{"id":51893,"date":"2024-11-03T14:14:59","date_gmt":"2024-11-03T08:44:59","guid":{"rendered":"https:\/\/yanamtakshashila.com\/?p=51893"},"modified":"2024-11-14T20:28:27","modified_gmt":"2024-11-14T14:58:27","slug":"board-exam-november-2024-applied-mathematics-ii-practicum-theory-model-question-paper-for-civil-branches-of-diploma","status":"publish","type":"post","link":"https:\/\/yanamtakshashila.com\/?p=51893","title":{"rendered":"BOARD EXAM &#8211; OCTOBER 2024 &#8211; APPLIED MATHEMATICS &#8211; II PRACTICUM (THEORY) MODEL QUESTION PAPER FOR ECE &amp; CE BRANCHES OF DIPLOMA"},"content":{"rendered":"\n<div data-wp-interactive=\"core\/file\" class=\"wp-block-file\"><object data-wp-bind--hidden=\"!state.hasPdfPreview\" hidden class=\"wp-block-file__embed\" data=\"https:\/\/yanamtakshashila.com\/wp-content\/uploads\/2024\/11\/Board-Practical-AM-2-Theory-2024-1.pdf\" type=\"application\/pdf\" style=\"width:100%;height:600px\" aria-label=\"Embed of Board Practical AM-2 - Theory - 2024.\"><\/object><a id=\"wp-block-file--media-6cd91abc-0dd7-447a-bd3b-76b84fa320b7\" href=\"https:\/\/yanamtakshashila.com\/wp-content\/uploads\/2024\/11\/Board-Practical-AM-2-Theory-2024-1.pdf\">Board Practical AM-2 &#8211; Theory &#8211; 2024<\/a><a href=\"https:\/\/yanamtakshashila.com\/wp-content\/uploads\/2024\/11\/Board-Practical-AM-2-Theory-2024-1.pdf\" class=\"wp-block-file__button wp-element-button\" download aria-describedby=\"wp-block-file--media-6cd91abc-0dd7-447a-bd3b-76b84fa320b7\">Download<\/a><\/div>\n\n\n\n<script async src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js?client=ca-pub-9453835310745500\"\n     crossorigin=\"anonymous\"><\/script>\n<!-- 350 x 250 ad -->\n<ins class=\"adsbygoogle\"\n     style=\"display:block\"\n     data-ad-client=\"ca-pub-9453835310745500\"\n     data-ad-slot=\"3583972194\"\n     data-ad-format=\"auto\"\n     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\">\\[\\color {purple} {1\\ .}\\ \\color {red} {Prove\\ that}\\ the\\ equation\\  x^2\\ +\\ 6\\ x\\ y\\ +\\ 9y^2\\ +\\ 4\\ x\\ +\\ 12\\ y\\ -\\ 5\\ =\\ 0\\ \\hspace{10cm}\\]\\[\\color {red} {is\\ a\\ parobala}\\ \\hspace{5cm}\\]<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\">\\[\\color {blue} {Soln:}\\ x^2\\ +\\ 6\\ x\\ y\\ +\\ 9y^2\\ +\\ 4\\ x\\ +\\ 12\\ y\\ -\\ 5\\ =\\ 0\\ &#8212;&#8212;&#8212;- (1)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ Condition\\ for\\ (1)\\ to\\ represent\\ parabola\\ is\\ h^2\\ =\\ ab\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Comparing\\ with\\ a\\ x^2\\ +\\ 2h\\ x\\ y\\ +\\   by^2\\ +\\  2\\ g\\ x\\ +\\ 2\\ f\\ y\\ +\\ c\\ = 0\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[We\\ get\\  a\\ =\\ 1,\\ b\\ =\\ 9\\  \\hspace{5cm}\\ 2h\\ =\\ 6,\\ \\implies\\ h\\ =\\ 3\\  \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[h^2\\ =\\ ab\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\therefore\\ (1)\\ represents\\ a\\ parabola\\ \\hspace{10cm}\\]<\/div>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/-_VhVPnPerY\" title=\"Conics - Part - 1\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {2\\ .