{"id":4550,"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-12\/"},"modified":"2018-05-04T15:00:06","modified_gmt":"2018-05-04T09:30:06","slug":"ncert-solutions-class-12-maths-miscellaneous-12","status":"publish","type":"post","link":"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-12\/","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-12\/#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-miscellaneous-12\/#2_Show_that_the_function_given_by_has_maximum_value_at\" >2. Show that the function given by \u00a0has maximum value at \u00a0<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-12\/#NCERT_Solutions_class_12_Maths_Miscellaneous\" >NCERT Solutions class 12 Maths Miscellaneous<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-12\/#3_The_two_equal_sides_of_an_isosceles_triangle_with_fixed_base_are_decreasing_at_the_rate_of_3_cm_per_second_How_fast_is_the_area_decreasing_when_the_two_equal_sides_are_equal_to_the_base\" >3. The two equal sides of an isosceles triangle with fixed base \u00a0are decreasing at the rate of 3 cm per second. How fast is the area decreasing when the two equal sides are equal to the base?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-12\/#NCERT_Solutions_class_12_Maths_Miscellaneous-2\" >NCERT Solutions class 12 Maths Miscellaneous<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-12\/#4_Find_the_equation_of_the_normal_to_the_curve_at_the_point_1_2\" >4. Find the equation of the normal to the curve \u00a0at the point (1, 2).<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-12\/#NCERT_Solutions_class_12_Maths_Miscellaneous-3\" >NCERT Solutions class 12 Maths Miscellaneous<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-12\/#5_Show_that_the_normal_at_any_point_to_the_curve_is_at_a_constant_distance_from_the_origin\" >5. Show that the normal at any point \u00a0to the curve \u00a0\u00a0is at a constant distance from the origin.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-12\/#NCERT_Solutions_class_12_Maths_Miscellaneous-4\" >NCERT Solutions class 12 Maths Miscellaneous<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-12\/#6_Find_the_intervals_in_which_the_function_given_by_is_i_increasing_ii_decreasing\" >6. Find the intervals in which the function \u00a0given by \u00a0is (i) increasing (ii) decreasing.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-11\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-12\/#7_Find_the_intervals_in_which_the_function_given_by_is_i_increasing_ii_decreasing\" >7. Find the intervals in which the function \u00a0given by \u00a0is (i) increasing (ii) decreasing.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-12\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-12\/#NCERT_Solutions_class_12_Maths_Miscellaneous-5\" >NCERT Solutions class 12 Maths Miscellaneous<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-13\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-12\/#8_Find_the_maximum_area_of_an_isosceles_triangle_inscribed_in_the_ellipse_with_its_vertex_at_one_end_of_the_major_axis\" >8. Find the maximum area of an isosceles triangle inscribed in the ellipse \u00a0with its vertex at one end of the major axis.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-14\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-12\/#NCERT_Solutions_class_12_Maths_Miscellaneous-6\" >NCERT Solutions class 12 Maths Miscellaneous<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-15\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-12\/#9_A_tank_with_rectangular_base_and_rectangular_sides_open_at_the_top_is_to_be_constructed_so_that_its_depth_is_2_m_and_volume_is_8_m3_If_building_of_tank_costs_70_per_sq_meter_for_the_base_and_45_per_square_meter_for_sides_What_is_the_cost_of_least_expensive_tank\" >9. A tank with rectangular base and rectangular sides, open at the top is to be constructed so that its depth is 2 m and volume is 8 m3. If building of tank costs ` 70 per sq. meter for the base and ` 45 per square meter for sides. What is the cost of least expensive tank?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-16\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-12\/#NCERT_Solutions_class_12_Maths_Miscellaneous-7\" >NCERT Solutions class 12 Maths Miscellaneous<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-17\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-12\/#10_The_sum_of_the_perimeter_of_a_circle_and_square_is_where_is_some_constant_Prove_that_the_sum_of_their_areas_is_least_when_the_side_of_square_is_double_the_radius_of_the_circle\" >10. The sum of the perimeter of a circle and square is \u00a0where \u00a0is some constant. Prove that the sum of their areas is least when the side of square is double the radius of the circle.<\/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-miscellaneous-12\/#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-3'><a class=\"ez-toc-link ez-toc-heading-19\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-12\/#11_A_window_is_in_the_form_of_a_rectangle_surmounted_by_a_semicircular_opening_The_total_perimeter_of_the_window_is_10_m_Find_the_dimensions_of_the_window_to_admit_maximum_light_through_the_whole_opening\" >11. A window is in the form of a rectangle surmounted by a semicircular opening. The total perimeter of the window is 10 m. Find the dimensions of the window to admit maximum light through the whole opening.<\/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-miscellaneous-12\/#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-3'><a class=\"ez-toc-link ez-toc-heading-21\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-12\/#12_A_point_on_the_hypotenuse_of_a_triangle_is_at_distances_and_from_the_sides_of_the_triangle_Show_that_the_maximum_length_of_the_hypotenuse_is\" >12. A point on the hypotenuse of a triangle is at distances \u00a0and \u00a0from the sides of the triangle. Show that the maximum length of the hypotenuse is \u00a0<\/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-miscellaneous-12\/#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-3'><a class=\"ez-toc-link ez-toc-heading-23\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-12\/#13_Find_the_points_at_which_the_function_given_by_has\" >13. Find the points at which the function \u00a0given by \u00a0has:<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-24\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-12\/#NCERT_Solutions_class_12_Maths_Miscellaneous-11\" >NCERT Solutions class 12 Maths Miscellaneous<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-25\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-12\/#14_Find_the_absolute_maximum_and_minimum_values_of_the_function_given_by\" >14. Find the absolute maximum and minimum values of the function \u00a0given by \u00a0<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-26\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-12\/#NCERT_Solutions_class_12_Maths_Miscellaneous-12\" >NCERT Solutions class 12 Maths Miscellaneous<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-27\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-12\/#15_Show_that_the_altitude_of_the_right_circular_cone_of_maximum_volume_that_can_be_inscribed_in_a_sphere_of_radius_is\" >15. Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius \u00a0is \u00a0<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-28\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-12\/#NCERT_Solutions_class_12_Maths_Miscellaneous-13\" >NCERT Solutions class 12 Maths Miscellaneous<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-29\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-12\/#16_Let_be_a_function_defined_on_such_that_for_all_Then_prove_that_is_an_increasing_function_on\" >16. Let \u00a0be a function defined on \u00a0such that \u00a0for all \u00a0Then prove that \u00a0is an increasing function on \u00a0<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-30\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-12\/#NCERT_Solutions_class_12_Maths_Miscellaneous-14\" >NCERT Solutions class 12 Maths Miscellaneous<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-31\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-12\/#17_Show_that_the_height_of_the_cylinder_of_maximum_volume_that_can_be_inscribed_in_a_sphere_of_radius_R_is_Also_find_the_maximum_volume\" >17. Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius R is \u00a0Also find the maximum volume.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-32\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-12\/#NCERT_Solutions_class_12_Maths_Miscellaneous-15\" >NCERT Solutions class 12 Maths Miscellaneous<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-33\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-12\/#18_Show_that_the_height_of_the_cylinder_of_greatest_volume_which_can_be_inscribed_in_a_right_circular_cone_of_height_and_having_semi-vertical_angle_is_one-third_that_of_the_cone_and_the_greatest_volume_of_the_cylinder_is\" >18. Show that the height of the cylinder of greatest volume which can be inscribed in a right circular cone of height \u00a0and having semi-vertical angle \u00a0is one-third that of the cone and the greatest volume of the cylinder is \u00a0<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-34\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-12\/#NCERT_Solutions_class_12_Maths_Miscellaneous-16\" >NCERT Solutions class 12 Maths Miscellaneous<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-35\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-12\/#19_A_cylindrical_tank_of_radius_10_m_is_being_filled_with_wheat_at_the_rate_of_314_cubic_meter_per_hour_Then_the_depth_of_wheat_is_increasing_at_the_rate_of\" >19. A cylindrical tank of radius 10 m is being filled with wheat at the rate of 314 cubic meter per hour. Then the depth of wheat is increasing at the rate of:<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-36\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-12\/#NCERT_Solutions_class_12_Maths_Miscellaneous-17\" >NCERT Solutions class 12 Maths Miscellaneous<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-37\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-12\/#20_The_slope_of_the_tangent_to_the_curve_at_the_point_is\" >20. The slope of the tangent to the curve \u00a0\u00a0at the point \u00a0is:<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-38\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-12\/#NCERT_Solutions_class_12_Maths_Miscellaneous-18\" >NCERT Solutions class 12 Maths Miscellaneous<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-39\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-12\/#21_The_line_is_a_tangent_to_the_curve_if_the_value_of_is\" >21. The line \u00a0is a tangent to the curve \u00a0if the value of \u00a0is:<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-40\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-12\/#NCERT_Solutions_class_12_Maths_Miscellaneous-19\" >NCERT Solutions class 12 Maths Miscellaneous<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-41\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-12\/#22_The_normal_at_the_point_1_1_on_the_curve_is\" >22. The normal at the point (1, 1) on the curve \u00a0is:<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-42\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-12\/#NCERT_Solutions_class_12_Maths_Miscellaneous-20\" >NCERT Solutions class 12 Maths Miscellaneous<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-43\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-12\/#23_The_normal_to_the_curve_passing_through_1_2_is\" >23. The normal to the curve \u00a0passing through (1, 2) is:<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-44\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-12\/#NCERT_Solutions_class_12_Maths_Miscellaneous-21\" >NCERT Solutions class 12 Maths Miscellaneous<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-45\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-12\/#NCERT_Solutions_class_12_Maths_Miscellaneous-22\" >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-46\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-12\/#CBSE_App_for_Class_12\" >CBSE App for Class 12<\/a><\/li><\/ul><\/nav><\/div>\n<p>NCERT Solutions class 12 Maths Miscellaneous Class 12 Maths book solutions are available in PDF format for free download. These ncert book chapter wise questions and answers are very helpful for CBSE board exam. CBSE recommends NCERT books and most of the questions in CBSE exam are asked from NCERT text books. Class 12 Maths chapter wise NCERT solution for Maths part 1 and Maths part 2 for all the chapters can be downloaded from our website and myCBSEguide mobile app for free.<\/p>\n<p><strong>Download\u00a0<a href=\"https:\/\/mycbseguide.com\/downloads\/cbse-class-12-mathematics-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 Miscellaneous\" width=\"175\" height=\"228\" \/><\/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 style=\"text-align: justify;\"><strong>1. Using differentials, find the approximate value of each of the following:<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(a) <img decoding=\"async\" style=\"height: 51px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image103.png\" \/>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(b) <img decoding=\"async\" style=\"height: 32px; width: 48px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image104.png\" \/>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. (a)<\/strong> <img decoding=\"async\" id=\"Picture 4989\" style=\"height: 55px; width: 44px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image105.png\" \/><\/p>\n<p style=\"text-align: justify;\">Let <img decoding=\"async\" id=\"Picture 4988\" style=\"height: 35px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image106.png\" \/>\u00a0\u2026\u2026\u2026.(i)<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4987\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/><img decoding=\"async\" id=\"Picture 4986\" style=\"height: 52px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image108.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4985\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4984\" style=\"height: 75px; width: 88px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image109.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 4983\" style=\"height: 75px; width: 56px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image110.png\" \/>\u00a0\u2026\u2026\u2026.(ii)<\/p>\n<p style=\"text-align: justify;\">Changing <img decoding=\"async\" id=\"Picture 4982\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image111.png\" \/>\u00a0to <img decoding=\"async\" id=\"Picture 4981\" style=\"height: 19px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image112.png\" \/>\u00a0and <img decoding=\"async\" id=\"Picture 4980\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image113.png\" \/>\u00a0to <img decoding=\"async\" id=\"Picture 4979\" style=\"height: 21px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image114.png\" \/>\u00a0in eq. (i), we have<\/p>\n<p style=\"text-align: justify;\">\u00a0<img decoding=\"async\" id=\"Picture 4978\" style=\"height: 55px; width: 265px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image115.png\" \/> \u2026\u2026\u2026.(iii)<\/p>\n<p style=\"text-align: justify;\">Here <img decoding=\"async\" id=\"Picture 4977\" style=\"height: 41px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image116.png\" \/>\u00a0and <img decoding=\"async\" id=\"Picture 4976\" style=\"height: 41px; width: 55px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image117.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4975\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image118.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4974\" style=\"height: 55px; width: 104px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image119.png\" \/><\/p>\n<p style=\"text-align: justify;\">From eq. (iii),<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4973\" style=\"height: 85px; width: 260px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image120.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4972\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4971\" style=\"height: 87px; width: 132px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image121.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4970\" style=\"height: 11px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image122.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4969\" style=\"height: 41px; width: 77px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image123.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4968\" style=\"height: 11px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image122.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4967\" style=\"height: 41px; width: 95px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image124.png\" \/>\u00a00.677<\/p>\n<p style=\"text-align: justify;\"><strong>(b)<\/strong> <img decoding=\"async\" id=\"Picture 4966\" style=\"height: 35px; width: 43px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image125.png\" \/><\/p>\n<p style=\"text-align: justify;\">Let <img decoding=\"async\" id=\"Picture 4965\" style=\"height: 35px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image126.png\" \/>\u00a0\u2026\u2026\u2026.(i)<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4964\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/><img decoding=\"async\" id=\"Picture 4963\" style=\"height: 52px; width: 65px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image127.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4962\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4960\" style=\"height: 75px; width: 87px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image128.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 4959\" style=\"height: 75px; width: 55px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image129.png\" \/>\u00a0\u2026\u2026\u2026.(ii)<\/p>\n<p style=\"text-align: justify;\">Changing <img decoding=\"async\" id=\"Picture 4958\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image111.png\" \/>\u00a0to <img decoding=\"async\" id=\"Picture 4957\" style=\"height: 19px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image112.png\" \/>\u00a0and <img decoding=\"async\" id=\"Picture 4956\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image113.png\" \/>\u00a0to <img decoding=\"async\" id=\"Picture 4955\" style=\"height: 21px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image114.png\" \/>\u00a0in eq. (i), we have<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4954\" style=\"height: 35px; width: 256px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image130.png\" \/>\u00a0\u2026\u2026.