding=\"async\" style=\"height: 41px; width: 23px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image027.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 27px; width: 92px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image045.png\" \/>\u00a0cm<sup>2<\/sup>\/sec<\/p>\n<p>Since <img decoding=\"async\" style=\"height: 41px; width: 23px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image027.png\" \/>\u00a0is positive, therefore area of circular wave is increasing at the rate of <img decoding=\"async\" style=\"height: 19px; width: 30px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image046.png\" \/>\u00a0cm<sup>2<\/sup>\/sec.<\/p>\n<hr \/>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_61-5\"><\/span>NCERT Solutions class 12 Maths Exercise 6.1<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"6_The_radius_of_a_circle_is_increasing_at_the_rate_of_07_cms_What_is_the_rate_of_its_circumference\"><\/span><strong>6. The radius of a circle is increasing at the rate of 0.7 cm\/s. What is the rate of its circumference?<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\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\/Ex6.1\/image006.png\" \/>\u00a0cm be the radius of the circle at time <img decoding=\"async\" style=\"height: 16px; width: 12px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image029.png\" \/><\/p>\n<p>Rate of increase of radius of circle = 0.7 cm\/sec<\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image016.png\" \/><img decoding=\"async\" style=\"height: 41px; width: 23px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image030.png\" \/>\u00a0is positive and = 0.7 cm\/sec<\/p>\n<p>Let <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image020.png\" \/>\u00a0be the circumference of the circle.<\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image016.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 56px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/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\/Ex6.1\/image010.png\" \/>Rate of change of circumference of circle = <img decoding=\"async\" style=\"height: 41px; width: 23px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image027.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 41px; width: 52px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image048.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 27px; width: 56px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image049.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 19px; width: 33px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image050.png\" \/>\u00a0cm\/sec<\/p>\n<hr \/>\n<h3><span class=\"ez-toc-section\" id=\"7_The_length_of_a_rectangle_is_decreasing_at_the_rate_of_5_cmminute_When_8_cm_and_6_cm_find_the_rates_of_change_of_a_the_perimeter_and_b_the_area_of_the_rectangle\"><\/span><strong>7. The length <img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image006.png\" \/>\u00a0of a rectangle is decreasing at the rate of 5 cm\/minute. When <img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image006.png\" \/>\u00a0= 8 cm and <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image020.png\" \/>\u00a0= 6 cm, find the rates of change of (a) the perimeter and (b) the area of the rectangle.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Ans. <\/strong>Given: Rate of decrease of length <img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image006.png\" \/>\u00a0of rectangle is 5 cm\/minute.<\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image016.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 23px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image030.png\" \/>\u00a0is negative = \u20135 cm\/minute<\/p>\n<p>Also, Rate of increase of width <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image020.png\" \/>\u00a0of rectangle is 4 cm\/minute<\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image016.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 23px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image027.png\" \/>\u00a0is positive<\/p>\n<p>= 4 cm\/minute<\/p>\n<p><strong>(a)<\/strong> Let <img decoding=\"async\" style=\"height: 13px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image051.png\" \/>\u00a0denotes the perimeter of rectangle.<\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 77px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image052.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image016.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 109px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image053.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 27px; width: 121px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image054.png\" \/>\u00a0is negative.<\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image010.png\" \/>\u00a0Perimeter of the rectangle is decreasing at the rate of 2 cm\/sec.<\/p>\n<p><strong>(b)<\/strong> Let <img decoding=\"async\" style=\"height: 13px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image051.png\" \/>\u00a0denotes the area of rectangle.<\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image016.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 17px; width: 44px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image055.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image016.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 111px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image056.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 27px; width: 111px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image057.png\" \/>\u00a0is positive.