{"id":4460,"date":"2016-05-16T11:49:00","date_gmt":"2016-05-16T06:19:00","guid":{"rendered":"http:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-1-2\/"},"modified":"2019-02-18T16:23:09","modified_gmt":"2019-02-18T10:53:09","slug":"ncert-solutions-class-12-maths-exercise-1-2","status":"publish","type":"post","link":"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-1-2\/","title":{"rendered":"NCERT Solutions Class 12 Maths Exercise 1.2"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_76 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 eztoc-toggle-hide-by-default' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-1-2\/#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-exercise-1-2\/#3_Prove_that_the_Greatest_integer_Function_given_by_is_neither_one-one_nor_onto_where_denotes_the_greatest_integer_less_than_or_equal_to\" >3. Prove that the Greatest integer Function , given by \u00a0is neither one-one nor onto, where \u00a0denotes the greatest integer less than or equal to \u00a0<\/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-exercise-1-2\/#4_Show_that_the_Modulus_Function_given_by_is_neither_one-one_nor_onto_where_is_if_is_positive_or_0_and_is_if_is_negative\" >4. Show that the Modulus Function , given by \u00a0is neither one-one nor onto, where is \u00a0if \u00a0is positive or 0 and \u00a0is \u00a0if \u00a0is negative.<\/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-exercise-1-2\/#5_Show_that_the_Signum_Function_given_by_is_neither_one-one_nor_onto\" >5. Show that the Signum Function , given by \u00a0is neither one-one nor onto.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-1-2\/#NCERT_Solutions_class_12_Maths_Exercise_12\" >NCERT Solutions class 12 Maths Exercise 1.2<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-1-2\/#6_Let_A_1_2_3_B_4_5_6_7_and_let_1_4_2_5_3_6_be_a_function_from_A_to_B_Show_that_is_one-one\" >6. Let A = {1, 2, 3}, B = {4, 5, 6, 7} and let \u00a0= {(1, 4), (2, 5), (3, 6)} be a function from A to B. Show that \u00a0is one-one.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-1-2\/#NCERT_Solutions_class_12_Maths_Exercise_12-2\" >NCERT Solutions class 12 Maths Exercise 1.2<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-1-2\/#7_In_each_of_the_following_cases_state_whether_the_function_is_one-one_onto_or_bijective_Justify_your_answer\" >7. In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-1-2\/#NCERT_Solutions_class_12_Maths_Exercise_12-3\" >NCERT Solutions class 12 Maths Exercise 1.2<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-1-2\/#8_Let_A_and_B_be_sets_Show_that_A_x_B_B_x_A_such_that_is_bijective_function\" >8. Let A and B be sets. Show that \u00a0: A x B \u00a0B x A such that \u00a0is bijective function.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-11\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-1-2\/#9_Let_be_defined_by_for_all_N\" >9. Let \u00a0be defined by \u00a0for all \u00a0N.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-12\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-1-2\/#NCERT_Solutions_class_12_Maths_Exercise_12-4\" >NCERT Solutions class 12 Maths Exercise 1.2<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-13\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-1-2\/#10_Let_A_R_%E2%80%93_3_and_B_R_%E2%80%93_1_Consider_the_function_A_B_defined_by_Is_one-one_and_onto_Justify_your_answer\" >10. Let A = R \u2013 {3} and B = R \u2013 {1}. Consider the function \u00a0: A \u00a0B defined by \u00a0Is \u00a0one-one and onto? Justify your answer.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-14\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-1-2\/#NCERT_Solutions_class_12_Maths_Exercise_12-5\" >NCERT Solutions class 12 Maths Exercise 1.2<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-15\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-1-2\/#11_Let_be_defined_as_Choose_the_correct_answer\" >11. Let \u00a0be defined as \u00a0Choose the correct answer:<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-16\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-1-2\/#NCERT_Solutions_class_12_Maths_Exercise_12-6\" >NCERT Solutions class 12 Maths Exercise 1.2<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-17\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-1-2\/#12_Let_be_defined_as_Choose_the_correct_answer\" >12. Let \u00a0be defined as \u00a0Choose the correct answer:<\/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-18\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-1-2\/#NCERT_Solutions_class_12_Maths_Exercise_12-7\" >NCERT Solutions class 12 Maths Exercise 1.2<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-19\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-1-2\/#CBSE_App_for_Students\" >CBSE App for Students<\/a><\/li><\/ul><\/nav><\/div>\n<p><strong>NCERT Solutions class 12 Maths Exercise 1.2<\/strong>\u00a0solutions 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 style=\"text-align: center;\"><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 Exercise 1.2\" width=\"104\" height=\"133\" \/><\/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. Show that the function <\/strong><strong><img decoding=\"async\" style=\"height: 25px; width: 85px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image001.png\" \/><\/strong><strong>\u00a0defined by <\/strong><strong><img decoding=\"async\" style=\"height: 41px; width: 65px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image002.png\" \/><\/strong><strong>\u00a0is one-one and onto, where <\/strong><strong><img decoding=\"async\" style=\"height: 25px; width: 21px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image003.png\" \/><\/strong><strong>\u00a0is the set of all non-zero real numbers. Is the result true, if the domain <\/strong><strong><img decoding=\"async\" style=\"height: 25px; width: 21px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image003.png\" \/><\/strong><strong>\u00a0is replaced by N with co-domain being same as <\/strong><strong><img decoding=\"async\" style=\"height: 25px; width: 21px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image003.png\" \/><\/strong><strong>?<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans.<\/strong> \u00a0<img decoding=\"async\" style=\"height: 41px; width: 153px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image004.png\" \/><\/p>\n<p style=\"text-align: justify;\"><strong>Part I<\/strong>: <img decoding=\"async\" style=\"height: 45px; width: 73px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image005.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 45px; width: 77px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image006.png\" \/><\/p>\n<p style=\"text-align: justify;\">If <img decoding=\"async\" style=\"height: 27px; width: 99px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image007.png\" \/>\u00a0then <img decoding=\"async\" style=\"height: 45px; width: 51px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image008.