(iii)<\/p>\n<p style=\"text-align: justify;\">Here <img decoding=\"async\" id=\"Picture 4953\" style=\"height: 19px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image131.png\" \/>\u00a0and <img decoding=\"async\" id=\"Picture 4952\" style=\"height: 19px; width: 43px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image132.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4951\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image118.png\" \/><img decoding=\"async\" id=\"Picture 4950\" style=\"height: 44px; width: 108px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image133.png\" \/><\/p>\n<p style=\"text-align: justify;\">From eq. (iii),<\/p>\n<p style=\"text-align: justify;\">\u00a0<img decoding=\"async\" id=\"Picture 4948\" style=\"height: 77px; width: 263px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image134.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4947\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/><img decoding=\"async\" id=\"Picture 4946\" style=\"height: 49px; width: 124px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image135.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4945\" style=\"height: 11px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image122.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4944\" style=\"height: 41px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image136.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4943\" style=\"height: 11px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image122.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4942\" style=\"height: 41px; width: 109px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image137.png\" \/>\u00a00.497<\/p>\n<hr \/>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"2_Show_that_the_function_given_by_has_maximum_value_at\"><\/span><strong>2. Show that the function given by <img decoding=\"async\" style=\"height: 41px; width: 88px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image138.png\" \/>\u00a0has maximum value at <img decoding=\"async\" style=\"height: 15px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image139.png\" \/>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Here <img decoding=\"async\" id=\"Picture 4941\" style=\"height: 41px; width: 128px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image140.png\" \/>\u00a0\u2026\u2026\u2026.(i)<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4939\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image118.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4938\" style=\"height: 60px; width: 208px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image141.png\" \/>\u00a0 \u2026\u2026..(ii)<\/p>\n<p style=\"text-align: justify;\">And <img decoding=\"async\" id=\"Picture 4937\" style=\"height: 64px; width: 341px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image142.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 4936\" style=\"height: 41px; width: 87px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image143.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4935\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4934\" style=\"height: 44px; width: 228px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image144.png\" \/>\u00a0\u2026\u2026\u2026.(iii)<\/p>\n<p style=\"text-align: justify;\">Now <img decoding=\"async\" id=\"Picture 4933\" style=\"height: 27px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image145.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4932\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4931\" style=\"height: 41px; width: 81px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image146.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4930\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4929\" style=\"height: 21px; width: 79px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image147.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4928\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 31\" style=\"height: 21px; width: 57px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image148.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 30\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 29\" style=\"height: 21px; width: 81px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image149.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 28\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 15px; width: 36px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image150.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image118.png\" \/>From eq. (iii),<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 25\" style=\"height: 41px; width: 124px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image151.png\" \/>\u00a0= <img decoding=\"async\" id=\"Picture 24\" style=\"height: 41px; width: 36px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image152.png\" \/>\u00a0= <img decoding=\"async\" id=\"Picture 23\" style=\"height: 41px; width: 23px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image153.png\" \/>\u00a0&lt; 0<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 22\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image118.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 15px; width: 36px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image150.png\" \/>\u00a0is a point of local maxima and maximum value of <img decoding=\"async\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image154.png\" \/>\u00a0is at <img decoding=\"async\" style=\"height: 15px; width: 36px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image150.png\" \/>.<\/p>\n<hr \/>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"3_The_two_equal_sides_of_an_isosceles_triangle_with_fixed_base_are_decreasing_at_the_rate_of_3_cm_per_second_How_fast_is_the_area_decreasing_when_the_two_equal_sides_are_equal_to_the_base\"><\/span><strong>3. The two equal sides of an isosceles triangle with fixed base <img decoding=\"async\" style=\"height: 19px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image155.png\" \/>\u00a0are decreasing at the rate of 3 cm per second. How fast is the area decreasing when the two equal sides are equal to the base?<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Let BC = <img decoding=\"async\" id=\"Picture 18\" style=\"height: 19px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image155.png\" \/>\u00a0be the fixed base and AB = AC = <img decoding=\"async\" id=\"Picture 17\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image111.png\" \/>\u00a0be<\/p>\n<p style=\"text-align: justify;\">the two equal sides of given isosceles triangle.<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 1\" style=\"height: 170px; width: 119px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image156.png\" \/><\/p>\n<p style=\"text-align: justify;\">Since <img decoding=\"async\" style=\"height: 41px; width: 55px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image157.png\" \/>\u00a0cm\/s\u00a0 \u2026\u2026\u2026.(i)<\/p>\n<p style=\"text-align: justify;\">Area of <img decoding=\"async\" id=\"Picture 15\" style=\"height: 41px; width: 73px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image158.png\" \/>\u00a0x BC x AM<\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 14\" style=\"height: 48px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image159.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 13\" style=\"height: 48px; width: 176px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image160.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 12\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image118.png\" \/><img decoding=\"async\" id=\"Picture 11\" style=\"height: 45px; width: 153px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image161.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 10\" style=\"height: 41px; width: 155px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image162.png\" \/>\u00a0[By chain rule]\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 9\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/><img decoding=\"async\" id=\"Picture 8\" style=\"height: 45px; width: 209px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image163.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 7\" style=\"height: 45px; width: 72px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image164.png\" \/>\u00a0cm<sup>2<\/sup>\/s<\/p>\n<p style=\"text-align: justify;\">Now when <img decoding=\"async\" id=\"Picture 6\" style=\"height: 21px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image165.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5\" style=\"height: 41px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image166.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4\" style=\"height: 48px; width: 172px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image167.png\" \/>\u00a0cm<sup>2<\/sup>\/s<\/p>\n<p style=\"text-align: justify;\">Therefore, the area is decreasing at the rate of <img decoding=\"async\" id=\"Picture 2\" style=\"height: 24px; width: 31px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image168.png\" \/>\u00a0cm<sup>2<\/sup>\/s.<\/p>\n<hr \/>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-2\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"4_Find_the_equation_of_the_normal_to_the_curve_at_the_point_1_2\"><\/span><strong>4. Find the equation of the normal to the curve <img decoding=\"async\" id=\"Picture 16\" style=\"height: 24px; width: 53px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image169.png\" \/>\u00a0at the point (1, 2).<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Equation of the curve is <img decoding=\"async\" id=\"Picture 5157\" style=\"height: 24px; width: 53px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image169.png\" \/>\u2026\u2026\u2026.(i)<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5156\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/><img decoding=\"async\" id=\"Picture 5155\" style=\"height: 41px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image170.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5154\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5153\" style=\"height: 44px; width: 51px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image171.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5152\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image118.png\" \/>\u00a0Slope of the tangent to the curve at the point (1, 2) to curve (i) is <img decoding=\"async\" id=\"Picture 5151\" style=\"height: 41px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image172.png\" \/>\u00a0=1<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5150\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image118.png\" \/>\u00a0Slope of the normal to the curve at (1, 2) is <img decoding=\"async\" id=\"Picture 5149\" style=\"height: 41px; width: 88px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image173.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5148\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image118.png\" \/>\u00a0Equation of the normal to the curve (i) at (1, 2) is \u00a0<img decoding=\"async\" id=\"Picture 5147\" style=\"height: 27px; width: 109px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image174.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5146\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5145\" style=\"height: 21px; width: 89px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image175.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5144\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5143\" style=\"height: 21px; width: 60px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image176.png\" \/><\/p>\n<hr \/>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-3\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"5_Show_that_the_normal_at_any_point_to_the_curve_is_at_a_constant_distance_from_the_origin\"><\/span><strong>5. Show that the normal at any point <img decoding=\"async\" id=\"Picture 19\" style=\"height: 19px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image177.png\" \/>\u00a0to the curve <img decoding=\"async\" id=\"Picture 20\" style=\"height: 21px; width: 139px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image178.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 21\" style=\"height: 21px; width: 136px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image179.png\" \/>\u00a0is at a constant distance from the origin.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>The parametric equations of the curve are<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5142\" style=\"height: 21px; width: 273px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image180.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5141\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5140\" style=\"height: 27px; width: 283px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image181.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5139\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image118.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5138\" style=\"height: 41px; width: 273px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image182.png\" \/><\/p>\n<p style=\"text-align: justify;\">And <img decoding=\"async\" id=\"Picture 5137\" style=\"height: 41px; width: 457px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image183.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5136\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image118.png\" \/>Slope of tangent at point <img decoding=\"async\" id=\"Picture 5135\" style=\"height: 27px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image184.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5134\" style=\"height: 41px; width: 204px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image185.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5133\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image118.png\" \/>Slope of normal at any point <img decoding=\"async\" id=\"Picture 5132\" style=\"height: 19px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image177.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5131\" style=\"height: 41px; width: 159px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image186.png\" \/><\/p>\n<p style=\"text-align: justify;\">And Equation of normal at any point <img decoding=\"async\" id=\"Picture 5130\" style=\"height: 19px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image177.png\" \/><\/p>\n<p style=\"text-align: justify;\">i.e., at <img decoding=\"async\" id=\"Picture 5129\" style=\"height: 27px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image184.png\" \/>\u00a0= <img decoding=\"async\" id=\"Picture 5128\" style=\"height: 29px; width: 247px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image187.png\" \/>\u00a0is <img decoding=\"async\" id=\"Picture 5127\" style=\"height: 41px; width: 349px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image188.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5126\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5125\" style=\"height: 24px; width: 435px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image189.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5124\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5123\" style=\"height: 29px; width: 232px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image190.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5122\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5121\" style=\"height: 21px; width: 128px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image191.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5120\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5119\" style=\"height: 21px; width: 149px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image192.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5118\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image118.png\" \/>Distance of normal from origin (0, 0)<\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5117\" style=\"height: 48px; width: 135px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image193.png\" \/>\u00a0which is a constant.<\/p>\n<hr \/>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-4\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"6_Find_the_intervals_in_which_the_function_given_by_is_i_increasing_ii_decreasing\"><\/span><strong>6. Find the intervals in which the function <img decoding=\"async\" id=\"Picture 26\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image194.png\" \/>\u00a0given by <img decoding=\"async\" id=\"Picture 27\" style=\"height: 41px; width: 183px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image195.png\" \/>\u00a0is (i) increasing (ii) decreasing.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Given: <img decoding=\"async\" id=\"Picture 5116\" style=\"height: 41px; width: 183px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image195.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5115\" style=\"height: 44px; width: 137px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image196.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5114\" style=\"height: 44px; width: 153px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image197.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5113\" style=\"height: 41px; width: 84px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image198.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5112\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image118.png\" \/><img decoding=\"async\" style=\"height: 67px; width: 361px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image199.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5111\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/><img decoding=\"async\" id=\"Picture 5110\" style=\"height: 52px; width: 309px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image200.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5109\" style=\"height: 52px; width: 189px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image201.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5108\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5107\" style=\"height: 49px; width: 153px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image202.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5106\" style=\"height: 56px; width: 157px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image203.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5105\" style=\"height: 52px; width: 209px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image204.