{"id":4485,"date":"2016-05-16T11:49:00","date_gmt":"2016-05-16T06:19:00","guid":{"rendered":"http:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-3\/"},"modified":"2018-10-11T11:23:45","modified_gmt":"2018-10-11T05:53:45","slug":"ncert-solutions-class-12-maths-miscellaneous-3","status":"publish","type":"post","link":"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-3\/","title":{"rendered":"NCERT Solutions class 12 Maths Miscellaneous"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_76 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 eztoc-toggle-hide-by-default' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-3\/#NCERT_Solutions_class_12_Maths_Matrices\" >NCERT Solutions class 12 Maths Matrices\u00a0<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-3\/#1_Let_A_show_that_where_I_is_the_identity_matrix_of_order_2_and_N\" >1. Let A = \u00a0show that \u00a0where I is the identity matrix of order 2 and \u00a0N.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-3\/#2_If_A_prove_that_A_N\" >2. If A = , prove that A\u2019\u2019 = \u00a0N.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-3\/#3_If_A_then_prove_that_A_where_is_any_positive_integer\" >3. If A = \u00a0then prove that A\u2019\u2019 = \u00a0where \u00a0is any positive integer.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-3\/#4_If_A_and_B_are_symmetric_matrices_prove_that_AB_%E2%80%93_BA_is_a_skew_symmetric_matrix\" >4. If A and B are symmetric matrices, prove that AB \u2013 BA is a skew symmetric matrix.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-3\/#5_Show_that_the_matrix_BAB_is_symmetric_or_skew_symmetric_according_as_A_is_symmetric_or_skew_symmetric\" >5. Show that the matrix B\u2019AB is symmetric or skew symmetric according as A is symmetric or skew symmetric.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-3\/#6_Find_the_values_of_if_the_matrix_A_satisfies_the_equation_AA_I\" >6. Find the values of \u00a0if the matrix A = \u00a0satisfies the equation A\u2019A = I.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-3\/#7_For_what_value_of\" >7. For what value of \u00a0\u00a0?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-3\/#8_If_A_show_that_A2_%E2%80%93_5A_7I_0\" >8. If A = \u00a0show that A2 \u2013 5A + 7I = 0.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-3\/#9_Find_if\" >9. Find \u00a0if \u00a0<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-11\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-3\/#10_A_manufacturer_produces_three_products_which_he_sells_in_two_markets_Annual_sales_are_indicated_below\" >10. A manufacturer produces three products, \u00a0which he sells in two markets. Annual sales are indicated below:<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-12\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-3\/#11_Find_the_matrix_X_so_that_X\" >11. Find the matrix X so that X \u00a0<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-13\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-3\/#12_If_A_and_B_are_square_matrices_of_the_same_order_such_that_AB_BA_then_prove_by_induction_that_AB_BA_Further_prove_that_AB_AB_for_all_N\" >12. If A and B are square matrices of the same order such that AB = BA, then prove by induction that AB\u2019\u2019 = B\u2019\u2019A. Further prove that (AB)\u2019\u2019 = A\u2019\u2019B\u2019\u2019 for all \u00a0N.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-14\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-3\/#13_If_A_is_such_that_A2_I_then\" >13. If A = \u00a0is such that A2 = I, then:<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-15\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-3\/#14_If_the_matrix_A_is_both_symmetric_and_skew_symmetric_then\" >14. If the matrix A is both symmetric and skew symmetric, then:<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-16\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-3\/#15_If_A_is_a_square_matrix_such_that_A2_A_then_I_A3_%E2%80%93_7A_is_equal_to\" >15. If A is a square matrix such that A2 = A, then (I + A)3 \u2013 7A is equal to:<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-17\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-3\/#NCERT_Solutions_class_12_Maths_Miscellaneous\" >NCERT Solutions class 12 Maths Miscellaneous<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-18\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-3\/#CBSE_App_for_Students\" >CBSE App for Students<\/a><\/li><\/ul><\/nav><\/div>\n<p>NCERT Solutions class 12 Maths Miscellaneous Class 12 Maths book solutions are available in PDF format for free download. These ncert book chapter wise questions and answers are very helpful for CBSE board exam. CBSE recommends NCERT books and most of the questions in CBSE exam are asked from NCERT text books. Class 12 Maths chapter wise NCERT solution for Maths part 1 and Maths part 2 for all the chapters can be downloaded from our website and myCBSEguide mobile app for free.<\/p>\n<p style=\"text-align: center;\"><b>NCERT solutions for Class 12 Maths Matrices\u00a0<\/b><a class=\"button\" style=\"font-weight: bold;\" href=\"https:\/\/mycbseguide.com\/downloads\/cbse-class-12-mathematics-matrices\/1287\/ncert-solutions\/5\/\">Download as PDF<\/a><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/media-mycbseguide.s3.ap-south-1.amazonaws.com\/images\/blog\/12%20Maths%20Book%20pic.jpg\" alt=\"NCERT Solutions class 12 Maths Miscellaneous\" width=\"120\" height=\"156\" \/><\/p>\n<h2><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Matrices\"><\/span>NCERT Solutions class 12 Maths Matrices<strong>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"1_Let_A_show_that_where_I_is_the_identity_matrix_of_order_2_and_N\"><\/span><strong>1. Let A = <img decoding=\"async\" style=\"height: 49px; width: 52px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image001.png\" \/>\u00a0show that <img decoding=\"async\" style=\"height: 30px; width: 173px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image002.png\" \/>\u00a0where I is the identity matrix of order 2 and <img decoding=\"async\" style=\"height: 15px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image003.png\" \/>\u00a0N.