{"id":44502,"date":"2023-11-05T20:09:09","date_gmt":"2023-11-05T14:39:09","guid":{"rendered":"https:\/\/yanamtakshashila.com\/?p=44502"},"modified":"2024-11-25T10:45:34","modified_gmt":"2024-11-25T05:15:34","slug":"october-2023-tamil-nadu-polytechnic-board-exam-basic-mathematics-iquestion-paper-with-solutions","status":"publish","type":"post","link":"https:\/\/yanamtakshashila.com\/?p=44502","title":{"rendered":"October-2023 TAMIL NADU POLYTECHNIC BOARD EXAM BASIC MATHEMATICS QUESTION PAPER WITH SOLUTIONS"},"content":{"rendered":"\n<p class=\"has-text-align-center\">1.    Answer any fifteen questions in <strong>PART- A. <\/strong>   All questions carry<\/p>\n\n\n\n<p class=\"has-text-align-center\">                                                         equal marks. (15 X 2 =30)<\/p>\n\n\n\n<p class=\"has-text-align-center\">2.   Answer all questions, choosing any two sub-divisions <\/p>\n\n\n\n<p class=\"has-text-align-center\">                      each question under Part-B.    All questions carry equal marks.  <\/p>\n\n\n\n<p class=\"has-text-align-center\">                                                       (5 X 14 = 70) ( 7 + 7)<\/p>\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}{If\\ A =\\begin{bmatrix}\n9 &amp; 10 \\\\\n13 &amp; 20\\\\\n\\end{bmatrix}\\  and\\ B =\\begin{bmatrix}\n3 &amp; 5 \\\\\n8 &amp; 9\\\\\n\\end{bmatrix}},\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {green}{\\ find}\\ 2A\\ +\\ B\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ 2A = 2\\begin{bmatrix}\n9 &amp; 10 \\\\\n13 &amp; 20\\\\\n\\end{bmatrix}\\  \\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2A = \\begin{bmatrix}\n18 &amp; 20 \\\\\n26 &amp; 40\\\\\n\\end{bmatrix}\\ &#8212;&#8212;&#8212;- (1)\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[B = \\begin{bmatrix}\n3 &amp; 5 \\\\\n8 &amp; 9\\\\\n\\end{bmatrix}\\ &#8212;&#8212;&#8212;- (2)\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2A\\ +\\ B = \\begin{bmatrix}\n18 &amp; 20 \\\\\n26 &amp; 40\\\\\n\\end{bmatrix}\\ +\\ \\begin{bmatrix}\n3 &amp; 5 \\\\\n8 &amp; 9\\\\\n\\end{bmatrix}\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\begin{bmatrix}\n18\\ +\\ 3 &amp; 20\\ +\\ 5 \\\\\n26\\ +\\ 8 &amp; 40\\  +\\ 9\\\\\n\\end{bmatrix}\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2A\\ +\\ B = \\begin{bmatrix}\n21 &amp; 25 \\\\\n34 &amp; 49\\\\\n\\end{bmatrix}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2. \\ \\color{green}{Find\\ the\\ value\\ of\\ x\\ if\\ \\begin{vmatrix}\n2 &amp; 3 \\\\\n4 &amp; x \\\\\n\\end{vmatrix}\\ = 0}\\ \\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\">\\[\\hspace{5cm}\\ \\begin{vmatrix}\n2 &amp; 3 \\\\\n4 &amp; x \\\\\n\\end{vmatrix}\\ = 0 \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2(x)\\ -\\ 4 (3) = 0\\ \\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2x\\ -\\ 12\\ =\\ 0\\ \\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2x\\  =\\  12\\ \\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[x\\  =\\ 6\\ \\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[3.\\ \\color{green}{Find\\ the\\ cofactor\\ of\\ -\\ 2\\ in}\\  \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\ \\color{green}{ \\begin{vmatrix}\n1 &amp; &#8211; 1 &amp; 1 \\\\\n2 &amp; 3 &amp; -\\ 3 \\\\\n6 &amp; -\\ 2 &amp; -\\ 1 \\\\\n\\end{vmatrix}}\\ \\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\">\\[cofactor\\ of\\ -\\ 2\\ =\\ (-1)^{3\\ +\\ 2}\\ \\begin{vmatrix}\n1 &amp; 1 \\\\\n2 &amp; -\\ 3 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1)^5 (-\\ 3\\ -\\ 2)\\ \\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\">\\[\\boxed{cofactor\\ of\\ -\\ 2 =\\  5}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">     <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[4.\\ \\color{green}{Find\\ the\\ inverse\\ matrix\\ of\\ \\begin{pmatrix}\n5 &amp; 2 \\\\\n-4 &amp; 3 \\\\\n\\end{pmatrix}}\\ \\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\">\\[\\color {blue}{Solution:}\\ Let\\ A\\ =\\begin{pmatrix}\n5 &amp; 2 \\\\\n-4 &amp; 3 \\\\\n\\end{pmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\begin{vmatrix}\nA \\\\\n\\end{vmatrix}\\ =\\ 5(3)\\ -\\ 2(-4) \\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 15\\ +\\ 8 \\hspace{12cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\begin{vmatrix} A \\\\ \\end{vmatrix}\\ =\\ 23\\ \\neq {0}\\ \\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\therefore\\ A^{-1}\\ 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\\ 5 = (-1)^{1\\ +\\ 1}\\ (3)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1)^2 (3)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 1 = 3\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 2 = (-1)^{1\\ +\\ 2}\\ (-4)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1)^3 (-4)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 2 = 4\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ -4 = (-1)^{2\\ +\\ 1}\\ (2)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1)^3 (2)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ &#8211; 4 =\\ -\\ 2\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 3\\ = (-1)^{2\\ +\\ 2}\\ (5)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1)^4 (5)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 3 =\\ 5\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\therefore\\ cofactor\\ matrix\\ = \\begin{pmatrix}\n3 &amp; 4 \\\\\n-2 &amp; 5 \\\\\n\\end{pmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Adj.\\ A = \\begin{pmatrix}\n3 &amp; -2 \\\\\n4 &amp; 5 \\\\\n\\end{pmatrix}\\ \\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}{23}\\ \\begin{pmatrix}\n3 &amp; -2 \\\\\n4 &amp; 5 \\\\\n\\end{pmatrix}\\ \\hspace{2cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[5.\\ \\color{green}{Convert\\ 240^0\\  to\\ equivalent\\ radians\\ value}\\ \\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\">\\[\\hspace{2cm}\\ 240^0\\ \\times \\frac{\\pi\\ radians}{180^0}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\hspace{2cm}\\ \\frac{4}{3}\\ \\pi\\  radians\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[6.