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5104\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5103\" style=\"height: 52px; width: 155px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image205.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" style=\"height: 52px; width: 113px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image206.png\" \/>\u2026\u2026\u2026.(i)<\/p>\n<p style=\"text-align: justify;\">Now <img decoding=\"async\" id=\"Picture 5102\" style=\"height: 19px; width: 83px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image207.png\" \/>\u00a0for all real <img decoding=\"async\" id=\"Picture 5101\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image111.png\" \/>\u00a0as <img decoding=\"async\" id=\"Picture 5100\" style=\"height: 19px; width: 88px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image208.png\" \/>. Also <img decoding=\"async\" id=\"Picture 5099\" style=\"height: 29px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image209.png\" \/>\u00a0&gt; 0<\/p>\n<p style=\"text-align: justify;\">(i) <img decoding=\"async\" id=\"Picture 5097\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image154.png\" \/>\u00a0is increasing if <img decoding=\"async\" id=\"Picture 5096\" style=\"height: 27px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image210.png\" \/>, i.e., from eq. (i), <img decoding=\"async\" id=\"Picture 5095\" style=\"height: 19px; width: 60px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image211.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5094\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/><img decoding=\"async\" id=\"Picture 5093\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image111.png\" \/>\u00a0lies in I and IV quadrants, i.e., <img decoding=\"async\" id=\"Picture 5092\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image154.png\" \/>\u00a0is increasing for <img decoding=\"async\" id=\"Picture 5091\" style=\"height: 41px; width: 65px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image212.png\" \/><\/p>\n<p style=\"text-align: justify;\">and <img decoding=\"async\" id=\"Picture 5090\" style=\"height: 41px; width: 83px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image213.png\" \/><\/p>\n<p style=\"text-align: justify;\">and (ii) <img decoding=\"async\" id=\"Picture 5089\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image154.png\" \/>\u00a0is decreasing if <img decoding=\"async\" id=\"Picture 5088\" style=\"height: 27px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image214.png\" \/>, i.e., from eq. (i), <img decoding=\"async\" id=\"Picture 5087\" style=\"height: 19px; width: 60px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image215.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5086\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5085\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image111.png\" \/>\u00a0lies in II and III quadrants, i.e., <img decoding=\"async\" id=\"Picture 5084\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image154.png\" \/>\u00a0is decreasing for <img decoding=\"async\" id=\"Picture 5083\" style=\"height: 41px; width: 77px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image216.png\" \/><\/p>\n<hr \/>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"7_Find_the_intervals_in_which_the_function_given_by_is_i_increasing_ii_decreasing\"><\/span><strong>7. Find the intervals in which the function <img decoding=\"async\" id=\"Picture 33\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image194.png\" \/>\u00a0given by <img decoding=\"async\" id=\"Picture 34\" style=\"height: 41px; width: 140px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image217.png\" \/>\u00a0is (i) increasing (ii) decreasing.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>Ans. (i)<\/strong> Given:\u00a0 <img decoding=\"async\" id=\"Picture 5082\" style=\"height: 41px; width: 140px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image217.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5081\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/><img decoding=\"async\" id=\"Picture 5080\" style=\"height: 27px; width: 101px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image218.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5079\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image118.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5078\" style=\"height: 45px; width: 209px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image219.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5077\" style=\"height: 51px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image220.png\" \/>\u00a0= <img decoding=\"async\" id=\"Picture 5076\" style=\"height: 41px; width: 97px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image221.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5075\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5074\" style=\"height: 41px; width: 203px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image222.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5073\" style=\"height: 41px; width: 184px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image223.png\" \/>\u2026\u2026\u2026.(i)<\/p>\n<p style=\"text-align: justify;\">Now <img decoding=\"async\" id=\"Picture 5072\" style=\"height: 27px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image145.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5071\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/><img decoding=\"async\" id=\"Picture 5070\" style=\"height: 41px; width: 184px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image223.png\" \/>\u00a0= 0<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5069\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/><img decoding=\"async\" id=\"Picture 5068\" style=\"height: 29px; width: 173px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image224.png\" \/>\u00a0= 0<\/p>\n<p style=\"text-align: justify;\">Here, <img decoding=\"async\" id=\"Picture 5067\" style=\"height: 29px; width: 89px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image225.png\" \/>\u00a0is positive for all real <img decoding=\"async\" id=\"Picture 5066\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image111.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5065\" style=\"height: 19px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image226.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5064\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image118.png\" \/><img decoding=\"async\" id=\"Picture 5063\" style=\"height: 19px; width: 56px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image227.png\" \/>\u00a0or <img decoding=\"async\" id=\"Picture 5062\" style=\"height: 19px; width: 56px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image228.png\" \/>\u00a0[Turning points]\n<p style=\"text-align: justify;\">Therefore, <img decoding=\"async\" id=\"Picture 5061\" style=\"height: 19px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image229.png\" \/>\u00a0or <img decoding=\"async\" id=\"Picture 5060\" style=\"height: 19px; width: 35px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image230.png\" \/>\u00a0divide the real line into three sub intervals <img decoding=\"async\" id=\"Picture 5059\" style=\"height: 27px; width: 105px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image231.png\" \/>\u00a0and <img decoding=\"async\" id=\"Picture 5058\" style=\"height: 27px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image232.png\" \/><\/p>\n<p style=\"text-align: justify;\">For <img decoding=\"async\" id=\"Picture 5057\" style=\"height: 27px; width: 59px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image233.png\" \/>, \u00a0from eq. (i) at <img decoding=\"async\" id=\"Picture 5056\" style=\"height: 19px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image234.png\" \/>\u00a0(say),<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5055\" style=\"height: 27px; width: 164px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image235.png\" \/><\/p>\n<p style=\"text-align: justify;\">\u00a0Therefore, <img decoding=\"async\" id=\"Picture 5054\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image154.png\" \/>\u00a0is increasing at <img decoding=\"async\" id=\"Picture 5053\" style=\"height: 19px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image229.png\" \/><\/p>\n<p style=\"text-align: justify;\">For <img decoding=\"async\" id=\"Picture 5052\" style=\"height: 27px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image236.png\" \/>from eq. (i) at <img decoding=\"async\" id=\"Picture 5051\" style=\"height: 41px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image237.png\" \/>\u00a0(say)<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5050\" style=\"height: 27px; width: 164px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image238.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, <img decoding=\"async\" id=\"Picture 5049\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image154.png\" \/>\u00a0is decreasing at <img decoding=\"async\" id=\"Picture 5048\" style=\"height: 21px; width: 56px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image239.png\" \/><\/p>\n<p style=\"text-align: justify;\">For <img decoding=\"async\" id=\"Picture 5047\" style=\"height: 27px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image240.png\" \/>from eq. (i) at <img decoding=\"async\" id=\"Picture 5046\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image241.png\" \/>\u00a0(say),<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5045\" style=\"height: 27px; width: 165px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image242.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, <img decoding=\"async\" id=\"Picture 5044\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image154.png\" \/>\u00a0is increasing at <img decoding=\"async\" id=\"Picture 5043\" style=\"height: 19px; width: 35px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image230.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, <img decoding=\"async\" id=\"Picture 5042\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image154.png\" \/>\u00a0is (i) an increasing function for <img decoding=\"async\" id=\"Picture 5041\" style=\"height: 19px; width: 44px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image243.png\" \/>\u00a0and for <img decoding=\"async\" id=\"Picture 5040\" style=\"height: 19px; width: 33px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image244.png\" \/>\u00a0and (ii) decreasing function for <img decoding=\"async\" id=\"Picture 5039\" style=\"height: 19px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image245.png\" \/><\/p>\n<hr \/>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-5\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"8_Find_the_maximum_area_of_an_isosceles_triangle_inscribed_in_the_ellipse_with_its_vertex_at_one_end_of_the_major_axis\"><\/span><strong>8. Find the maximum area of an isosceles triangle inscribed in the ellipse <img decoding=\"async\" id=\"Picture 37\" style=\"height: 44px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image246.png\" \/>\u00a0with its vertex at one end of the major axis.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Equation of the ellipse is <img decoding=\"async\" id=\"Picture 5038\" style=\"height: 44px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image246.png\" \/>\u00a0\u2026\u2026\u2026.(i)<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 160px; width: 290px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image247.png\" \/><\/p>\n<p style=\"text-align: justify;\">Comparing eq. (i) with <img decoding=\"async\" id=\"Picture 5037\" style=\"height: 24px; width: 120px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image248.png\" \/>\u00a0we have <img decoding=\"async\" id=\"Picture 5036\" style=\"height: 41px; width: 64px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image249.png\" \/>\u00a0and <img decoding=\"async\" id=\"Picture 5035\" style=\"height: 41px; width: 63px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image250.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5034\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/><img decoding=\"async\" id=\"Picture 5033\" style=\"height: 19px; width: 72px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image251.png\" \/>\u00a0and <img decoding=\"async\" id=\"Picture 5032\" style=\"height: 21px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image252.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5031\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image118.png\" \/>Any point on ellipse is P<img decoding=\"async\" id=\"Picture 5030\" style=\"height: 27px; width: 109px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image253.png\" \/><\/p>\n<p style=\"text-align: justify;\">Draw PM perpendicular to <img decoding=\"async\" id=\"Picture 5029\" style=\"height: 15px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image254.png\" \/>axis and produce it to meet the ellipse in the point Q.<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5028\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image118.png\" \/>\u00a0OM = <img decoding=\"async\" id=\"Picture 5027\" style=\"height: 19px; width: 48px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image255.png\" \/>\u00a0and PM = <img decoding=\"async\" id=\"Picture 5026\" style=\"height: 19px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image256.png\" \/><\/p>\n<p style=\"text-align: justify;\">We know that the ellipse (i) is symmetrical about <img decoding=\"async\" id=\"Picture 5025\" style=\"height: 15px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image254.png\" \/>axis, therefore, PM = QM and hence triangle APQ is isosceles.<\/p>\n<p style=\"text-align: justify;\">Area of <img decoding=\"async\" id=\"Picture 5024\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image257.png\" \/>APQ <img decoding=\"async\" id=\"Picture 5023\" style=\"height: 41px; width: 52px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image258.png\" \/>\u00a0x Base x Height<\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5022\" style=\"height: 41px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image259.png\" \/>\u00a0PQ.AM = <img decoding=\"async\" id=\"Picture 5021\" style=\"height: 41px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image259.png\" \/>\u00a0. 2PM.AM = PM (OA \u2013 OM)<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5020\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/><img decoding=\"async\" id=\"Picture 5019\" style=\"height: 27px; width: 148px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image260.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5018\" style=\"height: 27px; width: 144px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image261.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5017\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/><img decoding=\"async\" id=\"Picture 5016\" style=\"height: 41px; width: 189px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image262.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5015\" style=\"height: 41px; width: 131px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image263.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5014\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5013\" style=\"height: 41px; width: 181px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image264.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5012\" style=\"height: 27px; width: 123px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image265.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5011\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5010\" style=\"height: 44px; width: 183px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image266.png\" \/><\/p>\n<p style=\"text-align: justify;\">Now <img decoding=\"async\" id=\"Picture 5009\" style=\"height: 41px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image267.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5008\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/><img decoding=\"async\" id=\"Picture 5007\" style=\"height: 27px; width: 123px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image265.png\" \/>\u00a0= 0<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5006\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5005\" style=\"height: 19px; width: 116px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image268.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5004\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5003\" style=\"height: 19px; width: 93px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image269.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5002\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5001\" style=\"height: 27px; width: 149px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image270.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5000\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4999\" style=\"height: 19px; width: 47px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image271.png\" \/>\u00a0or <img decoding=\"async\" id=\"Picture 4998\" style=\"height: 19px; width: 91px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image272.png\" \/><\/p>\n<p style=\"text-align: justify;\">i.e., <img decoding=\"async\" id=\"Picture 4997\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image273.png\" \/>\u00a0or <img decoding=\"async\" id=\"Picture 4996\" style=\"height: 19px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image274.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4995\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4994\" style=\"height: 19px; width: 59px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image275.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4993\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image273.png\" \/>\u00a0is impossible<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4992\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image118.png\" \/><img decoding=\"async\" id=\"Picture 4927\" style=\"height: 19px; width: 59px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image275.png\" \/><\/p>\n<p style=\"text-align: justify;\">At <img decoding=\"async\" id=\"Picture 4926\" style=\"height: 19px; width: 59px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image275.png\" \/>, <img decoding=\"async\" id=\"Picture 4925\" style=\"height: 44px; width: 215px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image276.