{"id":25125,"date":"2022-01-16T12:40:00","date_gmt":"2022-01-16T07:10:00","guid":{"rendered":"https:\/\/yanamtakshashila.com\/?p=25125"},"modified":"2022-02-02T15:40:05","modified_gmt":"2022-02-02T10:10:05","slug":"solutions-to-seminar-1-for-engineering-mathematics-i","status":"publish","type":"post","link":"https:\/\/yanamtakshashila.com\/?p=25125","title":{"rendered":"SOLUTIONS TO SEMINAR &#8211; 1 FOR ENGINEERING MATHEMATICS &#8211; I"},"content":{"rendered":"\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black} {1.}\\ Solve\\ by\\ using\\ Cramers\\ Rule\\ \\hspace{20cm}\\]<script src=\"https:\/\/yanamtakshashila.com\/wp-includes\/js\/dist\/hooks.min.js?ver=dd5603f07f9220ed27f1\" id=\"wp-hooks-js\"><\/script>\n<script src=\"https:\/\/yanamtakshashila.com\/wp-includes\/js\/dist\/i18n.min.js?ver=c26c3dc7bed366793375\" id=\"wp-i18n-js\"><\/script>\n<script id=\"wp-i18n-js-after\">\nwp.i18n.setLocaleData( { 'text direction\\u0004ltr': [ 'ltr' ] } );\n\/\/# sourceURL=wp-i18n-js-after\n<\/script>\n<script  async src=\"https:\/\/cdnjs.cloudflare.com\/ajax\/libs\/mathjax\/2.7.7\/MathJax.js?config=TeX-MML-AM_CHTML\" id=\"mathjax-js\"><\/script>\n<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[3x &#8211; y + 2z=8,\\ x &#8211; y + z = 2\\ and\\ 2x + y &#8211; z = 1\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black}{Solution:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[3x &#8211; y + 2z=8\\ &#8212;&#8212;&#8212;&#8212;&#8212;&#8212;-(1)\\ \\hspace{6cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[x &#8211; y + z = 2\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2x + y &#8211; z = 1\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta = \\begin{vmatrix}\n3 &amp; -1 &amp; 2 \\\\\n1 &amp; -1 &amp; 1 \\\\\n2 &amp; 1 &amp; -1 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta =3\\begin{vmatrix}\n-1 &amp; 1 \\\\\n1 &amp; -1 \\\\\n\\end{vmatrix}\\ +\\ 1\\begin{vmatrix}\n1 &amp; 1 \\\\\n2 &amp; -1 \\\\\n\\end{vmatrix}\\ +\\ 2\\begin{vmatrix}\n1 &amp; -1\\\\\n2 &amp; 1 \\\\\n\\end{vmatrix}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta =3(1\\ -\\ 1)\\ + 1 (-1\\ -\\ 2) + 2(1\\ -\\ (-2))\\ \n\\hspace{9cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta =3(0)\\ + 1 (-3) + 2(3)\\ \n\\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta =0\\ &#8211; 3 + 6\\ \n\\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta =3\\ \n\\hspace{17cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_x = \\begin{vmatrix}\n8 &amp; -1 &amp; 2 \\\\\n2 &amp; -1 &amp; 1 \\\\\n1 &amp; 1 &amp; -1 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_x =8\\begin{vmatrix}\n-1 &amp; 1 \\\\\n1 &amp; -1 \\\\\n\\end{vmatrix}\\ +\\ 1\\begin{vmatrix}\n2 &amp; 1 \\\\\n1 &amp; -1 \\\\\n\\end{vmatrix}\\ +\\ 2\\begin{vmatrix}\n2 &amp; -1\\\\\n1 &amp; 1 \\\\\n\\end{vmatrix}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_x =8(1\\ -\\ 1) + 1 (-2\\ -\\ 1) + 2(2\\ -\\ (-1))\\ \n\\hspace{9cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_x =8(0)\\ + 1 (-3) + 2(3)\\ \n\\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_x = 0\\ &#8211; 3 + 6\\ \n\\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_x =3\\ \n\\hspace{17cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_y = \\begin{vmatrix}\n3 &amp; 8 &amp; 2 \\\\\n1 &amp; 2 &amp; 1 \\\\\n2 &amp; 1 &amp; -1 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_y =3\\begin{vmatrix}\n2 &amp;  1 \\\\\n1 &amp; -1 \\\\\n\\end{vmatrix}\\ -\\ 8\\begin{vmatrix}\n1 &amp; 1 \\\\\n2 &amp; -1 \\\\\n\\end{vmatrix}\\ +\\ 2\\begin{vmatrix}\n1 &amp; 2\\\\\n2 &amp;  1 \\\\\n\\end{vmatrix}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_y =3(-2\\ -\\ 1)\\ &#8211; 