{"id":4464,"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\/"},"modified":"2018-05-08T14:54:29","modified_gmt":"2018-05-08T09:24:29","slug":"ncert-solutions-class-12-maths-miscellaneous","status":"publish","type":"post","link":"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous\/","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\/#NCERT_Solutions_class_12_Maths_Relations_and_Functions\" >NCERT Solutions class 12 Maths Relations and Functions<\/a><ul class='ez-toc-list-level-4' ><li class='ez-toc-heading-level-4'><ul class='ez-toc-list-level-4' ><li class='ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous\/#NCERT_Solutions_class_12_Maths_Miscellaneous\" >NCERT Solutions class 12 Maths Miscellaneous<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous\/#3_If_is_defined_by_find\" >3. If \u00a0is defined by \u00a0find \u00a0<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous\/#NCERT_Solutions_class_12_Maths_Miscellaneous-2\" >NCERT Solutions class 12 Maths Miscellaneous<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous\/#4_Show_that_the_function_defined_by_R_is_one-one_and_onto_function\" >4. Show that the function \u00a0defined by \u00a0\u00a0R is one-one and onto function.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous\/#NCERT_Solutions_class_12_Maths_Miscellaneous-3\" >NCERT Solutions class 12 Maths Miscellaneous<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous\/#5_Show_that_the_function_given_by_is_injective\" >5. Show that the function \u00a0given by \u00a0is injective.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous\/#NCERT_Solutions_class_12_Maths_Miscellaneous-4\" >NCERT Solutions class 12 Maths Miscellaneous<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous\/#6_Give_examples_of_two_functions_and_such_that_is_injective_but_is_not_injective\" >6. Give examples of two functions \u00a0and \u00a0such that \u00a0is injective but \u00a0is not injective.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous\/#NCERT_Solutions_class_12_Maths_Miscellaneous-5\" >NCERT Solutions class 12 Maths Miscellaneous<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-11\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous\/#7_Give_examples_of_two_functions_and_such_that_is_onto_but_is_not_onto\" >7. Give examples of two functions \u00a0and \u00a0such that \u00a0is onto but \u00a0is not onto.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-12\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous\/#NCERT_Solutions_class_12_Maths_Miscellaneous-6\" >NCERT Solutions class 12 Maths Miscellaneous<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-13\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous\/#8_Given_a_non_empty_set_X_consider_P_X_which_is_the_set_of_all_subsets_of_X\" >8. Given a non empty set X, consider P (X) which is the set of all subsets of X.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-14\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous\/#NCERT_Solutions_class_12_Maths_Miscellaneous-7\" >NCERT Solutions class 12 Maths Miscellaneous<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-15\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous\/#11_Let_S_and_T_1_2_3_Find_of_the_following_functions_F_from_S_to_T_if_it_exists\" >11. Let S = \u00a0and T = {1, 2, 3}. Find \u00a0of the following functions F from S to T, if it exists.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-16\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous\/#NCERT_Solutions_class_12_Maths_Miscellaneous-8\" >NCERT Solutions class 12 Maths Miscellaneous<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-17\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous\/#12_Consider_the_binary_operation_R_x_R_R_and_o_R_x_R_R_defined_as_and_R_Show_that_is_commutative_but_not_associative_o_is_associative_but_not_commutative_Further_show_that_R_If_it_is_so_we_say_that_the_operation_distributes_over_the_operation_o_Does_o_distribute_over_Justify_your_answer\" >12. Consider the binary operation * : R x R \u00a0R and o = R x R \u00a0R defined as \u00a0and \u00a0R. Show that * is commutative but not associative, o is associative but not commutative. Further, show that \u00a0R, \u00a0[If it is so, we say that the operation * distributes over the operation o]. Does o distribute over *? Justify your answer.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-18\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous\/#NCERT_Solutions_class_12_Maths_Miscellaneous-9\" >NCERT Solutions class 12 Maths Miscellaneous<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-19\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous\/#13_Given_a_non-empty_set_X_let_P_X_x_P_X_P_X_be_defined_as_A_B_A_%E2%80%93_B_B_%E2%80%93_A_A_B_P_X_Show_that_the_empty_set_is_the_identity_for_the_operation_and_all_the_elements_A_of_P_X_are_invertible_with_A-1_A_Hint_and\" >13. Given a non-empty set X, let * : P (X) x P (X) \u00a0P (X) be defined as A * B = (A \u2013 B) \u00a0(B \u2013 A), \u00a0A, B \u00a0P (X). Show that the empty set \u00a0is the identity for the operation * and all the elements A of P (X) are invertible with A-1 = A. (Hint: \u00a0and )<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-20\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous\/#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-4'><a class=\"ez-toc-link ez-toc-heading-21\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous\/#14_Define_binary_operation_on_the_set_0_1_2_3_4_5_as\" >14. Define binary operation * on the set {0, 1, 2, 3, 4, 5} as \u00a0<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-22\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous\/#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-4'><a class=\"ez-toc-link ez-toc-heading-23\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous\/#15_Let_A_%E2%80%931_0_1_2_B_%E2%80%934_%E2%80%932_0_2_and_be_the_functions_defined_by_A_and_A_Are_and_equal_Justify_your_answer\" >15. Let A = {\u20131, 0, 1, 2}, B = {\u20134, \u20132, 0, 2} and \u00a0be the functions defined by \u00a0\u00a0A and \u00a0A. Are \u00a0and \u00a0equal? Justify your answer.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-24\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous\/#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-4'><a class=\"ez-toc-link ez-toc-heading-25\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous\/#16_Let_A_1_2_3_Then_number_of_relations_containing_1_2_and_1_3_which_are_reflexive_and_symmetric_but_not_transitive_is\" >16. Let A = {1, 2, 3}. Then number of relations containing (1, 2) and (1, 3) which are reflexive and symmetric but not transitive is:<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-26\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous\/#17_Let_A_1_2_3_Then_number_of_equivalence_relations_containing_1_2_is\" >17. Let A = {1, 2, 3}. Then number of equivalence relations containing (1, 2) is:<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-27\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous\/#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-4'><a class=\"ez-toc-link ez-toc-heading-28\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous\/#18_Let_be_the_Signum_Function_defined_as_and_be_the_Greatest_Function_given_by_where_is_greatest_integer_less_than_or_equal_to_Then_does_and_coincide_in_0_1\" >18. Let \u00a0be the Signum Function defined as \u00a0and \u00a0be the Greatest Function given by \u00a0where \u00a0is greatest integer less than or equal to \u00a0Then, does \u00a0and \u00a0coincide in (0, 1)?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-29\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous\/#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-4'><a class=\"ez-toc-link ez-toc-heading-30\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous\/#19_Number_of_binary_operation_on_the_set_are\" >19. Number of binary operation on the set \u00a0are:<\/a><\/li><\/ul><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-31\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous\/#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-2'><a class=\"ez-toc-link ez-toc-heading-32\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous\/#CBSE_App_for_Students\" >CBSE App for Students<\/a><\/li><\/ul><\/nav><\/div>\n<p>NCERT Solutions class 12 Maths Miscellaneous Class 12 Maths book solutions are available in PDF format for free download. These ncert book chapter wise questions and answers are very helpful for CBSE board exam. CBSE recommends NCERT books and most of the questions in CBSE exam are asked from NCERT text books. Class 12 Maths chapter wise NCERT solution for Maths part 1 and Maths part 2 for all the chapters can be downloaded from our website and myCBSEguide mobile app for free.