}\\ \\color {red} {Show\\ that}\\ the\\ equation\\  x^2\\ +\\ 4y^2\\ +\\ 4\\ x\\ +\\ 24\\ y\\ +\\ 31\\ =\\ 0\\ \\hspace{10cm}\\]\\[\\color {red} {represents\\ an\\ ellipse}\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ x^2\\ +\\ 4y^2\\ +\\ 4\\ x\\ +\\ 24\\ y\\ +\\ 31\\ =\\ 0\\ &#8212;&#8212;&#8212;- (1)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ Condition\\ for\\ (1)\\ to\\ represent\\ an\\ ellipse\\ is\\ h^2\\ -\\ ab\\ \\lt\\  0\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Comparing\\ with\\ a\\ x^2\\ +\\ 2h\\ x\\ y\\ +\\   by^2\\ +\\  2\\ g\\ x\\ +\\ 2\\ f\\ y\\ +\\ c\\ = 0\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[We\\ get\\  a\\ =\\ 1,\\ b\\ =\\ 4\\  \\hspace{5cm}\\ 2h\\ =\\ 0,\\ \\implies\\ h\\ =\\ 0\\  \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ h^2\\ -\\ ab\\ =\\ (0)^2\\ -\\ 1(4)\\  =\\ -\\ 4\\ \\lt\\  0\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\therefore\\ (1)\\ represents\\ an\\ ellipse\\ \\hspace{10cm}\\]<\/div>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/cym41qpKlGw\" title=\"CONICS - PART - 3\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {3\\ .}\\ \\color {red} {Check}\\ whether\\ the\\ conic\\ 2\\ x^2\\ -\\ 16\\ x\\ y\\ +\\   8\\ y^2\\ -\\ y\\ +\\ 3\\ = 0\\ \\color {red} {represent\\ a\\ hyperbola}\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ 2\\ x^2\\ -\\ 16\\ x\\ y\\ +\\   8\\ y^2\\ -\\ y\\ +\\ 3\\ &#8212;&#8212;&#8212;- (1)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ Condition\\ for\\ (1)\\ to\\ represent\\ hyperbola\\ is\\ h^2\\ -\\ ab\\ \\gt\\  0\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Comparing\\ with\\ a\\ x^2\\ +\\ 2h\\ x\\ y\\ +\\   by^2\\ +\\  2\\ g\\ x\\ +\\ 2\\ f\\ y\\ +\\ c\\ = 0\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[We\\ get\\  a\\ =\\ 2,\\ b\\ =\\ 8\\  \\hspace{5cm}\\ 2h\\ =\\ &#8211; 16,\\ \\implies\\ h\\ =\\ -\\ 8\\  \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ h^2\\ -\\ ab\\ =\\ (-8)^2\\ -\\ 2(4)\\ =\\ 64\\ -\\ 8\\ =\\ 56\\ \\gt\\  0\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\therefore\\ (1)\\ represents\\ a\\ hyperbola\\ \\hspace{10cm}\\]<\/div>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/rTVROmkF_kM\" title=\"CONICS - PART - 4\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {4\\ .}\\ \\color {red} {Find\\ the\\ equation\\ of\\ the\\ parabola}\\ with\\ focus\\ at\\ (1,\\ -1)\\ \\hspace{10cm}\\]\\[and\\ directrix\\ x\\ -\\ y\\ =\\ 0.\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ For\\ parabola\\ e\\ =\\ 1\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ Given\\ Focus\\ is\\ S(1,\\ -\\ 1)\\ and\\ directrix\\ is\\ x\\ -\\ y\\ =\\ 0.\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Always\\ \\frac{SP}{PM}\\ =\\ e\\ =\\ 1\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\\sqrt{(x\\ -\\ x_1)^2\\ +\\ (y\\ -\\ y_1)^2}}{\\pm\\ \\frac{a\\ x\\ +\\ b\\ y\\ +\\ c}{\\sqrt{a^2\\ +\\ b^2}}}\\ =\\ 1\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\\sqrt{(x\\ -\\ 1)^2\\ +\\ (y\\ +\\ 1)^2}}{\\pm\\ \\frac{x\\ -\\ y}{\\sqrt{(1)^2\\ +\\ (-1)^2}}}\\ =\\ 1\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\sqrt{(x\\ -\\ 1)^2\\ +\\ (y\\ +\\ 1)^2}\\ =\\ \\pm\\ \\frac{x\\ -\\ y}{\\sqrt{2}}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(x\\ -\\ 1)^2\\ +\\ (y\\ +\\ 1)^2\\ =\\ \\frac{(x\\ -\\ y)^2}{2}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ 2(x^2\\ -\\ 2\\ x\\ +\\ 1\\ +\\ y^2\\ +\\ 2\\ y\\ +\\ 1)\\ =\\ x^2\\ +\\ y^2\\ -\\ 2\\ x\\ y\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ 2\\ x^2\\ -\\ 4\\ x\\ +\\ 2\\ +\\ 2\\ y^2\\ +\\ 4\\ y\\ +\\ 2\\ -\\ x^2\\ -\\ y^2\\ +\\ 2\\ x\\ y\\ =\\ 0\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ 2\\ x^2\\ -\\ 4\\ x\\ +\\ 2\\ +\\ 2\\ y^2\\ +\\ 4\\ y\\ +\\ 2\\ -\\ x^2\\ -\\ y^2\\ +\\ 2\\ x\\ y\\ =\\ 0\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ \\boxed {x^2\\ +\\ 2\\ x\\ y\\ -\\ 4\\ x\\ +\\ y^2\\ +\\ 4\\ y\\ +\\ 4\\ =\\ 0}\\ \\hspace{8cm}\\]<\/div>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/cIi5PqPbBJc\" title=\"CONICS - PART - 5\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {5\\ .}\\ \\color {red} {Find\\ the\\ equation\\ of\\ the\\ parabola}\\ with\\ focus\\ at\\ (2,\\ 1)\\ \\hspace{10cm}\\]\\[and\\ directrix\\ 2x\\ +\\ y\\ +\\ 1\\ =\\ 0.\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ For\\ parabola\\ e\\ =\\ 1\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ Given\\ Focus\\ is\\ S(2,\\ 1)\\ and\\ directrix\\ is\\ 2\\ x\\ +\\ y\\ +\\ 1\\ =\\ 0.\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Always\\ \\frac{SP}{PM}\\ =\\ e\\ =\\ 1\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\\sqrt{(x\\ -\\ x_1)^2\\ +\\ (y\\ -\\ y_1)^2}}{\\pm\\ \\frac{a\\ x\\ +\\ b\\ y\\ +\\ c}{\\sqrt{a^2\\ +\\ b^2}}}\\ =\\ 1\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\\sqrt{(x\\ -\\ 2)^2\\ +\\ (y\\ -\\ 1)^2}}{\\pm\\ \\frac{2x\\ +\\ y\\ +\\ 1}{\\sqrt{(2)^2\\ +\\ (1)^2}}}\\ =\\ 1\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\sqrt{(x\\ -\\ 2)^2\\ +\\ (y\\ -\\ 1)^2}\\ =\\ \\pm\\ \\frac{2x\\ +\\ y\\ +\\ 1}{\\sqrt{5}}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(x\\ -\\ 2)^2\\ +\\ (y\\ -\\ 1)^2\\ =\\ \\frac{(2x\\ +\\ y\\ +\\ 1)^2}{5}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ 5(x^2\\ -\\ 4\\ x\\ +\\ 4\\ +\\ y^2\\ -\\ 2\\ y\\ +\\ 1)\\ =\\ 4x^2\\ +\\ y^2\\ +\\  1\\ +\\ 4\\ x\\ y\\ + 4\\ x\\ +\\ 2y\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ 5\\ x^2\\ -\\ 20\\ x\\ +\\ 20\\ +\\ 5\\ y^2\\ -\\ 10\\ y\\ +\\ 5\\ -\\ 4\\ x^2\\ -\\ y^2\\ -\\ 1\\ -\\ 4\\ x\\ y\\ -\\ 4x\\ -\\ 2y\\ =\\ 0\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ x^2\\ +\\  4\\ y^2\\ -\\ 4\\ x\\ y\\ -\\ 24\\ x\\   -\\ 12\\ y\\ +\\ 24\\ =\\ 0\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ \\boxed {x^2\\ +\\  4\\ y^2\\ -\\ 4\\ x\\ y\\ -\\ 24\\ x\\   -\\ 12\\ y\\ +\\ 24\\ =\\ 0}\\ \\hspace{8cm}\\]<\/div>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/QFvbqmEJSPg\" title=\"Conics - Part - 6\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {6.}\\ \\color {red} {What\\ is\\ the\\ principal\\ value\\ of}\\ Sin^{-1}\\ (\\frac{1}{2})\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ \\hspace{20cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Sin^{-1}\\ (\\frac{1}{2})\\ =\\ 30^{0}\\  \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {7.