{"id":4462,"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-1-4\/"},"modified":"2018-05-08T14:06:27","modified_gmt":"2018-05-08T08:36:27","slug":"ncert-solutions-class-12-maths-exercise-1-4","status":"publish","type":"post","link":"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-1-4\/","title":{"rendered":"NCERT Solutions class 12 Maths Exercise 1.4"},"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-1-4\/#NCERT_Solutions_class_12_Maths_Relations_and_Functions\" >NCERT Solutions class 12 Maths Relations and Functions\u00a0<\/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-1-4\/#NCERT_Solutions_class_12_Maths_Exercise_14\" >NCERT Solutions class 12 Maths Exercise 1.4<\/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-1-4\/#2_For_each_binary_operation_defined_below_determine_whether_is_commutative_or_associative\" >2. For each binary operation * defined below, determine whether * is commutative or associative:<\/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-1-4\/#NCERT_Solutions_class_12_Maths_Exercise_14-2\" >NCERT Solutions class 12 Maths Exercise 1.4<\/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-1-4\/#3_Consider_the_binary_operation_on_the_set_1_2_3_4_5_defined_by_Write_the_operation_table_of_the_operation\" >3. Consider the binary operation \u00a0on the set {1, 2, 3, 4, 5} defined by Write the operation table of the operation \u00a0<\/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-1-4\/#NCERT_Solutions_class_12_Maths_Exercise_14-3\" >NCERT Solutions class 12 Maths Exercise 1.4<\/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-1-4\/#4_Consider_a_binary_operation_on_the_set_1_2_3_4_5_given_by_the_following_multiplication_table_table_12\" >4. Consider a binary operation * on the set {1, 2, 3, 4, 5} given by the following multiplication table (table 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-1-4\/#NCERT_Solutions_class_12_Maths_Exercise_14-4\" >NCERT Solutions class 12 Maths Exercise 1.4<\/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-1-4\/#5_Let_be_the_binary_operation_on_the_set_1_2_3_4_5_defined_by_HCF_of_and_Is_the_operation_same_as_the_operation_defined_in_Exercise_4_above_Justify_your_answer\" >5. Let *\u2019 be the binary operation on the set {1, 2, 3, 4, 5} defined by \u00a0H.C.F. of \u00a0and \u00a0Is the operation *\u2019 same as the operation * defined in Exercise 4 above? Justify your answer.<\/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-1-4\/#NCERT_Solutions_class_12_Maths_Exercise_14-5\" >NCERT Solutions class 12 Maths Exercise 1.4<\/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-1-4\/#6_Let_be_the_binary_operation_on_N_given_by_LCM_of_and_Find\" >6. Let * be the binary operation on N given by \u00a0L.C.M. of \u00a0and \u00a0\u00a0Find:<\/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-1-4\/#NCERT_Solutions_class_12_Maths_Exercise_14-6\" >NCERT Solutions class 12 Maths Exercise 1.4<\/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-1-4\/#7_Is_defined_on_the_set_1_2_3_4_5_by_LCM_of_and_a_binary_operation_Justify_your_answer\" >7. Is * defined on the set {1, 2, 3, 4, 5} by \u00a0L.C.M. of \u00a0and \u00a0a binary operation? 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-1-4\/#NCERT_Solutions_class_12_Maths_Exercise_14-7\" >NCERT Solutions class 12 Maths Exercise 1.4<\/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-1-4\/#8_Let_be_the_binary_operation_on_N_defined_by_HCF_of_and_Is_commutative_Is_associative_Does_there_exist_identity_for_this_binary_operation_on_N\" >8. Let * be the binary operation on N defined by \u00a0H.C.F. of \u00a0and \u00a0Is * commutative? Is * associative? Does there exist identity for this binary operation on N?<\/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-1-4\/#NCERT_Solutions_class_12_Maths_Exercise_14-8\" >NCERT Solutions class 12 Maths Exercise 1.4<\/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-1-4\/#9_Let_be_a_binary_operation_on_the_set_Q_of_rational_numbers_as_follows\" >9. Let * be a binary operation on the set Q of rational numbers as follows:<\/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-exercise-1-4\/#NCERT_Solutions_class_12_Maths_Exercise_14-9\" >NCERT Solutions class 12 Maths Exercise 1.4<\/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-exercise-1-4\/#10_Show_that_none_of_the_operations_given_above_the_identity\" >10. Show that none of the operations given above the identity.<\/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-exercise-1-4\/#NCERT_Solutions_class_12_Maths_Exercise_14-10\" >NCERT Solutions class 12 Maths Exercise 1.4<\/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-exercise-1-4\/#11_Let_A_N_x_N_and_be_the_binary_operation_on_A_defined_by\" >11. Let A = N x N and * be the binary operation on A defined by \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-exercise-1-4\/#NCERT_Solutions_class_12_Maths_Exercise_14-11\" >NCERT Solutions class 12 Maths Exercise 1.4<\/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-exercise-1-4\/#12_State_whether_the_following_statements_are_true_or_false_Justify\" >12. State whether the following statements are true or false. Justify:<\/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-exercise-1-4\/#NCERT_Solutions_class_12_Maths_Exercise_14-12\" >NCERT Solutions class 12 Maths Exercise 1.4<\/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-exercise-1-4\/#13_Consider_a_binary_operation_on_N_defined_as_Choose_the_correct_answer\" >13. Consider a binary operation * on N defined as . Choose 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-26\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-1-4\/#NCERT_Solutions_class_12_Maths_Exercise_14-13\" >NCERT Solutions class 12 Maths Exercise 1.4<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-27\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-1-4\/#CBSE_App_for_Students\" >CBSE App for Students<\/a><\/li><\/ul><\/nav><\/div>\n<p>NCERT Solutions class 12 Maths Exercise 1.4 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 Exercise 1.4\" width=\"182\" height=\"237\" \/><\/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<strong>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p style=\"text-align: justify;\"><strong>1. Determine whether or not each of the definition of * given below gives a binary operation. In the event that * is not a binary operation, given justification for this.<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(i) On <\/strong><strong><img decoding=\"async\" style=\"height: 25px; width: 27px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image001.png\" \/><\/strong><strong>\u00a0define * by <\/strong><strong><img decoding=\"async\" style=\"height: 19px; width: 79px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image002.png\" \/><\/strong><strong>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(ii) On <\/strong><strong><img decoding=\"async\" style=\"height: 25px; width: 27px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image001.png\" \/><\/strong><strong>\u00a0define * by <\/strong><strong><img decoding=\"async\" style=\"height: 19px; width: 65px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image003.png\" \/><\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(iii) On R, define * by <\/strong><strong><img decoding=\"async\" style=\"height: 21px; width: 71px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image004.png\" \/><\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(iv) On <\/strong><strong><img decoding=\"async\" style=\"height: 25px; width: 27px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image001.png\" \/><\/strong><strong>\u00a0define * by <\/strong><strong><img decoding=\"async\" style=\"height: 27px; width: 85px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image005.png\" \/><\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(v) On <\/strong><strong><img decoding=\"async\" style=\"height: 25px; width: 27px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image001.png\" \/><\/strong><strong>\u00a0define * by <\/strong><strong><img decoding=\"async\" style=\"height: 19px; width: 57px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image006.png\" \/><\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong><strong>(i)<\/strong> On <img decoding=\"async\" style=\"height: 20px; width: 21px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image007.png\" \/>\u00a0= {1, 2, 3, \u2026..