<\/p>\n<p><strong>Download\u00a0<a href=\"https:\/\/mycbseguide.com\/downloads\/cbse-class-12-mathematics-relations-and-functions\/1285\/ncert-solutions\/5\/\">NCERT solutions for\u00a0Relations and Functions\u00a0<\/a>as PDF.<\/strong><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/media-mycbseguide.s3.ap-south-1.amazonaws.com\/images\/blog\/12%20Maths%20Book%20pic.jpg\" alt=\"NCERT Solutions class 12 Maths Miscellaneous\" width=\"169\" height=\"220\" \/><\/p>\n<h2><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Relations_and_Functions\"><\/span>NCERT Solutions class 12 Maths Relations and Functions<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p style=\"text-align: justify;\"><strong>1. Let <img decoding=\"async\" style=\"height: 21px; width: 85px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image001.png\" \/>\u00a0be defined as <img decoding=\"async\" style=\"height: 27px; width: 103px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image002.png\" \/>\u00a0Find the function <img decoding=\"async\" style=\"height: 21px; width: 81px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image003.png\" \/>\u00a0such that <img decoding=\"async\" style=\"height: 26px; width: 105px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image004.png\" \/>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Given: <img decoding=\"async\" style=\"height: 27px; width: 100px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image005.png\" \/><\/p>\n<p style=\"text-align: justify;\">Now \u00a0 <img decoding=\"async\" style=\"height: 29px; width: 107px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image006.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 29px; width: 191px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image007.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 19px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image008.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 152px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image009.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 19px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image008.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 87px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image010.png\" \/><\/p>\n<hr \/>\n<p style=\"text-align: justify;\"><strong>2. Let <img decoding=\"async\" style=\"height: 21px; width: 92px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image011.png\" \/>\u00a0be defined as <img decoding=\"async\" style=\"height: 27px; width: 85px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image012.png\" \/>\u00a0if <img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image013.png\" \/>\u00a0is odd and <img decoding=\"async\" style=\"height: 27px; width: 85px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image014.png\" \/>\u00a0if <img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image013.png\" \/>\u00a0is even. Show that <img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image015.png\" \/>\u00a0is invertible. Find the inverse of <img decoding=\"async\" style=\"height: 21px; width: 19px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image016.png\" \/>\u00a0Here, W is the set of all whole numbers.<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Given: <img decoding=\"async\" style=\"height: 21px; width: 92px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image011.png\" \/>\u00a0defined as <img decoding=\"async\" style=\"height: 49px; width: 188px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image017.png\" \/><\/p>\n<p style=\"text-align: justify;\">Injectivity: Let <img decoding=\"async\" style=\"height: 17px; width: 32px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image018.png\" \/>\u00a0be any two odd real numbers, then <img decoding=\"async\" style=\"height: 27px; width: 92px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image019.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 19px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image008.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 19px; width: 79px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image020.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 19px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image008.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 15px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image021.png\" \/><\/p>\n<p style=\"text-align: justify;\">Again,\u00a0 let <img decoding=\"async\" style=\"height: 17px; width: 32px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image018.png\" \/>\u00a0be any two even whole numbers, then <img decoding=\"async\" style=\"height: 27px; width: 92px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image019.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 19px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image008.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 19px; width: 80px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image022.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 19px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image008.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 15px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image021.png\" \/><\/p>\n<p style=\"text-align: justify;\">Is <img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image013.png\" \/>\u00a0is even and <img decoding=\"async\" style=\"height: 15px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image023.png\" \/>\u00a0is odd, then <img decoding=\"async\" style=\"height: 16px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image024.png\" \/><\/p>\n<p style=\"text-align: justify;\">Also, if <img decoding=\"async\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image025.png\" \/>\u00a0odd and <img decoding=\"async\" style=\"height: 27px; width: 43px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image026.png\" \/>\u00a0is even, then <img decoding=\"async\" style=\"height: 27px; width: 92px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image027.png\" \/><\/p>\n<p style=\"text-align: justify;\">Hence, <img decoding=\"async\" style=\"height: 16px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image024.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 19px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image008.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 92px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image027.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\/01%20Misc\/image028.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image015.png\" \/>\u00a0is an injective mapping.<\/p>\n<p style=\"text-align: justify;\">Surjectivity: Let <img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image013.png\" \/>\u00a0be an arbitrary whole number.<\/p>\n<p style=\"text-align: justify;\">If <img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image013.png\" \/>\u00a0is an odd number, then there exists an even whole number <img decoding=\"async\" style=\"height: 27px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image029.png\" \/>\u00a0such that<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 27px; width: 145px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image030.png\" \/><\/p>\n<p style=\"text-align: justify;\">If <img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image013.png\" \/>\u00a0is an even number, then there exists an odd whole number <img decoding=\"async\" style=\"height: 27px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image031.png\" \/>\u00a0such that<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 27px; width: 145px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image032.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, every <img decoding=\"async\" style=\"height: 15px; width: 26px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image033.png\" \/>\u00a0W has its pre-image in W.<\/p>\n<p style=\"text-align: justify;\">So, <img decoding=\"async\" style=\"height: 21px; width: 92px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image011.png\" \/>\u00a0is a surjective. Thus <img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image015.png\" \/>\u00a0is invertible and <img decoding=\"async\" style=\"height: 26px; width: 26px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image034.png\" \/>\u00a0exists.