}\\ \\color {red} {What\\ is\\ the\\ principal\\ value\\ of}\\ Cos^{-1}\\ (\\frac{1}{2})\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ \\hspace{20cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Cos^{-1}\\ (\\frac{1}{2})\\ =\\ 60^{0}\\  \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple}{8.}\\ \\color {red}{Find\\  the\\ value\\ of}\\ i^2\\ +\\ i^3\\ +\\ i^4\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ i^2\\ +\\ i^3\\ +\\ i^4\\ \\hspace{19cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ -1\\ +\\ i^2\\ i\\ +\\ (i^2)^2\\  \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ -1\\ +\\ (-1)\\ i\\ +\\ (-1)^2\\  \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ -1\\  -\\ i\\ +\\ 1\\  \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ -i\\  \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple}{9.}\\ If\\  Z_1 =  1 + i,\\  Z_2 =  3 + 2i,\\ \\color {red} {find\\  Z_1  +  Z_2}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ Z_1 =  1 + i,\\  Z_2 =  3 + 2i\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Z_1  +  Z_2 =    1 + i  + 3 + 2i\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=   1 + 3 + i (1 +2)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=   4 + 3i\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {10.}\\ If\\  Z_1 =  1\\ +\\ 4i,\\  Z_2\\ =\\  -\\ 3\\ +\\  6i,\\ \\color {red} {find\\  3Z_1\\  +\\ 2\\ Z_2}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ Z_1\\ =\\  1\\ +\\ 4i,\\   Z_2\\ =\\ -\\ 3\\ +\\  6i,\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[3Z_1\\  +\\  2\\ Z_2\\ =\\    3(1\\ +\\ 4i)\\  +\\ 2(-3\\ +\\ 6i)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =    3\\ +\\ 12i\\  -\\ 6\\ +\\ 12i\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=   3\\ -\\ 6\\ + i (12\\ +\\ 12)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\   -3\\ +\\ 24i\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {11.}\\ If\\  Z_1 =  1 + i,\\  Z_2 =  3 + 2i,\\ \\color {red} {find\\  Z_1  &#8211;  Z_2}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ Z_1 =  1 + i,\\  Z_2 =  3 + 2i\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Z_1  &#8211;  Z_2 =    1 + i  &#8211; (3 + 2i)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Z_1  &#8211;  Z_2 =    1 + i  &#8211; 3 &#8211; 2i\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Z_1  &#8211;  Z_2 =    1 &#8211; 3  + i &#8211; 2i\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Z_1  &#8211;  Z_2 =    &#8211; 2  &#8211; i\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {12.}\\ If\\  Z_1 =  2\\ +\\ 2i,\\  Z_2 =\\  3\\ +\\ 2i,\\ \\color {red} {find\\ the\\ value\\ of\\ z_1z_2}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ Z_1 =  2 + 2i,\\  Z_2\\ =\\  3\\ +\\ 2i\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Z_1  Z_2 =  (2 + 2i) (3\\ +\\ 2i)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Z_1  Z_2 =    6\\ +\\ 4i\\  +\\ 6i\\ +\\ 4 (-1)\\ \\hspace{5cm}\\ \\because i^2\\ =\\ -1\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =    6\\ +\\ 10i\\ &#8211; 4\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\    2\\ +\\ 10i\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {13.