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" style=\"height: 54px; width: 192px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image277.png\" \/>\u00a0[Negative]\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4924\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image118.png\" \/><img decoding=\"async\" id=\"Picture 4923\" style=\"height: 13px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image278.png\" \/>\u00a0is maximum at <img decoding=\"async\" id=\"Picture 4922\" style=\"height: 19px; width: 59px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image275.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4921\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image118.png\" \/>From eq. (i), Maximum area<\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 4920\" style=\"height: 41px; width: 163px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image279.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 4919\" style=\"height: 53px; width: 107px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image280.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 4918\" style=\"height: 53px; width: 71px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image281.png\" \/>\u00a0= <img decoding=\"async\" id=\"Picture 4916\" style=\"height: 45px; width: 52px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image282.png\" \/><\/p>\n<hr \/>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-6\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"9_A_tank_with_rectangular_base_and_rectangular_sides_open_at_the_top_is_to_be_constructed_so_that_its_depth_is_2_m_and_volume_is_8_m3_If_building_of_tank_costs_70_per_sq_meter_for_the_base_and_45_per_square_meter_for_sides_What_is_the_cost_of_least_expensive_tank\"><\/span><strong>9. A tank with rectangular base and rectangular sides, open at the top is to be constructed so that its depth is 2 m and volume is 8 m<sup>3<\/sup>. If building of tank costs ` 70 per sq. meter for the base and ` 45 per square meter for sides. What is the cost of least expensive tank?<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Given: Depth of tank = 2 m<\/p>\n<p style=\"text-align: justify;\">Let <img decoding=\"async\" id=\"Picture 4915\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image111.png\" \/>\u00a0m be the length and <img decoding=\"async\" id=\"Picture 4914\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image113.png\" \/>\u00a0m be the breadth of the base of the tank.<\/p>\n<p style=\"text-align: justify;\">Volume of tank = 8 cubic meters<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4913\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4912\" style=\"height: 21px; width: 60px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image283.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4911\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4910\" style=\"height: 21px; width: 44px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image284.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4909\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4907\" style=\"height: 41px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image285.png\" \/><\/p>\n<p style=\"text-align: justify;\">Cost of building the base of the tank at the rate of ` 70 per sq. meter is <img decoding=\"async\" id=\"Picture 4906\" style=\"height: 21px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image286.png\" \/><\/p>\n<p style=\"text-align: justify;\">And cost of building the four walls of the tank at the rate of ` 45 per sq. meter is<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4904\" style=\"height: 27px; width: 157px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image287.png\" \/><\/p>\n<p style=\"text-align: justify;\">= `<img decoding=\"async\" id=\"Picture 4903\" style=\"height: 27px; width: 93px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image288.png\" \/><\/p>\n<p style=\"text-align: justify;\">Let <img decoding=\"async\" id=\"Picture 4902\" style=\"height: 13px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image278.png\" \/>\u00a0be the total cost of building the tank.<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4901\" style=\"height: 21px; width: 151px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image289.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4900\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image118.png\" \/><img decoding=\"async\" id=\"Picture 4899\" style=\"height: 41px; width: 124px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image290.png\" \/>\u00a0and <img decoding=\"async\" id=\"Picture 4898\" style=\"height: 44px; width: 80px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image291.png\" \/><\/p>\n<p style=\"text-align: justify;\">Now <img decoding=\"async\" id=\"Picture 4897\" style=\"height: 41px; width: 47px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image292.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 2239\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 2238\" style=\"height: 41px; width: 91px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image293.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 2237\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/><img decoding=\"async\" style=\"height: 41px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image294.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 2236\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 2235\" style=\"height: 21px; width: 83px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image295.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 2234\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 2233\" style=\"height: 21px; width: 44px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image296.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 2232\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 2231\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image241.png\" \/>\u00a0[length cannot be negative]\n<p style=\"text-align: justify;\">At <img decoding=\"async\" id=\"Picture 2230\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image241.png\" \/><img decoding=\"async\" id=\"Picture 2229\" style=\"height: 44px; width: 117px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image297.png\" \/>\u00a0[Positive]\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 2228\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image118.png\" \/><img decoding=\"async\" id=\"Picture 2227\" style=\"height: 13px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image278.png\" \/>\u00a0is minimum at <img decoding=\"async\" id=\"Picture 2226\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image241.png\" \/>.<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 2225\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image118.png\" \/>Minimum cost = <img decoding=\"async\" id=\"Picture 2224\" style=\"height: 41px; width: 127px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image298.png\" \/><\/p>\n<p style=\"text-align: justify;\">= 280 + 360 + 360 = ` 1000<\/p>\n<hr \/>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-7\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"10_The_sum_of_the_perimeter_of_a_circle_and_square_is_where_is_some_constant_Prove_that_the_sum_of_their_areas_is_least_when_the_side_of_square_is_double_the_radius_of_the_circle\"><\/span><strong>10. The sum of the perimeter of a circle and square is <img decoding=\"async\" id=\"Picture 52\" style=\"height: 21px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image299.png\" \/>\u00a0where <img decoding=\"async\" id=\"Picture 53\" style=\"height: 19px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image300.png\" \/>\u00a0is some constant. Prove that the sum of their areas is least when the side of square is double the radius of the circle.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Let <img decoding=\"async\" id=\"Picture 2223\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image111.png\" \/>\u00a0be the radius of the circle and <img decoding=\"async\" id=\"Picture 2222\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image113.png\" \/>\u00a0be the side of square.<\/p>\n<p style=\"text-align: justify;\">According to question,\u00a0 Perimeter of circle + Perimeter of square = <img decoding=\"async\" id=\"Picture 2221\" style=\"height: 19px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image300.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 2220\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/><img decoding=\"async\" id=\"Picture 2219\" style=\"height: 21px; width: 88px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image301.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 2218\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 2217\" style=\"height: 21px; width: 88px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image302.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 2216\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 2176\" style=\"height: 41px; width: 81px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image303.png\" \/>\u2026\u2026\u2026.(i)<\/p>\n<p style=\"text-align: justify;\">Let <img decoding=\"async\" id=\"Picture 4799\" style=\"height: 13px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image278.png\" \/>\u00a0be the sum of areas of circle and square.<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4797\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image118.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4796\" style=\"height: 24px; width: 83px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image304.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4795\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4794\" style=\"height: 48px; width: 139px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image305.png\" \/>\u00a0[From eq. (i)]\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4793\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4792\" style=\"height: 44px; width: 203px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image306.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 4791\" style=\"height: 41px; width: 209px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image307.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4790\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image118.png\" \/><img decoding=\"async\" id=\"Picture 4789\" style=\"height: 41px; width: 208px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image308.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 44px; width: 152px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image309.png\" \/><\/p>\n<p style=\"text-align: justify;\">Now <img decoding=\"async\" id=\"Picture 4788\" style=\"height: 41px; width: 47px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image292.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4787\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4786\" style=\"height: 41px; width: 197px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image310.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4785\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4784\" style=\"height: 29px; width: 164px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image311.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4783\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4782\" style=\"height: 27px; width: 132px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image312.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4781\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4780\" style=\"height: 47px; width: 180px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image313.png\" \/><\/p>\n<p style=\"text-align: justify;\">At <img decoding=\"async\" id=\"Picture 4779\" style=\"height: 47px; width: 87px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image314.png\" \/><img decoding=\"async\" id=\"Picture 4778\" style=\"height: 44px; width: 152px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image309.png\" \/>\u00a0[Positive]\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4777\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image118.png\" \/><img decoding=\"async\" id=\"Picture 4776\" style=\"height: 13px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image278.png\" \/>\u00a0is minimum when <img decoding=\"async\" id=\"Picture 4775\" style=\"height: 47px; width: 87px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image314.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4774\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image118.png\" \/>\u00a0From eq. (i),<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4773\" style=\"height: 53px; width: 157px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image315.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 4772\" style=\"height: 45px; width: 89px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image316.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 4771\" style=\"height: 51px; width: 123px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image317.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4770\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4769\" style=\"height: 47px; width: 115px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image318.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 4768\" style=\"height: 47px; width: 63px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image319.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 4991\" style=\"height: 47px; width: 72px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image320.png\" \/>\u00a0= <img decoding=\"async\" id=\"Picture 4990\" style=\"height: 19px; width: 21px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image321.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, sum of areas is minimum when side of the square is double the radius of the circle.<\/p>\n<hr \/>\n<h3 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><\/h3>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"11_A_window_is_in_the_form_of_a_rectangle_surmounted_by_a_semicircular_opening_The_total_perimeter_of_the_window_is_10_m_Find_the_dimensions_of_the_window_to_admit_maximum_light_through_the_whole_opening\"><\/span><strong>11. A window is in the form of a rectangle surmounted by a semicircular opening. The total perimeter of the window is 10 m. Find the dimensions of the window to admit maximum light through the whole opening.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Let <img decoding=\"async\" id=\"Picture 5207\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image111.png\" \/>\u00a0m be the radius of the semi-circular opening of the window. Then one side of rectangle part of window is <img decoding=\"async\" id=\"Picture 5206\" style=\"height: 19px; width: 21px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image321.png\" \/>\u00a0and <img decoding=\"async\" id=\"Picture 5205\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image113.png\" \/>\u00a0m be the other side of rectangle.<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4908\" style=\"height: 174px; width: 120px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image322.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5204\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image118.png\" \/>Perimeter of window<\/p>\n<p style=\"text-align: justify;\">= Semi-circular arc AB + Length (AD + DC + BC)<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5203\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/><img decoding=\"async\" id=\"Picture 5202\" style=\"height: 41px; width: 164px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image323.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5201\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5200\" style=\"height: 21px; width: 116px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image324.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5199\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5198\" style=\"height: 21px; width: 116px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image325.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5197\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5196\" style=\"height: 44px; width: 116px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image326.png\" \/>\u2026\u2026\u2026.(i)<\/p>\n<p style=\"text-align: justify;\">Area of window <img decoding=\"async\" id=\"Picture 5195\" style=\"height: 27px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image327.png\" \/><\/p>\n<p style=\"text-align: justify;\">= Area of semi-circle + Area of rectangle<\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5194\" style=\"height: 41px; width: 93px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image328.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5193\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/><img decoding=\"async\" id=\"Picture 5192\" style=\"height: 51px; width: 199px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image329.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5191\" style=\"height: 41px; width: 181px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image330.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5190\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5189\" style=\"height: 41px; width: 204px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image331.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5188\" style=\"height: 41px; width: 143px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image332.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5187\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image118.png\" \/><img decoding=\"async\" id=\"Picture 5186\" style=\"height: 41px; width: 157px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image333.png\" \/>\u00a0and <img decoding=\"async\" id=\"Picture 5185\" style=\"height: 44px; width: 120px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image334.png\" \/>\u00a0= <img decoding=\"async\" id=\"Picture 5184\" style=\"height: 27px; width: 61px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image335.png\" \/><\/p>\n<p style=\"text-align: justify;\">Now <img decoding=\"async\" id=\"Picture 5183\" style=\"height: 41px; width: 47px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image292.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5182\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5181\" style=\"height: 41px; width: 147px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image336.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5180\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5179\" style=\"height: 19px; width: 124px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image337.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5178\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5177\" style=\"height: 27px; width: 117px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image338.