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image009.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 25px; width: 47px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image010.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\/Ex1.2\/image011.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image012.png\" \/>\u00a0is one-one.<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 41px; width: 65px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image002.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image009.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image013.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image009.png\" \/><img decoding=\"async\" style=\"height: 45px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image014.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image009.png\" \/><img decoding=\"async\" style=\"height: 48px; width: 71px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image015.png\" \/>\u00a0\u00a0<img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image011.png\" \/>\u00a0\u00a0<img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image012.png\" \/>\u00a0is onto.<\/p>\n<p style=\"text-align: justify;\"><strong>Part II<\/strong>:\u00a0 If <img decoding=\"async\" style=\"height: 27px; width: 99px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image007.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image009.png\" \/><img decoding=\"async\" style=\"height: 45px; width: 51px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image016.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image009.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 25px; width: 47px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image017.png\" \/>\u00a0where <img decoding=\"async\" style=\"height: 25px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image018.png\" \/>\u00a0N<\/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\/Ex1.2\/image011.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image012.png\" \/>\u00a0is one-one.<\/p>\n<p style=\"text-align: justify;\">But every real number belonging to co-domain may not have a pre-image in N.<\/p>\n<p style=\"text-align: justify;\">e.g. <img decoding=\"async\" style=\"height: 41px; width: 54px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image019.png\" \/>\u00a0N\u00a0 <img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image011.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image012.png\" \/>\u00a0is not onto.<\/p>\n<hr \/>\n<p style=\"text-align: justify;\"><strong>2. Check the injectivity and surjectivity of the following functions:<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(i) <\/strong><strong><img decoding=\"async\" style=\"height: 21px; width: 77px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image020.png\" \/><\/strong><strong>\u00a0given by <\/strong><strong><img decoding=\"async\" style=\"height: 27px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image021.png\" \/><\/strong><strong>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(ii) <\/strong><strong><img decoding=\"async\" style=\"height: 21px; width: 71px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image022.png\" \/><\/strong><strong>\u00a0given by <\/strong><strong><img decoding=\"async\" style=\"height: 27px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image021.png\" \/><\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(iii) <\/strong><strong><img decoding=\"async\" style=\"height: 21px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image023.png\" \/><\/strong><strong>\u00a0given by <\/strong><strong><img decoding=\"async\" style=\"height: 27px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image021.png\" \/><\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(iv) <\/strong><strong><img decoding=\"async\" style=\"height: 21px; width: 77px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image020.png\" \/><\/strong><strong>\u00a0given by <\/strong><strong><img decoding=\"async\" style=\"height: 27px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image024.png\" \/><\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(v) <\/strong><strong><img decoding=\"async\" style=\"height: 21px; width: 71px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image022.png\" \/><\/strong><strong>\u00a0given by <\/strong><strong><img decoding=\"async\" style=\"height: 27px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image024.png\" \/><\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. (i)<\/strong> <img decoding=\"async\" style=\"height: 21px; width: 77px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image020.png\" \/>\u00a0given by <img decoding=\"async\" style=\"height: 27px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image021.png\" \/><\/p>\n<p style=\"text-align: justify;\">If <img decoding=\"async\" style=\"height: 27px; width: 99px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image007.png\" \/>\u00a0then <img decoding=\"async\" style=\"height: 25px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image025.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image009.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 25px; width: 47px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image010.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\/Ex1.2\/image011.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image012.png\" \/>\u00a0is injective.<\/p>\n<p style=\"text-align: justify;\">There are such numbers of co-domain which have no image in domain N.<\/p>\n<p style=\"text-align: justify;\">e.g. 3<img decoding=\"async\" style=\"height: 13px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image026.png\" \/>\u00a0co-domain N, but there is no pre-image in domain of <img decoding=\"async\" style=\"height: 21px; width: 19px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image027.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\/Ex1.2\/image012.png\" \/>\u00a0is not onto. <img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image011.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image012.png\" \/>\u00a0is not surjective.<\/p>\n<p style=\"text-align: justify;\"><strong>(ii) <\/strong><img decoding=\"async\" style=\"height: 21px; width: 71px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image022.png\" \/>\u00a0given by <img decoding=\"async\" style=\"height: 27px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image021.png\" \/><\/p>\n<p style=\"text-align: justify;\">Since,\u00a0 Z = <img decoding=\"async\" style=\"height: 27px; width: 119px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image028.png\" \/>\u00a0therefore, <img decoding=\"async\" style=\"height: 27px; width: 113px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image029.