\\ \\color{green}{Write\\ any\\ two\\ characteristics\\ of\\ the\\ function\\ y\\ =\\ e^x}\\ \\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\">\\[\\hspace{2cm}\\ 1.\\ \\textbf{Exponential Growth}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{3cm}\\ The\\ function\\ y\\ =\\ e^x\\ exhibits\\ exponential\\ growth.\\ As\\ x\\ increases,\\ y\\ increases\\ very\\ rapidly\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{3cm}\\ Conversely,\\ as x\\ decreases,\\ y\\ approaches\\ zero\\ but\\ never\\ reaches\\ it\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ 2.\\ \\textbf{Derivative  and  Integral  properties}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{3cm}\\ The\\ function\\ y\\ =\\ e^x\\ \\text{is unique in that it is its own derivative and integral}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{3cm}\\ \\frac{d}{dx}\\ e^x\\ =\\ e^x\\ and\\ \\int{e^x}\\ dx\\ =\\ e^x\\ +\\ c\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[7.\\ \\color{green}{Evaluate\\  cos\\ {75}^0}\\  \\hspace{18cm}\\]<\/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\">\\[\\text{Given that}\\ 75^0\\ =\\ 30^0\\ +\\ 45^0\\ \\text{we use angle sum formula for cosine}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ W.\\ K.\\ T\\ cos( A + B )\\  =\\  cos A\\ Cos B\\ -\\  sin A sin B\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{3cm}\\ cos( 75^0 )\\  =\\  cos(30^0\\ +\\ cos(45^0)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{3cm}\\ =\\  c os\\ 30^0\\ cos\\ 45^0\\ -\\ sin\\ 30^0\\ sin\\ 45^0\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{3cm}\\ =\\  \\frac{\\sqrt{3}}{2}\\ \\frac{1}{\\sqrt{2}}\\ -\\ \\frac{1}{2}\\ \\frac{1}{\\sqrt{2}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{4cm}\\ =\\  \\frac{\\sqrt{3}\\ -\\ 1}{2\\ \\sqrt{2}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[8.\\ \\color{green}{If\\ tan\\ A\\ =\\ \\frac{1}{2},\\  find\\ tan\\ 2A}\\  \\hspace{18cm}\\]<\/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{2cm}\\ Given\\ tan\\ A\\ =\\  \\frac{1}{2}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ W.\\ K.\\ T\\ tan\\ 2A\\  =\\ \\frac{2\\ tan\\ A}{1\\ -\\ tan^2\\ A}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{3cm}\\ =\\ \\frac{2\\ \\times\\ \\frac{1}{2}}{1\\ -\\ (\\frac{1}{2})^2}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{3cm}\\ =\\ \\frac{1}{1\\ -\\ \\frac{1}{4}}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{3cm}\\ =\\ \\frac{1}{\\frac{4\\ -\\ 1}{4}}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{3cm}\\ =\\ \\frac{1}{\\frac{3}{4}}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{4cm}\\ =\\ \\frac{4}{3}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[9.\\ \\color{green}{If\\ \\overrightarrow{a}\\ =\\ 5\\overrightarrow{i}\\ +\\ 2\\overrightarrow{j}\\ -\\ 3\\overrightarrow{k}\\ and\\  \\overrightarrow{b}\\ = -\\ 3\\overrightarrow{i}\\ -\\ 2\\overrightarrow{j}\\ +\\ 5\\overrightarrow{k},\\ find\\   4\\overrightarrow{a}\\ +\\  6\\overrightarrow{b}}\\ \\hspace{10cm}\\]<\/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}= 5\\overrightarrow{i}\\ +\\ 2\\overrightarrow{j}\\ -\\ 3\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{b}\\ =\\ -\\ 3\\overrightarrow{i}\\ -\\ 2\\overrightarrow{j}\\ +\\ 5\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[4\\overrightarrow{a}\\ +\\  6\\overrightarrow{b}\\ =\\ 4(5\\overrightarrow{i}\\ +\\ 2\\overrightarrow{j}\\ -\\ 3\\overrightarrow{k})\\ +\\   6(  -\\ 3\\overrightarrow{i}\\ -\\ 2\\overrightarrow{j}\\ +\\ 5\\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 20\\overrightarrow{i}\\ +\\ 8\\overrightarrow{j}\\ -\\  12\\overrightarrow{k}\\ -\\   18\\overrightarrow{i}\\ -\\  12\\overrightarrow{j}\\ +\\ 30\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[4\\overrightarrow{a}\\ +\\  6\\overrightarrow{b}\\ =\\ 2\\overrightarrow{i}\\ -\\  4\\overrightarrow{j}\\ +\\ 18\\overrightarrow{k}\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[10.\\ \\color{green}{Find\\ the\\  Direction\\  cosines\\ of\\ the\\ vector\\  \\overrightarrow{i}\\ + 2\\overrightarrow{j}\\ -\\ 3\\overrightarrow{k}}\\ \\hspace{10cm}\\]<\/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}\\ + 2\\overrightarrow{j}\\ -\\ 3\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[r =\\overrightarrow{|a|} = \\sqrt{(1)^2 + (2)^2 + (-3)^2 }\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\sqrt{(1\\ +\\ 4\\ +\\ 9) }\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[r =\\sqrt{14}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ Direction\\  cosines\\  are \\frac{1}{\\sqrt(14)},  \\frac{2}{\\sqrt(14)},    \\frac{-3}{\\sqrt(14)}  \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\n\\[11.\\ \\color{green}{Show\\ that\\ the\\ vectors\\ \\overrightarrow{i}\\ -\\ 3\\overrightarrow{j}\\ +\\  5\\overrightarrow{k} and\\  -\\ 2\\overrightarrow{i}\\ +\\  6\\overrightarrow{j}\\ +\\ 4\\overrightarrow{k} are\\ mutually\\ perpendicular}\\ \\hspace{2cm}\\]<\/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} &#8211; 3\\overrightarrow{j} + 5\\overrightarrow{k} \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{b}= &#8211; 2\\overrightarrow{i}+ 6\\overrightarrow{j}+4\\overrightarrow{k} \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\overrightarrow{a}.\\overrightarrow{b}= (\\overrightarrow{i} &#8211; 3\\overrightarrow{j} + 5\\overrightarrow{k}) .(- 2\\overrightarrow{i}+ 6\\overrightarrow{j}+4\\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 1(-2)\\ +\\ -3(6)\\ +\\ 5 (4)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\  -\\ 2\\ -\\  18\\  +\\ 20\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"> \\[=\\   0\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{a}\\ and\\ \\overrightarrow{b}\\  are\\  perpendicular\\  vectors.\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/X-sFSLC6C3A\" title=\"Product of Vectors (Exercise) - 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\">\\[12.\\ \\color{green}{Find\\ the\\ values\\ of\\  \\overrightarrow{i}\\ . \\  \\overrightarrow{j}\\  and\\ \\overrightarrow{i}\\ \\times\\ \\overrightarrow{j}}\\ \\hspace{18cm}\\]<\/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{2cm}\\  \\overrightarrow{i}\\ . \\  \\overrightarrow{j}\\ =\\ 0\\  and\\  \\overrightarrow{i}\\ \\times\\ \\overrightarrow{j}\\ =\\ \\overrightarrow{k}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[13.