{"id":48272,"date":"2024-07-13T12:43:13","date_gmt":"2024-07-13T07:13:13","guid":{"rendered":"https:\/\/yanamtakshashila.com\/?p=48272"},"modified":"2024-07-14T19:26:46","modified_gmt":"2024-07-14T13:56:46","slug":"october-2023-tamil-nadu-polytechnic-board-exam-engineering-mathematics-ii40022question-paper-with-solutions","status":"publish","type":"post","link":"https:\/\/yanamtakshashila.com\/?p=48272","title":{"rendered":"October-2023 TAMIL NADU POLYTECHNIC BOARD EXAM ENGINEERING MATHEMATICS \u2013 II(40022)QUESTION PAPER WITH SOLUTIONS"},"content":{"rendered":"\n<ol class=\"wp-block-list\">\n<li>Answer all questions in <strong>PART- A. <\/strong>Each question carries one mark.&nbsp;<\/li>\n\n\n\n<li>Answer any ten questions in <strong>PART- B. <\/strong>Each question carries two marks.<\/li>\n\n\n\n<li>Answer all questions by selecting either A or B. Each question carries fifteen marks. (7 + 8)<br>Clarks Table and programmable calculators are not permitted.<\/li>\n<\/ol>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\underline{PART\\ -\\ A}\\]<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\">\\[1.\\ \\color{green}{Find\\ the\\ combined\\ equation\\ of\\ the\\ two\\ straight\\ lines\\ represented\\ by}\\ \\hspace{7cm}\\]\\[\\color{green}{2x\\ +\\ 3y\\ =\\ 0\\ and\\ 4x\\ -\\ 5y\\ =\\ 0}\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\  \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\ The\\ two\\ separate\\ lines\\ are\\ 2x\\ +\\ 3y\\ =0\\ and\\ 4x\\ -\\ 5y\\ =\\ 0\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[The\\ combined\\ equation\\ is\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ (2x\\ +\\ y) (4x\\ -\\ 5y)\\&nbsp; =\\ 0\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ 6x^2\\ -\\ 10xy\\ +\\ 4xy\\ -\\ 5y^2\\ =\\ 0\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ 6x^2\\ -\\ 6xy\\ -\\ 5y^2\\ =\\ 0\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2.\\ \\color{green}{Find\\ the\\  Unit\\ vector\\ parallel\\ to}\\ \\hspace{10cm}\\]\\[\\color{green}{2\\overrightarrow{i}\\ &#8211; \\overrightarrow{j}\\ +\\ 4\\overrightarrow{k}}\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\  \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{a}\\ =\\ 2\\overrightarrow{i}\\ -\\ \\overrightarrow{j}\\ +\\ 4\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{|a|} = \\sqrt{(2)^2 + (-1)^2+(4)^2 }\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= \\sqrt{(4\\ +\\ 1\\ +\\ 16)}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\sqrt{21}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{|a|}=\\sqrt{21}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Unit\\  vector\\  along\\ \\overrightarrow{a}=\\frac{\\overrightarrow{a}}{\\overrightarrow{|a|}}= \\frac{3\\overrightarrow{i}\\ + 4\\overrightarrow{j}- 5\\overrightarrow{k}}{\\sqrt{50}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[3.\\ \\color{green}{Find\\ the\\ value\\ of\\ [\\overrightarrow{i} + \\overrightarrow{j}\\ \\overrightarrow{j} + \\overrightarrow{k}\\ \\overrightarrow{k} + \\overrightarrow{i}]}\\ \\hspace{7cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\  \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{a}= \\overrightarrow{i}\\ + \\overrightarrow{j}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{b}= \\overrightarrow{j}\\ + \\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{c}= \\overrightarrow{k} + \\overrightarrow{i}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ [\\overrightarrow{a}\\   \\overrightarrow{b}    \\overrightarrow{c}] =\\begin{vmatrix}\n1 &amp; 1 &amp; 0\\\\\n0 &amp; 1 &amp; 1\\\\\n1 &amp; 0 &amp; 1\\\\\n\\end{vmatrix}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 1(1\\ -\\ 0)\\ -\\ 1(0\\ -\\ 1)\\ +\\ 0(0\\ -\\ 1)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 1(1)\\ -\\ 1(-1)\\ +\\ 0\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 1\\ +\\ 1\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 2\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[4.