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image009.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 17px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image030.png\" \/>\u00a0and 1 have same image.\u00a0\u00a0 <img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image011.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image012.png\" \/>\u00a0is not injective.<\/p>\n<p style=\"text-align: justify;\">There are such numbers of co-domain which have no image in domain Z.<\/p>\n<p style=\"text-align: justify;\">e.g. 3<img decoding=\"async\" style=\"height: 13px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image026.png\" \/>\u00a0co-domain, but <img decoding=\"async\" style=\"height: 25px; width: 36px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image031.png\" \/>\u00a0domain of <img decoding=\"async\" style=\"height: 21px; width: 19px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image027.png\" \/>\u00a0\u00a0<img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image011.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image012.png\" \/>\u00a0is not surjective.<\/p>\n<p style=\"text-align: justify;\"><strong>(iii)<\/strong> <img decoding=\"async\" style=\"height: 21px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image023.png\" \/>\u00a0given by <img decoding=\"async\" style=\"height: 27px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image021.png\" \/><\/p>\n<p style=\"text-align: justify;\">As <img decoding=\"async\" style=\"height: 27px; width: 113px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image029.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image009.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 17px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image030.png\" \/>\u00a0and 1 have same image.\u00a0 <img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image011.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image012.png\" \/>\u00a0is not injective.<\/p>\n<p style=\"text-align: justify;\">e.g. <img decoding=\"async\" style=\"height: 17px; width: 33px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image032.png\" \/>\u00a0co-domain, but <img decoding=\"async\" style=\"height: 23px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image033.png\" \/>\u00a0domain R of <img decoding=\"async\" style=\"height: 21px; width: 19px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image027.png\" \/>\u00a0\u00a0\u00a0 <img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image011.png\" \/>\u00a0\u00a0<img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image012.png\" \/>\u00a0is not surjective.<\/p>\n<p style=\"text-align: justify;\"><strong>(iv) <\/strong><img decoding=\"async\" style=\"height: 21px; width: 77px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image020.png\" \/>\u00a0given by <img decoding=\"async\" style=\"height: 27px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image024.png\" \/><\/p>\n<p style=\"text-align: justify;\">If <img decoding=\"async\" style=\"height: 27px; width: 99px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image007.png\" \/>\u00a0then <img decoding=\"async\" style=\"height: 25px; width: 48px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image034.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image009.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 25px; width: 47px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image010.png\" \/><\/p>\n<p style=\"text-align: justify;\">i.e., for every <img decoding=\"async\" style=\"height: 15px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image035.png\" \/>\u00a0N, has a unique image in its co-domain. \u00a0\u00a0<img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image011.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image012.png\" \/>\u00a0is injective.<\/p>\n<p style=\"text-align: justify;\">There are many such members of co-domain of <img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image012.png\" \/>\u00a0which do not have pre-image in its domain e.g., 2, 3, etc.<\/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\/Ex1.2\/image012.png\" \/>\u00a0is not onto. <img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image011.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image012.png\" \/>\u00a0is not surjective.<\/p>\n<p style=\"text-align: justify;\"><strong>(v)<\/strong><img decoding=\"async\" style=\"height: 21px; width: 71px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image022.png\" \/>\u00a0given by <img decoding=\"async\" style=\"height: 27px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image024.png\" \/><\/p>\n<p style=\"text-align: justify;\">If <img decoding=\"async\" style=\"height: 27px; width: 99px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image007.png\" \/>\u00a0then <img decoding=\"async\" style=\"height: 25px; width: 48px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image034.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image009.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 25px; width: 47px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image010.png\" \/><\/p>\n<p style=\"text-align: justify;\">i.e., for every <img decoding=\"async\" style=\"height: 15px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image035.png\" \/>\u00a0Z, has a unique image in its co-domain. \u00a0<img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image011.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image012.png\" \/>\u00a0is injective.<\/p>\n<p style=\"text-align: justify;\">There are many such members of co-domain of <img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image012.png\" \/>\u00a0which do not have pre-image in its domain.<\/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\/Ex1.2\/image012.png\" \/>\u00a0is not onto.<\/p>\n<hr \/>\n<h4 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"3_Prove_that_the_Greatest_integer_Function_given_by_is_neither_one-one_nor_onto_where_denotes_the_greatest_integer_less_than_or_equal_to\"><\/span><strong>3. Prove that the Greatest integer Function <\/strong><strong><img decoding=\"async\" style=\"height: 21px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image023.png\" \/><\/strong><strong>, given by <\/strong><strong><img decoding=\"async\" style=\"height: 27px; width: 79px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image036.png\" \/><\/strong><strong>\u00a0is neither one-one nor onto, where <\/strong><strong><img decoding=\"async\" style=\"height: 27px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image037.png\" \/><\/strong><strong>\u00a0denotes the greatest integer less than or equal to <\/strong><strong><img decoding=\"async\" style=\"height: 15px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image038.png\" \/><\/strong><strong>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Function <img decoding=\"async\" style=\"height: 21px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image023.png\" \/>, given by <img decoding=\"async\" style=\"height: 27px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image039.