<\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image010.png\" \/>\u00a0Area of the rectangle is increasing at the rate of 2 cm<sup>2<\/sup>\/sec.<\/p>\n<hr \/>\n<h3><span class=\"ez-toc-section\" id=\"8_A_balloon_which_always_remains_spherical_on_inflation_is_being_inflated_by_pumping_in_900_cubic_centimeters_of_gas_per_second_Find_the_rate_at_which_the_radius_of_the_balloon_increases_when_the_radius_is_15_cm\"><\/span><strong>8. A balloon, which always remains spherical on inflation, is being inflated by pumping in 900 cubic centimeters of gas per second. Find the rate at which the radius of the balloon increases when the radius is 15 cm.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\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\/Ex6.1\/image006.png\" \/>\u00a0cm be the radius of the spherical balloon at time <img decoding=\"async\" style=\"height: 16px; width: 12px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image029.png\" \/><\/p>\n<p>According to the question,\u00a0 <img decoding=\"async\" style=\"height: 45px; width: 115px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image058.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image016.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 101px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image059.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image016.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 113px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image060.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image016.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 112px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image061.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image016.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image062.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image016.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 49px; width: 95px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image063.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image016.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 47px; width: 97px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/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\/Ex6.1\/image016.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 104px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/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\/Ex6.1\/image010.png\" \/>\u00a0Radius of balloon is increasing at the rate of <img decoding=\"async\" style=\"height: 41px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image066.png\" \/>\u00a0cm sec.<\/p>\n<hr \/>\n<h3><span class=\"ez-toc-section\" id=\"9_A_balloon_which_always_remains_spherical_has_a_variables_radius_Find_the_rate_at_which_its_volume_is_increasing_with_the_radius_when_the_later_is_10_cm\"><\/span><strong>9. A balloon, which always remains spherical has a variables radius. Find the rate at which its volume is increasing with the radius when the later is 10 cm.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Ans. <\/strong>Since, V = <img decoding=\"async\" style=\"height: 41px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image067.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 45px; width: 116px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image068.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 41px; width: 52px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image069.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 21px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image070.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image016.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 148px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image071.png\" \/><\/p>\n<p>Therefore, the volume is increasing at the rate of <img decoding=\"async\" style=\"height: 19px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image072.png\" \/>\u00a0cm<sup>3<\/sup>\/sec.<\/p>\n<hr \/>\n<h3><span class=\"ez-toc-section\" id=\"10_A_ladder_5_cm_long_is_leaning_against_a_wall_The_bottom_of_the_ladder_is_pulled_along_the_ground_away_from_the_wall_at_the_rate_of_2_cms_How_fast_is_its_height_on_the_wall_decreasing_when_the_foot_of_the_ladder_is_4_m_away_from_the_wall\"><\/span><strong>10. A ladder 5 cm long is leaning against a wall. The bottom of the ladder is pulled along the ground, away from the wall, at the rate of 2 cm\/s. How fast is its height on the wall decreasing when the foot of the ladder is 4 m away from the wall?<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Ans. <\/strong>Let AB be the ladder and C is the junction of wall and ground, AB = 5 m \u00a0B<\/p>\n<p>Let CA = <img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image006.png\" \/>\u00a0meters, CB = <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image020.png\" \/>\u00a0meters<\/p>\n<p>According to the equation,<\/p>\n<p><img decoding=\"async\" id=\"AutoShape 1709\" style=\"height: 100px; width: 88px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image073.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image006.png\" \/>\u00a0increases, <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image020.png\" \/>\u00a0decreases<\/p>\n<p>and <img decoding=\"async\" style=\"height: 41px; width: 23px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image030.png\" \/>\u00a0= 2 cm\/s<\/p>\n<p>In <img decoding=\"async\" style=\"height: 21px; width: 51px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image074.png\" \/>AC<sup>2<\/sup> + BC<sup>2<\/sup> = AB<sup>2<\/sup>\u00a0 [Using Pythagoras theorem]\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image016.