\\ \\color {green}{Evaluate:  \\int\\ (x^3\\ -\\ x\\ -\\  2)\\ dx}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln\\ :}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int(x^3\\ -\\ x\\ -\\ 2)\\ dx = \\frac{x^4}{4}\\ -\\  \\frac{x^2}{2}\\ -\\ 2x\\ + c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[5.\\ \\color {green}{Evaluate:  \\int_1^3 \\frac{dx}{x}}\\ \\hspace{17cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln\\ :}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int_1^3 \\frac{1}{x}\\ dx = log\\ x \\Biggr]_{1}^{3}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=[log\\ 3 &#8211; log\\ 1]\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= log\\ 3 &#8211; 0\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= log\\ 3\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\int_1^2  \\frac{1}{x}\\ dx = log\\ 3}\\]<\/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:inline-block;width:350px;height:250px\"\n     data-ad-client=\"ca-pub-9453835310745500\"\n     data-ad-slot=\"3583972194\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\underline{PART\\ -\\ B}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[6.\\ \\color{green}{Find\\ the\\ equation\\ of\\ the\\ straight\\ line\\ parallel\\ to\\ the\\ line\\ 3x\\ +\\ 2y\\ -\\ 7\\ =\\ 0}\\ \\hspace{6cm}\\]\\[\\color{green}{and\\ passing\\ through\\ the\\ point\\ (1, -2)}\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\  \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Let\\ the\\ equation\\ of\\ line\\ parallel\\ to\\ 3x\\ +\\ 2y\\ -\\ 7\\ =0\\ &#8211; &#8211; -\\ (1)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[is\\ 3x\\ +\\ 2y\\ +\\ k\\ =0\\ &#8212;-\\ (2)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{Equation (2) passes through ( 1, -2 )}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{&nbsp;put x= 1, y =-2 in equation (2)}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{3(1) + 2(-2) +k = 0}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{3 &#8211; 4 + k = 0}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[k = 1\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{\u2234 Required line is 3x + 2y +1 = 0}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[7.\\ \\color{green}{Find\\ the\\ centre\\ and\\ radius\\ of\\ the\\ circle\\ x^2\\ +\\ y^2\\  -\\ 8\\ y\\ +\\ 3\\ =\\ 0}\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\  \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ Given\\ x^2\\ +\\ y^2\\ -\\ 8\\ y\\ +\\ 3\\ =\\ 0\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[We\\ know\\ that\\ the\\ equation\\ of\\ circle\\ is\\ x^2\\ +\\ y^2\\ +\\ 2\\ g\\ x\\ +\\ 2\\ f\\ y\\ +\\ c\\ =\\ 0\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2\\ g\\ =\\ 0\\ \\hspace{3cm}\\ 2\\ f\\ =\\ -\\ 8\\ \\hspace{3cm}\\ c\\ =\\ 3\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[g\\ =\\ 0\\ \\hspace{3cm}\\ f\\ =\\ -\\ 4\\ \\hspace{3cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[centre\\ =\\ (-\\ g,\\  -\\ f)\\ \\hspace{4cm}\\ r\\ =\\ \\sqrt{(g^2\\ +\\ f^2\\ -\\ c)}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[centre\\ =\\ (0,\\  4)\\ \\hspace{4cm}\\ r\\ =\\ \\sqrt{(0^2\\ +\\ (-4)^2\\ -\\ 3)}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{6cm}\\ r\\ =\\ \\sqrt{(0\\ +\\ 16\\ -\\ 3)}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{6cm}\\ r\\ =\\ \\sqrt{13}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[centre  =    (0,  4)\\ \\hspace{5cm}\\                           r\\ =\\ \\sqrt{13}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/nBvqNJIAneQ\" title=\"Analytical Geometry - 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\">\\[8.\\ \\color{green}{Show\\ that\\ the\\ equation\\  x^2\\ +\\ 4y^2\\ -\\ 4\\ x\\ +\\ 24\\ y\\ +\\ 31\\ =\\ 0}\\ \\hspace{10cm}\\]\\[\\color {green} {represents\\ an\\ ellipse}\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ \\hspace{18cm}\\]<\/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\">\\[9.