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\/Ex1.2\/image040.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 136px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image041.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\/Ex1.2\/image011.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 56px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image042.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 27px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image043.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\/Ex1.2\/image011.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image012.png\" \/>\u00a0is not one-one.<\/p>\n<p style=\"text-align: justify;\">All the images of <img decoding=\"async\" style=\"height: 16px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image044.png\" \/>\u00a0R belong to its domain have integers as the images in co-domain. But no fraction proper or improper belonging to co-domain of <img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image012.png\" \/>\u00a0has any pre-image in its domain.<\/p>\n<hr \/>\n<h4 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"4_Show_that_the_Modulus_Function_given_by_is_neither_one-one_nor_onto_where_is_if_is_positive_or_0_and_is_if_is_negative\"><\/span><strong>4. Show that the Modulus Function <\/strong><strong><img decoding=\"async\" style=\"height: 21px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image023.png\" \/><\/strong><strong>, given by <\/strong><strong><img decoding=\"async\" style=\"height: 27px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image045.png\" \/><\/strong><strong>\u00a0is neither one-one nor onto, where is <\/strong><strong><img decoding=\"async\" style=\"height: 17px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image046.png\" \/><\/strong><strong>\u00a0if <\/strong><strong><img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image047.png\" \/><\/strong><strong>\u00a0is positive or 0 and <\/strong><strong><img decoding=\"async\" style=\"height: 27px; width: 19px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image048.png\" \/><\/strong><strong>\u00a0is <\/strong><strong><img decoding=\"async\" style=\"height: 21px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image049.png\" \/><\/strong><strong>\u00a0if <\/strong><strong><img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image047.png\" \/><\/strong><strong>\u00a0is negative.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Modulus Function <img decoding=\"async\" style=\"height: 21px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image023.png\" \/>, given by <img decoding=\"async\" style=\"height: 27px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image050.png\" \/><\/p>\n<p style=\"text-align: justify;\">Now\u00a0\u00a0\u00a0 <img decoding=\"async\" style=\"height: 75px; width: 117px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image051.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image009.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image012.png\" \/>\u00a0contains <img decoding=\"async\" style=\"height: 27px; width: 171px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image052.png\" \/><\/p>\n<p style=\"text-align: justify;\">Thus negative integers are not images of any element.\u00a0\u00a0\u00a0\u00a0 <img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image011.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image012.png\" \/>\u00a0is not one-one.<\/p>\n<p style=\"text-align: justify;\">Also second set R contains some negative numbers which are not images of any real number.<\/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\/Ex1.2\/image011.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image012.png\" \/>\u00a0is not onto.<\/p>\n<hr \/>\n<h4 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"5_Show_that_the_Signum_Function_given_by_is_neither_one-one_nor_onto\"><\/span><strong>5. Show that the Signum Function <\/strong><strong><img decoding=\"async\" style=\"height: 21px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image023.png\" \/><\/strong><strong>, given by <\/strong><strong><img decoding=\"async\" style=\"height: 75px; width: 133px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image053.png\" \/><\/strong><strong>\u00a0is neither one-one nor onto.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Signum Function <img decoding=\"async\" style=\"height: 21px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image023.png\" \/>, given by <img decoding=\"async\" style=\"height: 75px; width: 133px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image053.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 27px; width: 105px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image054.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image009.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 120px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image055.png\" \/>\u00a0for <img decoding=\"async\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image056.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image009.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 25px; width: 47px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image057.png\" \/>\u00a0\u00a0 <img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image011.png\" \/>\u00a0\u00a0<img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image012.png\" \/>\u00a0is not one-one.<\/p>\n<p style=\"text-align: justify;\">Except <img decoding=\"async\" style=\"height: 21px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image058.png\" \/>\u00a0no other members of co-domain of <img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image012.png\" \/>\u00a0has any pre-image its domain.<\/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\/Ex1.2\/image011.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image012.png\" \/>\u00a0is not onto.<\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_12\"><\/span>NCERT Solutions class 12 Maths Exercise 1.2<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"6_Let_A_1_2_3_B_4_5_6_7_and_let_1_4_2_5_3_6_be_a_function_from_A_to_B_Show_that_is_one-one\"><\/span><strong>6. Let A = {1, 2, 3}, B = {4, 5, 6, 7} and let <\/strong><strong><img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image012.png\" \/><\/strong><strong>\u00a0= {(1, 4), (2, 5), (3, 6)} be a function from A to B. Show that <\/strong><strong><img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image012.png\" \/><\/strong><strong>\u00a0is one-one.