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Using Mathematical Induction, we see the result is true for <img decoding=\"async\" style=\"height: 21px; width: 36px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image004.png\" \/>\u00a0for<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 30px; width: 176px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image005.png\" \/><\/p>\n<p style=\"text-align: justify;\"><strong>Given:<\/strong> <img decoding=\"async\" style=\"height: 28px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image006.png\" \/>\u00a0is true, i.e. <img decoding=\"async\" style=\"height: 30px; width: 176px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image007.png\" \/><\/p>\n<p style=\"text-align: justify;\"><strong>To prove: <\/strong><img decoding=\"async\" style=\"height: 30px; width: 222px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image008.png\" \/><\/p>\n<p style=\"text-align: justify;\"><strong>Proof:<\/strong> L.H.S.\u00a0 = <img decoding=\"async\" style=\"height: 30px; width: 79px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image009.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 30px; width: 132px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image010.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 30px; width: 162px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image011.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" style=\"height: 21px; width: 252px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image012.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" style=\"height: 21px; width: 236px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image013.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" style=\"height: 28px; width: 125px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image014.png\" \/>\u00a0= R.H.S.<\/p>\n<p style=\"text-align: justify;\">Thus, <img decoding=\"async\" style=\"height: 28px; width: 57px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image015.png\" \/>\u00a0is true, therefore, <img decoding=\"async\" style=\"height: 28px; width: 36px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image016.png\" \/>\u00a0is true.<\/p>\n<hr \/>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"2_If_A_prove_that_A_N\"><\/span><strong>2. If A = <img decoding=\"async\" style=\"height: 75px; width: 64px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image017.png\" \/>, prove that A\u2019\u2019 = <img decoding=\"async\" style=\"height: 83px; width: 138px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image018.png\" \/>\u00a0N.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Given: A = <img decoding=\"async\" style=\"height: 75px; width: 64px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image017.png\" \/>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u2026\u2026\u2026.(i)<\/p>\n<p style=\"text-align: justify;\">Let <img decoding=\"async\" style=\"height: 28px; width: 44px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image019.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 83px; width: 151px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image020.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 15px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image021.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 28px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image022.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 83px; width: 116px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image023.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 75px; width: 64px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image017.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\/03%20Misc\/image024.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 28px; width: 35px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image025.png\" \/>\u00a0is true for <img decoding=\"async\" style=\"height: 19px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image026.png\" \/><\/p>\n<p style=\"text-align: justify;\">Now <img decoding=\"async\" style=\"height: 28px; width: 44px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image027.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 83px; width: 151px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image028.png\" \/>\u00a0\u2026\u2026\u2026.(ii)<\/p>\n<p style=\"text-align: justify;\">Multiplying eq. (ii) by eq. (i),\u00a0 <img decoding=\"async\" style=\"height: 83px; width: 223px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image029.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 15px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image021.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 83px; width: 132px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image030.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 28px; width: 57px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image015.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, <img decoding=\"async\" style=\"height: 28px; width: 36px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image016.png\" \/>\u00a0is true for all natural numbers by P.M.I.<\/p>\n<hr \/>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"3_If_A_then_prove_that_A_where_is_any_positive_integer\"><\/span><strong>3. If A = <img decoding=\"async\" style=\"height: 49px; width: 66px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image031.png\" \/>\u00a0then prove that A\u2019\u2019 = <img decoding=\"async\" style=\"height: 49px; width: 107px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image032.png\" \/>\u00a0where <img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image033.png\" \/>\u00a0is any positive integer.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Given: A\u2019\u2019 = <img decoding=\"async\" style=\"height: 49px; width: 107px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image032.png\" \/>\u00a0\u00a0\u00a0\u00a0<img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image024.png\" \/>\u00a0\u00a0<img decoding=\"async\" style=\"height: 49px; width: 141px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image034.