8 (-1\\ -\\ 2) + 2(1\\ -\\ (4))\\ \n\\hspace{9cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_y =3(- 3)\\ &#8211; 8 (-3) + 2(-3)\\ \n\\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_y = -9\\ + 24\\ &#8211; 6\\ \n\\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_y = 9\\ \n\\hspace{17cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_z = \\begin{vmatrix}\n3 &amp; -1 &amp; 8 \\\\\n1 &amp; -1 &amp; 2 \\\\\n2 &amp; 1 &amp; 1 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_z =3\\begin{vmatrix}\n-1 &amp; 2 \\\\\n1 &amp; 1 \\\\\n\\end{vmatrix}\\ +\\ 1\\begin{vmatrix}\n1 &amp; 2 \\\\\n2 &amp; 1 \\\\\n\\end{vmatrix}\\ +\\ 8\\begin{vmatrix}\n1 &amp; &#8211; 1\\\\\n2 &amp;  1 \\\\\n\\end{vmatrix}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_z =3(-1\\ -\\ 2)\\ + 1 (1\\ -\\ 4) + 8(1\\ -\\ (-2))\\ \n\\hspace{9cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_z =3(-3)\\ + 1 (-3) + 8(3)\\ \n\\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_z =-9\\  &#8211; 3 + 24\\ \n\\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_z =12\\ \n\\hspace{17cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[The\\ Solution\\ is\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[x=\\ \\frac{\\Delta_x}{\\Delta} =\\ \\frac{3}{3} =\\ 1\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[y=\\ \\frac{\\Delta_y}{\\Delta} =\\ \\frac{8}{4} =\\ 2\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[z=\\ \\frac{\\Delta_z}{\\Delta} =\\ \\frac{12}{4} =\\ 3\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[For\\ cross\\ verification\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Put\\ x =1\\ y = 2\\ z = 3\\ in\\ equation (1)\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[LHS = 3(1) &#8211; 3 + 2(4)\\]\\[ = 3 &#8211; 3 + 8 = 8\\]\\[ = RHS\\] <\/div>\n\n\n\n<script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js?client=ca-pub-9453835310745500\" crossorigin=\"anonymous\"><\/script>\n<!-- wide skyscraper -->\n<ins class=\"adsbygoogle\" style=\"display:inline-block;width:300px;height:600px\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"9987820756\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2.\\ Find\\ the\\ inverse\\ of\\ \\begin{bmatrix}\n3 &amp; 1 &amp; -1\\\\ \n2 &amp; -2 &amp; 0 \\\\\n1 &amp; 2 &amp; -1 \\\\ \\end{bmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black}{Solution:}\\ A\\ =\\begin{bmatrix}\n3 &amp; 1 &amp; -1 \\\\\n2 &amp; -2 &amp; 0 \\\\\n1 &amp; 2 &amp; -1 \\\\\n\\end{bmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\begin{vmatrix}\nA \\\\\n\\end{vmatrix}\\ =\\ 3\\begin{vmatrix}\n-2 &amp; 0 \\\\\n2 &amp; -1 \\\\\n\\end{vmatrix}\\ -\\ 1\\begin{vmatrix}\n2 &amp; 0 \\\\\n1 &amp; -1 \\\\\n\\end{vmatrix}\\ -\\ 1\\begin{vmatrix}\n2 &amp; -2\\\\\n1 &amp;  2 \\\\\n\\end{vmatrix}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ 3(2\\ -\\ 0)\\ &#8211; 1 (-2\\ -\\ 0) &#8211; 1(4\\ +\\ 2)\\ \n\\hspace{9cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =3(2)\\ &#8211; 1 (-2) &#8211; 1(6)\\ \n\\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ 6\\ +\\ 2\\ -\\  6\\ \n\\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\begin{vmatrix}\nA \\\\\n\\end{vmatrix}\\ =\\ 2\\ \\neq\\ 0\\\n\\hspace{17cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\therefore\\ Inverse\\ of\\ A\\ exist\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black}{Cofactors\\ of\\ Matrix\\ A:}\\ \\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 3 = (-1)^{1\\ +\\ 1}\\ \\begin{vmatrix}\n-2 &amp; 0 \\\\\n2 &amp; -1 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1)^2 (2 &#8211; 0)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (1) (2)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 