},\u00a0 <img decoding=\"async\" style=\"height: 19px; width: 79px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image002.png\" \/><\/p>\n<p style=\"text-align: justify;\">Let <img decoding=\"async\" style=\"height: 21px; width: 71px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image008.png\" \/>\u00a0\u00a0<img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image009.png\" \/><img decoding=\"async\" style=\"height: 19px; width: 119px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 20px; width: 21px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image007.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, operation * is not a binary operation on <img decoding=\"async\" style=\"height: 20px; width: 21px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image007.png\" \/>.<\/p>\n<p style=\"text-align: justify;\"><strong>(ii)<\/strong> On <img decoding=\"async\" style=\"height: 20px; width: 21px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image007.png\" \/>\u00a0= {1, 2, 3, \u2026..}, <img decoding=\"async\" style=\"height: 19px; width: 65px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image003.png\" \/><\/p>\n<p style=\"text-align: justify;\">Let <img decoding=\"async\" style=\"height: 21px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image011.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image009.png\" \/>\u00a0\u00a0<img decoding=\"async\" style=\"height: 19px; width: 112px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image012.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 20px; width: 21px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image007.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, operation * is a binary operation on <img decoding=\"async\" style=\"height: 20px; width: 21px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image007.png\" \/>.<\/p>\n<p style=\"text-align: justify;\"><strong>(iii)<\/strong> on R (set of real numbers) <img decoding=\"async\" style=\"height: 21px; width: 71px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image004.png\" \/><\/p>\n<p style=\"text-align: justify;\">Let <img decoding=\"async\" style=\"height: 21px; width: 85px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image013.png\" \/>\u00a0\u00a0<img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image009.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 29px; width: 152px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image014.png\" \/>R<\/p>\n<p style=\"text-align: justify;\">Therefore, operation * is a binary operation on R.<\/p>\n<p style=\"text-align: justify;\"><strong>(iv) <\/strong>On <img decoding=\"async\" style=\"height: 20px; width: 21px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image007.png\" \/>\u00a0= {1, 2, 3, \u2026..}, <img decoding=\"async\" style=\"height: 27px; width: 85px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image005.png\" \/><\/p>\n<p style=\"text-align: justify;\">Let <img decoding=\"async\" style=\"height: 21px; width: 73px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image015.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image009.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 119px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image016.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 20px; width: 21px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image007.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, operation * is a binary operation on <img decoding=\"async\" style=\"height: 20px; width: 21px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image007.png\" \/>.<\/p>\n<p style=\"text-align: justify;\"><strong>(v) <\/strong>On <img decoding=\"async\" style=\"height: 20px; width: 21px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image007.png\" \/>\u00a0= {1, 2, 3, \u2026..}, <img decoding=\"async\" style=\"height: 19px; width: 57px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image006.png\" \/><\/p>\n<p style=\"text-align: justify;\">Let <img decoding=\"async\" style=\"height: 21px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image009.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 19px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image018.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 20px; width: 21px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image007.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, operation * is a binary operation on <img decoding=\"async\" style=\"height: 20px; width: 21px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image007.png\" \/>.<\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_14\"><\/span>NCERT Solutions class 12 Maths Exercise 1.4<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"2_For_each_binary_operation_defined_below_determine_whether_is_commutative_or_associative\"><\/span><strong>2. For each binary operation * defined below, determine whether * is commutative or associative:<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p style=\"text-align: justify;\"><strong>(i) On <\/strong><strong><img decoding=\"async\" style=\"height: 21px; width: 19px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image019.png\" \/><\/strong><strong>\u00a0define <\/strong><strong><img decoding=\"async\" style=\"height: 19px; width: 79px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image002.png\" \/><\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(ii) On Q, define <\/strong><strong><img decoding=\"async\" style=\"height: 19px; width: 85px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image020.png\" \/><\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(iii) On Q, define <\/strong><strong><img decoding=\"async\" style=\"height: 41px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image021.png\" \/><\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(iv) On <\/strong><strong><img decoding=\"async\" style=\"height: 21px; width: 19px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image019.png\" \/><\/strong><strong>\u00a0define <\/strong><strong><img decoding=\"async\" style=\"height: 21px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image022.png\" \/><\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(v) On <\/strong><strong><img decoding=\"async\" style=\"height: 21px; width: 19px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image019.png\" \/><\/strong><strong>\u00a0define <\/strong><strong><img decoding=\"async\" style=\"height: 21px; width: 63px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image023.png\" \/><\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(vi) On R \u2013 {\u2013 1}, define <\/strong><strong><img decoding=\"async\" style=\"height: 41px; width: 77px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image024.png\" \/><\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong><strong>(i)<\/strong> For commutativity: <img decoding=\"async\" style=\"height: 19px; width: 79px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image002.