\\ \\color{green}{Find\\ the\\ projection\\ of\\ the\\ vector}\\ \\hspace{12cm}\\]\\[\\color{green}{3\\overrightarrow{i}+ 4\\overrightarrow{j}\\ +\\ 5\\overrightarrow{k} on\\  the\\ vector\\  \\overrightarrow{i}+ 2\\overrightarrow{j}\\ +\\ 6\\overrightarrow{k}}\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{a}= 3\\overrightarrow{i}+ 4\\overrightarrow{j}\\ +\\ 5\\overrightarrow{k} \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{b}= \\overrightarrow{i}\\ +\\ 2 \\overrightarrow{j}\\ + 6\\overrightarrow{k} \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Projection\\  of\\ \\overrightarrow{a} on \\overrightarrow{b} =\\frac{\\overrightarrow{a}.\\overrightarrow{b}}{\\overrightarrow{|b|}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\frac{(3\\overrightarrow{i}\\ +\\ 4\\overrightarrow{j}\\ +\\ 5\\overrightarrow{k}).(\\overrightarrow{i}+2\\overrightarrow{j}\\ +\\ 6\\overrightarrow{k})}{\\sqrt{(1)^2\\ +\\ (2)^2\\ +\\ (6)^2 }}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=  \\frac{3(1)\\ +\\ 4(2)\\ +\\ 5(6)}{\\sqrt{(1 + 4 + 36 }}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=  \\frac{3\\ +\\ 8\\ +\\ 30}{\\sqrt{41}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{Projection\\  of\\ \\overrightarrow{a} on \\overrightarrow{b} =\\frac{41}{\\sqrt{41}}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[10.\\ \\color{green}{Show\\ that\\ the\\ vectors}\\ \\hspace{15cm}\\]\\[\\color{green}{4\\overrightarrow{i}\\ +\\ \\overrightarrow{j}\\ -\\  3\\overrightarrow{k}\\ and\\  3\\overrightarrow{i}\\ +\\ 3\\overrightarrow{j}\\ +\\ 5\\overrightarrow{k} are\\ perpendicular}\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{a}\\ =\\ 4\\overrightarrow{i}\\ +\\ \\overrightarrow{j}\\ &#8211; 3\\ \\overrightarrow{k} \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{b}= 3\\overrightarrow{i}\\ +\\ 3\\overrightarrow{j}\\ +\\ 5\\overrightarrow{k} \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\overrightarrow{a}.\\overrightarrow{b}\\ =\\ (4\\overrightarrow{i}\\ +\\ \\overrightarrow{j}\\ -\\ 3\\overrightarrow{k}) .(3\\overrightarrow{i}\\ +\\ 3\\overrightarrow{j}\\ +\\ 5\\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 4(3)\\ +\\ 1(3)\\ -\\ 3(5)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 12\\ +\\ 3\\ -\\ 15\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 0\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\therefore\\ \\overrightarrow{a}\\ and\\ \\overrightarrow{b}\\  are\\  perpendicular\\  vectors.\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[11.\\ \\color{green}{Find\\  \\overrightarrow{a}\u00d7 \\overrightarrow{b}}\\ \\hspace{15cm}\\]\\[\\color{green}{if\\  \\overrightarrow{a}= \\overrightarrow{i}+ \\overrightarrow{j}+ \\overrightarrow{k} and \\overrightarrow{b}= 2\\overrightarrow{i}- \\overrightarrow{j}\\ +\\ \\overrightarrow{k}}\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{a}= \\overrightarrow{i}+ \\overrightarrow{j}+\\overrightarrow{k} \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{b}= 2\\overrightarrow{i}\\ -\\ \\overrightarrow{j}\\ +\\ \\overrightarrow{k} \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{a}\u00d7\\overrightarrow{b} =\\begin{vmatrix}\n\\overrightarrow{i} &amp; \\overrightarrow{j} &amp; \\overrightarrow{k}\\\\\n1 &amp; 1 &amp; 1\\\\\n2 &amp; -1 &amp; 1\\\\\n\\end{vmatrix}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\overrightarrow{i}( 1 + 1)\\ -\\ \\overrightarrow{j}(1\\ -\\ 2)\\ +\\ \\overrightarrow{k}(-1\\ -\\ 2)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\overrightarrow{i}(2)\\ -\\ \\overrightarrow{j}(-\\ 1)\\ +\\ \\overrightarrow{k}(-3)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{ \\overrightarrow{a}\u00d7 \\overrightarrow{b}\\ =\\ 2\\overrightarrow{i}\\ +\\ \\overrightarrow{j}\\ -3\\overrightarrow{k}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[12.