\\ \\color{green}{Calculate\\ the\\ arithmetic\\ mean\\ of\\  14,\\ 26,\\ 28,\\ 20,\\ 32,\\ 30}\\ \\hspace{10cm}\\]<\/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{2cm}\\  Given\\ n\\ =\\ 6\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Sigma x_i\\ =\\ 14\\ +\\ 26\\ +\\ +\\ 28\\ +\\ 20\\ +\\ 32\\ +\\ 30\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Sigma x_i\\ =\\ 150\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bar{x}\\ =\\ \\frac{\\Sigma x_i}{n}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{150}{6}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bar{x}\\ =\\ 25\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[14.\\ \\color{green}{The\\ arithmetic\\ mean\\ of\\ 10\\ numbers\\ is\\ 20\\ ,\\  find\\ the\\ total\\ of\\ the}\\ \\hspace{10cm}\\]\\[\\color{green}{numbers}\\ \\hspace{10cm}\\]<\/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{2cm}\\ Given\\ n\\ =\\ 10\\ and\\ \\bar{x}\\ =\\ 20\\ \\hspace{17cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bar{x} \\ =\\ \\frac{\\Sigma X}{n}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[20\\ =\\ \\frac{\\Sigma X}{10}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\therefore\\ \\Sigma X\\ =\\ 200\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[15.\\ \\color{green}{\\text{If the standard deviation of a data is 6.8,}}\\ \\color{green}{find\\ its\\ variance}\\ \\hspace{10cm}\\]<\/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{2cm}\\ Given\\ \\sigma\\ =\\ 6.8\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ W.\\ K.\\ T\\ variance\\ =\\ \\sigma^2\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{3cm}\\ =\\ (6.8)^2\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ \\boxed{variance\\ =\\ 46.24}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[16.\\ \\color{green}{Write\\ the\\ normal\\ equations\\ of\\ the\\ straight\\ line\\ y\\ =\\ a\\ x\\ +\\ b}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<p>The Normal equations are<\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[a\\ \\Sigma x_i\\ +\\ nb\\ =\\ \\Sigma y_i\\ &#8212;&#8212;&#8211;\\ (1)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[a\\ \\Sigma x_i^2\\ +\\ b\\ \\Sigma\\ x_i\\ =\\ \\Sigma x_i\\ y_i\\ &#8212;&#8212;&#8211;\\ (2)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[17.\\ \\color{green}{A\\ card\\ is\\ picked\\ randomly\\ from\\ a\\  pack\\ of\\ 52\\ cards.}\\ \\hspace{15cm}\\]\\[\\color {green} {Find\\ the\\ probability\\ of\\ getting\\ a\\ King}\\  \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<p>In a standard deck of 52 playing cards, there are 4 Kings (one from each suit: hearts, diamonds, clubs, and spades.<\/p>\n\n\n\n<p>The probability&nbsp;PP&nbsp;of drawing a King from a deck of 52 cards can be calculated as follows:<\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[P(King)\\ =\\ \\frac{Number\\ of\\ Kings}{Total\\ number\\ of\\ cards}\\ =\\ \\frac{4}{52}\\ =\\ \\frac{1}{13}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{So, the probability of picking a King from a standard deck of 52 cards is:}\\ \\boxed{\\frac{1}{13}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[18.\\ \\color{green}{A\\ die\\ is\\ rolled\\ once,}\\ \\hspace{15cm}\\]\\[\\color {green} {Find\\ the\\ probability\\ of\\ getting\\ an\\ odd\\ number.}\\ \\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\">\\[\\text{A standard die has 6 faces numbered from 1 to 6. The odd numbers on a die are 1, 3, and 5.}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{The probability P of rolling an odd number can be calculated as follows:}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[P(King)\\ =\\ \\frac{Number\\ of\\ Kings}{Odd\\ number}\\ =\\ \\frac{3}{6}\\ =\\ \\frac{1}{2}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{So, the probability of rolling an odd number on a standard die is:}\\ \\boxed{\\frac{1}{2}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[19.\\ \\color{green}{If\\ P(A)\\ =\\ 0.5,\\ P(B)\\ =\\ 0.3,\\ and\\ A\\ \\cap\\ B\\ is\\ empty,}\\ \\hspace{10cm}\\]\\[\\color {green} {find\\ P(A\\ \\cup\\ B)}\\ \\hspace{10cm}\\]<\/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\">\\[\\text{Given P(A) = 0.5,  P(B) = 0.3 and} A \\cap\\ B\\ is\\ empty\\  (\\text{ which means events A and B are mutually exclusive),}\\] \n<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[we\\ can\\ find\\ P(A\\ \\cup\\ B)\\ \\text{using the formula for the union of two mutually exclusive events:}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[P(A\\ \\cup\\ B\\ =\\ P(A)\\ +\\ P(B)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ =\\ 0.5\\ +\\ 0.3\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{P(A\\ \\cup\\ B\\ =\\ 0.8}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[20.\\ \\color{green}{f\\ P(A)\\ =\\ \\frac{1}{3},\\ P(B)\\ =\\ \\frac{3}{4},\\ P(A\\ \\cap\\ B)\\ =\\ \\frac{1}{6},}\\ \\hspace{10cm}\\]\\[\\color{green}{find\\ P(A\/B)\\ and\\ P(B\/A)}\\ \\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\">\\[To\\ find\\ the\\ conditional\\ probabilities\\ P(A\/B)\\ and\\ P(B\/A),\\ \\text{We use the definitions of conditional probability.}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[1.\\ P(A\/B)\\ =\\ \\frac{P(A\\ \\cap\\ B)}{P(B)}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2.