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 24px; width: 83px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image075.png\" \/>\u00a0 \u2026\u2026\u2026.(i)<\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image016.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 117px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image076.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image016.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 120px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image077.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image016.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 84px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image078.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image016.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 44px; width: 66px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image079.png\" \/> \u2026\u2026\u2026.(ii)<\/p>\n<p>When <img decoding=\"async\" style=\"height: 21px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image080.png\" \/><img decoding=\"async\" style=\"height: 24px; width: 81px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image081.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image016.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image082.png\" \/>[From eq. (i)]\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image010.png\" \/>\u00a0From eq. (ii),\u00a0 <img decoding=\"async\" style=\"height: 41px; width: 113px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image083.png\" \/>\u00a0cm\/s<\/p>\n<hr \/>\n<h3><span class=\"ez-toc-section\" id=\"11_A_particle_moves_along_the_curve_Find_the_points_on_the_curve_at_which_the_coordinate_is_changing_8_times_as_fast_as_the_coordinate\"><\/span><strong>11. A particle moves along the curve <img decoding=\"async\" style=\"height: 24px; width: 79px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image084.png\" \/>\u00a0Find the points on the curve at which the <img decoding=\"async\" style=\"height: 17px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image085.png\" \/>coordinate is changing 8 times as fast as the <img decoding=\"async\" style=\"height: 15px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image086.png\" \/>coordinate.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Ans. <\/strong>Given: Equation of the curve \u00a0<img decoding=\"async\" style=\"height: 24px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image087.png\" \/>\u2026\u2026\u2026.(i)<\/p>\n<p>Let <img decoding=\"async\" style=\"height: 27px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image088.png\" \/>\u00a0be the required point on curve (i)<\/p>\n<p>According to the question, <img decoding=\"async\" style=\"height: 41px; width: 47px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image089.png\" \/>\u00a0\u2026\u2026\u2026.(ii)<\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image010.png\" \/>From eq. (i), <img decoding=\"async\" style=\"height: 41px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image090.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image016.png\" \/><img decoding=\"async\" style=\"height: 21px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image091.png\" \/>\u00a0 [From eq. (ii)]\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image016.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 66px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image092.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image016.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 19px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image093.png\" \/><\/p>\n<p>Taking <img decoding=\"async\" style=\"height: 21px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image080.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image094.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image016.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 42px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image095.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image010.png\" \/>Required point is (4, 11).<\/p>\n<p>Taking <img decoding=\"async\" style=\"height: 21px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image096.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 85px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image097.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image016.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 56px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image098.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image010.png\" \/>Required point is <img decoding=\"async\" style=\"height: 45px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image099.png\" \/>.<\/p>\n<hr \/>\n<h3><span class=\"ez-toc-section\" id=\"12_The_radius_of_an_air_bubble_is_increasing_at_the_rate_of_cms_At_what_rate_is_the_volume_of_the_bubble_increasing_when_the_radius_is_1_cm\"><\/span><strong>12. The radius of an air bubble is increasing at the rate of <img decoding=\"async\" style=\"height: 41px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image100.png\" \/>\u00a0cm\/s. At what rate is the volume of the bubble increasing when the radius is 1 cm?<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\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\/Ex6.1\/image006.png\" \/>\u00a0cm be the radius of the air bubble at time <img decoding=\"async\" style=\"height: 16px; width: 12px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image029.png\" \/><\/p>\n<p>According to question,\u00a0 <img decoding=\"async\" style=\"height: 41px; width: 23px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image030.png\" \/>\u00a0is positive = <img decoding=\"async\" style=\"height: 41px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image100.png\" \/>\u00a0cm\/sec\u00a0 \u2026\u2026\u2026.