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5176\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5175\" style=\"height: 47px; width: 87px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image339.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5174\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5173\" style=\"height: 41px; width: 64px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image340.png\" \/><\/p>\n<p style=\"text-align: justify;\">At <img decoding=\"async\" id=\"Picture 5172\" style=\"height: 41px; width: 64px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image340.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5171\" style=\"height: 44px; width: 103px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image341.png\" \/>[Negative]\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5170\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image118.png\" \/><img decoding=\"async\" id=\"Picture 5169\" style=\"height: 13px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image278.png\" \/>\u00a0is maximum at <img decoding=\"async\" id=\"Picture 5168\" style=\"height: 41px; width: 64px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image340.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5167\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image118.png\" \/>From eq. (i), <img decoding=\"async\" id=\"Picture 5166\" style=\"height: 60px; width: 143px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image342.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5165\" style=\"height: 49px; width: 144px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image343.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5164\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/><img decoding=\"async\" id=\"Picture 5163\" style=\"height: 47px; width: 155px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image344.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5162\" style=\"height: 47px; width: 63px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image345.png\" \/>\u00a0= <img decoding=\"async\" id=\"Picture 5161\" style=\"height: 41px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image346.png\" \/>\u00a0m<\/p>\n<p style=\"text-align: justify;\">Therefore, Length of rectangle = <img decoding=\"async\" id=\"Picture 5160\" style=\"height: 41px; width: 72px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image347.png\" \/>\u00a0m and Width of rectangle = <img decoding=\"async\" id=\"Picture 5159\" style=\"height: 41px; width: 65px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image348.png\" \/>\u00a0m<\/p>\n<p style=\"text-align: justify;\">And Radius of semi-circle = <img decoding=\"async\" id=\"Picture 5158\" style=\"height: 41px; width: 64px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image340.png\" \/>\u00a0m<\/p>\n<hr \/>\n<h3 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><\/h3>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"12_A_point_on_the_hypotenuse_of_a_triangle_is_at_distances_and_from_the_sides_of_the_triangle_Show_that_the_maximum_length_of_the_hypotenuse_is\"><\/span><strong>12. A point on the hypotenuse of a triangle is at distances <img decoding=\"async\" id=\"Picture 71\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image349.png\" \/>\u00a0and <img decoding=\"async\" id=\"Picture 72\" style=\"height: 19px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image155.png\" \/>\u00a0from the sides of the triangle. Show that the maximum length of the hypotenuse is <img decoding=\"async\" id=\"Picture 73\" style=\"height: 45px; width: 91px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image350.png\" \/>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Let P be a point on the hypotenuse AC of a right triangle ABC such that PL<img decoding=\"async\" id=\"Picture 5264\" style=\"height: 17px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image351.png\" \/>\u00a0AB = <img decoding=\"async\" id=\"Picture 5263\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image349.png\" \/>\u00a0and PM<img decoding=\"async\" id=\"Picture 5262\" style=\"height: 17px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image351.png\" \/>\u00a0BC = <img decoding=\"async\" id=\"Picture 5261\" style=\"height: 19px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image155.png\" \/>\u00a0and let <img decoding=\"async\" id=\"Picture 5260\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image352.png\" \/>BAC = <img decoding=\"async\" id=\"Picture 5259\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image352.png\" \/>MPC = <img decoding=\"async\" id=\"Picture 5258\" style=\"height: 19px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image177.png\" \/>, then in right angled <img decoding=\"async\" id=\"Picture 5257\" style=\"height: 21px; width: 48px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image353.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5256\" style=\"height: 41px; width: 91px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image354.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 160px; width: 128px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image355.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5255\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image118.png\" \/>AP = PL<img decoding=\"async\" id=\"Picture 5254\" style=\"height: 19px; width: 51px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image356.png\" \/>\u00a0= <img decoding=\"async\" id=\"Picture 5253\" style=\"height: 19px; width: 61px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image357.png\" \/><\/p>\n<p style=\"text-align: justify;\">And in right angled <img decoding=\"async\" id=\"Picture 5252\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image257.png\" \/>PMC, <img decoding=\"async\" id=\"Picture 5251\" style=\"height: 41px; width: 79px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image358.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5250\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>PM = PM<img decoding=\"async\" id=\"Picture 5249\" style=\"height: 19px; width: 93px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image359.png\" \/><\/p>\n<p style=\"text-align: justify;\">Let AC = <img decoding=\"async\" id=\"Picture 5248\" style=\"height: 17px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image360.png\" \/>\u00a0then<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5247\" style=\"height: 13px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image278.png\" \/>\u00a0= AP + PC = <img decoding=\"async\" id=\"Picture 5246\" style=\"height: 41px; width: 184px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image361.png\" \/>\u2026\u2026\u2026.(i)<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5245\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image118.png\" \/><img decoding=\"async\" id=\"Picture 5244\" style=\"height: 41px; width: 229px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image362.png\" \/><\/p>\n<p style=\"text-align: justify;\">Now <img decoding=\"async\" id=\"Picture 5243\" style=\"height: 41px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image267.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5242\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/><img decoding=\"async\" id=\"Picture 5241\" style=\"height: 19px; width: 216px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image363.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5240\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5239\" style=\"height: 41px; width: 108px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image364.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5238\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5237\" style=\"height: 21px; width: 116px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image365.png\" \/><img decoding=\"async\" id=\"Picture 5236\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5235\" style=\"height: 44px; width: 73px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image366.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5234\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5233\" style=\"height: 41px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image367.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5232\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5231\" style=\"height: 55px; width: 85px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image368.png\" \/>\u00a0 \u2026\u2026\u2026.(ii)<\/p>\n<p style=\"text-align: justify;\">And <img decoding=\"async\" id=\"Picture 5230\" style=\"height: 44px; width: 568px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image369.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5229\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5228\" style=\"height: 44px; width: 383px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image370.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5227\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5226\" style=\"height: 44px; width: 56px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image371.png\" \/>\u00a0[<img decoding=\"async\" id=\"Picture 5225\" style=\"height: 21px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image372.png\" \/>\u00a0and <img decoding=\"async\" id=\"Picture 5224\" style=\"height: 19px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image177.png\" \/>\u00a0is +ve as <img decoding=\"async\" id=\"Picture 5223\" style=\"height: 41px; width: 65px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image373.png\" \/>\u00a0)<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5222\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image118.png\" \/><img decoding=\"async\" id=\"Picture 5221\" style=\"height: 13px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image278.png\" \/>\u00a0is minimum when <img decoding=\"async\" id=\"Picture 5220\" style=\"height: 55px; width: 85px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image368.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5219\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image118.png\" \/><img decoding=\"async\" id=\"Picture 5218\" style=\"height: 55px; width: 185px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image374.png\" \/>\u00a0= <img decoding=\"async\" id=\"Picture 5217\" style=\"height: 65px; width: 55px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image375.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5216\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5215\" style=\"height: 89px; width: 125px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image376.png\" \/><\/p>\n<p style=\"text-align: justify;\">Also <img decoding=\"async\" id=\"Picture 5214\" style=\"height: 55px; width: 199px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image377.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5213\" style=\"height: 65px; width: 55px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image378.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5212\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5211\" style=\"height: 89px; width: 140px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image379.png\" \/><\/p>\n<p style=\"text-align: justify;\">Putting these values in eq. (i),<\/p>\n<p style=\"text-align: justify;\">Minimum length of hypotenuse = <img decoding=\"async\" id=\"Picture 5210\" style=\"height: 89px; width: 191px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image380.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5209\" style=\"height: 60px; width: 140px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image381.png\" \/>\u00a0= <img decoding=\"async\" id=\"Picture 5208\" style=\"height: 60px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image382.png\" \/><\/p>\n<hr \/>\n<h3 style=\"text-align: center;\"><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><\/h3>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"13_Find_the_points_at_which_the_function_given_by_has\"><\/span><strong>13. Find the points at which the function <img decoding=\"async\" id=\"Picture 77\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image194.png\" \/>\u00a0given by <img decoding=\"async\" id=\"Picture 78\" style=\"height: 29px; width: 151px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image383.png\" \/>\u00a0has:<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>(i) local maxima <\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(ii) local minima <\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(iii) point of inflexion.<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Given: <img decoding=\"async\" id=\"Picture 5319\" style=\"height: 29px; width: 151px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image383.png\" \/>\u2026\u2026\u2026.(i)<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5318\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image118.png\" \/><img decoding=\"async\" id=\"Picture 5317\" style=\"height: 41px; width: 311px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image384.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5316\" style=\"height: 29px; width: 227px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image385.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5315\" style=\"height: 31px; width: 244px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image386.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5314\" style=\"height: 29px; width: 155px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image387.png\" \/><\/p>\n<p style=\"text-align: justify;\">Now <img decoding=\"async\" id=\"Picture 5313\" style=\"height: 27px; width: 63px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image388.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5312\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/><img decoding=\"async\" id=\"Picture 5311\" style=\"height: 29px; width: 155px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image387.png\" \/>= 0<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5310\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5309\" style=\"height: 19px; width: 59px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image389.png\" \/>\u00a0or <img decoding=\"async\" id=\"Picture 5308\" style=\"height: 19px; width: 56px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image227.png\" \/>\u00a0or <img decoding=\"async\" id=\"Picture 5307\" style=\"height: 19px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image390.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5306\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5305\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image241.png\" \/>\u00a0or <img decoding=\"async\" id=\"Picture 5304\" style=\"height: 19px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image229.png\" \/>\u00a0or <img decoding=\"async\" id=\"Picture 5303\" style=\"height: 41px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image391.png\" \/><\/p>\n<p style=\"text-align: justify;\">Now, for values of <img decoding=\"async\" id=\"Picture 5302\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image111.png\" \/>\u00a0close to <img decoding=\"async\" id=\"Picture 5301\" style=\"height: 41px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image392.png\" \/>\u00a0and to the left of <img decoding=\"async\" id=\"Picture 5300\" style=\"height: 41px; width: 85px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image393.png\" \/>. Also for values of <img decoding=\"async\" id=\"Picture 5299\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image111.png\" \/>\u00a0close to <img decoding=\"async\" id=\"Picture 5298\" style=\"height: 41px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image392.png\" \/>\u00a0and to the right of <img decoding=\"async\" id=\"Picture 5297\" style=\"height: 41px; width: 84px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image394.png\" \/>.<\/p>\n<p style=\"text-align: justify;\">Therefore, <img decoding=\"async\" id=\"Picture 5296\" style=\"height: 41px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image391.png\" \/>\u00a0is the point of local maxima.<\/p>\n<p style=\"text-align: justify;\">Now, for values of <img decoding=\"async\" id=\"Picture 5295\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image111.png\" \/>\u00a0close to 2 and to the left of <img decoding=\"async\" id=\"Picture 5294\" style=\"height: 27px; width: 80px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image395.png\" \/>. Also for values of <img decoding=\"async\" id=\"Picture 5293\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image111.png\" \/>\u00a0close to 2 and to the right of <img decoding=\"async\" id=\"Picture 5292\" style=\"height: 27px; width: 80px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image396.png\" \/>.<\/p>\n<p style=\"text-align: justify;\">Therefore, <img decoding=\"async\" id=\"Picture 5291\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image241.png\" \/>\u00a0is the point of local minima.<\/p>\n<p style=\"text-align: justify;\">Now as the values of <img decoding=\"async\" id=\"Picture 5290\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image111.png\" \/>\u00a0varies through <img decoding=\"async\" id=\"Picture 5289\" style=\"height: 27px; width: 64px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image397.png\" \/>\u00a0does not change its sign. Therefore, <img decoding=\"async\" id=\"Picture 5288\" style=\"height: 19px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image229.png\" \/>\u00a0is the point of inflexion.<\/p>\n<hr \/>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-11\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"14_Find_the_absolute_maximum_and_minimum_values_of_the_function_given_by\"><\/span><strong>14. Find the absolute maximum and minimum values of the function <img decoding=\"async\" id=\"Picture 81\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image194.png\" \/>\u00a0given by <img decoding=\"async\" id=\"Picture 82\" style=\"height: 27px; width: 204px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image398.png\" \/>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Given: <img decoding=\"async\" id=\"Picture 5287\" style=\"height: 27px; width: 200px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image399.png\" \/>\u00a0\u2026\u2026\u2026.(i)<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5286\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image118.png\" \/><img decoding=\"async\" id=\"Picture 5285\" style=\"height: 41px; width: 200px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image400.