<\/p>\n<p style=\"text-align: justify;\">For <img decoding=\"async\" style=\"height: 26px; width: 26px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image034.png\" \/>: <img decoding=\"async\" style=\"height: 21px; width: 57px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image035.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 19px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image008.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 57px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image036.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 21px; width: 57px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image037.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 16px; width: 19px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image008.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 57px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image038.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\/01%20Misc\/image028.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 49px; width: 201px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image039.png\" \/><\/p>\n<p style=\"text-align: justify;\">Hence, <img decoding=\"async\" style=\"height: 27px; width: 77px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image040.png\" \/><\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"3_If_is_defined_by_find\"><\/span><strong>3. If <img decoding=\"async\" style=\"height: 21px; width: 85px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image001.png\" \/>\u00a0is defined by <img decoding=\"async\" style=\"height: 27px; width: 125px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image041.png\" \/>\u00a0find <img decoding=\"async\" style=\"height: 27px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image042.png\" \/>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Given: <img decoding=\"async\" style=\"height: 27px; width: 120px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image043.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 19px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image008.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 29px; width: 175px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image044.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 19px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image008.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 32px; width: 216px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image045.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 32px; width: 201px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image046.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 19px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image008.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 130px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image047.png\" \/><\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-2\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"4_Show_that_the_function_defined_by_R_is_one-one_and_onto_function\"><\/span><strong>4. Show that the function <img decoding=\"async\" style=\"height: 27px; width: 186px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image048.png\" \/>\u00a0defined by <img decoding=\"async\" style=\"height: 47px; width: 96px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image049.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 15px; width: 26px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image050.png\" \/>\u00a0R is one-one and onto function.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong><img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image015.png\" \/>\u00a0is one-one: For any <img decoding=\"async\" style=\"height: 17px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image051.png\" \/>\u00a0R \u2013 {+1}, we have <img decoding=\"async\" style=\"height: 27px; width: 89px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image052.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 19px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image008.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 47px; width: 92px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image053.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 19px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image008.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 19px; width: 97px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image054.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 19px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image008.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 17px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image055.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, <img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image015.png\" \/>\u00a0is one-one function.<\/p>\n<p style=\"text-align: justify;\">If <img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image015.png\" \/>\u00a0is one-one, let <img decoding=\"async\" style=\"height: 17px; width: 27px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image056.png\" \/>\u00a0R \u2013 {1}, then <img decoding=\"async\" style=\"height: 27px; width: 64px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image057.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 19px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image008.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 60px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image058.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 19px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image008.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 45px; width: 60px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image059.png\" \/><\/p>\n<p style=\"text-align: justify;\">It is cleat that <img decoding=\"async\" style=\"height: 15px; width: 26px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image050.png\" \/>\u00a0R for all <img decoding=\"async\" style=\"height: 17px; width: 27px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image056.png\" \/>\u00a0R \u2013 {1}, also <img decoding=\"async\" style=\"height: 19px; width: 53px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image060.png\" \/><\/p>\n<p style=\"text-align: justify;\">Because <img decoding=\"async\" style=\"height: 19px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image061.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 19px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image008.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 45px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image062.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 19px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image008.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image063.png\" \/>\u00a0which is not possible.<\/p>\n<p style=\"text-align: justify;\">Thus for each R \u2013 {1} there exists <img decoding=\"async\" style=\"height: 45px; width: 70px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image064.png\" \/>\u00a0R \u2013 {1} such that<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 85px; width: 179px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image065.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore <img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image015.png\" \/>\u00a0is onto function.<\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-3\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"5_Show_that_the_function_given_by_is_injective\"><\/span><strong>5. Show that the function <img decoding=\"async\" style=\"height: 21px; width: 85px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image001.png\" \/>\u00a0given by <img decoding=\"async\" style=\"height: 27px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image066.png\" \/>\u00a0is injective.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Let <img decoding=\"async\" style=\"height: 26px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image067.png\" \/>\u00a0R be such that <img decoding=\"async\" style=\"height: 27px; width: 98px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image068.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 19px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image008.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 26px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image069.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 19px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image008.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 26px; width: 47px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image070.