(i)<\/p>\n<p>Volume of air bubble <img decoding=\"async\" style=\"height: 41px; width: 77px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image101.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image016.png\" \/><img decoding=\"async\" style=\"height: 41px; width: 91px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image102.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 41px; width: 73px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image103.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 45px; width: 66px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image104.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image016.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 71px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image105.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 30px; width: 84px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image106.png\" \/><\/p>\n<p>Therefore, required rate of increase of volume of air bubble is <img decoding=\"async\" style=\"height: 19px; width: 24px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image107.png\" \/>\u00a0cm<sup>3<\/sup>\/sec.<\/p>\n<hr \/>\n<h3><span class=\"ez-toc-section\" id=\"13_A_balloon_which_always_remains_spherical_has_a_variable_diameter_Find_the_rate_of_change_of_its_volume_with_respect_to\"><\/span><strong>13. A balloon which always remains spherical, has a variable diameter <img decoding=\"async\" style=\"height: 41px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image108.png\" \/>\u00a0Find the rate of change of its volume with respect to <img decoding=\"async\" style=\"height: 15px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image109.png\" \/>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Ans. <\/strong>Given: Diameter of the balloon = <img decoding=\"async\" style=\"height: 41px; width: 66px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image110.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image010.png\" \/>Radius of the balloon = <img decoding=\"async\" style=\"height: 41px; width: 66px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image111.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image010.png\" \/>\u00a0Volume of the balloon = <img decoding=\"async\" style=\"height: 49px; width: 111px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image112.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 41px; width: 80px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image113.png\" \/>\u00a0cu. units<\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image010.png\" \/>\u00a0Rate of change of volume w.r.t. <img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image006.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 41px; width: 28px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image114.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 41px; width: 172px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image115.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 41px; width: 104px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image116.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 41px; width: 90px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image117.png\" \/><\/p>\n<hr \/>\n<h3><span class=\"ez-toc-section\" id=\"14_Sand_is_pouring_from_a_pipe_at_the_rate_of_12_cm3s_The_falling_sand_forms_a_cone_on_the_ground_in_such_a_way_that_the_height_of_the_cone_is_always_one-sixth_of_the_radius_of_the_base_How_fast_is_the_height_of_the_sand_cone_increasing_when_the_height_is_4cm\"><\/span><strong>14. Sand is pouring from a pipe at the rate of 12 cm<sup>3<\/sup>\/s. The falling sand forms a cone on the ground in such a way that the height of the cone is always one-sixth of the radius of the base. How fast is the height of the sand cone increasing when the height is 4cm?<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Ans. <\/strong>Let the height and radius of the sand-cone formed at time <img decoding=\"async\" style=\"height: 16px; width: 9px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image118.png\" \/>\u00a0second be <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image020.png\" \/>\u00a0cm and <img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image006.png\" \/>\u00a0cm respectively.<\/p>\n<p>According to question,\u00a0 <img decoding=\"async\" style=\"height: 41px; width: 51px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image119.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image016.png\" \/><img decoding=\"async\" style=\"height: 21px; width: 47px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image120.png\" \/><\/p>\n<p>Volume of cone (V) = <img decoding=\"async\" style=\"height: 41px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image121.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 41px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image122.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 24px; width: 44px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image123.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image016.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 44px; width: 84px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image124.png\" \/><\/p>\n<p>Now, since <img decoding=\"async\" style=\"height: 41px; width: 56px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/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\/Ex6.1\/image016.