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>A = {1, 2, 3}, B = {4, 5, 6, 7} and <img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image012.png\" \/>\u00a0= {(1, 4), (2, 5), (3, 6)}<\/p>\n<p style=\"text-align: justify;\">Here,\u00a0\u00a0 <img decoding=\"async\" style=\"height: 27px; width: 120px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image059.png\" \/> and <img decoding=\"async\" style=\"height: 27px; width: 61px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image060.png\" \/><\/p>\n<p style=\"text-align: justify;\">Here, also distinct elements of A have distinct images in B.<\/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\/Ex1.2\/image012.png\" \/>\u00a0is not one-one and <img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image012.png\" \/>\u00a0is not bijective.<\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_12-2\"><\/span>NCERT Solutions class 12 Maths Exercise 1.2<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"7_In_each_of_the_following_cases_state_whether_the_function_is_one-one_onto_or_bijective_Justify_your_answer\"><\/span><strong>7. In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p style=\"text-align: justify;\"><strong>(i) <\/strong><strong><img decoding=\"async\" style=\"height: 21px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image023.png\" \/><\/strong><strong>\u00a0defined by <\/strong><strong><img decoding=\"async\" style=\"height: 27px; width: 92px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image061.png\" \/><\/strong><strong>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(ii) <\/strong><strong><img decoding=\"async\" style=\"height: 21px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image023.png\" \/><\/strong><strong>\u00a0defined by <\/strong><strong><img decoding=\"async\" style=\"height: 27px; width: 88px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image062.png\" \/><\/strong><strong>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. (i) <\/strong><img decoding=\"async\" style=\"height: 21px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image023.png\" \/>\u00a0defined by <img decoding=\"async\" style=\"height: 27px; width: 92px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image061.png\" \/><\/p>\n<p style=\"text-align: justify;\">Now,\u00a0\u00a0 if <img decoding=\"async\" style=\"height: 25px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image063.png\" \/>\u00a0R, then <img decoding=\"async\" style=\"height: 27px; width: 99px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image064.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 27px; width: 103px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image065.png\" \/><\/p>\n<p style=\"text-align: justify;\">And\u00a0\u00a0\u00a0\u00a0 if <img decoding=\"async\" style=\"height: 27px; width: 99px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image007.png\" \/>, then <img decoding=\"async\" style=\"height: 25px; width: 47px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image010.png\" \/>\u00a0\u00a0<img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image011.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image012.png\" \/>\u00a0is one-one.<\/p>\n<p style=\"text-align: justify;\">Again, if every element of Y (\u2013 R) is image of some element of X (R) under <img decoding=\"async\" style=\"height: 21px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image066.png\" \/>\u00a0i.e., for every <img decoding=\"async\" style=\"height: 17px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image067.png\" \/>\u00a0Y, there exists an element <img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image047.png\" \/>\u00a0in X such that <img decoding=\"async\" style=\"height: 27px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image068.png\" \/><\/p>\n<p style=\"text-align: justify;\">Now <img decoding=\"async\" style=\"height: 21px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image069.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image009.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 61px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image070.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\/Ex1.2\/image011.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 45px; width: 163px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image071.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image009.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 132px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image072.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\/Ex1.2\/image011.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image012.png\" \/>\u00a0is onto or bijective function.<\/p>\n<p style=\"text-align: justify;\"><strong>(ii) <\/strong><img decoding=\"async\" style=\"height: 21px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image023.png\" \/>\u00a0defined by <img decoding=\"async\" style=\"height: 27px; width: 88px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image062.png\" \/><\/p>\n<p style=\"text-align: justify;\">Now,\u00a0\u00a0 if <img decoding=\"async\" style=\"height: 25px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image063.png\" \/>\u00a0R, then <img decoding=\"async\" style=\"height: 27px; width: 92px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image073.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 27px; width: 93px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image074.png\" \/><\/p>\n<p style=\"text-align: justify;\">And\u00a0\u00a0\u00a0\u00a0 if <img decoding=\"async\" style=\"height: 27px; width: 99px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image007.png\" \/>, then <img decoding=\"async\" style=\"height: 25px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image025.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image009.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 25px; width: 56px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image075.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image011.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image012.png\" \/>\u00a0is not one-one.