}\\ \\color {red} {Find\\ the\\ Real\\ and\\ Imaginary\\ parts\\ of}\\ \\frac{2+ 3i}{4+ 5i}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ z = \\frac{2+ 3i}{4+ 5i}\\ \u00d7\\ \\frac{4- 5i}{4- 5i}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\frac{8 &#8211; 10i + 12i + 15}{(4)^2 + (5)^2}\\ \\hspace{3cm}\\ \\because [(a +ib)(a &#8211; ib)\\ =\\ (a)^2 + (b)^2]\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\frac{8 &#8211; 10i + 12i + 15}{16+ 25}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\frac{23 + 2i}{41}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\frac{23}{41}\\ +\\ i\\  \\frac{2}{41}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Re(Z)\\ =\\ \\frac{23}{41};\\  Im(Z)\\ =\\ \\frac{23}{41}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {14.}\\ \\color {red} {Find\\ the\\ modulus\\ and\\ amplitude\\ of}\\ 1 + \\sqrt{3}i\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:} \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Let\\ z = 1 + \\sqrt{3}i\\ = a\\ + ib\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[a = 1,\\ b\\ = \\sqrt{3}\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {brown} {T0\\ find\\ modulus}:\\ \\hspace{18cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[|z| = \\sqrt{a^2 + b^2}\\ \\hspace{12cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\sqrt{(1)^2 + (\\sqrt{3})^2}\\ \\hspace{12cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\sqrt{1 + 3}\\ =\\  \\sqrt{4}\\ \\hspace{12cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[|z| = 2\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {brown} {To\\ find\\ amplitude}:\\ \\hspace{18cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\u03b8  = tan^{-1} (\\frac{b}{a})\\ =\\ tan^{-1} \\frac{\\sqrt{3}}{1}\\ =\\  tan^{-1} \\sqrt{3}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\u03b8  = 60^0\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {15.}\\ \\color {red} {Find\\ the\\ modulus\\ and\\ amplitude\\ of}\\ \\frac{5 &#8211; i}{2- 3i}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:} \\hspace{22cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Let\\  z = \\frac{5 &#8211; i}{2- 3i}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= \\frac{5 &#8211; i}{2- 3i}\\ \u00d7\\ \\frac{2 + 3i}{2 + 3i}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\frac{10 + 15i -2i &#8211; 3i^2}{(3)^2 + (2)^2}\\ \\hspace{3cm}\\ \\because [(a +ib)(a &#8211; ib)\\ =\\ (a)^2 + (b)^2]\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\frac{10 + 13i + 3 }{9+ 4}\\ \\hspace{4cm}\\ \\because [i^2\\ =\\ -1]\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\frac{13 + 13i}{13}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ Z = \\frac{13}{13}\\ +\\ \\frac{13i}{13}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ z = 1 + \\ i\\ = a\\ + ib\\ \\hspace{14cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[a = 1,\\ b\\ = \\ 1\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {brown} {T0\\ find\\ modulus}:\\ \\hspace{18cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[|z| = \\sqrt{a^2 + b^2}\\ \\hspace{12cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\sqrt{(1)^2 + {1}^2}\\ \\hspace{12cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\sqrt{1 + 1}\\ =\\ \\sqrt{2}\\ \\hspace{12cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[|z| = \\sqrt{2}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {brown} {To\\ find\\ amplitude}:\\ \\hspace{18cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\u03b8  = tan^{-1} (\\frac{b}{a})\\ =\\ tan^{1} \\frac{1}{1}\\ =\\  tan^{-1} {1}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\u03b8  = 45^0\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {16.}\\ \\color {red} {Simplify}\\ (cos\\ 15^0  +  i sin\u2061\\ 15^0 )^6\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ (cos\\ 15^0  +  i sin\u2061\\ 15^0 )^6\\ =\\ cos\\ 90^0  +  i sin\u2061\\ 90^0 \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=   0 + i (1)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=   i\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{(cos\\ 15^0  +  i sin\u2061\\ 15^0 )^6\\ =\\ i}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {17.}\\ \\color {red} {Simplify:}\\  (cos\\ \u03b8 +  i sin\u2061\\ \u03b8 )^2\\ (cos\\ 3\u03b8 +  i sin\u2061\\ 3\u03b8 )\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ (cos\\ \u03b8 +  i sin\u2061\\ \u03b8 )^2\\ (cos\\ 3\u03b8 +  i sin\u2061\\ 3\u03b8 )\\ =\\ (cos\\ \u03b8 +  i sin\u2061\\ \u03b8 )^2\\ (cos\\ \u03b8 +  i sin\u2061\\ \u03b8 )^3\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=  (cos\\ \u03b8 +  i sin\u2061\\ \u03b8 )^{2 +3 }\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=  (cos\\ \u03b8 +  i sin\u2061\\ \u03b8 )^5\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=  cos\\ 5\u03b8 +  i sin\u2061\\ 5\u03b8\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{(cos\\ \u03b8 +  i sin\u2061\\ \u03b8 )^2\\ (cos\\ 3\u03b8 +  i sin\u2061\\ 3\u03b8 )\\ =\\ cos\\ 5\u03b8 +  i sin\u2061\\ 5\u03b8}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {18.}\\ If\\  a =  cos\u2061\\ x +  i sin\u2061\\ x,\\ b =  cos\u2061\\ y +  i sin\u2061\\ y,\\ \\color {red} {find\\ ab\\ and\\ \\frac{1}{ab}}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ a =  cos\u2061\\ x +  i sin\u2061\\ x,\\ b =  cos\u2061\\ y +  i sin\u2061\\ y\\ \\hspace{16cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ab =  (cos\u2061\\ x +  i sin\u2061\\ x)\\ (cos\u2061\\ y +  i sin\u2061\\ y)\\ \\hspace{12cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{ab\\ =  cos\u2061\\ (x + y) +  i sin\u2061\\ (x + y)}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{1}{ab}\\ =  \\frac{1}{cos\u2061\\ (x + y) +  i sin\u2061\\ (x + y)}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\frac{1}{ab}\\ =  cos\u2061\\ (x + y) &#8211;  i sin\u2061\\ (x + y)}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple}{19.