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 44px; width: 88px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/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\/Ex6.1\/image016.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 108px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image127.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image016.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 44px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image128.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image016.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 119px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image129.png\" \/>\u00a0cm\/sec<\/p>\n<hr \/>\n<h3><span class=\"ez-toc-section\" id=\"15_The_total_cost_C_in_rupees_associated_with_the_production_of_units_of_an_item_given_by_C_Find_the_marginal_cost_when_17_units_are_produced\"><\/span><strong>15. The total cost C<img decoding=\"async\" style=\"height: 27px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image130.png\" \/>\u00a0in rupees associated with the production of <img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image006.png\" \/>\u00a0units of an item given by \u00a0C<\/strong><img decoding=\"async\" style=\"height: 27px; width: 229px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image131.png\" \/><strong>\u00a0Find the marginal cost when 17 units are produced.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Ans. <\/strong>Marginal cost = <img decoding=\"async\" style=\"height: 41px; width: 27px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image132.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 41px; width: 236px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image133.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 21px; width: 141px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image134.png\" \/><\/p>\n<p>Now, when <img decoding=\"async\" style=\"height: 21px; width: 48px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image135.png\" \/>MC<\/p>\n<p>= <img decoding=\"async\" style=\"height: 30px; width: 180px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image136.png\" \/><\/p>\n<p>= 6.069 \u2013 0.102 + 15 = 20.967<\/p>\n<p>Therefore, required Marginal cost is ` 20.97.<\/p>\n<hr \/>\n<h3><span class=\"ez-toc-section\" id=\"16_The_total_revenue_in_rupees_received_from_the_sale_of_units_of_a_product_is_given_by\"><\/span><strong>16. The total revenue in rupees received from the sale of <img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image006.png\" \/>\u00a0units of a product is given by <\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>R <img decoding=\"async\" style=\"height: 27px; width: 141px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image137.png\" \/>\u00a0Find the marginal revenue when <img decoding=\"async\" style=\"height: 19px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image138.png\" \/>\u00a0<\/strong><\/p>\n<p><strong>Ans. <\/strong>Marginal Revenue (MR) = <img decoding=\"async\" style=\"height: 41px; width: 27px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image139.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 41px; width: 133px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image140.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 19px; width: 60px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image141.png\" \/><\/p>\n<p>Now, when <img decoding=\"async\" style=\"height: 21px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image142.png\" \/>MR = 26 x 7 + 26 = 208<\/p>\n<p>Therefore, the required marginal revenue is ` 208.<\/p>\n<hr \/>\n<p><strong>Choose the correct answer in Exercises 17 and 18.\u00a0<\/strong><\/p>\n<h3><span class=\"ez-toc-section\" id=\"17_The_rate_of_change_of_the_area_of_a_circle_with_respect_to_its_radius_at_6_cm_is\"><\/span><strong>17. The rate of change of the area of a circle with respect to its radius <img decoding=\"async\" style=\"height: 13px; width: 12px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image005.png\" \/>\u00a0at <img decoding=\"async\" style=\"height: 13px; width: 12px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image005.png\" \/>\u00a0=6 cm is:<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>(A) <img decoding=\"async\" style=\"height: 19px; width: 30px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image143.png\" \/><\/strong><\/p>\n<p><strong>(B) <img decoding=\"async\" style=\"height: 19px; width: 30px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image144.png\" \/>\u00a0<\/strong><\/p>\n<p><strong>(C) <img decoding=\"async\" style=\"height: 19px; width: 23px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image145.png\" \/>\u00a0<\/strong><\/p>\n<p><strong>(D) <img decoding=\"async\" style=\"height: 19px; width: 29px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image146.png\" \/>\u00a0<\/strong><\/p>\n<p><strong>Ans. <\/strong>Area of circle (A) = <img decoding=\"async\" style=\"height: 21px; width: 29px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image147.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image016.png\" \/><img decoding=\"async\" style=\"height: 41px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image148.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 19px; width: 84px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image149.png\" \/><\/p>\n<p>Therefore, option (B) is correct.