<\/p>\n<p style=\"text-align: justify;\">Again, if every element of Y (\u2013 R) is image of some element of X (R) under <img decoding=\"async\" style=\"height: 21px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image066.png\" \/>\u00a0i.e., for every <img decoding=\"async\" style=\"height: 17px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image067.png\" \/>\u00a0Y, there exists an element <img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image047.png\" \/>\u00a0in X such that <img decoding=\"async\" style=\"height: 27px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image068.png\" \/><\/p>\n<p style=\"text-align: justify;\">Now,\u00a0\u00a0 <img decoding=\"async\" style=\"height: 25px; width: 63px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image076.png\" \/><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image009.png\" \/><img decoding=\"async\" style=\"height: 27px; width: 79px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image077.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\/Ex1.2\/image011.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 31px; width: 219px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image078.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\/Ex1.2\/image011.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image012.png\" \/>\u00a0is not onto.<\/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\/Ex1.2\/image012.png\" \/>\u00a0is not bijective.<\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_12-3\"><\/span>NCERT Solutions class 12 Maths Exercise 1.2<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"8_Let_A_and_B_be_sets_Show_that_A_x_B_B_x_A_such_that_is_bijective_function\"><\/span><strong>8. Let A and B be sets. Show that <\/strong><strong><img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image012.png\" \/><\/strong><strong>\u00a0: A x B <\/strong><strong><img decoding=\"async\" style=\"height: 15px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image079.png\" \/><\/strong><strong>\u00a0B x A such that <\/strong><strong><img decoding=\"async\" style=\"height: 27px; width: 103px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image080.png\" \/><\/strong><strong>\u00a0is bijective function.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Injectivity: Let <img decoding=\"async\" style=\"height: 27px; width: 47px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image081.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 27px; width: 63px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image082.png\" \/>\u00a0A x B such that <img decoding=\"async\" style=\"height: 27px; width: 135px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image083.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image009.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 107px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image084.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image009.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 25px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image085.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 25px; width: 47px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image086.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image009.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 47px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image081.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 27px; width: 51px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image087.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image009.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 47px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image081.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 27px; width: 51px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image087.png\" \/>\u00a0for all <img decoding=\"async\" style=\"height: 27px; width: 112px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image088.png\" \/>\u00a0A x B<\/p>\n<p style=\"text-align: justify;\">So, <img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image012.png\" \/>\u00a0is injective.<\/p>\n<p style=\"text-align: justify;\">Surjectivity: Let <img decoding=\"async\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image089.png\" \/>\u00a0be an arbitrary element of B x A. Then <img decoding=\"async\" style=\"height: 19px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image090.png\" \/>\u00a0B and <img decoding=\"async\" style=\"height: 15px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image091.png\" \/>\u00a0A.<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image009.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 51px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image092.png\" \/>\u00a0A x B<\/p>\n<p style=\"text-align: justify;\">Thus, for all <img decoding=\"async\" style=\"height: 27px; width: 51px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image093.png\" \/>\u00a0B x A, their exists <img decoding=\"async\" style=\"height: 27px; width: 51px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image092.png\" \/>\u00a0A x B such that\u00a0\u00a0\u00a0\u00a0 <img decoding=\"async\" style=\"height: 27px; width: 103px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image080.png\" \/><\/p>\n<p style=\"text-align: justify;\">So, <img decoding=\"async\" style=\"height: 21px; width: 21px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image094.png\" \/>\u00a0A x B <img decoding=\"async\" style=\"height: 15px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image079.png\" \/>\u00a0B x A is an onto function, therefore <img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image012.png\" \/>\u00a0is bijective.<\/p>\n<hr \/>\n<h4 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"9_Let_be_defined_by_for_all_N\"><\/span><strong>9. Let <\/strong><strong><img decoding=\"async\" style=\"height: 21px; width: 77px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image020.png\" \/><\/strong><strong>\u00a0be defined by <\/strong><strong><img decoding=\"async\" style=\"height: 88px; width: 171px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image095.png\" \/><\/strong><strong>\u00a0for all <\/strong><strong><img decoding=\"async\" style=\"height: 15px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image096.png\" \/><\/strong><strong>\u00a0N.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p style=\"text-align: justify;\"><strong>State whether the function <\/strong><img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image012.png\" \/><strong>\u00a0is bijective. Justify your answer.<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong><img decoding=\"async\" style=\"height: 21px; width: 77px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image020.png\" \/>\u00a0be defined by <img decoding=\"async\" style=\"height: 88px; width: 171px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image095.png\" \/><\/p>\n<p style=\"text-align: justify;\">(a) <img decoding=\"async\" style=\"height: 41px; width: 172px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image097.