\\ P(B\/A)\\ =\\ \\frac{P(A\\ \\cap\\ B)}{P(A)}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Given\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bullet\\ P(A)\\ =\\ \\frac{1}{3}\\ \\hspace{10cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bullet\\ P(B)\\ =\\ \\frac{3}{4}\\ \\hspace{10cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bullet\\ P(A\\ \\cap\\ B)\\ =\\ \\frac{1}{6}\\ \\hspace{10cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[P(A\/B)\\ =\\ \\frac{P(A\\ \\cap\\ B)}{P(B)}\\ =\\ \\frac{\\frac{1}{6}} {\\frac{3}{4}}\\ =\\ \\frac{1}{6}\\ \\times\\ \\frac{4}{3}\\ =\\ \\frac{2}{9}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[P(B\/A)\\ =\\ \\frac{P(A\\ \\cap\\ B)}{P(A)}\\ =\\ \\frac{\\frac{1}{6}} {\\frac{1}{3}}\\ =\\ \\frac{1}{6}\\ \\times\\ \\frac{3}{1}\\ =\\ \\frac{1}{2}\\]<\/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\\ -\\ B}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[21\\ \\hspace{1cm}\\ (a)\\ \\hspace{1cm} \\color{green}{Verify\\ (AB)^T\\ =\\ B^T\\ A^T\\ If\\ A =\\begin{bmatrix}\n1 &amp; 0 &amp; 3 \\\\\n2 &amp; 1 &amp; &#8211; 1 \\\\\n1 &amp; -1 &amp; 1 \\\\\n\\end{bmatrix}\\ and\\ \\ B =\\begin{bmatrix}\n1 &amp; 0 &amp; 2 \\\\\n0 &amp; 1 &amp; 2 \\\\\n1 &amp; 2 &amp; 0 \\\\\n\\end{bmatrix}}\\ \\hspace{12cm}\\]<\/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\\ A =\\begin{bmatrix}\n1 &amp; 0 &amp; 3 \\\\\n2 &amp; 1 &amp; &#8211; 1 \\\\\n1 &amp; -1 &amp; 1 \\\\\n\\end{bmatrix}\\ and\\ \\ B =\\begin{bmatrix}\n1 &amp; 0 &amp; 2 \\\\\n0 &amp; 1 &amp; 2 \\\\\n1 &amp; 2 &amp; 0 \\\\\n\\end{bmatrix}\\ \\hspace{12cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[AB =\\begin{bmatrix}\n1 &amp; 0 &amp; 3 \\\\\n2 &amp; 1 &amp; &#8211; 1 \\\\\n1 &amp; -1 &amp; 1 \\\\\n\\end{bmatrix}\\  \\begin{bmatrix}\n1 &amp; 0 &amp; 2 \\\\\n0 &amp; 1 &amp; 2 \\\\\n1 &amp; 2 &amp; 0 \\\\\n\\end{bmatrix}\\  \\hspace{12cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[  = \\begin{bmatrix}\n1\\ \u00d7\\ 1\\ +\\ 0 \u00d7\\ 0\\ +\\ 3\\ \u00d7\\ 1&amp; 1\\ \u00d7\\ 0\\ +\\ 0 \u00d7\\ 1\\ +\\ 3\\ \u00d7\\ 2 &amp; 1\\ \u00d7\\ 2\\ +\\ 0 \u00d7\\ 2\\ +\\ 3\\ \u00d7\\ 0\\\\\n2\\ \u00d7\\ 1\\ +\\ 1 \u00d7\\ 0\\ +\\ -1\\ \u00d7\\ 1\\ &amp; 2\\ \u00d7\\ 0\\ +\\ 1 \u00d7\\ 1\\ +\\ -1\\ \u00d7\\ 2\\ &amp; 2\\ \u00d7\\ 2\\ +\\ 1 \u00d7\\ 2\\ +\\ -1\\ \u00d7\\ 0\\\\\n1\\ \u00d7\\ 1\\ +\\ -1 \u00d7\\ 0\\ +\\ 1\\ \u00d7\\ 1\\ &amp; 1\\ \u00d7\\ 0\\ +\\ -1 \u00d7\\ 1\\ +\\ -1\\ \u00d7\\ 2\\ &amp; 1\\ \u00d7\\ 2\\ +\\ -1 \u00d7\\ 2\\ +\\ 1\\ \u00d7\\ 0\\\\\n\\end{bmatrix}\\  \\hspace{9cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[  = \\begin{bmatrix}\n1\\ +\\ 0\\ +\\ 3\\ &amp; 0\\ +\\ 0\\ +\\ 6 &amp; 2\\ +\\ 0  +\\ 0\\\\\n2\\ +\\ 0\\ +\\ -1\\ &amp; 0\\ +\\ 1\\ -\\ 2\\ &amp; 4\\ +\\ 2\\ +\\ 0\\\\\n1\\ +\\ 0\\ +\\ 1\\ &amp; 0\\ -\\   1\\ -\\ 2\\ &amp; 2\\ -\\ 2\\ +\\ 0\\\\\n\\end{bmatrix}\\  \\hspace{9cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ AB = \\begin{bmatrix}\n4 &amp; 6 &amp; 2 \\\\\n1 &amp; -1 &amp; 6 \\\\\n2 &amp; -3 &amp; 0 \\\\\n\\end{bmatrix}\\  \\hspace{12cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(AB)^T\\ = \\begin{bmatrix}\n4 &amp; 1 &amp; 2 \\\\\n6 &amp; -1 &amp; &#8211; 3 \\\\\n2 &amp; 6 &amp; 0 \\\\\n\\end{bmatrix}\\  &#8212;&#8212;&#8212;&#8212;&#8211; (1)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Let\\ B^T =\\begin{bmatrix}\n1 &amp; 0 &amp; 1 \\\\\n0 &amp; 1 &amp; 2 \\\\\n2 &amp; 2 &amp; 0 \\\\\n\\end{bmatrix}\\ and\\ \\ A^T\\ =\\begin{bmatrix}\n1 &amp; 2 &amp; 1 \\\\\n0 &amp; 1 &amp; &#8211; 1 \\\\\n3 &amp; -1 &amp; 1 \\\\\n\\end{bmatrix}\\ \\hspace{12cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[B^T\\ A^T\\ =\\begin{bmatrix}\n1 &amp; 0 &amp; 1 \\\\\n0 &amp; 1 &amp; 2 \\\\\n2 &amp; 2 &amp; 0 \\\\\n\\end{bmatrix}\\ \\begin{bmatrix}\n1 &amp; 2 &amp; 1 \\\\\n0 &amp; 1 &amp; &#8211; 1 \\\\\n3 &amp; -1 &amp; 1 \\\\\n\\end{bmatrix}\\  \\hspace{12cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[  = \\begin{bmatrix}\n1\\ \u00d7\\ 1\\ +\\ 0 \u00d7\\ 0\\ +\\ 1\\ \u00d7\\ 3&amp; 1\\ \u00d7\\ 2\\ +\\ 0 \u00d7\\ 1\\ +\\ 1\\ \u00d7\\ -1 &amp; 1\\ \u00d7\\ 1\\ +\\ 0 \u00d7\\ -1\\ +\\ 1\\ \u00d7\\ 1\\\\\n0\\ \u00d7\\ 1\\ +\\ 1 \u00d7\\ 0\\ +\\ -2\\ \u00d7\\ 3\\ &amp; 0\\ \u00d7\\ 2\\ +\\ 1 \u00d7\\ 1\\ +\\ 2\\ \u00d7\\ -1\\ &amp; 0\\ \u00d7\\ 1\\ +\\ 1 \u00d7\\ -1\\ +\\ -2\\ \u00d7\\ 1\\\\\n2\\ \u00d7\\ 1\\ +\\ 2 \u00d7\\ 0\\ +\\ 0\\ \u00d7\\ 3\\ &amp; 2\\ \u00d7\\ 2\\ +\\ 2 \u00d7\\ 1\\ +\\ 0\\ \u00d7\\ -1\\ &amp; 2\\ \u00d7\\ 1\\ +\\ 2 \u00d7\\ -1\\ +\\ 0\\ \u00d7\\ 1\\\\\n\\end{bmatrix}\\  \\hspace{9cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[  = \\begin{bmatrix}\n1\\ +\\ 0\\ +\\ 3\\ &amp; 2\\ +\\ 0\\ +\\ -1 &amp; 1\\ +\\ 0  +\\ 1\\\\\n0\\ +\\ 0\\ +\\ 6\\ &amp; 0\\ +\\ 1\\ -\\ 2\\ &amp; 0\\ -\\ 1\\ -\\ 2\\\\\n2\\ +\\ 0\\ +\\ 0\\ &amp; 4\\ +\\   2\\ +\\ 0\\ &amp; 2\\ -\\ 2\\ +\\ 0\\\\\n\\end{bmatrix}\\  \\hspace{9cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[B^T\\ A^T\\ = \\begin{bmatrix}\n4 &amp; 1 &amp; 2 \\\\\n6 &amp; -1 &amp; &#8211; 3 \\\\\n2 &amp; 6 &amp; 0 \\\\\n\\end{bmatrix}\\  &#8212;&#8212;&#8212;&#8212;&#8211; (2)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[From\\ (1)\\ and\\ (2)\\ (AB)^T\\ =\\ B^T\\ A^T\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ (b)\\ \\hspace{1cm} \\color{green}{Solve\\ the\\ following\\ equations\\ using\\ Cramers\\ Rule}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{4cm} \\color{green}{3\\ x\\ +\\ y\\ -\\  z\\ =\\ 2,\\ 2\\ x\\ -\\ y\\ +\\  2\\ z\\ =\\  6\\ and\\  2\\ x\\ +\\ y\\ -\\ 2\\ z\\ =\\  -\\ 2}\\ \\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\">\\[3\\ x\\ +\\ y\\ -\\  z\\ =\\ 2\\ &#8212;&#8212;&#8212;&#8212;&#8212;&#8212;-(1)\\ \\hspace{6cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2\\ x\\ -\\ y\\ +\\  2\\ z\\ =\\  6\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2\\ x\\ +\\ y\\ -\\ 2\\ z\\ =\\  -\\ 2\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta = \\begin{vmatrix}\n3 &amp; 1 &amp; -1 \\\\\n2 &amp; -1 &amp; 2\\\\\n2 &amp; 1 &amp; -2 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta =\\ 3\\begin{vmatrix}\n-1 &amp; 2 \\\\\n1 &amp; -2 \\\\\n\\end{vmatrix}\\ -\\ 1\\begin{vmatrix}\n2 &amp; 2 \\\\\n2 &amp; -2 \\\\\n\\end{vmatrix}\\ -\\ 1\\begin{vmatrix}\n2 &amp; -1\\\\\n2 &amp; 1 \\\\\n\\end{vmatrix}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta =\\ 3(2\\ -\\ 2)\\ &#8211; 1 (-\\ 4\\ -\\  4)\\ -\\ 1(2\\ +\\ 2)\\ \n\\hspace{9cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta\\ =\\ 3(0)\\ &#8211; 1 (-8)\\ -\\ 1(4)\\ \n\\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta =\\ 0\\ +\\ 8\\ -\\  4\\ \n\\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\Delta\\ =\\ 4}\\ \n\\hspace{17cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_x = \\begin{vmatrix}\n2 &amp; 1 &amp; -1 \\\\\n6 &amp; -1 &amp; 2 \\\\\n-2 &amp; 1 &amp; -2 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_x\\ =\\ 2\\begin{vmatrix}\n-1 &amp; 2 \\\\\n1 &amp; -2 \\\\\n\\end{vmatrix}\\ -\\ 1\\begin{vmatrix}\n6 &amp; 2 \\\\\n-2 &amp; -2 \\\\\n\\end{vmatrix}\\ -\\ 1\\begin{vmatrix}\n6 &amp; -1\\\\\n-2 &amp; 1 \\\\\n\\end{vmatrix}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_x\\ =\\ 2(2\\ -\\ 2) &#8211; 1 (-12\\ +\\ 4)\\ -\\ 1(6\\ -\\ 2)\\ \n\\hspace{9cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_x\\ =\\ 2(0)\\ &#8211; 1 (-8)\\ -\\ 1(4)\\ \n\\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_x =\\ 0\\ +\\ 8\\ -\\ 4\\ \n\\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\Delta_x\\ =\\ 4}\\ \n\\hspace{17cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_y = \\begin{vmatrix}\n3 &amp; 2 &amp; -1 \\\\\n2 &amp; 6 &amp; 2 \\\\\n2 &amp; -2 &amp; -2 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_y\\ =\\ 3\\begin{vmatrix}\n6 &amp;  2 \\\\\n-2 &amp; -2\\\\\n\\end{vmatrix}\\ -\\ 2\\begin{vmatrix}\n2 &amp; 2 \\\\  2 &amp; -2 \\\\\n\\end{vmatrix}\\ -\\ 1\\begin{vmatrix}\n2 &amp; 6\\\\\n2 &amp;  -2 \\\\\n\\end{vmatrix}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_y\\ =\\ 3(-\\ 12\\ +\\ 4)\\ -\\ 2 (-4\\ -\\ 4)\\ -\\ 1(-4\\ -\\ 12)\\ \n\\hspace{9cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_y\\ =\\ 3(-8)\\ -\\ 2 (-8)\\ -\\ 1(-16)\\ \n\\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_y\\ =\\ -\\ 24\\ +\\ 16\\ +\\ 16\\ \n\\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\Delta_y\\ =\\ 8}\\ \n\\hspace{17cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_z = \\begin{vmatrix}\n3 &amp; 1 &amp; 2 \\\\\n2 &amp; -1 &amp; 6 \\\\\n2 &amp; 1 &amp; -2 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_z\\ =\\ 3\\begin{vmatrix}\n-1 &amp; 6 \\\\\n1 &amp; -2 \\\\\n\\end{vmatrix}\\ -\\ 1\\begin{vmatrix}\n2 &amp; 6 \\\\\n2 &amp; -2 \\\\\n\\end{vmatrix}\\ +\\ 2\\begin{vmatrix}\n2 &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(2\\ -\\ 6)\\ -\\ 1 (-\\ 4\\ -\\ 12)\\ +\\ 2(2\\ +\\ 2)\\ \n\\hspace{9cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_z\\ =\\ 3(-4)\\ -\\ 1 (-16)\\ +\\ 2(4)\\ \n\\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_z\\ =\\ -\\ 12\\  +\\  16\\ +\\  8\\ \n\\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\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{4}{4} =\\ 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\\ and\\ z = 3\\ in\\ equation (1)\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[LHS\\ =\\ 3(1)\\ +\\ 2 -\\ 3\\]\\[ =\\ 3\\ +\\ 2\\ &#8211;\n=\\ 2\\]\\[ = RHS\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ (c)\\ \\hspace{1cm} \\color{green}{Find\\ the\\ inverse\\ of\\ \\begin{bmatrix}\n1 &amp; -1 &amp; 1 \\\\\n2 &amp; -3 &amp; -3 \\\\\n6 &amp; -2 &amp;  -1 \\\\\n\\end{bmatrix}}\\ \\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\">\\[\\hspace{2cm}\\ Let\\ A\\ =\\begin{bmatrix}\n1 &amp; -1 &amp; 1 \\\\\n2 &amp; -3 &amp; -3 \\\\\n6 &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}\\ =1\\begin{vmatrix}\n-3 &amp; -3 \\\\\n-2 &amp; -1 \\\\\n\\end{vmatrix}\\ +\\ 1\\begin{vmatrix}\n2 &amp; -3 \\\\\n6 &amp; -1 \\\\\n\\end{vmatrix}\\ +\\ 1\\begin{vmatrix}\n2 &amp; -3\\\\\n6 &amp;  -2 \\\\\n\\end{vmatrix}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =1(3\\ -\\ 6)\\ + 1 (-2\\ +\\ 18) + 1(-4\\ +\\ 18)\\ \n\\hspace{9cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =1(-3)\\ + 1 (16) + 1(14)\\ \n\\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = -3\\ + 16 + 14\\ \n\\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\begin{vmatrix}\nA \\\\\n\\end{vmatrix}\\ = 27\\ \\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\\ 1 = (-1)^{1\\ +\\ 1}\\ \\begin{vmatrix}\n-3 &amp; -3 \\\\\n-2 &amp; -1 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1)^2 (3 &#8211; 6)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (1) (-3)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 1 = -3\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ -1 = (-1)^{1\\ +\\ 2}\\ \\begin{vmatrix}\n2 &amp; -3 \\\\\n6 &amp; -1 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1)^3 (-2 + 18)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1) (16)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ -1 = -16\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 1 = (-1)^{1\\ +\\ 3}\\ \\begin{vmatrix}\n2 &amp; -3 \\\\\n6 &amp; -2 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1)^4 (-4 + 18)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (1) (14)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 1 = 14\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 2 = (-1)^{2\\ +\\ 1}\\ \\begin{vmatrix}\n-1 &amp; 1 \\\\\n-2 &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) (3)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 2 = -3\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ -3 = (-1)^{2\\ +\\ 2}\\ \\begin{vmatrix}\n1 &amp; 1 \\\\\n6 &amp; -1 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1)^4 (-1- 6)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (1) (-7)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ -3 = -7\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ -3 = (-1)^{2\\ +\\ 3}\\ \\begin{vmatrix}\n1 &amp; -1 \\\\\n6 &amp; -2 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1)^5 (-2+ 6)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1) (4)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ -3 = -4\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 6 = (-1)^{3\\ +\\ 1}\\ \\begin{vmatrix}\n-1 &amp; 1 \\\\\n-3 &amp; -3 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1)^4 (3 + 3)\\ \\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\\ 6 = 6\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ -2 = (-1)^{3\\ +\\ 2}\\ \\begin{vmatrix}\n1 &amp; 1 \\\\\n2 &amp; -3 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1)^5 (-3- 2)\\ \\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\\ -2 = 5\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ -1 = (-1)^{3\\ +\\ 3}\\ \\begin{vmatrix}\n1 &amp; -1 \\\\\n2 &amp; -3 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1)^6 (-3+ 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\\ -1 = -1\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Cofactor\\ matrix=\\begin{bmatrix}\n-3 &amp; -16 &amp; 14 \\\\\n-3 &amp; -7 &amp; -4 \\\\\n6 &amp; 5 &amp; -1 \\\\\n\\end{bmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Adj.