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, <img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image015.png\" \/>\u00a0is one-one function, hence <img decoding=\"async\" style=\"height: 27px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image066.png\" \/>\u00a0is injective.<\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-4\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"6_Give_examples_of_two_functions_and_such_that_is_injective_but_is_not_injective\"><\/span><strong>6. Give examples of two functions <img decoding=\"async\" style=\"height: 21px; width: 83px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image071.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 21px; width: 77px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image072.png\" \/>\u00a0such that <img decoding=\"async\" style=\"height: 21px; width: 29px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image073.png\" \/>\u00a0is injective but <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image074.png\" \/>\u00a0is not injective.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p style=\"text-align: justify;\"><strong>(Hint: Consider <\/strong><img decoding=\"async\" style=\"height: 27px; width: 63px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image075.png\" \/><strong>\u00a0and <\/strong><img decoding=\"async\" style=\"height: 27px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image076.png\" \/><strong>\u00a0)<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Given: two functions <img decoding=\"async\" style=\"height: 21px; width: 83px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image071.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 21px; width: 77px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image072.png\" \/><\/p>\n<p style=\"text-align: justify;\">Let <img decoding=\"async\" style=\"height: 27px; width: 63px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image075.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 27px; width: 61px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image077.png\" \/> <img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image028.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 29px; width: 191px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image078.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, <img decoding=\"async\" style=\"height: 21px; width: 29px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image073.png\" \/>\u00a0is injective but <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image074.png\" \/>\u00a0is not injective.<\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-5\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"7_Give_examples_of_two_functions_and_such_that_is_onto_but_is_not_onto\"><\/span><strong>7. Give examples of two functions <img decoding=\"async\" style=\"height: 21px; width: 85px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image079.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 21px; width: 81px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image080.png\" \/>\u00a0such that <img decoding=\"async\" style=\"height: 21px; width: 29px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image073.png\" \/>\u00a0is onto but <img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image015.png\" \/>\u00a0is not onto.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p style=\"text-align: justify;\"><strong>(Hint: Consider <\/strong><img decoding=\"async\" style=\"height: 27px; width: 83px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image081.png\" \/><strong>\u00a0and <\/strong><img decoding=\"async\" style=\"height: 49px; width: 136px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image082.png\" \/><strong>\u00a0)<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Let <img decoding=\"async\" style=\"height: 27px; width: 83px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image081.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\/01%20Misc\/image028.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 49px; width: 156px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image083.png\" \/><\/p>\n<p style=\"text-align: justify;\">These are two examples in which <img decoding=\"async\" style=\"height: 21px; width: 29px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image073.png\" \/>\u00a0is onto but <img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image015.png\" \/>\u00a0is not onto.<\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-6\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"8_Given_a_non_empty_set_X_consider_P_X_which_is_the_set_of_all_subsets_of_X\"><\/span><strong>8. Given a non empty set X, consider P (X) which is the set of all subsets of X.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p style=\"text-align: justify;\"><strong>Define the relation AR in P (X) as follows:<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>For subsets A, B in P (X), ARB if and only if A<\/strong><img decoding=\"async\" style=\"height: 13px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image084.png\" \/><strong>B. Is R an equivalence relation on P (X)? Justify your answer.<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>(i) A <img decoding=\"async\" style=\"height: 13px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image084.png\" \/>\u00a0A\u00a0 <img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image028.png\" \/> R is reflexive.<\/p>\n<p style=\"text-align: justify;\">(ii) A <img decoding=\"async\" style=\"height: 13px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image084.png\" \/>\u00a0B <img decoding=\"async\" style=\"height: 15px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image085.png\" \/>\u00a0B <img decoding=\"async\" style=\"height: 13px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image084.png\" \/>\u00a0A <img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image028.png\" \/>\u00a0R is not commutative.<\/p>\n<p style=\"text-align: justify;\">(iii) If A <img decoding=\"async\" style=\"height: 13px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image084.png\" \/>\u00a0B, B <img decoding=\"async\" style=\"height: 13px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image084.png\" \/>\u00a0C, then A <img decoding=\"async\" style=\"height: 13px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image084.png\" \/>\u00a0C\u00a0 <img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image028.png\" \/>\u00a0R is transitive.<\/p>\n<p style=\"text-align: justify;\">Therefore, R is not equivalent relation.<\/p>\n<hr \/>\n<p style=\"text-align: justify;\"><strong>9. Given a non-empty set X, consider the binary operation * : P (X) x P (X) <img decoding=\"async\" style=\"height: 15px; width: 19px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image086.png\" \/>\u00a0P (X) given by A * B = A <img decoding=\"async\" style=\"height: 13px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image087.png\" \/>\u00a0B <img decoding=\"async\" style=\"height: 17px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image088.png\" \/>\u00a0A, B in P (X), where P (X) is the power set of X. Show that X is the identity element for this operation and X is the only invertible element in P (X) with respect to the operation *.<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Let S be a non-empty set and P(S) be its power set. Let any two subsets A and B of S.<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 19px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image008.png\" \/>\u00a0A <img decoding=\"async\" style=\"height: 13px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image089.png\" \/>\u00a0B <img decoding=\"async\" style=\"height: 13px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image084.png\" \/>\u00a0S<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 19px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image008.png\" \/>\u00a0A <img decoding=\"async\" style=\"height: 13px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image089.png\" \/>\u00a0B <img decoding=\"async\" style=\"height: 13px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image090.png\" \/>\u00a0P(S)<\/p>\n<p style=\"text-align: justify;\">Therefore, <img decoding=\"async\" style=\"height: 17px; width: 26px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image091.png\" \/>\u00a0is an binary operation on P(S).