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 19px; width: 79px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image025.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 27px; width: 103px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image026.png\" \/><\/p>\n<p style=\"text-align: justify;\">For associativity: <img decoding=\"async\" style=\"height: 27px; width: 141px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image027.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 27px; width: 150px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image028.png\" \/><\/p>\n<p style=\"text-align: justify;\">Also, <img decoding=\"async\" style=\"height: 27px; width: 141px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image029.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 27px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image030.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\/Ex1.4\/image009.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 137px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image031.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, the operation * is neither commutative nor associative.<\/p>\n<p style=\"text-align: justify;\"><strong>(ii) <\/strong>For commutativity: <img decoding=\"async\" style=\"height: 19px; width: 85px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image020.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 19px; width: 179px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image032.png\" \/><\/p>\n<p style=\"text-align: justify;\">For associativity: <img decoding=\"async\" style=\"height: 27px; width: 145px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image033.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 27px; width: 163px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image034.png\" \/><\/p>\n<p style=\"text-align: justify;\">Also, <img decoding=\"async\" style=\"height: 27px; width: 156px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image035.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 19px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image036.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\/Ex1.4\/image009.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 137px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image031.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, the operation * is commutative but not associative.<\/p>\n<p style=\"text-align: justify;\"><strong>(iii) <\/strong>For commutativity: <img decoding=\"async\" style=\"height: 41px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image021.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 41px; width: 102px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image037.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 19px; width: 31px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image038.png\" \/><\/p>\n<p style=\"text-align: justify;\">For associativity: <img decoding=\"async\" style=\"height: 45px; width: 132px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image039.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 51px; width: 85px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image040.png\" \/><\/p>\n<p style=\"text-align: justify;\">Also, <img decoding=\"async\" style=\"height: 45px; width: 132px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image041.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 51px; width: 85px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image040.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\/Ex1.4\/image009.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 137px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image042.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, the operation * is commutative and associative.<\/p>\n<p style=\"text-align: justify;\"><strong>(iv)<\/strong> For commutativity: <img decoding=\"async\" style=\"height: 21px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image022.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 21px; width: 147px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image043.png\" \/><\/p>\n<p style=\"text-align: justify;\">For associativity: <img decoding=\"async\" style=\"height: 27px; width: 116px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image044.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 27px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image045.png\" \/><\/p>\n<p style=\"text-align: justify;\">Also, <img decoding=\"async\" style=\"height: 29px; width: 131px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image046.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 21px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image047.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\/Ex1.4\/image009.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 137px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image031.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, the operation * is commutative but not associative.<\/p>\n<p style=\"text-align: justify;\"><strong>(v) <\/strong>For commutativity: <img decoding=\"async\" style=\"height: 21px; width: 63px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image023.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 21px; width: 61px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image048.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\/Ex1.4\/image049.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 19px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image050.png\" \/><\/p>\n<p style=\"text-align: justify;\">For associativity: <img decoding=\"async\" style=\"height: 27px; width: 112px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image051.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 31px; width: 35px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image052.png\" \/><\/p>\n<p style=\"text-align: justify;\">Also, <img decoding=\"async\" style=\"height: 29px; width: 125px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image053.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 21px; width: 23px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image054.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\/Ex1.4\/image009.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 137px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image031.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, the operation * is neither commutative nor associative.<\/p>\n<p style=\"text-align: justify;\"><strong>(vi) <\/strong>For commutativity: <img decoding=\"async\" style=\"height: 41px; width: 77px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image024.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 41px; width: 79px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image055.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image049.png\" \/>\u00a0\u00a0<img decoding=\"async\" style=\"height: 19px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image050.png\" \/><\/p>\n<p style=\"text-align: justify;\">For associativity: <img decoding=\"async\" style=\"height: 45px; width: 144px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image056.