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 41px; width: 113px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image098.png\" \/><\/p>\n<p style=\"text-align: justify;\">The elements 1, 2, belonging to domain of <img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image012.png\" \/>\u00a0have the same image 1 in its co-domain.<\/p>\n<p style=\"text-align: justify;\">So, <img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image012.png\" \/>\u00a0is not one-one, therefore, <img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image012.png\" \/>\u00a0is not injective.<\/p>\n<p style=\"text-align: justify;\">(b) Every number of co-domain has pre-image in its domain e.g., 1 has two pre-images 1 and 2.<\/p>\n<p style=\"text-align: justify;\">So, <img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image012.png\" \/>\u00a0is onto, therefore, <img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image012.png\" \/>\u00a0is not bijective.<\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_12-4\"><\/span>NCERT Solutions class 12 Maths Exercise 1.2<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"10_Let_A_R_%E2%80%93_3_and_B_R_%E2%80%93_1_Consider_the_function_A_B_defined_by_Is_one-one_and_onto_Justify_your_answer\"><\/span><strong>10. Let A = R \u2013 {3} and B = R \u2013 {1}. Consider the function <\/strong><strong><img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image012.png\" \/><\/strong><strong>\u00a0: A <\/strong><strong><img decoding=\"async\" style=\"height: 15px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image079.png\" \/><\/strong><strong>\u00a0B defined by <\/strong><strong><img decoding=\"async\" style=\"height: 45px; width: 107px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image099.png\" \/><\/strong><strong>\u00a0Is <\/strong><strong><img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image012.png\" \/><\/strong><strong>\u00a0one-one and onto? Justify your answer.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>A = R \u2013 {3} and B = R \u2013 {1} and <img decoding=\"async\" style=\"height: 41px; width: 88px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image100.png\" \/><\/p>\n<p style=\"text-align: justify;\">Let <img decoding=\"async\" style=\"height: 25px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image063.png\" \/>\u00a0A, then <img decoding=\"async\" style=\"height: 45px; width: 96px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image101.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 45px; width: 99px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image102.png\" \/><\/p>\n<p style=\"text-align: justify;\">Now, for <img decoding=\"async\" style=\"height: 27px; width: 99px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image007.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image009.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 45px; width: 96px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image103.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image009.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 25px; width: 196px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image104.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image009.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 25px; width: 257px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image105.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image009.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 22px; width: 141px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image106.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image009.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 25px; width: 47px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image011.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image012.png\" \/>\u00a0is one-one function.<\/p>\n<p style=\"text-align: justify;\">Now <img decoding=\"async\" style=\"height: 41px; width: 63px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image107.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image009.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 104px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image108.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image009.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 97px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image109.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image009.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 111px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image110.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image009.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 45px; width: 71px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image111.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\/Ex1.2\/image011.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 75px; width: 136px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image112.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 37px; width: 104px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image113.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image009.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 65px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image114.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\/Ex1.2\/image012.png\" \/>\u00a0is an onto function.<\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_12-5\"><\/span>NCERT Solutions class 12 Maths Exercise 1.2<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"11_Let_be_defined_as_Choose_the_correct_answer\"><\/span><strong>11. Let <\/strong><strong><img decoding=\"async\" style=\"height: 21px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image023.png\" \/><\/strong><strong>\u00a0be defined as <\/strong><strong><img decoding=\"async\" style=\"height: 27px; width: 71px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image115.png\" \/><\/strong><strong>\u00a0Choose the correct answer:<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p style=\"text-align: justify;\"><strong>(A) <\/strong><strong><img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image012.png\" \/><\/strong><strong>\u00a0is one-one onto<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(B) <\/strong><strong><img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image012.png\" \/><\/strong><strong>\u00a0is many-one onto<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(C) <\/strong><strong><img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image012.png\" \/><\/strong><strong>\u00a0is one-one but not onto<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(D) <\/strong><strong><img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image012.png\" \/><\/strong><strong>\u00a0is neither one-one nor onto<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong><img decoding=\"async\" style=\"height: 27px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image116.