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5284\" style=\"height: 19px; width: 132px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image401.png\" \/>\u00a0= <img decoding=\"async\" id=\"Picture 5283\" style=\"height: 27px; width: 117px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image402.png\" \/><\/p>\n<p style=\"text-align: justify;\">Now <img decoding=\"async\" id=\"Picture 5282\" style=\"height: 27px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image145.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5281\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/><img decoding=\"async\" id=\"Picture 5280\" style=\"height: 27px; width: 117px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image402.png\" \/>\u00a0= 0<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5279\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5278\" style=\"height: 19px; width: 60px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image403.png\" \/>\u00a0or <img decoding=\"async\" id=\"Picture 5277\" style=\"height: 19px; width: 96px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image404.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5276\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5275\" style=\"height: 41px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image405.png\" \/>\u00a0or <img decoding=\"async\" id=\"Picture 5274\" style=\"height: 19px; width: 65px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image406.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5273\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5272\" style=\"height: 41px; width: 61px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image407.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5271\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5270\" style=\"height: 41px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image408.png\" \/> [Turning points]\n<p style=\"text-align: justify;\">Now <img decoding=\"async\" id=\"Picture 5269\" style=\"height: 45px; width: 153px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image409.png\" \/><\/p>\n<p style=\"text-align: justify;\">= 0 + 1 = 1<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5268\" style=\"height: 56px; width: 316px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image410.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5267\" style=\"height: 27px; width: 148px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image411.png\" \/>1 + 0 = 1<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5266\" style=\"height: 29px; width: 215px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image412.png\" \/>\u00a0= 1<\/p>\n<p style=\"text-align: justify;\">Therefore, absolute maximum is <img decoding=\"async\" id=\"Picture 5265\" style=\"height: 41px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image413.png\" \/>\u00a0and absolute minimum is 1.<\/p>\n<hr \/>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-12\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"15_Show_that_the_altitude_of_the_right_circular_cone_of_maximum_volume_that_can_be_inscribed_in_a_sphere_of_radius_is\"><\/span><strong>15. Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius <img decoding=\"async\" id=\"Picture 87\" style=\"height: 13px; width: 12px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image414.png\" \/>\u00a0is <img decoding=\"async\" id=\"Picture 88\" style=\"height: 41px; width: 27px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image415.png\" \/>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Let <img decoding=\"async\" id=\"Picture 5377\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image111.png\" \/>\u00a0be the radius of base of cone and <img decoding=\"async\" id=\"Picture 5376\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image113.png\" \/>\u00a0be the height of the cone inscribed in a sphere of radius <img decoding=\"async\" id=\"Picture 5375\" style=\"height: 15px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image416.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4940\" style=\"height: 127px; width: 121px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image417.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5374\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image118.png\" \/>OD = AD \u2013 AO = <img decoding=\"async\" id=\"Picture 5373\" style=\"height: 17px; width: 36px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image418.png\" \/><\/p>\n<p style=\"text-align: justify;\">In right angled triangle OBD,<\/p>\n<p style=\"text-align: justify;\">OD<sup>2<\/sup> + BD<sup>2<\/sup> = OB<sup>2<\/sup><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5372\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/><img decoding=\"async\" id=\"Picture 5371\" style=\"height: 29px; width: 112px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image419.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5370\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5369\" style=\"height: 24px; width: 144px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image420.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5368\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5367\" style=\"height: 24px; width: 87px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image421.png\" \/> \u2026\u2026\u2026.(i)<\/p>\n<p style=\"text-align: justify;\">Volume of cone (V) = <img decoding=\"async\" id=\"Picture 5366\" style=\"height: 41px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image422.png\" \/>\u00a0= <img decoding=\"async\" id=\"Picture 5365\" style=\"height: 41px; width: 104px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image423.png\" \/>[From eq. (i)]\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5364\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0V = <img decoding=\"async\" id=\"Picture 5363\" style=\"height: 41px; width: 91px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image424.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5362\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5361\" style=\"height: 44px; width: 129px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image425.png\" \/>and <img decoding=\"async\" id=\"Picture 5360\" style=\"height: 47px; width: 125px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image426.png\" \/><\/p>\n<p style=\"text-align: justify;\">Now <img decoding=\"async\" id=\"Picture 5359\" style=\"height: 44px; width: 51px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image427.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5358\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/><img decoding=\"async\" id=\"Picture 5357\" style=\"height: 41px; width: 115px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image428.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5356\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/><img decoding=\"async\" id=\"Picture 5355\" style=\"height: 41px; width: 111px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image429.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5354\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5353\" style=\"height: 21px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image430.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5352\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5351\" style=\"height: 41px; width: 48px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image431.png\" \/><\/p>\n<p style=\"text-align: justify;\">At <img decoding=\"async\" id=\"Picture 5350\" style=\"height: 41px; width: 48px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image431.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5349\" style=\"height: 47px; width: 123px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image432.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5348\" style=\"height: 41px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image433.png\" \/>\u00a0[Negative]\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5347\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image118.png\" \/>Volume is maximum at <img decoding=\"async\" id=\"Picture 5346\" style=\"height: 41px; width: 48px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image431.png\" \/><\/p>\n<hr \/>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-13\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"16_Let_be_a_function_defined_on_such_that_for_all_Then_prove_that_is_an_increasing_function_on\"><\/span><strong>16. Let <img decoding=\"async\" id=\"Picture 92\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image194.png\" \/>\u00a0be a function defined on <img decoding=\"async\" id=\"Picture 93\" style=\"height: 27px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image434.png\" \/>\u00a0such that <img decoding=\"async\" id=\"Picture 94\" style=\"height: 27px; width: 71px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image435.png\" \/>\u00a0for all <img decoding=\"async\" id=\"Picture 95\" style=\"height: 27px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image436.png\" \/>\u00a0Then prove that <img decoding=\"async\" id=\"Picture 96\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image194.png\" \/>\u00a0is an increasing function on <img decoding=\"async\" id=\"Picture 97\" style=\"height: 27px; width: 44px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image437.png\" \/>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Let I be the interval <img decoding=\"async\" id=\"Picture 5345\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image438.png\" \/><\/p>\n<p style=\"text-align: justify;\">Given: <img decoding=\"async\" id=\"Picture 5344\" style=\"height: 27px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image439.png\" \/>\u00a0for all <img decoding=\"async\" id=\"Picture 5343\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image111.png\" \/>\u00a0in an interval I. Let <img decoding=\"async\" id=\"Picture 5342\" style=\"height: 24px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image440.png\" \/>\u00a0I with <img decoding=\"async\" id=\"Picture 5341\" style=\"height: 24px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image441.png\" \/><\/p>\n<p style=\"text-align: justify;\">By Lagrange\u2019s Mean Value Theorem, we have,<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5340\" style=\"height: 48px; width: 157px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image442.png\" \/>\u00a0where <img decoding=\"async\" id=\"Picture 5339\" style=\"height: 24px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image443.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5338\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/><img decoding=\"async\" id=\"Picture 5337\" style=\"height: 27px; width: 204px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image444.png\" \/>\u00a0where <img decoding=\"async\" id=\"Picture 5336\" style=\"height: 24px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image443.png\" \/><\/p>\n<p style=\"text-align: justify;\">Now <img decoding=\"async\" id=\"Picture 5335\" style=\"height: 24px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image441.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5334\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5333\" style=\"height: 24px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image445.png\" \/> \u2026\u2026\u2026.(i)<\/p>\n<p style=\"text-align: justify;\">Also, <img decoding=\"async\" id=\"Picture 5332\" style=\"height: 27px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image439.png\" \/>\u00a0for all <img decoding=\"async\" id=\"Picture 5331\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image111.png\" \/>\u00a0in an interval I<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5330\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5329\" style=\"height: 27px; width: 65px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image446.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5328\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image118.png\" \/>From eq. (i), <img decoding=\"async\" id=\"Picture 5327\" style=\"height: 27px; width: 120px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image447.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5326\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5325\" style=\"height: 27px; width: 99px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image448.png\" \/><\/p>\n<p style=\"text-align: justify;\">Thus, for every pair of points <img decoding=\"async\" id=\"Picture 5324\" style=\"height: 24px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image440.png\" \/>\u00a0I with <img decoding=\"async\" id=\"Picture 5323\" style=\"height: 24px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image441.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5322\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5321\" style=\"height: 27px; width: 99px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image448.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, <img decoding=\"async\" id=\"Picture 5320\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image154.png\" \/>\u00a0is strictly increasing in I.<\/p>\n<hr \/>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-14\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"17_Show_that_the_height_of_the_cylinder_of_maximum_volume_that_can_be_inscribed_in_a_sphere_of_radius_R_is_Also_find_the_maximum_volume\"><\/span><strong>17. Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius R is <img decoding=\"async\" id=\"Picture 99\" style=\"height: 44px; width: 32px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image449.png\" \/>\u00a0Also find the maximum volume.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Let <img decoding=\"async\" id=\"Picture 5452\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image111.png\" \/>\u00a0be the radius and <img decoding=\"async\" id=\"Picture 5451\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image113.png\" \/>\u00a0be the height of the cylinder inscribed in a sphere having centre O and radius R. <img decoding=\"async\" id=\"Picture 5450\" style=\"height: 27px; width: 88px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image450.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4949\" style=\"height: 126px; width: 126px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image451.png\" \/><\/p>\n<p style=\"text-align: justify;\">In right triangle OAM,\u00a0 AM<sup>2<\/sup> + OM<sup>2<\/sup> = OA<sup>2<\/sup><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5449\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5448\" style=\"height: 49px; width: 96px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image452.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5447\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5446\" style=\"height: 44px; width: 87px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image453.png\" \/> \u2026\u2026\u2026.(i)<\/p>\n<p style=\"text-align: justify;\">Volume of cylinder (V) = <img decoding=\"async\" id=\"Picture 5445\" style=\"height: 24px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image454.png\" \/>\u00a0\u2026\u2026\u2026.(ii)<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5444\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0V = <img decoding=\"async\" id=\"Picture 5443\" style=\"height: 51px; width: 95px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image455.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5442\" style=\"height: 51px; width: 92px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image456.png\" \/>\u00a0\u2026\u2026\u2026.(iii)<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5441\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image118.png\" \/><img decoding=\"async\" id=\"Picture 5440\" style=\"height: 51px; width: 129px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image457.png\" \/>and <img decoding=\"async\" id=\"Picture 5439\" style=\"height: 47px; width: 165px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image458.png\" \/><\/p>\n<p style=\"text-align: justify;\">Now <img decoding=\"async\" id=\"Picture 5438\" style=\"height: 44px; width: 51px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image427.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5437\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/><img decoding=\"async\" id=\"Picture 5436\" style=\"height: 51px; width: 115px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image459.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5435\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/><img decoding=\"async\" id=\"Picture 5434\" style=\"height: 44px; width: 87px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image460.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5433\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/><img decoding=\"async\" id=\"Picture 5432\" style=\"height: 44px; width: 65px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image461.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5431\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/><img decoding=\"async\" id=\"Picture 5430\" style=\"height: 44px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image462.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5429\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5428\" style=\"height: 44px; width: 52px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image463.png\" \/><\/p>\n<p style=\"text-align: justify;\">At <img decoding=\"async\" id=\"Picture 5427\" style=\"height: 44px; width: 52px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image463.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5426\" style=\"height: 48px; width: 127px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image464.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5425\" style=\"height: 24px; width: 55px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image465.png\" \/>[Negative]\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5424\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image118.png\" \/>V is maximum at <img decoding=\"async\" id=\"Picture 5423\" style=\"height: 44px; width: 52px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image463.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5422\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image118.png\" \/>From eq. (iii),<\/p>\n<p style=\"text-align: justify;\">Maximum value of cylinder = <img decoding=\"async\" id=\"Picture 5421\" style=\"height: 51px; width: 165px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image466.