\\ A=\\begin{bmatrix}\n-3 &amp; -3 &amp; 6 \\\\\n-16 &amp; -7 &amp; 5 \\\\\n14 &amp; -4 &amp; -1 \\\\\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}{27}\\ \\begin{bmatrix}\n-3 &amp; -3 &amp; 6 \\\\\n-16 &amp; -7 &amp; 5 \\\\\n14 &amp; -4 &amp; -1 \\\\\n\\end{bmatrix}\\ \\hspace{2cm}\\]<\/div>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/eJ8q9ayyRkc\" title=\"Inverse of a Matrix - Example - 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\">\\[22\\ \\hspace{1cm}\\ (a)\\ \\hspace{1cm} \\color{green}{If\\ cos\\ \\theta\\ =\\ \\frac{3}{5},\\  then\\ find\\ \\text{the values of other five trigonometric}}\\ \\hspace{10cm}\\]\\[\\color{green}{ratios}\\ \\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\">\\[\\hspace{2cm}\\ Given\\ cos\\ \\theta\\ =\\ \\frac{3}{5}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ sin^2\u03b8\\ +\\  cos^2\u03b8\\ =\\  1\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ sin^2\u03b8\\ =\\  1\\ -\\  cos^2\u03b8\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{3cm}\\ =\\  1\\ -\\  (\\frac{3}{5})^2\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{3cm}\\ =\\  1\\ -\\  \\frac{9}{25}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{3cm}\\ =\\ \\frac{25\\ -\\ 9}{25}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{3cm}\\ =\\ \\frac{16}{25}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ \\boxed{sin\\ \\theta\\ =\\ \\pm\\ \\frac{4}{5}}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ tan\\ \u03b8\\ =\\  \\frac{sin\\ \\theta}{cos\\ \\theta}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{3cm}\\ =\\   \\frac{\\pm\\ \\frac{4}{5}}{\\frac{3}{5}}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ \\boxed{tan\\ \\theta\\ =\\ \\pm\\ \\frac{4}{5}}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ cot\\ \u03b8\\ =\\  \\frac{1}{tan\\ \\theta}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{3cm}\\ =\\ \\frac{1}{\\pm\\ \\frac{4}{5}}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ \\boxed{cot\\ \\theta\\ =\\ \\pm\\ \\frac{5}{4}}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ sec\\ \u03b8\\ =\\  \\frac{1}{cos\\ \\theta}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{3cm}\\ =\\   \\frac{1}{\\frac{3}{5}}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ \\boxed{sec\\ \\theta\\ =\\ \\frac{5}{3}}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ cosec\\ \u03b8\\ =\\  \\frac{1}{sin\\ \\theta}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{3cm}\\ =\\   \\frac{1}{\\pm\\ \\frac{4}{5}}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ \\boxed{sec\\ \\theta\\ =\\ \\pm\\ \\frac{5}{4}}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ (b)\\ \\hspace{1cm} \\color{green}{If\\ A\\ +\\ B\\ =\\ {45}^0 \\ Prove\\ that\\ (1\\ +\\ tan\\ A)\\ (1\\ +\\ tan\\ B)\\ =\\ 2}\\ \\hspace{15cm}\\]\\[\\hspace{3cm}\\ \\color{green} {Hence\\ deduce\\ the\\ value\\ of\\ tan\\ 22\\ {\\frac{1}{2}}^0}\\ \\hspace{13cm}\\]<\/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{2cm}\\ Given\\ A\\ +\\ B\\ =\\ 45^{0}\\  \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Taking\\ Tan\\ on\\ both\\ sides\\  \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[tan\\ (A\\ +\\ B)\\ =\\ tan\\ 45^{0}\\  \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{tan A\\ +\\ tan B}{1\\ -\\ tan A\\ tan B}\\ =\\ 1\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[tan\\ A\\ +\\ tan\\ B\\ =\\ 1\\ -\\ tan\\ A\\ tan\\ B\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[tan\\ A\\ +\\ tan\\ B\\ +\\ tan\\ A\\ tan\\ B\\ =\\ 1\\ &#8212;&#8212;&#8212;&#8212;\\ (1)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[L.\\ H.\\ S\\ =\\ (1\\ +\\ tan\\ A)\\ (1\\ +\\ tan\\ B)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 1\\ +\\ tan\\ B\\ +\\ tan\\ A\\ +\\ tan\\ A\\ tan\\ B\\  \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 1\\ +\\ 1\\  \\hspace{5cm}\\ using\\ (1)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 2\\ =\\ R.\\ H.\\ S\\  \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(1\\ +\\ tan\\ A)\\ (1\\ +\\ tan\\ B)\\ =\\ 2\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Put\\ B\\ =\\ A\\  \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[A\\ +\\ A\\ =\\ 45^{0}\\  \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2A\\ =\\ 45^{0}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[A\\ =\\ 22\\ {\\frac{1}{2}}^0\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(1\\ +\\ tan\\ 22\\ {\\frac{1}{2}}^0)\\ (1\\ +\\ tan\\ 22\\ {\\frac{1}{2}}^0)\\ =\\ 2\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(1\\ +\\ tan\\ 22\\ {\\frac{1}{2}}^0)^2\\  =\\ 2\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[1\\ +\\ tan\\ 22\\ {\\frac{1}{2}}^0\\  =\\ \\sqrt{2}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed {tan\\ 22\\ {\\frac{1}{2}}^0\\  =\\ \\sqrt{2}\\  -\\ 1}\\hspace{10cm}\\]<\/div>\n\n\n<p><iframe width=\"853\" height=\"480\" src=\"https:\/\/www.youtube.com\/embed\/InSRI_8U_RM\" title=\"TRIGONOMETRICAL IDENTITIES - PART - 2\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ (c)\\ \\hspace{1cm} \\color{green}{Show\\ that\\  \\frac{sin\\ A\\ +\\ sin\\ 2A}{1\\ +\\ cos\\ A\\ +\\ cos\\ 2A}\\ =\\ tan\\ A}\\   \\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\">\\[\\hspace{2cm}\\ LHS\\ =\\ \\frac{sin\\ A\\ +\\ sin\\ 2A}{1\\ +\\ Cos\\ A\\ +\\ cos\\ 2A}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{sin\\ A\\ +\\ 2\\ sin\\ A\\ Cos\\ A}{2\\ cos^2\\ A\\ +\\ Cos\\ A}\\ \\hspace{2cm}\\ W.\\ K.\\ T.