<\/p>\n<p style=\"text-align: justify;\">Similarly, if A, B <img decoding=\"async\" style=\"height: 13px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image090.png\" \/>\u00a0P(S) and A \u2013 B <img decoding=\"async\" style=\"height: 13px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image090.png\" \/>\u00a0P(S), then the intersection of sets <img decoding=\"async\" style=\"height: 13px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image087.png\" \/>\u00a0and difference of sets are also binary operation on P(S) and A <img decoding=\"async\" style=\"height: 13px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image087.png\" \/>\u00a0S = A = S <img decoding=\"async\" style=\"height: 13px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image087.png\" \/>\u00a0A for every subset A of sets<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 19px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image008.png\" \/>\u00a0A <img decoding=\"async\" style=\"height: 13px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image087.png\" \/>\u00a0S = A = S <img decoding=\"async\" style=\"height: 13px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image087.png\" \/>\u00a0A for all A <img decoding=\"async\" style=\"height: 13px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image090.png\" \/>\u00a0P(S)<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 19px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image008.png\" \/>\u00a0S is the identity element for intersection <img decoding=\"async\" style=\"height: 27px; width: 29px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image092.png\" \/>\u00a0on P(S).<\/p>\n<hr \/>\n<p style=\"text-align: justify;\"><strong>10. Find the number of all onto functions from the set {1, 2, 3, \u2026\u2026., <img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image013.png\" \/>} to itself.<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>The number of onto functions that can be defined from a finite set A containing <img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image013.png\" \/>\u00a0elements onto a finite set B containing <img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image013.png\" \/>\u00a0elements = <img decoding=\"async\" style=\"height: 21px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image093.png\" \/><\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-7\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"11_Let_S_and_T_1_2_3_Find_of_the_following_functions_F_from_S_to_T_if_it_exists\"><\/span><strong>11. Let S = <img decoding=\"async\" style=\"height: 27px; width: 53px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image094.png\" \/>\u00a0and T = {1, 2, 3}. Find <img decoding=\"async\" style=\"height: 19px; width: 26px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image095.png\" \/>\u00a0of the following functions F from S to T, if it exists.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p style=\"text-align: justify;\"><strong>(i) F = <img decoding=\"async\" style=\"height: 27px; width: 131px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image096.png\" \/>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(ii) F = <img decoding=\"async\" style=\"height: 27px; width: 128px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image097.png\" \/><\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>S = <img decoding=\"async\" style=\"height: 27px; width: 53px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image094.png\" \/>\u00a0and T = {1, 2, 3}<\/p>\n<p style=\"text-align: justify;\">(i) F = <img decoding=\"async\" style=\"height: 27px; width: 131px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image096.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 19px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image008.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 176px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image098.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 19px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image008.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 211px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image099.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\/01%20Misc\/image028.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 166px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image100.png\" \/><\/p>\n<p style=\"text-align: justify;\">(ii) <img decoding=\"async\" style=\"height: 27px; width: 128px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image097.png\" \/><\/p>\n<p style=\"text-align: justify;\">F is not one-one function, since element <img decoding=\"async\" style=\"height: 19px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image101.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 15px; width: 11px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image102.png\" \/>\u00a0have the same image 1.<\/p>\n<p style=\"text-align: justify;\">Therefore, F is not one-one function.<\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-8\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"12_Consider_the_binary_operation_R_x_R_R_and_o_R_x_R_R_defined_as_and_R_Show_that_is_commutative_but_not_associative_o_is_associative_but_not_commutative_Further_show_that_R_If_it_is_so_we_say_that_the_operation_distributes_over_the_operation_o_Does_o_distribute_over_Justify_your_answer\"><\/span><strong>12. Consider the binary operation * : R x R <img decoding=\"async\" style=\"height: 15px; width: 19px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image086.png\" \/>\u00a0R and o = R x R <img decoding=\"async\" style=\"height: 15px; width: 19px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image086.png\" \/>\u00a0R defined as <img decoding=\"async\" style=\"height: 27px; width: 85px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image103.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 21px; width: 105px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image104.png\" \/>\u00a0R. Show that * is commutative but not associative, o is associative but not commutative. Further, show that <img decoding=\"async\" style=\"height: 21px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image105.png\" \/>\u00a0R, <img decoding=\"async\" style=\"height: 27px; width: 173px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image106.png\" \/>\u00a0[If it is so, we say that the operation * distributes over the operation o]. Does o distribute over *? Justify your answer.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p style=\"text-align: justify;\"><strong>Ans. Part I<\/strong>: \u00a0 <img decoding=\"async\" style=\"height: 27px; width: 85px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image103.png\" \/>\u00a0also <img decoding=\"async\" style=\"height: 27px; width: 143px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image107.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image028.png\" \/>\u00a0operation * is commutative.<\/p>\n<p style=\"text-align: justify;\">Now, <img decoding=\"async\" style=\"height: 29px; width: 290px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image108.png\" \/><\/p>\n<p style=\"text-align: justify;\">And <img decoding=\"async\" style=\"height: 27px; width: 204px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image109.png\" \/><\/p>\n<p style=\"text-align: justify;\">Here, <img decoding=\"async\" style=\"height: 27px; width: 137px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image110.png\" \/> <img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image028.png\" \/>\u00a0operation * is not associative.<\/p>\n<p style=\"text-align: justify;\"><strong>Part II<\/strong>: <img decoding=\"async\" style=\"height: 21px; width: 98px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image111.png\" \/>\u00a0R<\/p>\n<p style=\"text-align: justify;\">And, <img decoding=\"async\" style=\"height: 19px; width: 53px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image112.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 19px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image008.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 19px; width: 70px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image113.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image028.png\" \/>\u00a0operation <img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image114.png\" \/>\u00a0is not commutative.<\/p>\n<p style=\"text-align: justify;\">Now <img decoding=\"async\" style=\"height: 27px; width: 125px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image115.