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 15px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image021.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 49px; width: 202px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image035.png\" \/>\u00a0which is true for <img decoding=\"async\" style=\"height: 19px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image026.png\" \/><\/p>\n<p style=\"text-align: justify;\">Now, <img decoding=\"async\" style=\"height: 49px; width: 142px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image036.png\" \/>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u2026\u2026\u2026.(i)<\/p>\n<p style=\"text-align: justify;\">Again <img decoding=\"async\" style=\"height: 55px; width: 231px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image037.png\" \/>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u2026\u2026\u2026.(ii)<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 15px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image021.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 55px; width: 296px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image038.png\" \/>\u00a0\u00a0[From eq. (i)]\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 15px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image021.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 49px; width: 242px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image039.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 49px; width: 131px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image040.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 15px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image021.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 55px; width: 236px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image041.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, the result is true for <img decoding=\"async\" style=\"height: 19px; width: 61px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image042.png\" \/><\/p>\n<p style=\"text-align: justify;\">Hence, by the principal of mathematical induction, the result is true for all positive integers <img decoding=\"async\" style=\"height: 15px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image043.png\" \/><\/p>\n<hr \/>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"4_If_A_and_B_are_symmetric_matrices_prove_that_AB_%E2%80%93_BA_is_a_skew_symmetric_matrix\"><\/span><strong>4. If A and B are symmetric matrices, prove that AB \u2013 BA is a skew symmetric matrix.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>A and B are symmetric matrices.\u00a0 <img decoding=\"async\" style=\"height: 15px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image021.png\" \/>\u00a0\u00a0A\u2019 = A and B\u2019 = B\u00a0 \u2026\u2026\u2026.(i)<\/p>\n<p style=\"text-align: justify;\">Now, (AB \u2013 BA)\u2019 = (AB)\u2019 \u2013 (BA)\u2019\u00a0\u00a0 <img decoding=\"async\" style=\"height: 15px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image021.png\" \/>\u00a0(AB \u2013 BA)\u2019 = B\u2019A\u2019 \u2013 A\u2019B\u2019 [Reversal law]\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 15px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image021.png\" \/>\u00a0(AB \u2013 BA)\u2019 = BA \u2013 AB\u00a0\u00a0 [Using eq. (i)]\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 15px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image021.png\" \/>\u00a0(AB \u2013 BA)\u2019 = \u2013 (AB \u2013 BA)<\/p>\n<p style=\"text-align: justify;\">Therefore, (AB \u2013 BA) is a skew symmetric.<\/p>\n<hr \/>\n<p style=\"text-align: center;\">NCERT Solutions class 12 Maths Miscellaneous<\/p>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"5_Show_that_the_matrix_BAB_is_symmetric_or_skew_symmetric_according_as_A_is_symmetric_or_skew_symmetric\"><\/span><strong>5. Show that the matrix B\u2019AB is symmetric or skew symmetric according as A is symmetric or skew symmetric.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>(B\u2019AB)\u2019 = [B\u2019(AB]\u2019 = (AB)\u2019 (B\u2019)\u2019\u00a0\u00a0 [<img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image044.png\" \/>\u00a0(CD)\u2019 = D\u2019C\u2019]\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 15px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image021.png\" \/>\u00a0 (B\u2019AB)\u2019 = B\u2019A\u2019B\u00a0\u00a0 \u2026\u2026\u2026.(i)<\/p>\n<p style=\"text-align: justify;\">Case I: A is a symmetric matrix, then\u00a0 <img decoding=\"async\" style=\"height: 15px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image021.png\" \/>\u00a0A\u2019 = A<\/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\/03%20Misc\/image024.png\" \/>\u00a0 From eq. (i)\u00a0 (B\u2019AB)\u2019 = B\u2019AB<\/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\/03%20Misc\/image024.png\" \/>\u00a0B\u2019AB is a symmetric matrix.<\/p>\n<p style=\"text-align: justify;\">Case II: A is a skew symmetric matrix. <img decoding=\"async\" style=\"height: 15px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image021.png\" \/>\u00a0A\u2019 = \u2013 A<\/p>\n<p style=\"text-align: justify;\">Putting A\u2019 = \u2013 A in eq. (i),\u00a0 (B\u2019AB)\u2019 = B\u2019(\u2013 A)B = \u2013 B\u2019AB<\/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\/03%20Misc\/image024.png\" \/>\u00a0B\u2019AB is a skew symmetric matrix.<\/p>\n<hr \/>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"6_Find_the_values_of_if_the_matrix_A_satisfies_the_equation_AA_I\"><\/span><strong>6. Find the values of <img decoding=\"async\" style=\"height: 17px; width: 42px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image045.png\" \/>\u00a0if the matrix A = <img decoding=\"async\" style=\"height: 75px; width: 93px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image046.png\" \/>\u00a0satisfies the equation A\u2019A = I.