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 65px; width: 127px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image057.png\" \/><\/p>\n<p style=\"text-align: justify;\">Also, <img decoding=\"async\" style=\"height: 45px; width: 143px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image058.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 61px; width: 144px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image059.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\/Ex1.4\/image009.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 137px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image031.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, the operation * is neither commutative nor associative.<\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_14-2\"><\/span>NCERT Solutions class 12 Maths Exercise 1.4<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"3_Consider_the_binary_operation_on_the_set_1_2_3_4_5_defined_by_Write_the_operation_table_of_the_operation\"><\/span><strong>3. Consider the binary operation <\/strong><strong><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image060.png\" \/><\/strong><strong>\u00a0on the set {1, 2, 3, 4, 5} defined by <\/strong><strong><img decoding=\"async\" style=\"height: 27px; width: 119px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image061.png\" \/><\/strong><strong>Write the operation table of the operation <\/strong><strong><img decoding=\"async\" style=\"height: 15px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image062.png\" \/><\/strong><strong>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>\u00a0Let A = {1, 2, 3, 4, 5} defined by <img decoding=\"async\" style=\"height: 27px; width: 115px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image063.png\" \/>\u00a0i.e., minimum of <img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image064.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 19px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image065.png\" \/><\/p>\n<table style=\"height: 336px;\" border=\"1\" width=\"745\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td style=\"width: .4in;\">\n<p style=\"text-align: center;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image060.png\" \/><\/p>\n<\/td>\n<td style=\"width: .4in;\">\n<p style=\"text-align: center;\">1<\/p>\n<\/td>\n<td style=\"width: .4in;\">\n<p style=\"text-align: center;\">2<\/p>\n<\/td>\n<td style=\"width: .4in;\">\n<p style=\"text-align: center;\">3<\/p>\n<\/td>\n<td style=\"width: .4in;\">\n<p style=\"text-align: center;\">4<\/p>\n<\/td>\n<td style=\"width: .4in;\">\n<p style=\"text-align: center;\">5<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: .4in;\">\n<p style=\"text-align: center;\">1<\/p>\n<\/td>\n<td style=\"width: .4in;\">\n<p style=\"text-align: center;\">1<\/p>\n<\/td>\n<td style=\"width: .4in;\">\n<p style=\"text-align: center;\">1<\/p>\n<\/td>\n<td style=\"width: .4in;\">\n<p style=\"text-align: center;\">1<\/p>\n<\/td>\n<td style=\"width: .4in;\">\n<p style=\"text-align: center;\">1<\/p>\n<\/td>\n<td style=\"width: .4in;\">\n<p style=\"text-align: center;\">1<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: .4in;\">\n<p style=\"text-align: center;\">2<\/p>\n<\/td>\n<td style=\"width: .4in;\">\n<p style=\"text-align: center;\">1<\/p>\n<\/td>\n<td style=\"width: .4in;\">\n<p style=\"text-align: center;\">2<\/p>\n<\/td>\n<td style=\"width: .4in;\">\n<p style=\"text-align: center;\">2<\/p>\n<\/td>\n<td style=\"width: .4in;\">\n<p style=\"text-align: center;\">2<\/p>\n<\/td>\n<td style=\"width: .4in;\">\n<p style=\"text-align: center;\">2<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: .4in;\">\n<p style=\"text-align: center;\">3<\/p>\n<\/td>\n<td style=\"width: .4in;\">\n<p style=\"text-align: center;\">1<\/p>\n<\/td>\n<td style=\"width: .4in;\">\n<p style=\"text-align: center;\">2<\/p>\n<\/td>\n<td style=\"width: .4in;\">\n<p style=\"text-align: center;\">3<\/p>\n<\/td>\n<td style=\"width: .4in;\">\n<p style=\"text-align: center;\">3<\/p>\n<\/td>\n<td style=\"width: .4in;\">\n<p style=\"text-align: center;\">3<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: .4in;\">\n<p style=\"text-align: center;\">4<\/p>\n<\/td>\n<td style=\"width: .4in;\">\n<p style=\"text-align: center;\">1<\/p>\n<\/td>\n<td style=\"width: .4in;\">\n<p style=\"text-align: center;\">2<\/p>\n<\/td>\n<td style=\"width: .4in;\">\n<p style=\"text-align: center;\">3<\/p>\n<\/td>\n<td style=\"width: .4in;\">\n<p style=\"text-align: center;\">4<\/p>\n<\/td>\n<td style=\"width: .4in;\">\n<p style=\"text-align: center;\">4<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: .4in;\">\n<p style=\"text-align: center;\">5<\/p>\n<\/td>\n<td style=\"width: .4in;\">\n<p style=\"text-align: center;\">1<\/p>\n<\/td>\n<td style=\"width: .4in;\">\n<p style=\"text-align: center;\">2<\/p>\n<\/td>\n<td style=\"width: .4in;\">\n<p style=\"text-align: center;\">3<\/p>\n<\/td>\n<td style=\"width: .4in;\">\n<p style=\"text-align: center;\">4<\/p>\n<\/td>\n<td style=\"width: .4in;\">\n<p style=\"text-align: center;\">5<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_14-3\"><\/span>NCERT Solutions class 12 Maths Exercise 1.4<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"4_Consider_a_binary_operation_on_the_set_1_2_3_4_5_given_by_the_following_multiplication_table_table_12\"><\/span><strong>4. Consider a binary operation * on the set {1, 2, 3, 4, 5} given by the following multiplication table (table 1.2).<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p style=\"text-align: justify;\"><strong>(i) Compute (2 * 3) * 4 and 2 * (3 * 4)<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(ii) Is * commutative?<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(iii) Compute (2 * 3) * (4 * 5)<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(Hint: Use the following table)<\/strong><\/p>\n<p><strong>Table 1.2<\/strong><\/p>\n<table border=\"1\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td style=\"width: .4in;\">*<\/td>\n<td style=\"width: .4in;\">1<\/td>\n<td style=\"width: .4in;\">2<\/td>\n<td style=\"width: .4in;\">3<\/td>\n<td style=\"width: .4in;\">4<\/td>\n<td style=\"width: .4in;\">5<\/td>\n<\/tr>\n<tr>\n<td style=\"width: .4in;\">1<\/td>\n<td style=\"width: .4in;\">1<\/td>\n<td style=\"width: .4in;\">1<\/td>\n<td style=\"width: .4in;\">1<\/td>\n<td style=\"width: .4in;\">1<\/td>\n<td style=\"width: .4in;\">1<\/td>\n<\/tr>\n<tr>\n<td style=\"width: .4in;\">2<\/td>\n<td style=\"width: .4in;\">1<\/td>\n<td style=\"width: .4in;\">2<\/td>\n<td style=\"width: .4in;\">1<\/td>\n<td style=\"width: .4in;\">2<\/td>\n<td style=\"width: .4in;\">1<\/td>\n<\/tr>\n<tr>\n<td style=\"width: .4in;\">3<\/td>\n<td style=\"width: .4in;\">1<\/td>\n<td style=\"width: .4in;\">1<\/td>\n<td style=\"width: .4in;\">3<\/td>\n<td style=\"width: .4in;\">1<\/td>\n<td style=\"width: .4in;\">1<\/td>\n<\/tr>\n<tr>\n<td style=\"width: .4in;\">4<\/td>\n<td style=\"width: .4in;\">1<\/td>\n<td style=\"width: .4in;\">2<\/td>\n<td style=\"width: .4in;\">1<\/td>\n<td style=\"width: .4in;\">4<\/td>\n<td style=\"width: .4in;\">1<\/td>\n<\/tr>\n<tr>\n<td style=\"width: .4in;\">5<\/td>\n<td style=\"width: .4in;\">1<\/td>\n<td style=\"width: .4in;\">1<\/td>\n<td style=\"width: .4in;\">1<\/td>\n<td style=\"width: .4in;\">1<\/td>\n<td style=\"width: .4in;\">5<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\"><strong>Ans. (i)<\/strong> 2 * 3 = 1 and 3 * 4 = 1<\/p>\n<p style=\"text-align: justify;\">Now (2 * 3) * 4 = 1 * 4 = 1 and 2 * (3 * 4) = 2 * 1 = 1<\/p>\n<p style=\"text-align: justify;\"><strong>(ii) <\/strong>2 * 3 = 1 and 3 * 4 = 1<\/p>\n<p style=\"text-align: justify;\">2 * 3 = 3 * 2 and other element of the given set.