\\ \\color{green}{Find\\ the\\ value\\ of\\ &#8216;m&#8217;\\ so\\ that\\ the\\ vectors}\\ \\hspace{10cm}\\]\\[\\color{green}{2\\ \\overrightarrow{i}\\ &#8211; \\overrightarrow{j}\\ +\\  \\overrightarrow{k},\\ \\overrightarrow{i}\\ +\\ 2\\ \\overrightarrow{j}\\ -\\ 3\\ \\overrightarrow{k}\\ ,\\ 3\\overrightarrow{i}\\ +\\ m\\ \\overrightarrow{j}\\ +\\ 5\\ \\overrightarrow{k}\\ coplanar}\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{a}= 2\\overrightarrow{i}- \\overrightarrow{j}+ \\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{b}= \\overrightarrow{i}+ 2\\overrightarrow{j}- 3\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{c}= 3\\overrightarrow{i}+ m\\overrightarrow{j}+ 5\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Given\\  \\overrightarrow{a},\\overrightarrow{b} and\\  \\overrightarrow{c} are\\ coplanar\\ \\implies  [\\overrightarrow{a}\\   \\overrightarrow{b}\\    \\overrightarrow{c}] = 0\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\begin{vmatrix}\n2 &amp;- 1 &amp; 1\\\\\n1 &amp; 2 &amp; -3\\\\\n3 &amp; m &amp; 5\\\\\n\\end{vmatrix}=0\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2(10\\ +\\ 3\\ m)\\ +\\ 1(5\\ +\\ 9)\\ +\\ 1(m\\ -\\ 6)\\ =\\ 0\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[20\\ +\\ 6m\\ +\\ 14\\ +\\ m\\ -\\ 6\\ =\\ 0\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[28\\ +\\ 7m\\ =\\ 0\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[7m\\ =\\ -\\ 28\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[m\\ =\\ -\\ 4\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[13.\\ \\color{green}{Find\\ the\\ gradient\\ of\\ \\phi = xyz}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\phi = xyz\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\nabla\\ \\phi=(\\frac{\\partial\\ \\phi}{\\partial\\ x})\\overrightarrow{i} + (\\frac{\\partial\\ \\phi}{\\partial\\ y})\\overrightarrow{j} + (\\frac{\\partial\\ \\phi}{\\partial\\ z})\\overrightarrow{k}&#8212;&#8212;&#8212;&#8212;(1)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\\partial\\ \\phi}{\\partial\\ x} = yz \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\\partial\\ \\phi}{\\partial\\ y} = xz\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\\partial\\ \\phi}{\\partial\\ z} = xy\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Equation\\ (1)\\ becomes\\ \\hspace{10cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\nabla\\ \\phi= yz\\overrightarrow{i} + xz\\overrightarrow{j} + xy\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[14.\\ \\color{green}{Find\\ div\\ \\overrightarrow{F}}\\ \\hspace{18cm}\\]\\[\\color{green}{if\\ \\overrightarrow{F}\\ =\\ 3\\ x^2\\ \\overrightarrow{i}\\  +\\ 5\\ x\\ y^2\\ \\overrightarrow{j}\\ +\\ x\\ y\\ z^3\\  \\overrightarrow{k}\\ at\\ the\\ point\\ (1,\\ 2,\\ 3)}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\overrightarrow{F}\\ =\\ 3\\ x^2\\ \\overrightarrow{i}\\  +\\ 5\\ x\\ y^2\\ \\overrightarrow{j}\\ +\\ x\\ y\\ z^3\\ \\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\nabla\\ .\\ F\\ =\\ \\frac{\\partial\\ F}{\\partial\\ x}\\ +\\ \\frac{\\partial\\ F}{\\partial\\ y}\\ +\\ \\frac{\\partial\\ F}{\\partial\\ z}\\ &#8212;&#8212;&#8212;&#8212;(1)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\\partial\\ F}{\\partial\\ x}\\ =\\ 6\\ x\\ +\\ 5\\ y^2\\ +\\ y\\ z^3\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\\partial\\ F}{\\partial\\ y}\\ =\\ 10\\ x\\ y\\ +\\ 5\\ y^2\\ +\\ x\\ z^3\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\\partial\\ F}{\\partial\\ z}\\ =\\ 3\\ x\\ y\\ z^2\\]<\/div>\n\n\n\n<p>Equation (1) becomes<\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\nabla\\ .\\ F\\ =\\ 6\\ x\\ +\\ 5\\ y^2\\ +\\ y\\ z^3 \\ +\\ 10\\ x\\ y\\ +\\ 5\\ y^2\\ +\\ x\\ z^3\\ +\\ 3\\ x\\ y\\ z^2\\]<\/div>\n\n\n\n<p>At (1, 2, 3)<\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\nabla\\ .