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Given: A = <img decoding=\"async\" style=\"height: 75px; width: 93px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image046.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 15px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image021.png\" \/>\u00a0\u00a0A\u2019 = <img decoding=\"async\" style=\"height: 75px; width: 109px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image047.png\" \/><\/p>\n<p style=\"text-align: justify;\">Now A\u2019A = I\u00a0 <img decoding=\"async\" style=\"height: 15px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image021.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 75px; width: 277px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image048.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 15px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image021.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 83px; width: 383px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image049.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 15px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image021.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 83px; width: 192px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image050.png\" \/><\/p>\n<p style=\"text-align: justify;\">Equating corresponding entries, we have<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 21px; width: 50px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image051.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 15px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image021.png\" \/>\u00a0\u00a0<img decoding=\"async\" style=\"height: 42px; width: 46px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image052.png\" \/>\u00a0\u00a0<img decoding=\"async\" style=\"height: 15px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image021.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 44px; width: 63px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image053.png\" \/><\/p>\n<p style=\"text-align: justify;\">And <img decoding=\"async\" style=\"height: 24px; width: 51px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image054.png\" \/>\u00a0\u00a0<img decoding=\"async\" style=\"height: 15px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image021.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 42px; width: 46px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image055.png\" \/>\u00a0\u00a0<img decoding=\"async\" style=\"height: 15px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image021.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 44px; width: 64px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image056.png\" \/><\/p>\n<p style=\"text-align: justify;\">And <img decoding=\"async\" style=\"height: 21px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image057.png\" \/>\u00a0\u00a0<img decoding=\"async\" style=\"height: 15px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image021.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 42px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image058.png\" \/>\u00a0\u00a0<img decoding=\"async\" style=\"height: 15px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image021.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 44px; width: 61px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/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\/03%20Misc\/image024.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 44px; width: 63px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image053.png\" \/>, <img decoding=\"async\" style=\"height: 44px; width: 64px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image056.png\" \/>, <img decoding=\"async\" style=\"height: 44px; width: 61px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image059.png\" \/><\/p>\n<hr \/>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"7_For_what_value_of\"><\/span><strong>7. For what value of <img decoding=\"async\" style=\"height: 17px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image060.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 75px; width: 172px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image061.png\" \/>\u00a0?<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Given: <img decoding=\"async\" style=\"height: 75px; width: 172px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image061.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 15px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image021.png\" \/>\u00a0\u00a0<img decoding=\"async\" style=\"height: 75px; width: 238px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image062.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 15px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image021.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 75px; width: 111px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image063.png\" \/>\u00a0\u00a0\u00a0<img decoding=\"async\" style=\"height: 15px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image021.png\" \/>\u00a0\u00a0<img decoding=\"async\" style=\"height: 28px; width: 159px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image064.png\" \/><\/p>\n<p style=\"text-align: justify;\">Equating corresponding entries, we have<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 19px; width: 89px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image065.png\" \/>\u00a0\u00a0 <img decoding=\"async\" style=\"height: 15px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image021.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 19px; width: 54px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image066.png\" \/>\u00a0\u00a0<img decoding=\"async\" style=\"height: 15px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image021.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 19px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image067.png\" \/><\/p>\n<hr \/>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"8_If_A_show_that_A2_%E2%80%93_5A_7I_0\"><\/span><strong>8. If A = <img decoding=\"async\" style=\"height: 49px; width: 61px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image068.png\" \/>\u00a0show that A<sup>2<\/sup> \u2013 5A + 7I = 0.