<\/p>\n<p style=\"text-align: justify;\">Hence the operation is commutative.<\/p>\n<p style=\"text-align: justify;\"><strong>(iii) <\/strong>(2 * 3) * (4 * 5) = 1 * 1 = 1<\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_14-4\"><\/span>NCERT Solutions class 12 Maths Exercise 1.4<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"5_Let_be_the_binary_operation_on_the_set_1_2_3_4_5_defined_by_HCF_of_and_Is_the_operation_same_as_the_operation_defined_in_Exercise_4_above_Justify_your_answer\"><\/span><strong>5. Let *\u2019 be the binary operation on the set {1, 2, 3, 4, 5} defined by <\/strong><strong><img decoding=\"async\" style=\"height: 19px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image066.png\" \/><\/strong><strong>\u00a0H.C.F. of <\/strong><strong><img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image064.png\" \/><\/strong><strong>\u00a0and <\/strong><strong><img decoding=\"async\" style=\"height: 19px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image065.png\" \/><\/strong><strong>\u00a0Is the operation *\u2019 same as the operation * defined in Exercise 4 above? Justify your answer.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Let A = {1, 2, 3, 4, 5} and <img decoding=\"async\" style=\"height: 19px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image066.png\" \/>\u00a0H.C.F. of <img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image064.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 19px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image065.png\" \/><\/p>\n<table border=\"1\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td style=\"width: .4in;\">\n<p style=\"text-align: center;\">*\u2019<\/p>\n<\/td>\n<td style=\"width: .4in;\">\n<p style=\"text-align: center;\">1<\/p>\n<\/td>\n<td style=\"width: .4in;\">\n<p style=\"text-align: center;\">2<\/p>\n<\/td>\n<td style=\"width: .4in;\">\n<p style=\"text-align: center;\">3<\/p>\n<\/td>\n<td style=\"width: .4in;\">\n<p style=\"text-align: center;\">4<\/p>\n<\/td>\n<td style=\"width: .4in;\">\n<p style=\"text-align: center;\">5<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: .4in;\">\n<p style=\"text-align: center;\">1<\/p>\n<\/td>\n<td style=\"width: .4in;\">\n<p style=\"text-align: center;\">1<\/p>\n<\/td>\n<td style=\"width: .4in;\">\n<p style=\"text-align: center;\">1<\/p>\n<\/td>\n<td style=\"width: .4in;\">\n<p style=\"text-align: center;\">1<\/p>\n<\/td>\n<td style=\"width: .4in;\">\n<p style=\"text-align: center;\">1<\/p>\n<\/td>\n<td style=\"width: .4in;\">\n<p style=\"text-align: center;\">1<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: .4in;\">\n<p style=\"text-align: center;\">2<\/p>\n<\/td>\n<td style=\"width: .4in;\">\n<p style=\"text-align: center;\">1<\/p>\n<\/td>\n<td style=\"width: .4in;\">\n<p style=\"text-align: center;\">2<\/p>\n<\/td>\n<td style=\"width: .4in;\">\n<p style=\"text-align: center;\">1<\/p>\n<\/td>\n<td style=\"width: .4in;\">\n<p style=\"text-align: center;\">2<\/p>\n<\/td>\n<td style=\"width: .4in;\">\n<p style=\"text-align: center;\">1<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: .4in;\">\n<p style=\"text-align: center;\">3<\/p>\n<\/td>\n<td style=\"width: .4in;\">\n<p style=\"text-align: center;\">1<\/p>\n<\/td>\n<td style=\"width: .4in;\">\n<p style=\"text-align: center;\">1<\/p>\n<\/td>\n<td style=\"width: .4in;\">\n<p style=\"text-align: center;\">3<\/p>\n<\/td>\n<td style=\"width: .4in;\">\n<p style=\"text-align: center;\">1<\/p>\n<\/td>\n<td style=\"width: .4in;\">\n<p style=\"text-align: center;\">1<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: .4in;\">\n<p style=\"text-align: center;\">4<\/p>\n<\/td>\n<td style=\"width: .4in;\">\n<p style=\"text-align: center;\">1<\/p>\n<\/td>\n<td style=\"width: .4in;\">\n<p style=\"text-align: center;\">2<\/p>\n<\/td>\n<td style=\"width: .4in;\">\n<p style=\"text-align: center;\">1<\/p>\n<\/td>\n<td style=\"width: .4in;\">\n<p style=\"text-align: center;\">4<\/p>\n<\/td>\n<td style=\"width: .4in;\">\n<p style=\"text-align: center;\">1<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: .4in;\">\n<p style=\"text-align: center;\">5<\/p>\n<\/td>\n<td style=\"width: .4in;\">\n<p style=\"text-align: center;\">1<\/p>\n<\/td>\n<td style=\"width: .4in;\">\n<p style=\"text-align: center;\">1<\/p>\n<\/td>\n<td style=\"width: .4in;\">\n<p style=\"text-align: center;\">1<\/p>\n<\/td>\n<td style=\"width: .4in;\">\n<p style=\"text-align: center;\">1<\/p>\n<\/td>\n<td style=\"width: .4in;\">\n<p style=\"text-align: center;\">5<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_14-5\"><\/span>NCERT Solutions class 12 Maths Exercise 1.4<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"6_Let_be_the_binary_operation_on_N_given_by_LCM_of_and_Find\"><\/span><strong>6. Let * be the binary operation on N given by <\/strong><strong><img decoding=\"async\" style=\"height: 19px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image067.png\" \/><\/strong><strong>\u00a0L.C.M. of <\/strong><strong><img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image064.png\" \/><\/strong><strong>\u00a0and <\/strong><strong><img decoding=\"async\" style=\"height: 19px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image065.png\" \/><\/strong><strong>\u00a0\u00a0Find:<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p style=\"text-align: justify;\"><strong>(i) 5 * 7,\u00a0 10 * 16<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(ii) Is * commutative?<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(iii) Is * associative?<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(iv) Find the identity of * in N.<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(v) Which elements of N are invertible for the operation *?<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong><img decoding=\"async\" style=\"height: 19px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image067.png\" \/>\u00a0L.C.M. of <img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image064.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 19px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image068.png\" \/><\/p>\n<p style=\"text-align: justify;\"><strong>(i)<\/strong> 5 * 7 = L.C.M. of 5 and 7 = 35<\/p>\n<p style=\"text-align: justify;\">20 * 16 = L.C.M. of 20 and 16 = 80<\/p>\n<p style=\"text-align: justify;\"><strong>(ii)<\/strong> <img decoding=\"async\" style=\"height: 19px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image067.png\" \/>\u00a0L.C.M. of <img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image064.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 19px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image068.png\" \/>\u00a0= L.C.M. of <img decoding=\"async\" style=\"height: 19px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image069.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image064.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 19px; width: 31px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image070.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, operation * is commutative.<\/p>\n<p style=\"text-align: justify;\"><strong>(iii)<\/strong> <img decoding=\"async\" style=\"height: 27px; width: 227px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image071.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 27px; width: 240px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image072.