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5420\" style=\"height: 45px; width: 103px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image467.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5419\" style=\"height: 47px; width: 44px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image468.png\" \/><\/p>\n<hr \/>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-15\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"18_Show_that_the_height_of_the_cylinder_of_greatest_volume_which_can_be_inscribed_in_a_right_circular_cone_of_height_and_having_semi-vertical_angle_is_one-third_that_of_the_cone_and_the_greatest_volume_of_the_cylinder_is\"><\/span><strong>18. Show that the height of the cylinder of greatest volume which can be inscribed in a right circular cone of height <img decoding=\"async\" id=\"Picture 105\" style=\"height: 19px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image469.png\" \/>\u00a0and having semi-vertical angle <img decoding=\"async\" id=\"Picture 106\" style=\"height: 15px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image470.png\" \/>\u00a0is one-third that of the cone and the greatest volume of the cylinder is <img decoding=\"async\" id=\"Picture 107\" style=\"height: 41px; width: 95px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image471.png\" \/>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Let <img decoding=\"async\" id=\"Picture 5418\" style=\"height: 13px; width: 12px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image414.png\" \/>\u00a0be the radius of the right circular cone of height <img decoding=\"async\" id=\"Picture 5417\" style=\"height: 19px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image472.png\" \/>\u00a0Let the radius of the inscribed cylinder be <img decoding=\"async\" id=\"Picture 5416\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image111.png\" \/>\u00a0and height <img decoding=\"async\" id=\"Picture 5415\" style=\"height: 17px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image473.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 4961\" style=\"height: 146px; width: 144px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image474.png\" \/><\/p>\n<p style=\"text-align: justify;\">In similar triangles APQ and ARC, <img decoding=\"async\" id=\"Picture 5414\" style=\"height: 41px; width: 71px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image475.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5413\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5412\" style=\"height: 41px; width: 65px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image476.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5411\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5410\" style=\"height: 21px; width: 79px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image477.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5409\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5408\" style=\"height: 27px; width: 147px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image478.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5407\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5406\" style=\"height: 41px; width: 85px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image479.png\" \/><\/p>\n<p style=\"text-align: justify;\">Volume of cylinder (V) = <img decoding=\"async\" id=\"Picture 5405\" style=\"height: 24px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image454.png\" \/>\u2026\u2026\u2026.(ii)<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5404\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0V = <img decoding=\"async\" id=\"Picture 5403\" style=\"height: 41px; width: 87px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image480.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5402\" style=\"height: 41px; width: 89px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image481.png\" \/>\u2026\u2026\u2026.(iii)<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5401\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image118.png\" \/><img decoding=\"async\" id=\"Picture 5400\" style=\"height: 44px; width: 137px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image482.png\" \/>\u00a0and <img decoding=\"async\" id=\"Picture 5399\" style=\"height: 47px; width: 132px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image483.png\" \/><\/p>\n<p style=\"text-align: justify;\">Now <img decoding=\"async\" id=\"Picture 5398\" style=\"height: 44px; width: 51px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image427.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5397\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5396\" style=\"height: 41px; width: 121px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image484.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5395\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/><img decoding=\"async\" id=\"Picture 5394\" style=\"height: 21px; width: 87px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image485.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5393\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/><img decoding=\"async\" id=\"Picture 5392\" style=\"height: 19px; width: 73px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image486.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5391\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/><img decoding=\"async\" id=\"Picture 5390\" style=\"height: 41px; width: 47px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image487.png\" \/><\/p>\n<p style=\"text-align: justify;\">At <img decoding=\"async\" id=\"Picture 5389\" style=\"height: 41px; width: 47px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image487.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5388\" style=\"height: 47px; width: 143px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image488.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5387\" style=\"height: 41px; width: 116px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image489.png\" \/>[Negative]\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5386\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image118.png\" \/>V is maximum at <img decoding=\"async\" id=\"Picture 5385\" style=\"height: 41px; width: 47px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image487.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5384\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image118.png\" \/>From eq. (iii),<\/p>\n<p style=\"text-align: justify;\">Maximum value of cylinder = <img decoding=\"async\" id=\"Picture 5383\" style=\"height: 51px; width: 116px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image490.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5382\" style=\"height: 45px; width: 101px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image491.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5381\" style=\"height: 41px; width: 60px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image492.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5380\" style=\"height: 41px; width: 107px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image493.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" id=\"Picture 5379\" style=\"height: 41px; width: 92px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image494.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5378\" style=\"height: 45px; width: 92px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image495.png\" \/><\/p>\n<hr \/>\n<p style=\"text-align: justify;\"><strong>Choose the correct answer in the Exercises 19 to 24:<\/strong><\/p>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-16\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"19_A_cylindrical_tank_of_radius_10_m_is_being_filled_with_wheat_at_the_rate_of_314_cubic_meter_per_hour_Then_the_depth_of_wheat_is_increasing_at_the_rate_of\"><\/span><strong>19. A cylindrical tank of radius 10 m is being filled with wheat at the rate of 314 cubic meter per hour. Then the depth of wheat is increasing at the rate of:<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>(A) 1 m\/h\u00a0 <\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(B) 0.1 m\/h\u00a0 <\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(C) 1.1 m\/h\u00a0 <\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(D) 0.5 m\/h<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Let <img decoding=\"async\" id=\"Picture 5499\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image113.png\" \/>\u00a0be the depth of the wheat in the cylindrical tank of radius 10 m at time <img decoding=\"async\" id=\"Picture 5498\" style=\"height: 16px; width: 12px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image496.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5497\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image118.png\" \/>\u00a0 V = Volume of wheat in cylindrical tank at time <img decoding=\"async\" id=\"Picture 5496\" style=\"height: 29px; width: 140px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image497.png\" \/>\u00a0cu. m<\/p>\n<p style=\"text-align: justify;\">It is given that <img decoding=\"async\" id=\"Picture 5495\" style=\"height: 41px; width: 28px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image498.png\" \/>\u00a0= 314 cu. m\/hr<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5494\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5493\" style=\"height: 41px; width: 104px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image499.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5492\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5491\" style=\"height: 21px; width: 87px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image500.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5490\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5489\" style=\"height: 27px; width: 120px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image501.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5488\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5487\" style=\"height: 17px; width: 27px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image502.png\" \/>\u00a01 m\/h<\/p>\n<p style=\"text-align: justify;\">Therefore, option (A) is correct.<\/p>\n<hr \/>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-17\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"20_The_slope_of_the_tangent_to_the_curve_at_the_point_is\"><\/span><strong>20. The slope of the tangent to the curve <img decoding=\"async\" id=\"Picture 115\" style=\"height: 24px; width: 91px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image503.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 116\" style=\"height: 24px; width: 99px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image504.png\" \/>\u00a0at the point <img decoding=\"async\" id=\"Picture 117\" style=\"height: 27px; width: 47px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image505.png\" \/>\u00a0is:<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>(A) <img decoding=\"async\" id=\"Picture 118\" style=\"height: 41px; width: 24px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image506.png\" \/><\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(B) <img decoding=\"async\" id=\"Picture 119\" style=\"height: 41px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image507.png\" \/><\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(C) <img decoding=\"async\" id=\"Picture 120\" style=\"height: 41px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image508.png\" \/><\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(D) <img decoding=\"async\" id=\"Picture 121\" style=\"height: 41px; width: 24px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image509.png\" \/>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Equation of the curves are <img decoding=\"async\" id=\"Picture 5486\" style=\"height: 21px; width: 88px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image510.png\" \/>\u2026..(i) and <img decoding=\"async\" id=\"Picture 5485\" style=\"height: 24px; width: 99px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image504.png\" \/>\u2026..(ii)<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5484\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image118.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5483\" style=\"height: 41px; width: 73px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image511.png\" \/>\u00a0and <img decoding=\"async\" id=\"Picture 5482\" style=\"height: 41px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image512.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5481\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image118.png\" \/>\u00a0Slope of the tangent to the given curve at point <img decoding=\"async\" id=\"Picture 5480\" style=\"height: 27px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image184.png\" \/>\u00a0= <img decoding=\"async\" id=\"Picture 5479\" style=\"height: 41px; width: 135px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image513.png\" \/>\u00a0\u2026..(iii)<\/p>\n<p style=\"text-align: justify;\">At the given point <img decoding=\"async\" id=\"Picture 5478\" style=\"height: 27px; width: 52px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image514.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5477\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image241.png\" \/>\u00a0and <img decoding=\"async\" id=\"Picture 5476\" style=\"height: 21px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image515.png\" \/><\/p>\n<p style=\"text-align: justify;\">At <img decoding=\"async\" id=\"Picture 5475\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image241.png\" \/>,\u00a0 from eq. (i),\u00a0 <img decoding=\"async\" id=\"Picture 5474\" style=\"height: 21px; width: 88px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image516.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5473\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5472\" style=\"height: 21px; width: 95px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image517.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5471\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5470\" style=\"height: 27px; width: 109px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image518.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5469\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5468\" style=\"height: 21px; width: 77px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image519.png\" \/><\/p>\n<p style=\"text-align: justify;\">At <img decoding=\"async\" id=\"Picture 5467\" style=\"height: 21px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image515.png\" \/>, from eq. (ii),\u00a0 <img decoding=\"async\" id=\"Picture 5466\" style=\"height: 21px; width: 105px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image520.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5465\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5464\" style=\"height: 21px; width: 99px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image521.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5463\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5462\" style=\"height: 21px; width: 81px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image522.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5461\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5460\" style=\"height: 27px; width: 107px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image523.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5459\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5458\" style=\"height: 21px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image524.png\" \/><\/p>\n<p style=\"text-align: justify;\">Here, common value of <img decoding=\"async\" id=\"Picture 5457\" style=\"height: 16px; width: 9px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image525.png\" \/>\u00a0in the two sets of values is <img decoding=\"async\" id=\"Picture 5456\" style=\"height: 19px; width: 33px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image526.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5455\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image118.png\" \/>\u00a0 From eq. (iii),<\/p>\n<p style=\"text-align: justify;\">Slope of the tangent to the given curve at point <img decoding=\"async\" id=\"Picture 5454\" style=\"height: 27px; width: 47px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image505.png\" \/>\u00a0= <img decoding=\"async\" id=\"Picture 5453\" style=\"height: 49px; width: 87px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image527.png\" \/><\/p>\n<p style=\"text-align: justify;\">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_Miscellaneous-18\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"21_The_line_is_a_tangent_to_the_curve_if_the_value_of_is\"><\/span><strong>21. The line <img decoding=\"async\" id=\"Picture 125\" style=\"height: 21px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image528.png\" \/>\u00a0is a tangent to the curve <img decoding=\"async\" id=\"Picture 126\" style=\"height: 24px; width: 53px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image169.png\" \/>\u00a0if the value of <img decoding=\"async\" id=\"Picture 127\" style=\"height: 15px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image529.png\" \/>\u00a0is:<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>(A) 1 <\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(B) 2 <\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(C) 3 <\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(D) <img decoding=\"async\" id=\"Picture 128\" style=\"height: 41px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image259.png\" \/>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Equation of the curve is <img decoding=\"async\" id=\"Picture 5540\" style=\"height: 24px; width: 53px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image169.png\" \/>\u2026\u2026\u2026.(i)<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5539\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image118.png\" \/>\u00a0 <img decoding=\"async\" id=\"Picture 5538\" style=\"height: 41px; width: 77px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image530.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5537\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5536\" style=\"height: 44px; width: 51px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image171.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5535\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image118.png\" \/>\u00a0 Slope of the tangent to the given curve at point <img decoding=\"async\" id=\"Picture 5534\" style=\"height: 27px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image184.png\" \/>\u00a0= <img decoding=\"async\" id=\"Picture 5533\" style=\"height: 44px; width: 51px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image171.