\\ F\\ =\\ 6\\ (1)\\ +\\ 5\\ (2)^2\\ +\\ 2\\ (3)^3 \\ +\\ 10\\ (1)\\ (2)\\ +\\ 5\\ (2)^2\\ +\\ 1\\ (3)^3\\ +\\ 3\\ (1)\\ (2)\\ (3)^2\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 6\\ +\\ 5\\ (4)\\ +\\ 2\\ (9) \\ +\\ 10\\ (2)\\ +\\ 5\\ (4)\\ +\\ 1\\ (3)\\ +\\ 3\\ (18)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 6\\ +\\ 20\\ +\\ 18 \\ +\\ 20\\ +\\ 20\\ +\\ 3\\ +\\ 54\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\nabla\\ .\\ F\\ =\\ 141\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[15.\\ \\color {green}{Evaluate:  \\int(x + 3 ) ( x + 2) \\ dx}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln\\ :}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int(x + 3 ) ( x + 2)\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= \\int( x^2  +\\ 3x\\  +\\  2x\\ +\\ 6)\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= \\int(x^2\\ +\\  5x\\ +\\ 6)\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ \\frac{x^3}{3}\\ +\\  5\\frac{x^2}{2}\\ +\\ 6x + c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ \\frac{x^3}{3}\\ +\\ \\frac{5}{2}\\ x^2\\ +\\ 6x + c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[16.\\ \\color {green}{Evaluate:  \\int\\ (3 sin x + 9)\\ dx}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln\\ :}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int(3 sin x\\ +\\ 9)\\ dx\\ =\\ 3\\ \\int sin x\\ dx\\ +\\ 9\\ \\int 1\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=- 2 cos x\\ +\\ 9x + c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\therefore\\ \\boxed{\\int(3 sin x\\ +\\ 9)\\ dx\\ =\\ &#8211; 2\\ cos x\\ + 9\\ x + c}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[17.\\ \\color {green}{Evaluate:  \\int\\ \\frac{2x}{x^2 +\\ 4}\\ dx}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Soln:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[put\\   u\\ =\\ x^2\\ +\\ 4\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{du}{dx}= \\ 2x\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[du\\ =\\ 2\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int\\ \\frac{2x}{x^2\\ +\\ 4}\\ dx= \\int\\ \\frac{du}{u}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ log\\ u\\ +\\ c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= log(x^2\\ +\\ 4) + c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\int\\ \\frac{2x}{x^2\\ +\\ 4}\\ dx= log(x^2\\ +\\ 4) + c}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[18.\\ \\color {green}{Evaluate:  \\int\\ x^2\\ sin\\ 2x\\ dx}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Soln:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[  ILATE  \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[u= x^2\\ \\hspace{2cm}\\  dv = sin\\ 2x\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[u^! = 2x\\ \\hspace{2cm}\\  v = &#8211; \\frac{cos\\ 2x}{2}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[u^{!!} = 2\\ \\hspace{2cm}\\  v_1 = &#8211; \\frac{sin\\ 2x}{4}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{3cm}\\  v_2 =  \\frac{cos\\ 2x}{8}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int u\\ dv = uv &#8211; u^!v_1 + u^{!!}v_2 \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int x^2\\ sin\\ 2x\\ dx = x^2(-\\frac{cos\\ 2x}{2}) &#8211; 2x (-\\frac{sin\\ 2x}{4}) + 2(\\frac{cos\\ 2x}{8}) + c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = &#8211; x^2\\frac{cos\\ 2x}{2} + x \\frac{sin\\ 2x}{2} + \\frac{cos\\ 2x}{4} + c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[19.\\ \\color {green}{Evaluate:  \\int\\ log\\ x\\ dx}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Soln:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int log\\ x\\ dx = \\int 1.