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" style=\"height: 21px; width: 133px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image073.png\" \/><\/p>\n<p style=\"text-align: justify;\">Similarly, <img decoding=\"async\" style=\"height: 27px; width: 208px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image074.png\" \/><\/p>\n<p style=\"text-align: justify;\">Thus, <img decoding=\"async\" style=\"height: 27px; width: 137px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image042.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, the operation is associative.<\/p>\n<p style=\"text-align: justify;\"><strong>(iv) <\/strong>Identity of * in N = 1 because <img decoding=\"async\" style=\"height: 19px; width: 29px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image075.png\" \/>\u00a0= L.C.M. of <img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image064.png\" \/>\u00a0and 1 = <img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image064.png\" \/><\/p>\n<p style=\"text-align: justify;\"><strong>(v) <\/strong>Only the element 1 in N is invertible for the operation * because <img decoding=\"async\" style=\"height: 41px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image076.png\" \/><\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_14-6\"><\/span>NCERT Solutions class 12 Maths Exercise 1.4<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"7_Is_defined_on_the_set_1_2_3_4_5_by_LCM_of_and_a_binary_operation_Justify_your_answer\"><\/span><strong>7. Is * defined on the set {1, 2, 3, 4, 5} by <\/strong><strong><img decoding=\"async\" style=\"height: 19px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image067.png\" \/><\/strong><strong>\u00a0L.C.M. of <\/strong><strong><img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image064.png\" \/><\/strong><strong>\u00a0and <\/strong><strong><img decoding=\"async\" style=\"height: 19px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image069.png\" \/><\/strong><strong>\u00a0a binary operation? Justify your answer.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Let A = {1, 2, 3, 4, 5} and <img decoding=\"async\" style=\"height: 19px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image066.png\" \/>\u00a0L.C.M. of <img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image064.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 19px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image065.png\" \/><\/p>\n<div>\n<table border=\"1\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td style=\"width: .4in;\">*<\/td>\n<td style=\"width: .4in;\">1<\/td>\n<td style=\"width: .4in;\">2<\/td>\n<td style=\"width: .4in;\">3<\/td>\n<td style=\"width: .4in;\">4<\/td>\n<td style=\"width: .4in;\">5<\/td>\n<\/tr>\n<tr>\n<td style=\"width: .4in;\">1<\/td>\n<td style=\"width: .4in;\">1<\/td>\n<td style=\"width: .4in;\">2<\/td>\n<td style=\"width: .4in;\">3<\/td>\n<td style=\"width: .4in;\">4<\/td>\n<td style=\"width: .4in;\">5<\/td>\n<\/tr>\n<tr>\n<td style=\"width: .4in;\">2<\/td>\n<td style=\"width: .4in;\">2<\/td>\n<td style=\"width: .4in;\">2<\/td>\n<td style=\"width: .4in;\">6<\/td>\n<td style=\"width: .4in;\">4<\/td>\n<td style=\"width: .4in;\">10<\/td>\n<\/tr>\n<tr>\n<td style=\"width: .4in;\">3<\/td>\n<td style=\"width: .4in;\">3<\/td>\n<td style=\"width: .4in;\">x<\/td>\n<td style=\"width: .4in;\">3<\/td>\n<td style=\"width: .4in;\">12<\/td>\n<td style=\"width: .4in;\">15<\/td>\n<\/tr>\n<tr>\n<td style=\"width: .4in;\">4<\/td>\n<td style=\"width: .4in;\">4<\/td>\n<td style=\"width: .4in;\">4<\/td>\n<td style=\"width: .4in;\">12<\/td>\n<td style=\"width: .4in;\">4<\/td>\n<td style=\"width: .4in;\">20<\/td>\n<\/tr>\n<tr>\n<td style=\"width: .4in;\">5<\/td>\n<td style=\"width: .4in;\">5<\/td>\n<td style=\"width: .4in;\">x<\/td>\n<td style=\"width: .4in;\">15<\/td>\n<td style=\"width: .4in;\">20<\/td>\n<td style=\"width: .4in;\">5<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p style=\"text-align: justify;\">Here, 2 * 3 = 6<img decoding=\"async\" style=\"height: 16px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image077.png\" \/>\u00a0A<\/p>\n<p style=\"text-align: justify;\">Therefore, the operation * is not a binary operation.<\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_14-7\"><\/span>NCERT Solutions class 12 Maths Exercise 1.4<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"8_Let_be_the_binary_operation_on_N_defined_by_HCF_of_and_Is_commutative_Is_associative_Does_there_exist_identity_for_this_binary_operation_on_N\"><\/span><strong>8. Let * be the binary operation on N defined by <\/strong><strong><img decoding=\"async\" style=\"height: 19px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image067.png\" \/><\/strong><strong>\u00a0H.C.F. of <\/strong><strong><img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image064.png\" \/><\/strong><strong>\u00a0and <\/strong><strong><img decoding=\"async\" style=\"height: 19px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image065.png\" \/><\/strong><strong>\u00a0Is * commutative? Is * associative? Does there exist identity for this binary operation on N?<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong><img decoding=\"async\" style=\"height: 19px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image067.png\" \/>\u00a0H.C.F. of <img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image064.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 19px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image065.png\" \/><\/p>\n<p style=\"text-align: justify;\"><strong>(i) <\/strong><img decoding=\"async\" style=\"height: 19px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image067.png\" \/>\u00a0H.C.F. of <img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image064.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 19px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image068.png\" \/>\u00a0= H.C.F. of <img decoding=\"async\" style=\"height: 19px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image069.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image064.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 19px; width: 31px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image070.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, operation * is commutative.<\/p>\n<p style=\"text-align: justify;\"><strong>(ii)<\/strong> <img decoding=\"async\" style=\"height: 27px; width: 223px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image078.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 27px; width: 236px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image079.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" style=\"height: 27px; width: 203px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image080.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, the operation is associative.<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 19px; width: 95px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image081.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, there does not exist any identity element.