1 =\\ 2\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 1\\ =\\ (-1)^{1\\ +\\ 2}\\ \\begin{vmatrix}\n2 &amp; 0 \\\\\n1 &amp; -1 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1)^3 (-\\ 2 -\\ 0)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1) (-2)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 1\\ =\\ 2\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ -1 = (-1)^{1\\ +\\ 3}\\ \\begin{vmatrix}\n2 &amp; -2 \\\\\n1 &amp; 2 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1)^4 (4\\ +\\ 2)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (1) (6)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ -1\\ =\\ 6\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 2\\ =\\ (-1)^{2\\ +\\ 1}\\ \\begin{vmatrix}\n1 &amp; -1 \\\\\n2 &amp; -1 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1)^3 (-1 +\\  2)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1) (1)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 2 =\\ -1\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ -\\ 2 = (-1)^{2\\ +\\ 2}\\ \\begin{vmatrix}\n3 &amp; -1 \\\\\n1 &amp; -\\ 1 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1)^4 (-3 +\\  1)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (1) (-2)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ -2 =\\ -2\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 0\\ =\\ (-1)^{2\\ +\\ 3}\\ \\begin{vmatrix}\n3 &amp; 1 \\\\\n1 &amp; 2 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1)^5 (6\\ -\\ 1)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1) (5)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 0\\ = -5\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 1\\ = (-1)^{3\\ +\\ 1}\\ \\begin{vmatrix}\n1 &amp; -1 \\\\\n-2 &amp; 0 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1)^4 (0 &#8211; 2)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (1) (-2)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 1 =\\ -2\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 2\\ = (-1)^{3\\ +\\ 2}\\ \\begin{vmatrix}\n3 &amp; -1 \\\\\n2 &amp; 0 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1)^5 (0 + 2)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1) (2)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 2 = -2\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ -1 = (-1)^{3\\ +\\ 3}\\ \\begin{vmatrix}\n3 &amp; 1 \\\\\n2 &amp; -2 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1)^6 (-6 &#8211; 2)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (1) (-8)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ -1 =\\ -8\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Cofactor\\ matrix=\\begin{bmatrix}\n2 &amp; 2 &amp; 6 \\\\\n-1 &amp; -2 &amp; -5 \\\\\n-2 &amp; -2 &amp; -8 \\\\\n\\end{bmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Adj.\\ A=\\begin{bmatrix}\n2 &amp; -1 &amp; -2 \\\\\n2 &amp; -2 &amp; -2 \\\\\n6 &amp; -5 &amp; -8 \\\\\n\\end{bmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[A^{-1} = \\frac{1}{\\begin{vmatrix} A \\\\ \\end{vmatrix}}\\ adj.\\ A\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[A^{-1} = \\frac{1}{2}\\ \\begin{bmatrix}\n2 &amp; -1 &amp; -2 \\\\\n2 &amp; -2 &amp; -2 \\\\\n6 &amp; -5 &amp; -8 \\\\\n\\end{bmatrix}\\ \\hspace{2cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[3.\\ Find\\ the\\ rank\\ of\\ the\\ matrix\\ \\begin{bmatrix}\n1 &amp; 2 &amp; 3  &amp; 1 \\\\\n2 &amp; 3 &amp; 4 &amp; 2 \\\\\n3 &amp; 4 &amp; 5 &amp; 1 \\\\\n\\end{bmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black}{Solution:}\\ A\\ =\\begin{bmatrix}\n1 &amp; 2 &amp; 3  &amp; 1 \\\\\n2 &amp; 3 &amp; 4 &amp; 2 \\\\\n3 &amp; 4 &amp; 5 &amp; 1 \\\\\n\\end{bmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Order\\ of\\ A = 3 \u00d7 4\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\therefore \\ rank\\ of\\ A=\\rho(A)\\ \\leq\\ Min{3,4} =3\\ . \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[The\\ highest\\ order\\ of\\ minors\\ of\\ A = 3.