\\ log\\ x\\ dx\\]<\/div>\n\n\n\n<p class=\"has-text-align-center\"><strong>ILATE<\/strong><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[u= log\\ x\\ \\hspace{2cm}\\  dv = 1\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{du}{dx} = \\frac{d}{dx} ( log\\ x)\\ \\hspace{2cm}\\ \\int dv = \\int 1\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{du}{dx} = \\frac{1}{x}\\ \\hspace{2cm}\\  v = x\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\ du = \\frac{1}{x}\\ dx\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int u\\ dv = uv &#8211; \\int v\\ du\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int log\\ x\\ dx = x\\ log\\ x- \\int  x\\ (\\frac{1}{x})\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\  x\\ log\\ x- \\int  1\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\  x\\ log\\ x- x + c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\int log\\ x\\ dx = x\\ log\\ x- x +c}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/sLgzncTxqx0\" title=\"Integration By Parts - Part - 7\" 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\">\\[20.\\ \\color {green}{Evaluate:  \\int\\ x\\ e^{3x}\\ dx}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Soln:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ ILATE \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[u= x\\ \\hspace{2cm}\\  dv = e^{3x}\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{du}{dx} = \\frac{d}{dx} (x)\\ \\hspace{2cm}\\ \\int dv = \\int e^{3x}\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{du}{dx} = 1\\ \\hspace{2cm}\\  v = \\frac{e^{3x}}{3}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\ du = dx\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int u\\ dv = uv &#8211; \\int v\\ du\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\int x\\ e^{3x} dx = x\\ \\frac{e^{3x}}{3} &#8211; \\int \\frac{e^{3x}}{3}\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= x\\ \\frac{e^{3x}}{3} &#8211;  \\frac{e^{3x}}{9}\\ + c\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\int x\\ e^{3x}\\ dx = x\\ \\frac{e^{3x}}{3} &#8211;  \\frac{e^{3x}}{9} +c}\\]<\/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<!-- Ad1 -->\n<ins class=\"adsbygoogle\"\n     style=\"display:block\"\n     data-ad-client=\"ca-pub-9453835310745500\"\n     data-ad-slot=\"8240817448\"\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\">\\[\\underline{PART\\ -\\ C}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[21.\\ A)\\ i.\\ \\color{green}{Find\\ the\\ equation\\ of\\ the\\ circle\\ on\\ the\\ line\\ joining\\ the\\ points\\ (2,3),\\ (-\\ 4,\\ 5)\\ as\\ diameter.}\\ \\hspace{5cm}\\]\\[\\color{green}{Aslo\\ find\\ the\\ centre\\ and\\ radius\\ of\\ the\\ circle}\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{1cm}\\ ii.\\ \\color{green}{Prove\\ that\\ the\\ circles\\ x^2\\ +\\ y^2\\ -\\ 4x\\ -\\ 6y\\ +\\ 9\\ = 0}\\ \\hspace{5cm}\\]\\[\\color{green}{and\\ x^2\\ +\\  y^2\\ +\\ 2x\\ +\\ 2y\\ -\\ 7\\ = 0\\ touch\\ each\\ other.}\\ \\hspace{5cm}\\]\\[\\color{green}{\\text{Find the co-ordinates of the point of contact.}}\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Given\\ x^2 + y^2 -\\ 4x\\ -\\ 6y\\ +\\ 9\\ = 0 &#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212; (1)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Given\\ x^2\\ +\\  y^2\\ +\\ 2x\\ +\\ 2y\\ -\\ 7\\ = 0 &#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212; (2)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[From\\ (1)\\ \\hspace 10cm\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2g_1 =\\ -\\ 4\\ \\hspace 2cm\\  2f_1\\ =\\ -\\ 6\\ \\hspace 2cm\\ c_1 =\\ 9\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[g_1 =\\ -\\ 2\\ \\hspace 2cm\\  f_1 =\\ -\\ 3\\ \\hspace 2cm\\ c_1 =\\ 9\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Centre\\ is\\  C_1 = (-g_1,\\ -f_1)\\ \\hspace 10cm\\ r_1 = \\sqrt{g_1^2 + f_1^2 -c_1}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ C_1 = (2,\\ \\ 3)\\ \\hspace 10cm\\ r_1 = \\sqrt{(-2)^2\\ +\\ (-3)^2\\  -\\  9}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\hspace 10cm\\ r_1 = \\sqrt{4\\ +\\ 9\\  -\\ 9}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\hspace 10cm\\ r_1 = \\sqrt{4}\\ =\\  2\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{C_1 = ( 2, 3)\\  and\\ r_1 =\\ 2}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[From\\ (2)\\ \\hspace 10cm\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2g_2 =\\ 2\\ \\hspace 2cm\\  2f_2\\ =\\ 2\\ \\hspace 2cm\\ c_2 =\\ -\\ 7\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[g_2 =\\ 1\\ \\hspace 2cm\\  f_2 =\\ 1\\ \\hspace 2cm\\ c_2 =\\  -\\ 7\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Centre\\ is\\  C_2 = (-g_2,\\ -f_2)\\ \\hspace 10cm\\ r_2 = \\sqrt{g_2^2 + f_2^2 -c_2}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ C_2 = (-\\ 1\\ , -\\ 1)\\ \\hspace 10cm\\ r_2 = \\sqrt{(1)^2\\ +\\ (1)^2\\  +\\ 7}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\hspace 10cm\\ r_2\\ = \\sqrt{1\\ + 1\\  +\\ 7}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\hspace 10cm\\ r_2 = \\sqrt{9} =\\ 3\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{C_2 = (-\\ 1, -\\ 1)\\  and\\ r_2 =\\ 3}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[C_1C_2 = \\sqrt{(-1\\ -\\ 2)^2\\ +\\ (-1\\ -\\  3)^2}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[C_1C_2 = \\sqrt{(-3)^2\\ +\\ (-4)^2}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[C_1C_2 = \\sqrt{9\\ +\\ 16}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[C_1C_2 = \\sqrt{25}\\ =\\ 5\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[r_1\\ +\\  r_2\\ =\\ 2\\ +\\ 3\\ =\\ 5\\ =\\  C_1C_2\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{C_1C_2 = r_1 + r_2}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[The\\ given\\ circles\\ touch\\ each\\ other\\ externally\\]<\/div>\n\n\n\n<p>To find the point of contact:<\/p>\n\n\n\n<p>For externally touching circles, the point of contact lies on the line segment joining the centers of the two circles. We can find the point of contact using the section formula, dividing the line segment joining&nbsp;C1&nbsp;and&nbsp;C2&nbsp;in the ratio of the radii&nbsp;r1:r2.<\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{C_1 = (2, 3)\\ ,\\ C_2 =\\ (-\\ 1, -\\ 1)\\ and\\  r_1\\ =\\ 2.\\ r_2\\ =\\ 3}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Here\\ x_1\\ =\\ 2\\ ,\\ y_1\\ =\\ 3\\ and\\ x_2\\ =\\ -1\\ ,\\ y_2\\ =\\ -1\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[P(x\\ ,\\ y)\\ =\\ (\\frac{r_1\\ x_2\\ +\\ r_2\\ x_1}{r_1\\ +\\ r_2},\\ \\frac{r_1\\ y_2\\ +\\ r_2\\ y_1}{r_1\\ +\\ r_2})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ (\\frac{2(-1)\\ +\\ 3(2)}{2\\ +\\ 3},\\ \\frac{2(-1)\\ +\\ 3(3)}{2\\ +\\ 3})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ (\\frac{-2\\ +\\ 6}{5},\\ \\frac{-2\\ +\\ 9}{5})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[P(x\\ ,\\ y)\\ =\\ (\\frac{4}{5},\\ \\frac{7}{5})\\]<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Equation (1) becomes At (1, 2, 3) ILATE To find the point of contact: For externally touching circles, the point of contact lies on the line segment joining the centers of the two circles. We can find the point of contact using the section formula, dividing the line segment joining&nbsp;C1&nbsp;and&nbsp;C2&nbsp;in the ratio of the radii&nbsp;r1:r2.<\/p>\n","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":[1],"tags":[],"class_list":["post-48272","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.5 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>October-2023 TAMIL NADU POLYTECHNIC BOARD EXAM ENGINEERING MATHEMATICS \u2013 II(40022)QUESTION PAPER WITH SOLUTIONS - YANAMTAKSHASHILA<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/yanamtakshashila.com\/?p=48272\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"October-2023 TAMIL NADU POLYTECHNIC BOARD EXAM ENGINEERING MATHEMATICS \u2013 II(40022)QUESTION PAPER WITH SOLUTIONS - YANAMTAKSHASHILA\" \/>\n<meta property=\"og:description\" content=\"Equation (1) becomes At (1, 2, 3) ILATE To find the point of contact: For externally touching circles, the point of contact lies on the line segment joining the centers of the two circles. 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