\\ Sin\\ 2A\\ =\\  2\\ Sin\\ A\\ Cos\\ A\\ , and\\ 1\\ +\\ cos\\ 2A\\ =\\ 2\\ cos^2\\ A\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{Sin\\ A(1\\ +\\ 2\\ Cos\\ A)}{Cos\\ A(1\\ +\\ 2\\ Cos\\ A)}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{Sin\\ A}{CosS\\ A}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ tan\\ A\\ =\\ R.H.S\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[23\\ \\hspace{1cm}\\ (a)\\ \\hspace{1cm} \\color{green}{Prove\\ that\\ the\\ points}\\ \\hspace{15cm}\\]\\[\\color{green}{5\\overrightarrow{i}\\ + 6\\overrightarrow{j}\\ +\\ 7\\overrightarrow{k},  6\\overrightarrow{i}\\ +\\ 7\\overrightarrow{j}\\ +\\ 5\\overrightarrow{k} and\\ 7\\overrightarrow{i}\\ +\\ 5 \\overrightarrow{j}\\ +\\ 6\\overrightarrow{k}\\  form\\  an\\ equilateral\\  triangle}\\]<\/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{2cm}\\ Given\\ \\hspace{17cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{OA}\\ =\\ 5\\overrightarrow{i}\\ + 6\\overrightarrow{j}\\ +\\ 7\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{OB}\\ =\\ 6\\overrightarrow{i}\\ +\\ 7\\overrightarrow{j}\\ +\\ 5\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{OC}\\ =\\ 7\\overrightarrow{i}\\ +\\ 5 \\overrightarrow{j}\\ +\\ 6\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{AB} = \\overrightarrow{OB}-\\overrightarrow{OA}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 6\\overrightarrow{i}\\ +\\ 7\\overrightarrow{j}\\ +\\ 5\\overrightarrow{k}\\ -\\ (5\\overrightarrow{i}\\ + 6\\overrightarrow{j}\\ +\\ 7\\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 6\\overrightarrow{i}\\ +\\ 7\\overrightarrow{j}\\ +\\  5\\overrightarrow{k}\\ -\\ 5\\overrightarrow{i}\\ -\\ 6\\overrightarrow{j}\\ &#8211; 7\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{AB}\\ =\\  \\overrightarrow{i}\\ +\\ \\overrightarrow{j}\\ &#8211; 2\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[AB =\\overrightarrow{|AB|} = \\sqrt{(1)^2 + (1)^2 +(-2)^2 }\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\sqrt{1 + 1 +4 }\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[AB = \\sqrt{6}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{BC} = \\overrightarrow{OC}-\\overrightarrow{OB}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 7\\overrightarrow{i}\\ +\\ 5 \\overrightarrow{j}\\ +\\ 6\\overrightarrow{k}\\ -\\ (6\\overrightarrow{i}\\ +\\ 7\\overrightarrow{j}\\ +\\ 5\\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 7\\overrightarrow{i}\\ +\\ 5 \\overrightarrow{j}\\ +\\ 6\\overrightarrow{k}\\ &#8211; 6\\overrightarrow{i}\\ &#8211; 7\\overrightarrow{j}\\ &#8211; 5\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{BC}=  \\overrightarrow{i}\\ -\\ 2\\overrightarrow{j}\\  +\\ \\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[BC =\\overrightarrow{|BC|} = \\sqrt{(1)^2 + (-2)^2 +(1)^2 }\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\sqrt{1 + 4 + 1}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[BC = \\sqrt{6}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{AC} = \\overrightarrow{OC}-\\overrightarrow{OA}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 7\\overrightarrow{i}\\ +\\ 5 \\overrightarrow{j}\\ +\\ 6\\overrightarrow{k}\\ -\\ (5\\overrightarrow{i}\\ + 6\\overrightarrow{j}\\ +\\ 7\\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 7\\overrightarrow{i}\\ +\\ 5 \\overrightarrow{j}\\ +\\ 6\\overrightarrow{k}\\ &#8211; 5\\overrightarrow{i}\\ -\\ 6\\overrightarrow{j}\\ -\\ 7\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{AC}\\ =\\  2\\overrightarrow{i}\\ -\\ \\overrightarrow{j}\\ -\\ \\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[AC =\\overrightarrow{|AC|} = \\sqrt{(2)^2 + (-1)^2 +(-1)^2 }\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\sqrt{4 + 1 + 1}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[AC = \\sqrt{6}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[AB = BC = AC = \\sqrt{6}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[The\\ given\\ triangle\\ is\\ an\\ equilateral\\ triangle\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ (b)\\ \\hspace{1cm} \\color{green}{Prove\\ that\\ \\overrightarrow{i}+2\\overrightarrow{j}+ \\overrightarrow{k},\\  \\overrightarrow{i}  + \\overrightarrow{j}- 3\\overrightarrow{k}\\ and\\ 7\\overrightarrow{i}-4\\overrightarrow{j}+\\overrightarrow{k} are\\ mutually\\ orthogonal.}\\ \\hspace{13cm}\\]<\/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}+2\\overrightarrow{j}+ \\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\overrightarrow{b}= \\overrightarrow{i}  + \\overrightarrow{j}- 3\\overrightarrow{k} \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{c}= 7\\overrightarrow{i}-4\\overrightarrow{j}+\\overrightarrow{k} \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\overrightarrow{a}.\\overrightarrow{b}= (\\overrightarrow{i}+2\\overrightarrow{j}+ \\overrightarrow{k}) .(\\overrightarrow{i}  + \\overrightarrow{j}- 3\\overrightarrow{k})\\]<\/div>\n\n\n\n<p class=\"has-text-align-center\">   = 1(1) + 2(1) + 1(-3)<\/p>\n\n\n\n<p class=\"has-text-align-center\">=   0 <\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{a}\\ and\\ \\overrightarrow{b}\\  are\\  perpendicular\\  vectors.\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\overrightarrow{b}.\\overrightarrow{c}= (\\overrightarrow{i}  + \\overrightarrow{j}- 3\\overrightarrow{k}) .( 7\\overrightarrow{i}-4\\overrightarrow{j}+\\overrightarrow{k})\\]<\/div>\n\n\n\n<p class=\"has-text-align-center\"> = 1(7) + 1(-4) &#8211; 3(1)<\/p>\n\n\n\n<p class=\"has-text-align-center\">= 7 &#8211; 4 &#8211; 3<\/p>\n\n\n\n<p class=\"has-text-align-center\"> =   0 <\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{b} and\\ \\overrightarrow{c}\\  are\\  perpendicular\\  vectors.\\]<\/div>\n\n\n\n<p class=\"has-text-align-center\">  =   7(1) -4 (2) + 1 (1)  <\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\overrightarrow{c}.\\overrightarrow{a}= (7\\overrightarrow{i}-4\\overrightarrow{j}+\\overrightarrow{k}) .(\\overrightarrow{i}+2\\overrightarrow{j}+ \\overrightarrow{k})\\]<\/div>\n\n\n\n<p class=\"has-text-align-center\"> = 7 &#8211; 8 + 1<\/p>\n\n\n\n<p class=\"has-text-align-center\">  =   0  <\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{c} and\\ \\overrightarrow{a}\\  are\\  perpendicular\\  vectors.\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ The\\ three\\ vectors\\  are\\ mutually\\ perpendicular.