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 27px; width: 125px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image116.png\" \/><\/p>\n<p style=\"text-align: justify;\">Here <img decoding=\"async\" style=\"height: 27px; width: 131px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image117.png\" \/> \u00a0 <img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image028.png\" \/>\u00a0operation <img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image114.png\" \/>\u00a0is associative.<\/p>\n<p style=\"text-align: justify;\"><strong>Part III<\/strong>: L.H.S.\u00a0 <img decoding=\"async\" style=\"height: 27px; width: 60px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image118.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 27px; width: 85px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image103.png\" \/><\/p>\n<p style=\"text-align: justify;\">R.H.S. <img decoding=\"async\" style=\"height: 27px; width: 96px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image119.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 27px; width: 153px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image120.png\" \/>\u00a0= L.H.S. \u00a0 Proved.<\/p>\n<p style=\"text-align: justify;\">Now, another distribution law: <img decoding=\"async\" style=\"height: 27px; width: 164px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image121.png\" \/><\/p>\n<p style=\"text-align: justify;\">L.H.S. <img decoding=\"async\" style=\"height: 29px; width: 168px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image122.png\" \/><\/p>\n<p style=\"text-align: justify;\">R.H.S. <img decoding=\"async\" style=\"height: 27px; width: 211px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image123.png\" \/><\/p>\n<p style=\"text-align: justify;\">As L.H.S. <img decoding=\"async\" style=\"height: 15px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image085.png\" \/>\u00a0R.H.S.<\/p>\n<p style=\"text-align: justify;\">Therefore, the operation <img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image114.png\" \/>\u00a0does not distribute over.<\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-9\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"13_Given_a_non-empty_set_X_let_P_X_x_P_X_P_X_be_defined_as_A_B_A_%E2%80%93_B_B_%E2%80%93_A_A_B_P_X_Show_that_the_empty_set_is_the_identity_for_the_operation_and_all_the_elements_A_of_P_X_are_invertible_with_A-1_A_Hint_and\"><\/span><strong>13. Given a non-empty set X, let * : P (X) x P (X) <img decoding=\"async\" style=\"height: 15px; width: 19px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image086.png\" \/>\u00a0P (X) be defined as A * B = (A \u2013 B) <img decoding=\"async\" style=\"height: 13px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image089.png\" \/>\u00a0(B \u2013 A), <img decoding=\"async\" style=\"height: 17px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image088.png\" \/>\u00a0A, B <img decoding=\"async\" style=\"height: 13px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image090.png\" \/>\u00a0P (X). Show that the empty set <img decoding=\"async\" style=\"height: 21px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image124.png\" \/>\u00a0is the identity for the operation * and all the elements A of P (X) are invertible with A<sup>-1<\/sup> = A. (Hint: <img decoding=\"async\" style=\"height: 27px; width: 145px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image125.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 27px; width: 203px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image126.png\" \/>)<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>For every A <img decoding=\"async\" style=\"height: 13px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image090.png\" \/>\u00a0P(X), we have<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 27px; width: 166px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image127.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 21px; width: 70px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image128.png\" \/><\/p>\n<p style=\"text-align: justify;\">And <img decoding=\"async\" style=\"height: 27px; width: 166px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image129.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 21px; width: 70px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image130.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 19px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image008.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image124.png\" \/>\u00a0is the identity element for the operation * on P(X).<\/p>\n<p style=\"text-align: justify;\">Also A * A = (A \u2013 A) <img decoding=\"async\" style=\"height: 13px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image089.png\" \/>\u00a0(A \u2013 A) = <img decoding=\"async\" style=\"height: 21px; width: 64px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image131.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 19px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image008.png\" \/>\u00a0Every element A of P(X) is invertible with <img decoding=\"async\" style=\"height: 19px; width: 27px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image132.png\" \/>\u00a0= A.<\/p>\n<hr \/>\n<h4 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><\/h4>\n<h4 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"14_Define_binary_operation_on_the_set_0_1_2_3_4_5_as\"><\/span><strong>14. Define binary operation * on the set {0, 1, 2, 3, 4, 5} as <img decoding=\"async\" style=\"height: 49px; width: 181px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image133.png\" \/>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p style=\"text-align: justify;\"><strong>Show that zero is the identity for this operation and each element <\/strong><img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image134.png\" \/><strong>\u00a0of the set is invertible with <\/strong><img decoding=\"async\" style=\"height: 19px; width: 35px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image135.png\" \/><strong>\u00a0being the inverse of <\/strong><img decoding=\"async\" style=\"height: 15px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image136.png\" \/><strong>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>A binary operation (or composition) * on a (non-empty) set is a function * : A x A <img decoding=\"async\" style=\"height: 15px; width: 19px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image086.png\" \/>A. We denote <img decoding=\"async\" style=\"height: 27px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image137.png\" \/>\u00a0by <img decoding=\"async\" style=\"height: 19px; width: 32px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image138.png\" \/>\u00a0for every ordered pair <img decoding=\"async\" style=\"height: 27px; width: 51px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image139.png\" \/>\u00a0A x A.<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 19px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image008.png\" \/>\u00a0A binary operation on a no-empty set A is a rule that associates with every ordered pair of elements <img decoding=\"async\" style=\"height: 21px; width: 27px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image140.png\" \/>\u00a0(distinct or equal) of A some unique element <img decoding=\"async\" style=\"height: 19px; width: 32px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image138.png\" \/>\u00a0of A.