\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[A\\ has\\ the\\ following\\ minors\\ of\\ order 3.\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ A_1\\ =\\begin{vmatrix}\n1 &amp; 2 &amp; 3 \\\\\n2 &amp; 3 &amp; 4 \\\\\n3 &amp; 4 &amp; 5 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=1\\begin{vmatrix}\n3 &amp; 4 \\\\\n4 &amp; 5 \\\\\n\\end{vmatrix}\\ -\\ 2\\begin{vmatrix}\n2 &amp; 4 \\\\\n3 &amp; 5 \\\\\n\\end{vmatrix}\\ +\\ 3\\begin{vmatrix}\n2 &amp; 3\\\\\n3 &amp;  4 \\\\\n\\end{vmatrix}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ 1(15\\ -\\ 16)\\ &#8211; 2 (10\\ -\\ 12) +\\ 3(8\\ -\\ 9)\\ \n\\hspace{9cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ 1(-1)\\ &#8211; 2 (-2) + 3(-1)\\ \n\\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ -1\\ +\\ 4 -\\ 3\\ =\\ 12\\ =\\ 0\\\n\\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ A_2\\ =\\begin{vmatrix}\n1 &amp; 2 &amp; 1 \\\\\n2 &amp; 3 &amp; 2 \\\\\n3 &amp; 4 &amp; 1 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=1\\begin{vmatrix}\n3 &amp; 2 \\\\\n4 &amp; 1\\\\\n\\end{vmatrix}\\ -\\ 2\\begin{vmatrix}\n2 &amp; 2 \\\\\n3 &amp; 1 \\\\\n\\end{vmatrix}\\ +\\ 1\\begin{vmatrix}\n2 &amp; 3\\\\\n3 &amp;  4 \\\\\n\\end{vmatrix}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =1(3\\ -\\ 8)\\ &#8211; 2 (2\\ -\\ 6) +\\ 1(8\\ -\\ 9)\\ \n\\hspace{9cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =1(-5)\\ &#8211; 2 (-4) + 1(-1)\\ \n\\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = 5\\ +\\ 8 &#8211; 1\\ =\\ 2\\ \\neq  0\\\n\\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ A_3\\ =\\begin{bmatrix}\n1 &amp; 3 &amp; 1 \\\\\n2 &amp; 4 &amp; 2 \\\\\n3 &amp; 5 &amp; 1 \\\\\n\\end{bmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=1\\begin{vmatrix}\n4 &amp; 2 \\\\\n5 &amp; 1 \\\\\n\\end{vmatrix}\\ -\\ 3\\begin{vmatrix}\n2 &amp; 2 \\\\\n3 &amp; 1 \\\\\n\\end{vmatrix}\\ +\\ 1\\begin{vmatrix}\n2 &amp; 4\\\\\n3 &amp;  5 \\\\\n\\end{vmatrix}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =1(4\\ -\\ 10)\\ &#8211; 3 (2\\ -\\ 6) + 1(10\\ -\\ 12)\\ \n\\hspace{9cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =1(-6)\\ &#8211; 3 (-4) + 1(-2)\\ \n\\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = -6\\ +\\ 12 &#8211; 2\\ =\\ -6\\ \\neq  0\\\n\\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ A_4\\ =\\begin{bmatrix}\n2 &amp; 3 &amp; 1 \\\\\n3 &amp; 4 &amp; 2 \\\\\n4 &amp; 5 &amp; 1 \\\\\n\\end{bmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=2\\begin{vmatrix}\n4 &amp; 2 \\\\\n5 &amp; 1 \\\\\n\\end{vmatrix}\\ -\\ 3\\begin{vmatrix}\n3 &amp; 2 \\\\\n4 &amp; 1 \\\\\n\\end{vmatrix}\\ +\\ 1\\begin{vmatrix}\n3 &amp; 4\\\\\n4 &amp; 5  \\\\\n\\end{vmatrix}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =2(4\\ -\\ 10)\\ &#8211; 3 (3\\ -\\ 8) + 1(15\\ -\\ 16)\\ \n\\hspace{9cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =2(-6)\\ &#8211; 3 (-5) + 1(-1)\\ \n\\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = -12\\ + 15 &#8211; 1\\ =\\ 2\\ \\neq  0\\ \n\\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\therefore\\ \\rho(A)\\  =\\ 3\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js?client=ca-pub-9453835310745500\" crossorigin=\"anonymous\"><\/script>\n<!-- Leader board 1 -->\n<ins class=\"adsbygoogle\" style=\"display:block\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"8769628924\" data-ad-format=\"auto\" data-full-width-responsive=\"true\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[4.