}\\ \\color {red} {Simplify\\ :}\\ \\frac{cos\u2061\\ 5 \u03b8 +  i sin\u2061\\ 5 \u03b8} {cos\u2061\\ 3 \u03b8 +  i sin\u2061\\ 3 \u03b8}\\  \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ \\frac{cos\u2061\\ 5 \u03b8 +  i sin\u2061\\ 5 \u03b8} {cos\u2061\\ 3 \u03b8 +  i sin\u2061\\ 3 \u03b8}=  \\frac{(cos\u2061\\  \u03b8 +  i sin\u2061\\  \u03b8)^5} {(cos\u2061\\  \u03b8 +  i sin\u2061\\  \u03b8)^3} \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=  (cos\\ \u03b8 +  i sin\u2061\\ \u03b8 )^{5 -3 }\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=  (cos\\ \u03b8 +  i sin\u2061\\ \u03b8 )^2\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=  cos\\ 2\u03b8 +  i sin\u2061\\ 2\u03b8\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\frac{cos\u2061\\ 5 \u03b8 +  i sin\u2061\\ 5 \u03b8} {cos\u2061\\ 3 \u03b8 +  i sin\u2061\\ 3 \u03b8}\\ =\\ cos\\ 2\u03b8 +  i sin\u2061\\ 2\u03b8}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {20.}\\ \\color {red} {Simplify\\ using\\ DeMoivre\u2019s\\ theorem:}\\ \\frac{(cos\u2061\\  3\u03b8 &#8211;  i sin\u2061\\  3\u03b8)^5\\ (cos\u2061\\  4\u03b8 +  i sin\u2061\\  4\u03b8)^4} {(cos\u2061\\  2\u03b8 + i sin\u2061\\  2\u03b8)^7\\ (cos\u2061\\  3\u03b8 &#8211;  i sin\u2061\\  3\u03b8)^6}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ \\frac{(cos\u2061\\  3\u03b8 &#8211;  i sin\u2061\\  3\u03b8)^5\\ (cos\u2061\\  4\u03b8 +  i sin\u2061\\  4\u03b8)^4} {(cos\u2061\\  2\u03b8 + i sin\u2061\\  2\u03b8)^7\\ (cos\u2061\\  3\u03b8 &#8211;  i sin\u2061\\  3\u03b8)^6}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= \\frac{(cos\u2061\\  \u03b8 +  i sin\u2061\\  \u03b8)^{5 \\times -3}\\ (cos\u2061\\  \u03b8 +  i sin\u2061\\  \u03b8)^{4 \\times 4}} {(cos\u2061\\  \u03b8 + i sin\u2061\\  \u03b8)^{2 \\times 7}\\ (cos\u2061\\  \u03b8 +  i sin\u2061\\  \u03b8)^{6 \\times -3 }}\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= \\frac{(cos\u2061\\  \u03b8 +  i sin\u2061\\  \u03b8)^{-15}\\ (cos\u2061\\  \u03b8 +  i sin\u2061\\  \u03b8)^{16}} {(cos\u2061\\  \u03b8 + i sin\u2061\\  \u03b8)^{14}\\ (cos\u2061\\  \u03b8 +  i sin\u2061\\  \u03b8)^{-18}}\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=  (cos\\ \u03b8 +  i sin\u2061\\ \u03b8 )^{-15 + 16 + 14 &#8211; 18 }\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=  (cos\\ \u03b8 +  i sin\u2061\\ \u03b8 )^5\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=  cos\\ 5\u03b8 +  i sin\u2061\\ 5\u03b8\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\frac{(cos\u2061\\  3\u03b8 &#8211;  i sin\u2061\\  3\u03b8)^5\\ (cos\u2061\\  4\u03b8 +  i sin\u2061\\  4\u03b8)^4} {(cos\u2061\\  2\u03b8 + i sin\u2061\\  2\u03b8)^7\\ (cos\u2061\\  3\u03b8 &#8211;  i sin\u2061\\  3\u03b8)^6}\\ =\\ cos\\ 5\u03b8 +  i sin\u2061\\ 5\u03b8}\\]<\/div>\n\n\n\n<p><\/p>\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":"set","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":false,"jetpack_social_options":{"image_generator_settings":{"template":"highway","enabled":false},"version":2},"_wpas_customize_per_network":false,"jetpack_post_was_ever_published":false},"categories":[711788742],"tags":[],"class_list":["post-51893","post","type-post","status-publish","format-standard","hentry","category-applied-mathematics-ii"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>BOARD EXAM - OCTOBER 2024 - APPLIED MATHEMATICS - II PRACTICUM (THEORY) MODEL QUESTION PAPER FOR ECE &amp; 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