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Given: A = <img decoding=\"async\" style=\"height: 49px; width: 55px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image069.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\/03%20Misc\/image024.png\" \/>\u00a0 A<sup>2<\/sup> \u2013 5A + 7I = <img decoding=\"async\" style=\"height: 49px; width: 256px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image070.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" style=\"height: 49px; width: 253px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image071.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 49px; width: 195px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image072.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" style=\"height: 49px; width: 179px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image073.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 49px; width: 140px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image074.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 49px; width: 120px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image075.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" style=\"height: 49px; width: 46px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image076.png\" \/>\u00a0= 0 = R.H.S.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Proved.<\/p>\n<hr \/>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"9_Find_if\"><\/span><strong>9. Find <img decoding=\"async\" style=\"height: 17px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image060.png\" \/>\u00a0if <img decoding=\"async\" style=\"height: 75px; width: 205px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image077.png\" \/>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Given: <img decoding=\"async\" style=\"height: 75px; width: 203px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image078.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 15px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image021.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 75px; width: 258px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image079.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 15px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image021.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 75px; width: 188px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image080.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 15px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image021.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 30px; width: 217px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image081.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 15px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image021.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 30px; width: 171px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image082.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 15px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image021.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 31px; width: 124px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image083.png\" \/><\/p>\n<p style=\"text-align: justify;\">Equating corresponding entries, we have<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 21px; width: 72px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image084.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 15px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image021.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 52px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image085.png\" \/>\u00a0\u00a0<img decoding=\"async\" style=\"height: 15px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image021.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 24px; width: 65px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image086.png\" \/><\/p>\n<hr \/>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"10_A_manufacturer_produces_three_products_which_he_sells_in_two_markets_Annual_sales_are_indicated_below\"><\/span><strong>10. A manufacturer produces three products, <img decoding=\"async\" style=\"height: 17px; width: 42px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image045.png\" \/>\u00a0which he sells in two markets. Annual sales are indicated below:<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<table border=\"1\" cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td style=\"width: 174.6pt;\">\n<p style=\"text-align: justify;\"><strong>\u00a0<\/strong><\/p>\n<\/td>\n<td style=\"width: 174.6pt;\">\n<p style=\"text-align: justify;\"><strong>Market<\/strong><\/p>\n<\/td>\n<td style=\"width: 174.6pt;\">\n<p style=\"text-align: justify;\"><strong>Products<\/strong><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 174.6pt;\">\n<p style=\"text-align: justify;\"><strong>I. 10,000<\/strong><\/p>\n<\/td>\n<td style=\"width: 174.6pt;\">\n<p style=\"text-align: justify;\"><strong>2,000<\/strong><\/p>\n<\/td>\n<td style=\"width: 174.6pt;\">\n<p style=\"text-align: justify;\"><strong>18,000<\/strong><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 174.6pt;\">\n<p style=\"text-align: justify;\"><strong>II. 6,000<\/strong><\/p>\n<\/td>\n<td style=\"width: 174.6pt;\">\n<p style=\"text-align: justify;\"><strong>20,000<\/strong><\/p>\n<\/td>\n<td style=\"width: 174.6pt;\">\n<p style=\"text-align: justify;\"><strong>8,000<\/strong><\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\"><strong>\u00a0(a) If unit sales prices of <img decoding=\"async\" style=\"height: 17px; width: 29px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image087.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 13px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image088.png\" \/>\u00a0are ` 2.50, ` 1.50 and ` 1.00 respectively, find the total revenue in each market with the help of matrix algebra.