<\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_14-8\"><\/span>NCERT Solutions class 12 Maths Exercise 1.4<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"9_Let_be_a_binary_operation_on_the_set_Q_of_rational_numbers_as_follows\"><\/span><strong>9. Let * be a binary operation on the set Q of rational numbers as follows:<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p style=\"text-align: justify;\"><strong>(i) <\/strong><strong><img decoding=\"async\" style=\"height: 19px; width: 79px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image002.png\" \/><\/strong><strong>\u00a0\u00a0\u00a0\u00a0\u00a0(ii)\u00a0 <\/strong><strong><img decoding=\"async\" style=\"height: 21px; width: 90px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image082.png\" \/><\/strong><strong>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(iii) <\/strong><strong><img decoding=\"async\" style=\"height: 19px; width: 88px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image083.png\" \/><\/strong><strong>\u00a0\u00a0\u00a0\u00a0\u00a0(iv) <\/strong><strong><img decoding=\"async\" style=\"height: 29px; width: 96px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image084.png\" \/><\/strong><strong>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(v) <\/strong><strong><img decoding=\"async\" style=\"height: 41px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image085.png\" \/><\/strong><strong>\u00a0\u00a0\u00a0\u00a0\u00a0(vi) <\/strong><strong><img decoding=\"async\" style=\"height: 21px; width: 71px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image004.png\" \/><\/strong><strong>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Find which of the binary operations are commutative and which are associative.<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. (i)<\/strong> <img decoding=\"async\" style=\"height: 27px; width: 201px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image086.png\" \/>\u00a0\u00a0<img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image009.png\" \/>\u00a0operation * is not commutative.<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 27px; width: 288px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image087.png\" \/><\/p>\n<p style=\"text-align: justify;\">And <img decoding=\"async\" style=\"height: 27px; width: 288px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image088.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\/Ex1.4\/image089.png\" \/>\u00a0\u00a0<img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image009.png\" \/>\u00a0operation * is not associative.<\/p>\n<p style=\"text-align: justify;\"><strong>(ii) <\/strong><img decoding=\"async\" style=\"height: 21px; width: 196px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image090.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image009.png\" \/>\u00a0operation * is commutative.<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 29px; width: 340px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image091.png\" \/><\/p>\n<p style=\"text-align: justify;\">And <img decoding=\"async\" style=\"height: 31px; width: 260px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image092.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\/Ex1.4\/image089.png\" \/>\u00a0\u00a0<img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image009.png\" \/>\u00a0operation * is not associative.<\/p>\n<p style=\"text-align: justify;\"><strong>(iii) <\/strong><img decoding=\"async\" style=\"height: 27px; width: 153px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image093.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 27px; width: 196px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image094.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, operation * is not commutative.<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 27px; width: 291px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image095.png\" \/><\/p>\n<p style=\"text-align: justify;\">And <img decoding=\"async\" style=\"height: 27px; width: 247px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image096.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\/Ex1.4\/image089.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image009.png\" \/>\u00a0operation * is not associative.<\/p>\n<p style=\"text-align: justify;\"><strong>(iv)<\/strong> <img decoding=\"async\" style=\"height: 29px; width: 208px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image097.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image009.png\" \/>\u00a0operation * is commutative.<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 37px; width: 252px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image098.png\" \/><\/p>\n<p style=\"text-align: justify;\">And <img decoding=\"async\" style=\"height: 37px; width: 252px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image099.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\/Ex1.4\/image089.png\" \/>\u00a0\u00a0<img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image009.png\" \/>\u00a0operation * is not associative.<\/p>\n<p style=\"text-align: justify;\"><strong>(v) <\/strong><img decoding=\"async\" style=\"height: 41px; width: 145px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image100.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image009.png\" \/>\u00a0operation * is commutative.<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 60px; width: 204px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image101.png\" \/>\u00a0And <img decoding=\"async\" style=\"height: 60px; width: 204px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image102.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\/Ex1.4\/image103.png\" \/>\u00a0\u00a0<img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image009.png\" \/>\u00a0operation * is associative.<\/p>\n<p style=\"text-align: justify;\"><strong>(vi) <\/strong><img decoding=\"async\" style=\"height: 21px; width: 71px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image004.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 21px; width: 115px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image104.png\" \/>\u00a0\u00a0<img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image009.png\" \/>\u00a0operation * is not commutative.<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 29px; width: 251px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image105.png\" \/><\/p>\n<p style=\"text-align: justify;\">And <img decoding=\"async\" style=\"height: 31px; width: 251px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image106.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\/Ex1.4\/image089.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image009.png\" \/>\u00a0operation * is not associative.<\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_14-9\"><\/span>NCERT Solutions class 12 Maths Exercise 1.4<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"10_Show_that_none_of_the_operations_given_above_the_identity\"><\/span><strong>10. Show that none of the operations given above the identity.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Let the identity be I.<\/p>\n<p style=\"text-align: justify;\"><strong>(i)<\/strong> <img decoding=\"async\" style=\"height: 19px; width: 107px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image107.png\" \/><\/p>\n<p style=\"text-align: justify;\"><strong>(ii)<\/strong> <img decoding=\"async\" style=\"height: 21px; width: 120px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image108.png\" \/><\/p>\n<p style=\"text-align: justify;\"><strong>(iii)<\/strong> <img decoding=\"async\" style=\"height: 19px; width: 116px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image109.png\" \/><\/p>\n<p style=\"text-align: justify;\"><strong>(iv) <\/strong><img decoding=\"async\" style=\"height: 29px; width: 127px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image110.png\" \/><\/p>\n<p style=\"text-align: justify;\"><strong>(v) <\/strong><img decoding=\"async\" style=\"height: 41px; width: 96px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image111.png\" \/><\/p>\n<p style=\"text-align: justify;\"><strong>(vi)<\/strong> <img decoding=\"async\" style=\"height: 21px; width: 100px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image112.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, none of the operations given above has identity.