<\/p>\n<div>\n<table style=\"height: 328px;\" border=\"1\" width=\"764\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td style=\"height: .3in; width: .4in;\">\n<p style=\"text-align: center;\">*<\/p>\n<\/td>\n<td style=\"height: .3in; width: .4in;\">\n<p style=\"text-align: center;\">0<\/p>\n<\/td>\n<td style=\"height: .3in; width: .4in;\">\n<p style=\"text-align: center;\">1<\/p>\n<\/td>\n<td style=\"height: .3in; width: .4in;\">\n<p style=\"text-align: center;\">2<\/p>\n<\/td>\n<td style=\"height: .3in; width: .4in;\">\n<p style=\"text-align: center;\">3<\/p>\n<\/td>\n<td style=\"height: .3in; width: .4in;\">\n<p style=\"text-align: center;\">4<\/p>\n<\/td>\n<td style=\"height: .3in; width: .4in;\">\n<p style=\"text-align: center;\">5<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"height: .3in; width: .4in;\">\n<p style=\"text-align: center;\">0<\/p>\n<\/td>\n<td style=\"height: .3in; width: .4in;\">\n<p style=\"text-align: center;\">0<\/p>\n<\/td>\n<td style=\"height: .3in; width: .4in;\">\n<p style=\"text-align: center;\">1<\/p>\n<\/td>\n<td style=\"height: .3in; width: .4in;\">\n<p style=\"text-align: center;\">2<\/p>\n<\/td>\n<td style=\"height: .3in; width: .4in;\">\n<p style=\"text-align: center;\">3<\/p>\n<\/td>\n<td style=\"height: .3in; width: .4in;\">\n<p style=\"text-align: center;\">4<\/p>\n<\/td>\n<td style=\"height: .3in; width: .4in;\">\n<p style=\"text-align: center;\">5<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"height: .3in; width: .4in;\">\n<p style=\"text-align: center;\">1<\/p>\n<\/td>\n<td style=\"height: .3in; width: .4in;\">\n<p style=\"text-align: center;\">1<\/p>\n<\/td>\n<td style=\"height: .3in; width: .4in;\">\n<p style=\"text-align: center;\">2<\/p>\n<\/td>\n<td style=\"height: .3in; width: .4in;\">\n<p style=\"text-align: center;\">3<\/p>\n<\/td>\n<td style=\"height: .3in; width: .4in;\">\n<p style=\"text-align: center;\">4<\/p>\n<\/td>\n<td style=\"height: .3in; width: .4in;\">\n<p style=\"text-align: center;\">5<\/p>\n<\/td>\n<td style=\"height: .3in; width: .4in;\">\n<p style=\"text-align: center;\">0<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"height: .3in; width: .4in;\">\n<p style=\"text-align: center;\">2<\/p>\n<\/td>\n<td style=\"height: .3in; width: .4in;\">\n<p style=\"text-align: center;\">2<\/p>\n<\/td>\n<td style=\"height: .3in; width: .4in;\">\n<p style=\"text-align: center;\">3<\/p>\n<\/td>\n<td style=\"height: .3in; width: .4in;\">\n<p style=\"text-align: center;\">4<\/p>\n<\/td>\n<td style=\"height: .3in; width: .4in;\">\n<p style=\"text-align: center;\">5<\/p>\n<\/td>\n<td style=\"height: .3in; width: .4in;\">\n<p style=\"text-align: center;\">0<\/p>\n<\/td>\n<td style=\"height: .3in; width: .4in;\">\n<p style=\"text-align: center;\">1<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"height: .3in; width: .4in;\">\n<p style=\"text-align: center;\">3<\/p>\n<\/td>\n<td style=\"height: .3in; width: .4in;\">\n<p style=\"text-align: center;\">3<\/p>\n<\/td>\n<td style=\"height: .3in; width: .4in;\">\n<p style=\"text-align: center;\">4<\/p>\n<\/td>\n<td style=\"height: .3in; width: .4in;\">\n<p style=\"text-align: center;\">5<\/p>\n<\/td>\n<td style=\"height: .3in; width: .4in;\">\n<p style=\"text-align: center;\">0<\/p>\n<\/td>\n<td style=\"height: .3in; width: .4in;\">\n<p style=\"text-align: center;\">1<\/p>\n<\/td>\n<td style=\"height: .3in; width: .4in;\">\n<p style=\"text-align: center;\">2<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"height: .3in; width: .4in;\">\n<p style=\"text-align: center;\">4<\/p>\n<\/td>\n<td style=\"height: .3in; width: .4in;\">\n<p style=\"text-align: center;\">4<\/p>\n<\/td>\n<td style=\"height: .3in; width: .4in;\">\n<p style=\"text-align: center;\">5<\/p>\n<\/td>\n<td style=\"height: .3in; width: .4in;\">\n<p style=\"text-align: center;\">0<\/p>\n<\/td>\n<td style=\"height: .3in; width: .4in;\">\n<p style=\"text-align: center;\">1<\/p>\n<\/td>\n<td style=\"height: .3in; width: .4in;\">\n<p style=\"text-align: center;\">2<\/p>\n<\/td>\n<td style=\"height: .3in; width: .4in;\">\n<p style=\"text-align: center;\">3<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"height: .3in; width: .4in;\">\n<p style=\"text-align: center;\">5<\/p>\n<\/td>\n<td style=\"height: .3in; width: .4in;\">\n<p style=\"text-align: center;\">5<\/p>\n<\/td>\n<td style=\"height: .3in; width: .4in;\">\n<p style=\"text-align: center;\">0<\/p>\n<\/td>\n<td style=\"height: .3in; width: .4in;\">\n<p style=\"text-align: center;\">1<\/p>\n<\/td>\n<td style=\"height: .3in; width: .4in;\">\n<p style=\"text-align: center;\">2<\/p>\n<\/td>\n<td style=\"height: .3in; width: .4in;\">\n<p style=\"text-align: center;\">3<\/p>\n<\/td>\n<td style=\"height: .3in; width: .4in;\">\n<p style=\"text-align: center;\">4<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p style=\"text-align: justify;\">For all <img decoding=\"async\" style=\"height: 16px; width: 26px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image141.png\" \/>\u00a0A, we have <img decoding=\"async\" style=\"height: 19px; width: 32px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image142.png\" \/>\u00a0(mod 6) = 0<\/p>\n<p style=\"text-align: justify;\">And <img decoding=\"async\" style=\"height: 27px; width: 132px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image143.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 27px; width: 157px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image144.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 19px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image008.png\" \/>\u00a00 is the identity element for the operation.<\/p>\n<p style=\"text-align: justify;\">Also on 0 = 0 \u2013 0 = 0 *<\/p>\n<p style=\"text-align: justify;\">2 * 1 = 3 = 1 * 2<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 21px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image145.png\" \/> \u00a0 <img decoding=\"async\" style=\"height: 21px; width: 47px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image146.png\" \/><\/p>\n<hr \/>\n<h4 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><\/h4>\n<h4 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"15_Let_A_%E2%80%931_0_1_2_B_%E2%80%934_%E2%80%932_0_2_and_be_the_functions_defined_by_A_and_A_Are_and_equal_Justify_your_answer\"><\/span><strong>15. Let A = {\u20131, 0, 1, 2}, B = {\u20134, \u20132, 0, 2} and <img decoding=\"async\" style=\"height: 21px; width: 98px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image147.png\" \/>\u00a0be the functions defined by <img decoding=\"async\" style=\"height: 27px; width: 51px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image148.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 26px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image149.png\" \/>\u00a0A and <img decoding=\"async\" style=\"height: 45px; width: 147px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image150.png\" \/>\u00a0A. Are <img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image015.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image074.png\" \/>\u00a0equal? Justify your answer.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p style=\"text-align: justify;\"><strong>(Hint: One may note that two functions <\/strong><img decoding=\"async\" style=\"height: 21px; width: 83px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image151.png\" \/><strong>\u00a0and <\/strong><img decoding=\"async\" style=\"height: 21px; width: 80px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image152.png\" \/><strong>\u00a0such that <\/strong><img decoding=\"async\" style=\"height: 27px; width: 87px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image153.png\" \/><strong>\u00a0<\/strong><img decoding=\"async\" style=\"height: 17px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image088.png\" \/><strong>\u00a0<\/strong><img decoding=\"async\" style=\"height: 15px; width: 26px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image154.png\" \/><strong>\u00a0A, are called equal functions).<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>When <img decoding=\"async\" style=\"height: 19px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image061.png\" \/>\u00a0then <img decoding=\"async\" style=\"height: 27px; width: 109px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image155.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 45px; width: 152px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image156.png\" \/><\/p>\n<p style=\"text-align: justify;\">At <img decoding=\"async\" style=\"height: 21px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image157.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 61px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image158.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 45px; width: 199px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image159.png\" \/><\/p>\n<p style=\"text-align: justify;\">At <img decoding=\"async\" style=\"height: 21px; width: 36px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image160.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 107px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image161.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 45px; width: 203px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image162.png\" \/><\/p>\n<p style=\"text-align: justify;\">At <img decoding=\"async\" style=\"height: 21px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image163.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 115px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image164.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 45px; width: 185px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image165.png\" \/><\/p>\n<p style=\"text-align: justify;\">Thus for each <img decoding=\"async\" style=\"height: 15px; width: 26px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image154.png\" \/>\u00a0A, <img decoding=\"async\" style=\"height: 27px; width: 87px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image153.