<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(b) If the unit costs of the above three commodities are ` 2.00, ` 1.00 and 50 paise respectively. Find the gross profit.<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>According to question, the matrix \u00a0A = <img decoding=\"async\" style=\"height: 68px; width: 181px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image089.png\" \/><\/p>\n<p style=\"text-align: justify;\"><strong>(a)<\/strong> Let B be the column matrix representing sale price of each unit of products <img decoding=\"async\" style=\"height: 17px; width: 44px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image090.png\" \/><\/p>\n<p style=\"text-align: justify;\">Then B = <img decoding=\"async\" style=\"height: 75px; width: 52px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image091.png\" \/><\/p>\n<p style=\"text-align: justify;\">Now Revenue = Sale price c Number of items sold<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 15px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image021.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 75px; width: 217px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image092.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 49px; width: 173px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image093.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 49px; width: 64px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image094.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, the revenue collected by sale of all items in Market I = ` 46,000 and the revenue collected by sale of all items in Market II = ` 53,000.<\/p>\n<p style=\"text-align: justify;\"><strong>(b)<\/strong> Let C be the column matrix representing cost price of each unit of products <img decoding=\"async\" style=\"height: 17px; width: 44px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image090.png\" \/><\/p>\n<p style=\"text-align: justify;\">Then C = <img decoding=\"async\" style=\"height: 75px; width: 52px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image095.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\/03%20Misc\/image024.png\" \/>\u00a0 Total cost = AC = <img decoding=\"async\" style=\"height: 75px; width: 217px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image096.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" style=\"height: 49px; width: 176px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image097.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 49px; width: 64px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image094.png\" \/><img decoding=\"async\" style=\"height: 49px; width: 64px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image098.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\/03%20Misc\/image024.png\" \/>\u00a0 The profit collected in two markets is given in matrix form as<\/p>\n<p style=\"text-align: justify;\">Profit matrix = Revenue matrix \u2013 Cost matrix<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 15px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image021.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 49px; width: 212px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image099.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, the gross profit in both the markets = ` 15000 + ` 17000 = ` 32,000.<\/p>\n<hr \/>\n<p style=\"text-align: center;\">NCERT Solutions class 12 Maths Miscellaneous<\/p>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"11_Find_the_matrix_X_so_that_X\"><\/span><strong>11. Find the matrix X so that X <img decoding=\"async\" style=\"height: 49px; width: 188px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image100.png\" \/>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Given: X<img decoding=\"async\" style=\"height: 49px; width: 184px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image101.png\" \/>\u00a0\u00a0\u2026\u2026\u2026.(i)<\/p>\n<p style=\"text-align: justify;\">Putting X = <img decoding=\"async\" style=\"height: 49px; width: 57px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image102.png\" \/>\u00a0in eq. (i),\u00a0 <img decoding=\"async\" style=\"height: 49px; width: 237px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image103.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 15px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image021.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 49px; width: 313px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image104.png\" \/><\/p>\n<p style=\"text-align: justify;\">Equating corresponding entries, we have<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 19px; width: 77px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image105.png\" \/>\u00a0\u00a0 \u2026..(ii)\u00a0\u00a0 <img decoding=\"async\" style=\"height: 19px; width: 84px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image106.png\" \/>\u00a0\u00a0\u00a0\u2026..(iii)<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 19px; width: 84px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image107.png\" \/>\u00a0\u2026..(iv)\u00a0\u00a0 <img decoding=\"async\" style=\"height: 19px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image108.png\" \/>\u00a0\u00a0\u00a0\u2026..(v)<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 19px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image109.png\" \/>\u2026..(vi)\u00a0\u00a0 <img decoding=\"async\" style=\"height: 19px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image110.png\" \/>\u00a0\u00a0\u00a0\u2026..(vi)<\/p>\n<p style=\"text-align: justify;\">Solving eq. (ii) and (iii), we have\u00a0\u00a0 <img decoding=\"async\" style=\"height: 19px; width: 35px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image111.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 19px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image112.png\" \/><\/p>\n<p style=\"text-align: justify;\">Solving eq. (v) and (vi), we have\u00a0\u00a0 <img decoding=\"async\" style=\"height: 19px; width: 35px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image113.