\\ Find\\ the\\ coefficient\\ of\\ x^{30}\\ in\\ the\\ expansion\\ of\\ (x^4\\ +\\ \\frac{1}{x^6})^{15}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black}{Solution:}\\ T_{r + 1} = nC_rx^{n-r} a^r \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Here\\ X\\ =\\ x^4,\\  a\\ =\\ \\frac{1}{x^6},\\ n\\ =\\ 15,\\  r\\ =\\ r\\ \\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[T_{r + 1} = 15C_r\\ (x^4)^{15-r}\\ (\\frac{1}{x^6})^r \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = 15C_r\\ x^{60 &#8211; 4r}\\ \\frac{1^r}{x^{6r}} \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = 15C_r\\ x^{60 &#8211; 4r &#8211; 6r} \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[T_{r + 1}  = 15C_r\\ x^{60 &#8211; 10r}\\ &#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212; (1)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[We\\ assume\\ that\\ x^{30}\\ occurs\\ in\\ 15C_r\\ x^{60 &#8211; 10r}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\therefore\\  x^{60 &#8211; 10r}\\ =\\ x^{32}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Equating\\ the\\ indices\\ 60 &#8211; 10r\\ =\\ 30\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[-10r\\ =\\ 30\\ -\\ 60\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[-10r\\ = -\\ 30\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[r\\ = 3\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Put\\ r\\ =\\ 3\\ in\\ (1)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[T_{3 + 1}  = 15C_3\\ 1^3\\ x^{60\\ -\\ 30}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[T_4  = 15C_3\\ x^{30}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[The\\ coefficient\\ of\\ x^{30}\\ =\\ 15\\ C_3\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[5.\\ Find\\ the\\ term\\ independent\\ of\\ x\\ in\\ the\\ expansion\\ of\\ (3\\sqrt{x} &#8211; \\frac{2}{x^2})^{10}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black}{Solution:}\\ T_{r + 1} = nC_rx^{n-r} a^r \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Here\\ X\\ =\\ 3\\sqrt{x},\\  a\\ =\\ \\frac{-2}{x^2},\\ n\\ =\\ 10,\\  r\\ =\\ r\\ \\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[T_{r + 1} = 10C_r\\ (3\\sqrt{x})^{10-r}\\ (-\\frac{2}{x^2})^r \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = 10C_r\\ 3^{10\\ -\\ r}\\ x^(\\frac{10-r}{2})\\ \\frac{-2^r}{x^{2r}} \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = 10C_r\\ x^(\\frac{10-r}{2})\\  {-2}^r\\ \\ x^{-2r}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = 10C_r\\ x^(\\frac{10-r}{2}\\ -\\ 2r)\\ {-2}^r\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = 10C_r\\ x^(\\frac{10-r &#8211; 4r}{2})\\ {-2}^r\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[T_{r + 1} = 10C_r\\ x^(\\frac{10-5r}{2})\\ {-2}^r\\ &#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212; (1)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[To\\ find\\ the\\ independent\\ term\\ of\\ x,\\ find\\ the\\ coefficient\\ of\\ x^0.\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\therefore\\  x^{\\frac{10-5r}{2}}\\ =\\ x^0\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{10-5r}{2}\\ =\\ 0\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[10 &#8211; 5r\\ =\\ 0\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[-5r\\ = -\\ 10\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[r\\ = 2\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Put\\ r\\ =\\ 2\\ in\\ (1)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\therefore\\  independent\\ term\\ of\\ x =\\ 10C_2\\ 3^{10\\ -\\ 2}\\ (-2)^2\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 10C_2\\ 3^8\\ (4)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[6.