<\/p>\n<hr \/>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_61-6\"><\/span>NCERT Solutions class 12 Maths Exercise 6.1<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>18. The total revenue in Rupees received from the sale of <img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image006.png\" \/>\u00a0units of a product is given by <\/strong><\/p>\n<p><strong>R <img decoding=\"async\" style=\"height: 27px; width: 126px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image150.png\" \/>\u00a0The marginal revenue, when <img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image006.png\" \/>\u00a0= 15 is:<\/strong><\/p>\n<p><strong>(A) 116 <\/strong><\/p>\n<p><strong>(B) 96 <\/strong><\/p>\n<p><strong>(C) 90 <\/strong><\/p>\n<p><strong>(D) 126<\/strong><\/p>\n<p><strong>Ans. <\/strong>Total revenue <img decoding=\"async\" style=\"height: 27px; width: 137px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image151.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image010.png\" \/>Marginal revenue = <img decoding=\"async\" style=\"height: 41px; width: 121px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.1\/image152.png\" \/><\/p>\n<p>= 6 x 15 + 36 = 126<\/p>\n<p>Therefore, option (D) is correct.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_61-7\"><\/span>NCERT Solutions class 12 Maths Exercise 6.1<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>NCERT Solutions Class 12 Maths PDF (Download) Free from myCBSEguide app and myCBSEguide website. Ncert solution class 12 Maths includes text book solutions from both part 1 and part 2. NCERT Solutions for CBSE Class 12 Maths have total 20 chapters. 12 Maths NCERT Solutions in PDF for free Download on our website. Ncert Maths class 12 solutions PDF and Maths ncert class 12 PDF solutions with latest modifications and as per the latest CBSE syllabus are only available in myCBSEguide<\/p>\n<h2><span class=\"ez-toc-section\" id=\"NCERT_Solutions_for_Mathematics_Class_3rd_to_12th\"><\/span>NCERT Solutions for Mathematics Class 3rd to 12th<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<ul>\n<li><a href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-for-class-12-mathematics\/\">NCERT Solutions for Class 12 Mathematics<\/a><\/li>\n<li><a href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-for-class-11-mathematics\/\">NCERT Solutions for Class 11 Mathematics<\/a><\/li>\n<li><a href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-for-class-10-maths\/\">NCERT Solutions for Class 10 Mathematics<\/a><\/li>\n<li><a href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-for-class-9-maths\/\">NCERT Solutions for Class 9 Mathematics<\/a><\/li>\n<li><a href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-for-class-8-mathematics\/\">NCERT Solutions for Class 8 Mathematics<\/a><\/li>\n<li><a href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-for-class-7-mathematics\/\">NCERT Solutions for Class 7 Mathematics<\/a><\/li>\n<li><a href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-6-maths\/\">NCERT Solutions for Class 6 Mathematics<\/a><\/li>\n<li><a href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-for-class-5-mathematics\/\">NCERT Solutions for Class 5 Mathematics<\/a><\/li>\n<li><a href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-for-class-4-mathematics\/\">NCERT Solutions for Class 4 Mathematics<\/a><\/li>\n<li><a href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-for-class-3-mathematics\/\">NCERT Solutions for Class 3 Mathematics<\/a><\/li>\n<\/ul>\n<h2><span class=\"ez-toc-section\" id=\"CBSE_App_for_Class_12\"><\/span>CBSE App for Class 12<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>To download\u00a0NCERT Solutions for class 12 Physics, Chemistry, Biology, History, Political Science, Economics, Geography, Computer Science, Home Science, Accountancy, Business Studies and Home Science; do check myCBSEguide app or website. myCBSEguide provides sample papers with solution, test papers for chapter-wise practice, NCERT solutions, NCERT Exemplar solutions, quick revision notes for ready reference, CBSE guess papers and CBSE important question papers. Sample Paper all are made available through\u00a0<a href=\"https:\/\/play.google.com\/store\/apps\/details?id=in.techchefs.MyCBSEGuide&amp;referrer=utm_source%3Dmycbse_bottom%26utm_medium%3Dtext%26utm_campaign%3Dmycbseads\"><strong>the best app for CBSE students<\/strong><\/a>\u00a0and myCBSEguide website.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>NCERT Solutions class 12 Maths Exercise 6.1 Class 12 Maths book solutions are available in PDF format for free download. These ncert book chapter wise questions and answers are very helpful for CBSE board exam. CBSE recommends NCERT books and most of the questions in CBSE exam are asked from NCERT text books. Class 12 &#8230; <a title=\"NCERT Solutions class 12 Maths Exercise 6.1\" class=\"read-more\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-1\/\" aria-label=\"More on NCERT Solutions class 12 Maths Exercise 6.1\">Read more<\/a><\/p>\n","protected":false},"author":1,"featured_media":-1,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1472,281],"tags":[283,1342,321,216],"class_list":["post-4544","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mathematics-class-12","category-ncert-solutions","tag-cbse-study-material","tag-class-12","tag-mathematics","tag-ncert-solutions"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.0 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>NCERT Solutions class 12 Maths Exercise 6.1 | myCBSEguide<\/title>\n<meta name=\"description\" content=\"NCERT Solutions class 12 Maths Exercise 6.1 in PDF format for free download. 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