<\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_14-10\"><\/span>NCERT Solutions class 12 Maths Exercise 1.4<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"11_Let_A_N_x_N_and_be_the_binary_operation_on_A_defined_by\"><\/span><strong>11. Let A = N x N and * be the binary operation on A defined by <\/strong><strong><img decoding=\"async\" style=\"height: 27px; width: 181px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image113.png\" \/><\/strong><strong>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p style=\"text-align: justify;\"><strong>Show that * is commutative and associative. Find the identity element for * on A, if any.<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>A = N x N and * is a binary operation defined on A.<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 27px; width: 373px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image114.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image009.png\" \/>\u00a0The operation is commutative<\/p>\n<p style=\"text-align: justify;\">Again,\u00a0<img decoding=\"async\" style=\"height: 29px; width: 433px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image115.png\" \/><\/p>\n<p style=\"text-align: justify;\">And\u00a0<img decoding=\"async\" style=\"height: 29px; width: 421px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image116.png\" \/><\/p>\n<p style=\"text-align: justify;\">Here,\u00a0<img decoding=\"async\" style=\"height: 29px; width: 296px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image117.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image009.png\" \/>\u00a0The operation is associative.<\/p>\n<p style=\"text-align: justify;\">Let identity function be <img decoding=\"async\" style=\"height: 27px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image118.png\" \/>, then\u00a0<img decoding=\"async\" style=\"height: 27px; width: 184px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image119.png\" \/><\/p>\n<p style=\"text-align: justify;\">For identity function <img decoding=\"async\" style=\"height: 16px; width: 59px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image120.png\" \/> <img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image049.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 19px; width: 36px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image121.png\" \/><\/p>\n<p style=\"text-align: justify;\">And for <img decoding=\"async\" style=\"height: 21px; width: 61px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image122.png\" \/><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image049.png\" \/><img decoding=\"async\" style=\"height: 21px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image123.png\" \/><\/p>\n<p style=\"text-align: justify;\">As 0<img decoding=\"async\" style=\"height: 15px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image124.png\" \/>\u00a0N, therefore, identity-element does not exist.<\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_14-11\"><\/span>NCERT Solutions class 12 Maths Exercise 1.4<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"12_State_whether_the_following_statements_are_true_or_false_Justify\"><\/span><strong>12. State whether the following statements are true or false. Justify:<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p style=\"text-align: justify;\"><strong>(i) For an arbitrary binary operation * on a set N, <\/strong><strong><img decoding=\"async\" style=\"height: 19px; width: 107px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image125.png\" \/><\/strong><strong>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(ii) If * is a commutative binary operation on N, then <\/strong><strong><img decoding=\"async\" style=\"height: 27px; width: 141px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image126.png\" \/><\/strong><strong>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. (i) <\/strong>* being a binary operation on N, is defined as <img decoding=\"async\" style=\"height: 19px; width: 107px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image125.png\" \/><\/p>\n<p style=\"text-align: justify;\">Hence operation * is not defined, therefore, the given statement is false.<\/p>\n<p style=\"text-align: justify;\"><strong>(ii) <\/strong>* being a binary operation on N.<\/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\/Ex1.4\/image009.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 19px; width: 73px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image127.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image009.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 212px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image128.png\" \/><\/p>\n<p style=\"text-align: justify;\">Thus,\u00a0\u00a0 <img decoding=\"async\" style=\"height: 27px; width: 137px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image129.png\" \/>, therefore the given statement is true.<\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_14-12\"><\/span>NCERT Solutions class 12 Maths Exercise 1.4<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"13_Consider_a_binary_operation_on_N_defined_as_Choose_the_correct_answer\"><\/span><strong>13. Consider a binary operation * on N defined as <\/strong><strong><img decoding=\"async\" style=\"height: 21px; width: 90px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image130.png\" \/><\/strong><strong>. Choose the correct answer:<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p style=\"text-align: justify;\"><strong>(A) Is * both associative and commutative?<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(B) Is * commutative but not associative?<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(C) Is * associative but not commutative?<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(D) Is * neither commutative nor associative?<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. \u00a0<\/strong><img decoding=\"async\" style=\"height: 21px; width: 193px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image131.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image009.png\" \/>\u00a0The operation * is commutative.<\/p>\n<p style=\"text-align: justify;\">Again,\u00a0 <img decoding=\"async\" style=\"height: 31px; width: 247px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image132.png\" \/><\/p>\n<p style=\"text-align: justify;\">And\u00a0<img decoding=\"async\" style=\"height: 31px; width: 333px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex1.4\/image133.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\/Ex1.4\/image009.png\" \/>\u00a0The operation * is not associative.<\/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_Exercise_14-13\"><\/span>NCERT Solutions class 12 Maths Exercise 1.4<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 1.4 Class 12 Maths book solutions are available in PDF format for free download. These ncert book chapter wise questions and answers are very helpful for CBSE board exam. CBSE recommends NCERT books and most of the questions in CBSE exam are asked from NCERT text books. Class 12 &#8230; <a title=\"NCERT Solutions class 12 Maths Exercise 1.4\" class=\"read-more\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-1-4\/\" aria-label=\"More on NCERT Solutions class 12 Maths Exercise 1.4\">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-4462","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 1.4 | myCBSEguide<\/title>\n<meta name=\"description\" content=\"NCERT Solutions class 12 Maths Exercise 1.4 in PDF format for free download. 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