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 23px; width: 51px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image117.png\" \/><\/p>\n<p style=\"text-align: justify;\">Let <img decoding=\"async\" style=\"height: 25px; width: 65px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image118.png\" \/>, then <img decoding=\"async\" style=\"height: 27px; width: 71px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image119.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 27px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image120.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\/Ex1.2\/image011.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 25px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image121.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image009.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 25px; width: 65px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image122.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\/Ex1.2\/image012.png\" \/>\u00a0is not one-one function.<\/p>\n<p style=\"text-align: justify;\">Now, <img decoding=\"async\" style=\"height: 25px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image123.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image009.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 35px; width: 55px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image124.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\/Ex1.2\/image011.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 51px; width: 111px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image125.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 51px; width: 127px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image126.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\/Ex1.2\/image012.png\" \/>\u00a0is not onto function.<\/p>\n<p style=\"text-align: justify;\">Therefore, option (D) is correct.<\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_12-6\"><\/span>NCERT Solutions class 12 Maths Exercise 1.2<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"12_Let_be_defined_as_Choose_the_correct_answer\"><\/span><strong>12. Let <\/strong><strong><img decoding=\"async\" style=\"height: 21px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image023.png\" \/><\/strong><strong>\u00a0be defined as <\/strong><strong><img decoding=\"async\" style=\"height: 27px; width: 73px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image127.png\" \/><\/strong><strong>\u00a0Choose the correct answer:<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p style=\"text-align: justify;\"><strong>(A) <\/strong><strong><img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image012.png\" \/><\/strong><strong>\u00a0is one-one onto<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(B) <\/strong><strong><img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image012.png\" \/><\/strong><strong>\u00a0is many-one onto<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(C) <\/strong><strong><img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image012.png\" \/><\/strong><strong>\u00a0is one-one but not onto<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(D) <\/strong><strong><img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image012.png\" \/><\/strong><strong>\u00a0is neither one-one nor onto<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans.\u00a0 <\/strong>Let <img decoding=\"async\" style=\"height: 25px; width: 56px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image128.png\" \/>\u00a0R such that <img decoding=\"async\" style=\"height: 27px; width: 99px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image007.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image009.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 25px; width: 60px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image129.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image009.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 25px; width: 47px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image010.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\/Ex1.2\/image012.png\" \/>\u00a0is one-one function.<\/p>\n<p style=\"text-align: justify;\">Now, consider <img decoding=\"async\" style=\"height: 17px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image067.png\" \/>\u00a0R (co-domain of <img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image012.png\" \/>) certainly <img decoding=\"async\" style=\"height: 36px; width: 47px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image130.png\" \/>\u00a0R (domain of <img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image012.png\" \/>)<\/p>\n<p style=\"text-align: justify;\">Thus for all <img decoding=\"async\" style=\"height: 17px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image067.png\" \/>\u00a0R (co-domain of <img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image012.png\" \/>) there exists <img decoding=\"async\" style=\"height: 37px; width: 48px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image131.png\" \/>\u00a0R (domain of <img decoding=\"async\" style=\"height: 21px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image012.png\" \/>) such that<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 45px; width: 161px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.2\/image132.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\/Ex1.2\/image012.png\" \/>\u00a0is onto function.<\/p>\n<p style=\"text-align: justify;\">Therefore, option (A) is correct.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_12-7\"><\/span>NCERT Solutions class 12 Maths Exercise 1.2<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 Exercise 1.2\u00a0solutions 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 &#8230; <a title=\"NCERT Solutions Class 12 Maths Exercise 1.2\" class=\"read-more\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-1-2\/\" aria-label=\"More on NCERT Solutions Class 12 Maths Exercise 1.2\">Read more<\/a><\/p>\n","protected":false},"author":1,"featured_media":-1,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1432,281],"tags":[283,1342,321,216],"class_list":["post-4460","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mathematics-cbse-class-12","category-ncert-solutions","tag-cbse-study-material","tag-class-12","tag-mathematics","tag-ncert-solutions"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.0 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>NCERT Solutions Class 12 Maths Exercise 1.2 | myCBSEguide<\/title>\n<meta name=\"description\" content=\"NCERT Solutions class 12 Maths Exercise 1.2 in PDF format for free download. 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