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, <img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image015.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image074.png\" \/>\u00a0are equal function.<\/p>\n<hr \/>\n<h4 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><\/h4>\n<h4 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"16_Let_A_1_2_3_Then_number_of_relations_containing_1_2_and_1_3_which_are_reflexive_and_symmetric_but_not_transitive_is\"><\/span><strong>16. Let A = {1, 2, 3}. Then number of relations containing (1, 2) and (1, 3) which are reflexive and symmetric but not transitive is:<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\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) 4<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>It is clear that 1 is reflexive and symmetric but not transitive.<\/p>\n<p style=\"text-align: justify;\">Therefore, option (A) is correct.<\/p>\n<hr \/>\n<h4 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"17_Let_A_1_2_3_Then_number_of_equivalence_relations_containing_1_2_is\"><\/span><strong>17. Let A = {1, 2, 3}. Then number of equivalence relations containing (1, 2) is:<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\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) 4<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>2<\/p>\n<p style=\"text-align: justify;\">Therefore, option (B) is correct.<\/p>\n<hr \/>\n<h4 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><\/h4>\n<h4 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"18_Let_be_the_Signum_Function_defined_as_and_be_the_Greatest_Function_given_by_where_is_greatest_integer_less_than_or_equal_to_Then_does_and_coincide_in_0_1\"><\/span><strong>18. Let <img decoding=\"async\" style=\"height: 21px; width: 85px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image001.png\" \/>\u00a0be the Signum Function defined as <img decoding=\"async\" style=\"height: 75px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image166.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 21px; width: 83px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image167.png\" \/>\u00a0be the Greatest Function given by <img decoding=\"async\" style=\"height: 27px; width: 77px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image168.png\" \/>\u00a0where <img decoding=\"async\" style=\"height: 27px; width: 26px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image169.png\" \/>\u00a0is greatest integer less than or equal to <img decoding=\"async\" style=\"height: 15px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image170.png\" \/>\u00a0Then, does <img decoding=\"async\" style=\"height: 21px; width: 29px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image171.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 21px; width: 29px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image073.png\" \/>\u00a0coincide in (0, 1)?<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>It is clear that <img decoding=\"async\" style=\"height: 21px; width: 97px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image172.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 21px; width: 97px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image173.png\" \/><\/p>\n<p style=\"text-align: justify;\">Consider <img decoding=\"async\" style=\"height: 41px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image174.png\" \/>\u00a0which lie on (0, # 1)<\/p>\n<p style=\"text-align: justify;\">Now, <img decoding=\"async\" style=\"height: 49px; width: 251px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image175.png\" \/><\/p>\n<p style=\"text-align: justify;\">And <img decoding=\"async\" style=\"height: 49px; width: 296px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image176.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 19px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image008.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 70px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image177.png\" \/>\u00a0in (0, 1]\n<p style=\"text-align: justify;\">Therefore, option (B) is correct.<\/p>\n<hr \/>\n<h4 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><\/h4>\n<h4 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"19_Number_of_binary_operation_on_the_set_are\"><\/span><strong>19. Number of binary operation on the set <img decoding=\"async\" style=\"height: 27px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image178.png\" \/>\u00a0are:<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p style=\"text-align: justify;\"><strong>(A) 10<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(b) 16<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(C) 20<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(D) 8<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>A = <img decoding=\"async\" style=\"height: 27px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image178.png\" \/><\/p>\n<p style=\"text-align: justify;\">A x A = <img decoding=\"async\" style=\"height: 27px; width: 173px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image179.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\/01%20Misc\/image028.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 64px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image180.png\" \/>\u00a0= 4<\/p>\n<p style=\"text-align: justify;\">Number of subsets = <img decoding=\"async\" style=\"height: 19px; width: 19px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/01%20Misc\/image181.png\" \/>\u00a0= 16<\/p>\n<p style=\"text-align: justify;\">Hence number of binary operation is 16.<\/p>\n<p style=\"text-align: justify;\">Therefore, option (B) is correct.<\/p>\n<h2><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><\/h2>\n<p>NCERT Solutions Class 12 Maths PDF (Download) Free from myCBSEguide app and myCBSEguide website. Ncert solution class 12 Maths includes text book solutions from both part 1 and part 2. NCERT Solutions for CBSE Class 12 Maths have total 13 chapters. 12 Maths NCERT Solutions in PDF for free Download on our website. Ncert Maths class 12 solutions PDF and Maths ncert class 12 PDF solutions with latest modifications and as per the latest CBSE syllabus are only available in myCBSEguide<\/p>\n<h2><span class=\"ez-toc-section\" id=\"CBSE_App_for_Students\"><\/span>CBSE App for Students<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>To download\u00a0NCERT Solutions for class 12 Physics, Chemistry, Biology, History, Political Science, Economics, Geography, Computer Science, Home Science, Accountancy, Business Studies and Home Science; do check myCBSEguide app or website. myCBSEguide provides sample papers with solution, test papers for chapter-wise practice, NCERT solutions, NCERT Exemplar solutions, quick revision notes for ready reference, CBSE guess papers and CBSE important question papers. Sample Paper all are made available through\u00a0<a href=\"https:\/\/play.google.com\/store\/apps\/details?id=in.techchefs.MyCBSEGuide&amp;referrer=utm_source%3Dmycbse_bottom%26utm_medium%3Dtext%26utm_campaign%3Dmycbseads\"><strong>the best app for CBSE students<\/strong><\/a>\u00a0and myCBSEguide website.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>NCERT Solutions class 12 Maths Miscellaneous Class 12 Maths book solutions are available in PDF format for free download. These ncert book chapter wise questions and answers are very helpful for CBSE board exam. CBSE recommends NCERT books and most of the questions in CBSE exam are asked from NCERT text books. Class 12 Maths &#8230; <a title=\"NCERT Solutions class 12 Maths Miscellaneous\" class=\"read-more\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous\/\" aria-label=\"More on NCERT Solutions class 12 Maths Miscellaneous\">Read more<\/a><\/p>\n","protected":false},"author":1,"featured_media":-1,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1432,281],"tags":[283,1342,321,216],"class_list":["post-4464","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mathematics-cbse-class-12","category-ncert-solutions","tag-cbse-study-material","tag-class-12","tag-mathematics","tag-ncert-solutions"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.0 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>NCERT Solutions class 12 Maths Miscellaneous | myCBSEguide<\/title>\n<meta name=\"description\" content=\"NCERT Solutions class 12 Maths Miscellaneous in PDF format for free download. 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