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 19px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image114.png\" \/><\/p>\n<p style=\"text-align: justify;\">Putting these values in X = <img decoding=\"async\" style=\"height: 49px; width: 57px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image102.png\" \/>,\u00a0\u00a0\u00a0 X = <img decoding=\"async\" style=\"height: 49px; width: 64px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image115.png\" \/><\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><\/h4>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"12_If_A_and_B_are_square_matrices_of_the_same_order_such_that_AB_BA_then_prove_by_induction_that_AB_BA_Further_prove_that_AB_AB_for_all_N\"><\/span><strong>12. If A and B are square matrices of the same order such that AB = BA, then prove by induction that AB\u2019\u2019 = B\u2019\u2019A. Further prove that (AB)\u2019\u2019 = A\u2019\u2019B\u2019\u2019 for all <img decoding=\"async\" style=\"height: 15px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image003.png\" \/>\u00a0N.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Given: AB = BA \u2026..(i)<\/p>\n<p style=\"text-align: justify;\">Let <img decoding=\"async\" style=\"height: 28px; width: 44px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image019.png\" \/>\u00a0\u00a0\u00a0<img decoding=\"async\" style=\"height: 20px; width: 81px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image116.png\" \/>\u00a0\u00a0\u00a0\u2026\u2026(ii)<\/p>\n<p style=\"text-align: justify;\">For <img decoding=\"async\" style=\"height: 21px; width: 36px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image004.png\" \/>\u00a0\u00a0<img decoding=\"async\" style=\"height: 28px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image022.png\" \/>\u00a0\u00a0becomes AB = BA<\/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\/03%20Misc\/image024.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 28px; width: 35px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image025.png\" \/>\u00a0is true for <img decoding=\"async\" style=\"height: 19px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image026.png\" \/><\/p>\n<p style=\"text-align: justify;\">For <img decoding=\"async\" style=\"height: 21px; width: 42px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image117.png\" \/>\u00a0\u00a0<img decoding=\"async\" style=\"height: 28px; width: 44px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image027.png\" \/>\u00a0\u00a0\u00a0<img decoding=\"async\" style=\"height: 20px; width: 81px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image118.png\" \/><\/p>\n<p style=\"text-align: justify;\">Multiplying both sides by B,\u00a0 <img decoding=\"async\" style=\"height: 20px; width: 99px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image119.png\" \/>\u00a0\u00a0<img decoding=\"async\" style=\"height: 15px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image021.png\" \/>\u00a0\u00a0<img decoding=\"async\" style=\"height: 20px; width: 97px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image120.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 15px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image021.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 20px; width: 96px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image121.png\" \/>\u00a0\u00a0\u00a0[From eq. (i)]\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\/03%20Misc\/image024.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 28px; width: 57px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image015.png\" \/>\u00a0is also true.<\/p>\n<p style=\"text-align: justify;\">Therefore, <img decoding=\"async\" style=\"height: 28px; width: 36px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image016.png\" \/>\u00a0is true for all <img decoding=\"async\" style=\"height: 15px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image003.png\" \/>\u00a0N by P.M.I.<\/p>\n<hr \/>\n<p style=\"text-align: center;\">NCERT Solutions class 12 Maths Miscellaneous<\/p>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"13_If_A_is_such_that_A2_I_then\"><\/span><strong>13. If A = <img decoding=\"async\" style=\"height: 49px; width: 66px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image122.png\" \/>\u00a0is such that A<sup>2<\/sup> = I, then:<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>(A) <img decoding=\"async\" style=\"height: 24px; width: 99px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image123.png\" \/>\u00a0\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(B) <img decoding=\"async\" style=\"height: 24px; width: 99px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image124.png\" \/>\u00a0\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(C) <img decoding=\"async\" style=\"height: 24px; width: 99px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image125.png\" \/>\u00a0\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(D) <img decoding=\"async\" style=\"height: 24px; width: 99px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image126.png\" \/>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Given: A = <img decoding=\"async\" style=\"height: 49px; width: 66px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image122.png\" \/>\u00a0and A<sup>2<\/sup> = I<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 15px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image021.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 49px; width: 193px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image127.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 15px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image021.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 53px; width: 205px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image128.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 15px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image021.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 53px; width: 203px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image129.png\" \/><\/p>\n<p style=\"text-align: justify;\">Equating corresponding entries, we have<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 24px; width: 73px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image130.