\\ Show\\ that\\ the\\ complex\\ numbers\\ 2\\ -\\ 2i,\\ 8\\ +\\ 4i,\\ 5\\ +\\ 7i\\ and\\ -1\\ +\\ i\\ form\\ a\\ rectangle\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black}{Solution:} \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Let\\ A = 2\\ -\\ 2i\\ = (2, &#8211; 2)\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ B = 8\\ +\\ 4i\\ = (8, 4)\\ \\hspace{13cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ C = 5\\ +\\ 7i\\ = (5, 7)\\ \\hspace{13cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ D = -1\\ +\\ i\\ = (-1, 1)\\ \\hspace{13cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[AB\\ =\\ \\sqrt{ (x_1-  x_2)^2  +  (y_1- y_2)^2 }\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\sqrt{ (2\\ -\\  8)^2  +\\  (-2\\ -\\ 4)^2 }\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ \\sqrt{ ((-6)^2  +  (-6)^2 )}\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ \\sqrt{ (36  +  36 )}\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[AB = \\sqrt {72}\\  \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[BC\\ =\\ \\sqrt{ (x_1-  x_2)^2  +  (y_1- y_2)^2 }\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\sqrt{ (8\\ -\\  5)^2  +\\  (4\\ -\\ 7)^2 }\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ \\sqrt{ ((3)^2  +  (-3)^2 )}\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ \\sqrt{ (9  +  9 )}\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[BC = \\sqrt{18}\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[CD\\ =\\ \\sqrt{ (x_1-  x_2)^2  +  (y_1- y_2)^2 }\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\sqrt{ (5\\ +\\  1)^2  +\\  (7\\ -\\ 1)^2 }\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ \\sqrt{ ((6)^2  +  (6)^2 )}\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ \\sqrt{ (36  +  36 )}\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[CD = \\sqrt{72}\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[DA\\ =\\ \\sqrt{ (x_1-  x_2)^2  +  (y_1- y_2)^2 }\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\sqrt{ (-1\\ -\\  2)^2  +\\  (1\\ +\\ 2)^2 }\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ \\sqrt{ ((-3)^2  +  (3)^2 )}\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ \\sqrt{ (9  +  9 )}\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[DA = \\sqrt{18}\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\u2234\\  AB = CD \\ and\\   BC = DA\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[AC\\ =\\ \\sqrt{ (x_1-  x_2)^2  +  (y_1- y_2)^2 }\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\sqrt{ (2\\ -\\  5)^2  +  (-2\\ -\\ 7)^2 }\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ \\sqrt{ ((-3)^2  +  (-9)^2 )}\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ \\sqrt{ (9  +  81 )}\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[AC = \\sqrt{90}\\  \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[BD\\ =\\ \\sqrt{ (x_1-  x_2)^2  +  (y_1- y_2)^2 }\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\sqrt{ (8\\ +\\  1)^2  +  (4\\ &#8211; 1)^2 }\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ \\sqrt{ ((9)^2  +  (3)^2 )}\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ \\sqrt{ (81 +  9 )}\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[BD = \\sqrt{90}\\  \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\u2234\\ AC = BD\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\u2234\\ The\\ given\\ complex\\ numbers\\ form\\ a\\ rectangle\\]<\/div>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":187055548,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center 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