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5532\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image118.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5531\" style=\"height: 44px; width: 44px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image531.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5530\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5529\" style=\"height: 41px; width: 44px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image532.png\" \/>\u00a0 \u2026\u2026\u2026.(ii)<\/p>\n<p style=\"text-align: justify;\">Now <img decoding=\"async\" id=\"Picture 5528\" style=\"height: 21px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image528.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5527\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5526\" style=\"height: 41px; width: 73px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image533.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5525\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5524\" style=\"height: 41px; width: 73px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image534.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5523\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5522\" style=\"height: 41px; width: 65px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image535.png\" \/>\u00a0 \u2026..(iii)<\/p>\n<p style=\"text-align: justify;\">Putting the values of <img decoding=\"async\" id=\"Picture 5521\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image111.png\" \/>\u00a0and <img decoding=\"async\" id=\"Picture 5520\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image113.png\" \/>\u00a0in eq. (i), <img decoding=\"async\" id=\"Picture 5519\" style=\"height: 44px; width: 100px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image536.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5518\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5517\" style=\"height: 19px; width: 60px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image537.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5516\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5515\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image538.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, option (A) is correct.<\/p>\n<hr \/>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-19\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"22_The_normal_at_the_point_1_1_on_the_curve_is\"><\/span><strong>22. The normal at the point (1, 1) on the curve <img decoding=\"async\" id=\"Picture 134\" style=\"height: 24px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image539.png\" \/>\u00a0is:<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>(A) <img decoding=\"async\" id=\"Picture 135\" style=\"height: 21px; width: 60px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image540.png\" \/>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(B) <img decoding=\"async\" id=\"Picture 136\" style=\"height: 21px; width: 60px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image541.png\" \/>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(C) <img decoding=\"async\" id=\"Picture 137\" style=\"height: 21px; width: 80px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image542.png\" \/><\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(D) <img decoding=\"async\" id=\"Picture 138\" style=\"height: 21px; width: 56px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image543.png\" \/>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Equation of the given curve is <img decoding=\"async\" id=\"Picture 5514\" style=\"height: 24px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image539.png\" \/>\u00a0\u2026\u2026\u2026.(i)<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5513\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image118.png\" \/>\u00a0 <img decoding=\"async\" id=\"Picture 5512\" style=\"height: 41px; width: 88px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image544.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5511\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5510\" style=\"height: 41px; width: 57px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image545.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5509\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image118.png\" \/>\u00a0 Slope of the tangent to the given curve at point (1, 1) = <img decoding=\"async\" id=\"Picture 5508\" style=\"height: 41px; width: 116px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image546.png\" \/>\u00a0(say)<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5507\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image118.png\" \/>\u00a0Slope of the normal = <img decoding=\"async\" id=\"Picture 5506\" style=\"height: 41px; width: 77px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image547.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5505\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image118.png\" \/>\u00a0Equation of the normal at (1, 1) is <img decoding=\"async\" id=\"Picture 5504\" style=\"height: 27px; width: 96px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image548.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5503\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5502\" style=\"height: 21px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image549.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5501\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5500\" style=\"height: 21px; width: 60px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image541.png\" \/><\/p>\n<p style=\"text-align: justify;\">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_Miscellaneous-20\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"23_The_normal_to_the_curve_passing_through_1_2_is\"><\/span><strong>23. The normal to the curve <img decoding=\"async\" id=\"Picture 141\" style=\"height: 24px; width: 53px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image550.png\" \/>\u00a0passing through (1, 2) is:<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>(A) <img decoding=\"async\" id=\"Picture 142\" style=\"height: 21px; width: 60px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image176.png\" \/>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(B) <img decoding=\"async\" id=\"Picture 143\" style=\"height: 21px; width: 60px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image551.png\" \/>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(C) <img decoding=\"async\" id=\"Picture 144\" style=\"height: 21px; width: 56px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image552.png\" \/>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(D) <img decoding=\"async\" id=\"Picture 145\" style=\"height: 21px; width: 56px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image543.png\" \/><\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Equation of the curve is <img decoding=\"async\" id=\"Picture 5571\" style=\"height: 24px; width: 53px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image550.png\" \/>\u2026\u2026\u2026..(i)<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5570\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image118.png\" \/>\u00a0 <img decoding=\"async\" id=\"Picture 5569\" style=\"height: 41px; width: 65px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image553.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5568\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5567\" style=\"height: 41px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image554.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5566\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image118.png\" \/>\u00a0Slope of the normal at <img decoding=\"async\" id=\"Picture 5565\" style=\"height: 27px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image184.png\" \/>\u00a0= <img decoding=\"async\" id=\"Picture 5564\" style=\"height: 44px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image555.png\" \/>\u2026\u2026\u2026.(ii)<\/p>\n<p style=\"text-align: justify;\">Again slope of normal at given point (1, 2) = <img decoding=\"async\" id=\"Picture 5563\" style=\"height: 41px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image556.png\" \/>\u2026\u2026\u2026.(iii)<\/p>\n<p style=\"text-align: justify;\">From eq. (ii) and (iii), we have\u00a0 <img decoding=\"async\" id=\"Picture 5562\" style=\"height: 41px; width: 73px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image557.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5561\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5560\" style=\"height: 21px; width: 115px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image558.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5559\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5558\" style=\"height: 21px; width: 44px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image559.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5557\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5556\" style=\"height: 41px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image560.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5555\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image118.png\" \/>\u00a0From eq. (i), <img decoding=\"async\" id=\"Picture 5554\" style=\"height: 41px; width: 47px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image561.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5553\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5552\" style=\"height: 21px; width: 43px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image562.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5551\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5550\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image241.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5549\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image118.png\" \/>\u00a0 <img decoding=\"async\" id=\"Picture 5548\" style=\"height: 41px; width: 88px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image563.png\" \/><\/p>\n<p style=\"text-align: justify;\">Now, at point (2, 1), slope of the normal from eq. (ii) = <img decoding=\"async\" id=\"Picture 5547\" style=\"height: 41px; width: 91px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image564.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5546\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image118.png\" \/>\u00a0 Equation of the normal is <img decoding=\"async\" id=\"Picture 5545\" style=\"height: 27px; width: 109px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image565.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5544\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5543\" style=\"height: 21px; width: 89px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image566.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 5542\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 5541\" style=\"height: 21px; width: 60px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image176.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, option (A) is correct.<\/p>\n<hr \/>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-21\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>24. The points on the curve <img decoding=\"async\" id=\"Picture 152\" style=\"height: 24px; width: 63px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image567.png\" \/>\u00a0where the normal to the curve make equal intercepts with axes are:<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(A) <img decoding=\"async\" id=\"Picture 153\" style=\"height: 45px; width: 55px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image568.png\" \/>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(B) <img decoding=\"async\" id=\"Picture 154\" style=\"height: 45px; width: 55px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image569.png\" \/><\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(C) <img decoding=\"async\" id=\"Picture 155\" style=\"height: 45px; width: 55px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image570.png\" \/><\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(D) <img decoding=\"async\" id=\"Picture 156\" style=\"height: 45px; width: 52px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image571.png\" \/><\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Equation of the curve is <img decoding=\"async\" id=\"Picture 157\" style=\"height: 24px; width: 56px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image572.png\" \/>\u2026\u2026\u2026.(i)<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 158\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image118.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 159\" style=\"height: 41px; width: 85px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image573.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 160\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 161\" style=\"height: 47px; width: 101px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image574.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 162\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image118.png\" \/>\u00a0Slope of the tangent to curve (i) at any point <img decoding=\"async\" id=\"Picture 163\" style=\"height: 27px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image184.png\" \/>\u00a0= <img decoding=\"async\" id=\"Picture 164\" style=\"height: 47px; width: 59px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image575.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 165\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image118.png\" \/>\u00a0Slope of the normal = negative reciprocal = <img decoding=\"async\" id=\"Picture 166\" style=\"height: 41px; width: 66px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image576.png\" \/><\/p>\n<p style=\"text-align: justify;\">\u00a0 [<img decoding=\"async\" id=\"Picture 167\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image577.png\" \/>\u00a0Slopes of lines making equal intercepts on the axes are <img decoding=\"async\" id=\"Picture 168\" style=\"height: 17px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image578.png\" \/>]\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 169\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 170\" style=\"height: 24px; width: 71px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image579.png\" \/><\/p>\n<p style=\"text-align: justify;\">Taking positive sign, <img decoding=\"async\" id=\"Picture 171\" style=\"height: 24px; width: 61px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image580.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 172\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 173\" style=\"height: 44px; width: 56px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image581.png\" \/> \u2026\u2026\u2026.(ii)<\/p>\n<p style=\"text-align: justify;\">From eq. (i) and (ii), we have <img decoding=\"async\" id=\"Picture 174\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image582.png\" \/>\u00a0and <img decoding=\"async\" id=\"Picture 175\" style=\"height: 41px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image583.png\" \/><\/p>\n<p style=\"text-align: justify;\">Taking positive sign, <img decoding=\"async\" id=\"Picture 176\" style=\"height: 24px; width: 71px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image584.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 177\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 178\" style=\"height: 44px; width: 48px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image585.png\" \/> \u2026\u2026\u2026.(ii)<\/p>\n<p style=\"text-align: justify;\">From eq. (i) and (ii), we have <img decoding=\"async\" id=\"Picture 179\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image582.png\" \/>\u00a0and <img decoding=\"async\" id=\"Picture 180\" style=\"height: 41px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image586.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 181\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image118.png\" \/>\u00a0Required points are <img decoding=\"async\" id=\"Picture 182\" style=\"height: 45px; width: 55px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/06%20Misc\/image568.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, option (A) is correct.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-22\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>NCERT Solutions Class 12 Maths PDF (Download) Free from myCBSEguide app and myCBSEguide website. Ncert solution class 12 Maths includes text book solutions from both part 1 and part 2. NCERT Solutions for CBSE Class 12 Maths have total 20 chapters. 12 Maths NCERT Solutions in PDF for free Download on our website. Ncert Maths class 12 solutions PDF and Maths ncert class 12 PDF solutions with latest modifications and as per the latest CBSE syllabus are only available in myCBSEguide<\/p>\n<h2><span class=\"ez-toc-section\" id=\"CBSE_App_for_Class_12\"><\/span>CBSE App for Class 12<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>To download\u00a0NCERT Solutions for class 12 Physics, Chemistry, Biology, History, Political Science, Economics, Geography, Computer Science, Home Science, Accountancy, Business Studies and Home Science; do check myCBSEguide app or website. myCBSEguide provides sample papers with solution, test papers for chapter-wise practice, NCERT solutions, NCERT Exemplar solutions, quick revision notes for ready reference, CBSE guess papers and CBSE important question papers. Sample Paper all are made available through\u00a0<a href=\"https:\/\/play.google.com\/store\/apps\/details?id=in.techchefs.MyCBSEGuide&amp;referrer=utm_source%3Dmycbse_bottom%26utm_medium%3Dtext%26utm_campaign%3Dmycbseads\"><strong>the best app for CBSE students<\/strong><\/a>\u00a0and myCBSEguide website.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>NCERT Solutions class 12 Maths Miscellaneous Class 12 Maths book solutions are available in PDF format for free download. These ncert book chapter wise questions and answers are very helpful for CBSE board exam. CBSE recommends NCERT books and most of the questions in CBSE exam are asked from NCERT text books. Class 12 Maths &#8230; <a title=\"NCERT Solutions class 12 Maths Miscellaneous\" class=\"read-more\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-12\/\" 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":[],"class_list":["post-4550","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mathematics-cbse-class-12","category-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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