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 15px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image021.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 24px; width: 99px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image125.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, option (C) is correct.<\/p>\n<hr \/>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"14_If_the_matrix_A_is_both_symmetric_and_skew_symmetric_then\"><\/span><strong>14. If the matrix A is both symmetric and skew symmetric, then:<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>(A) A is a diagonal matrix\u00a0\u00a0 <\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(B) A is a zero matrix<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(C) A is a square matrix\u00a0\u00a0 <\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(D) None of these<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Since, A is symmetric, therefore, A\u2019 = A \u2026\u2026..(i)<\/p>\n<p style=\"text-align: justify;\">And A is skew-symmetric, therefore, A\u2019 = \u2013 A<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 15px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image021.png\" \/>\u00a0 A = \u2013 A\u00a0 [From eq. (i)]\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 15px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image021.png\" \/>\u00a0 A + A = 0 <img decoding=\"async\" style=\"height: 15px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image021.png\" \/>\u00a02A = 0\u00a0 <img decoding=\"async\" style=\"height: 15px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image021.png\" \/>\u00a0A = 0<\/p>\n<p style=\"text-align: justify;\">Therefore, A is zero matrix.<\/p>\n<p style=\"text-align: justify;\">Therefore, option (B) is correct.<\/p>\n<hr \/>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"15_If_A_is_a_square_matrix_such_that_A2_A_then_I_A3_%E2%80%93_7A_is_equal_to\"><\/span><strong>15. If A is a square matrix such that A<sup>2<\/sup> = A, then (I + A)<sup>3<\/sup> \u2013 7A is equal to:<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>(A) A\u00a0\u00a0 <\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(B) I \u2013 A\u00a0\u00a0 <\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(C) I\u00a0\u00a0 <\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(D) 3A<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Given: A<sup>2<\/sup> = A\u00a0\u00a0\u00a0 \u2026..(i)<\/p>\n<p style=\"text-align: justify;\">Multiplying both sides by A,\u00a0 A<sup>3<\/sup> = A<sup>2<\/sup> = A [From eq. (i)]\u00a0 \u2026\u2026(ii)<\/p>\n<p style=\"text-align: justify;\">Also given (I + A)<sup>3<\/sup> \u2013 7A = I<sup>3<\/sup> + A<sup>3<\/sup> + 3I<sup>2<\/sup>A + 3IA<sup>2<\/sup> \u2013 7A<\/p>\n<p style=\"text-align: justify;\">Putting A<sup>2<\/sup> = A [from eq. (i)] and A<sup>3<\/sup> = A [from eq. (ii)],<\/p>\n<p style=\"text-align: justify;\">= I + A + 3IA + 3IA \u2013 7A = I + A + 3A + 3A \u2013 7A\u00a0\u00a0 [<img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/03%20Misc\/image044.png\" \/>\u00a0IA = A]\n<p style=\"text-align: justify;\">= I + 7A \u2013 7A = I<\/p>\n<p style=\"text-align: justify;\">Therefore, option (C) is correct.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>NCERT Solutions class 12 Maths Miscellaneous PDF (Download) Free from myCBSEguide app and myCBSEguide website. Ncert solution class 12 Maths includes text book solutions from both part 1 and part 2. NCERT Solutions for CBSE Class 12 Maths have total 13 chapters. 12 Maths NCERT Solutions in PDF for free Download on our website. Ncert Maths class 12 solutions PDF and Maths ncert class 12 PDF solutions with latest modifications and as per the latest CBSE syllabus are only available in myCBSEguide<\/p>\n<h2><span class=\"ez-toc-section\" id=\"CBSE_App_for_Students\"><\/span>CBSE App for Students<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>To download\u00a0NCERT Solutions for class 12 Physics, Chemistry, Biology, History, Political Science, Economics, Geography, Computer Science, Home Science, Accountancy, Business Studies and Home Science; do check myCBSEguide app or website. myCBSEguide provides sample papers with solution, test papers for chapter-wise practice, NCERT solutions, NCERT Exemplar solutions, quick revision notes for ready reference, CBSE guess papers and CBSE important question papers. Sample Paper all are made available through\u00a0<a href=\"https:\/\/play.google.com\/store\/apps\/details?id=in.techchefs.MyCBSEGuide&amp;referrer=utm_source%3Dmycbse_bottom%26utm_medium%3Dtext%26utm_campaign%3Dmycbseads\"><strong>the best app for CBSE students<\/strong><\/a>\u00a0and myCBSEguide website.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>NCERT Solutions class 12 Maths Miscellaneous Class 12 Maths book solutions are available in PDF format for free download. These ncert book chapter wise questions and answers are very helpful for CBSE board exam. CBSE recommends NCERT books and most of the questions in CBSE exam are asked from NCERT text books. Class 12 Maths &#8230; <a title=\"NCERT Solutions class 12 Maths Miscellaneous\" class=\"read-more\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-3\/\" aria-label=\"More on NCERT Solutions class 12 Maths Miscellaneous\">Read more<\/a><\/p>\n","protected":false},"author":1,"featured_media":-1,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1432,281],"tags":[283,1342,321,216],"class_list":["post-4485","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mathematics-cbse-class-12","category-ncert-solutions","tag-cbse-study-material","tag-class-12","tag-mathematics","tag-ncert-solutions"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.0 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>NCERT Solutions class 12 Maths Miscellaneous | myCBSEguide<\/title>\n<meta name=\"description\" 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