\\]<\/div>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/5m31gltcYek\" title=\"Product of Vectors (Exercise) - Part - 19\" 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\">\\[\\hspace{2cm}\\ (c)\\ \\hspace{1cm} \\color{green}{Find\\ the\\ area\\ of\\ the\\ triangle\\ formed\\ by\\ the\\ points\\ whose\\ position\\ vectors\\ are}\\ \\hspace{8cm}\\]\\[\\color{green}{\\overrightarrow{i}\\ +\\ 3\\overrightarrow{j}\\ +\\ 2\\overrightarrow{k}\\ ,  2\\overrightarrow{i}\\ -\\ \\overrightarrow{j}\\ +\\ \\overrightarrow{k}\\ and\\  -\\overrightarrow{i}\\ + 2\\overrightarrow{j}\\ +\\ 3\\overrightarrow{k}}\\  \\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{OA}\\ =\\ \\overrightarrow{i}\\ +\\ 3\\overrightarrow{j}\\ +\\ 2\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{OB}\\ =\\ 2\\overrightarrow{i}\\ -\\ \\overrightarrow{j}\\ +\\ \\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{OC}\\ =\\ -\\overrightarrow{i}\\ + 2\\overrightarrow{j}\\ +\\ 3\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{AB} = \\overrightarrow{OB}-\\overrightarrow{OA}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 2\\overrightarrow{i}\\ -\\ \\overrightarrow{j}\\ +\\ \\overrightarrow{k}\\ &#8211; (\\overrightarrow{i}\\ +\\ 3\\overrightarrow{j}\\ +\\ 2\\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 2\\overrightarrow{i}\\ -\\ \\overrightarrow{j}\\ +\\ \\overrightarrow{k}\\ -\\  \\overrightarrow{i}\\ &#8211; 3\\overrightarrow{j}\\ &#8211; 2\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\overrightarrow{AB}\\ =\\  \\overrightarrow{i}\\ &#8211; 4\\overrightarrow{j}\\ -\\ \\overrightarrow{k}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{BC} = \\overrightarrow{OC}-\\overrightarrow{OB}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ -\\overrightarrow{i}\\ + 2\\overrightarrow{j}\\ +\\ 3\\overrightarrow{k}\\ &#8211; (2\\overrightarrow{i}\\ -\\ \\overrightarrow{j}\\ +\\ \\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ -\\overrightarrow{i}\\ + 2\\overrightarrow{j}\\ +\\ 3\\overrightarrow{k}\\ &#8211; 2\\overrightarrow{i}\\ +\\ \\overrightarrow{j}\\ -\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\overrightarrow{BC}\\ =\\  -\\ 3\\overrightarrow{i}\\ +\\ 3\\overrightarrow{j}\\ +\\ 2\\overrightarrow{k}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{AB}\u00d7\\overrightarrow{BC} =\\ \\begin{vmatrix}\n\\overrightarrow{i} &amp; \\overrightarrow{j} &amp;\\overrightarrow{k}\\\\\n1 &amp; -4 &amp; -1\\\\\n-3 &amp; 3 &amp; 2\\\\\n\\end{vmatrix}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\overrightarrow{i}( -8 + 3)\\ -\\ \\overrightarrow{j}(2 &#8211; 3)\\ +\\ \\overrightarrow{k}(3 &#8211; 12)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\overrightarrow{i}(-5) -\\overrightarrow{j}(-1)+\\overrightarrow{k}(-9)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\overrightarrow{AB}\u00d7 \\overrightarrow{BC}\\ =\\ -5\\overrightarrow{i}\\ +\\ \\overrightarrow{j}\\ -\\ 9\\overrightarrow{k}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[|\\overrightarrow{AB} \u00d7 \\overrightarrow{BC}| =  \\sqrt{(-5 )^2\\ +\\ (1)^2\\ + (-9)^2 }=\\sqrt{25 + 1 + 81 }=\\sqrt{107}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ Area\\ of \\ triangle = \\frac{1}{2}|\\overrightarrow{AB} \u00d7 \\overrightarrow{BC}|\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{Area\\ of \\ triangle\\ =\\frac{\\sqrt{107}}{2}\\ sq. units}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[24\\ \\hspace{1cm}\\ (a)\\ \\hspace{1cm} \\color{green}{Find\\ the\\ arithmetic\\ mean\\ of\\ the\\ following\\ data}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<figure class=\"wp-block-table\"><table><tbody><tr><td>Class-interval<\/td><td>0-10<\/td><td>10-20<\/td><td>20-30<\/td><td>30-40<\/td><td>40-50<\/td><td>50-60<\/td><td>60-70<\/td><\/tr><tr><td>Frequency<\/td><td>3<\/td><td>5<\/td><td>16<\/td><td>18<\/td><td>12<\/td><td>7<\/td><td>4<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<figure class=\"wp-block-table is-style-regular has-medium-font-size\" style=\"margin-top:0;margin-right:0;margin-bottom:0;margin-left:0;padding-top:0;padding-bottom:0\"><table style=\"border-width:1px\"><tbody><tr><td>Marks<\/td><td>No. of students<br>f<sub>i<\/sub><\/td><td class=\"has-text-align-left\" data-align=\"left\">Mid-value <br>      x<sub>i<\/sub><\/td><td>f<sub>i<\/sub>x<sub>i<\/sub><\/td><\/tr><tr><td>0-10<\/td><td>3<\/td><td class=\"has-text-align-left\" data-align=\"left\">5<\/td><td>15<\/td><\/tr><tr><td>10-20<\/td><td>5<\/td><td class=\"has-text-align-left\" data-align=\"left\">15<\/td><td>75<\/td><\/tr><tr><td>20-30<\/td><td>16<\/td><td class=\"has-text-align-left\" data-align=\"left\">25<\/td><td>400<\/td><\/tr><tr><td>30-40<\/td><td>18<\/td><td class=\"has-text-align-left\" data-align=\"left\">35<\/td><td>630<\/td><\/tr><tr><td>40-50<\/td><td>12<\/td><td class=\"has-text-align-left\" data-align=\"left\">45<\/td><td>540<\/td><\/tr><tr><td>50-60<\/td><td>7<\/td><td class=\"has-text-align-left\" data-align=\"left\">55<\/td><td>385<\/td><\/tr><tr><td>60-70<\/td><td>4<\/td><td class=\"has-text-align-left\" data-align=\"left\">65<\/td><td>260<\/td><\/tr><tr><td><\/td><td>N = 65<\/td><td class=\"has-text-align-left\" data-align=\"left\"><\/td><td><mathml>\\[\\Sigma f_i\\ x_i\\ =\\ 2305\\]<\/mathml><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\therefore\\ \\bar{x}\\ =\\ \\frac{\\Sigma f_i\\ x_i}{N}\\ =\\ \\frac{2305}{65}\\ =\\ 35.46\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ (b)\\ \\hspace{1cm} \\color{green}{\\text{Calculate the standard deviation for the following data}}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<figure class=\"wp-block-table\"><table><tbody><tr><td>Items<\/td><td>5<\/td><td>15<\/td><td>25<\/td><td>35<\/td><\/tr><tr><td>Frequency<\/td><td>2<\/td><td>1<\/td><td>1<\/td><td>3<\/td><\/tr><\/tbody><\/table><\/figure>\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{2cm}\\ Given\\ n\\ =\\ 4\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\sigma\\ =\\ \\sqrt{\\frac{\\Sigma f_i\\ d_i^2}{N}\\ -\\ (\\frac{\\Sigma f_i\\ d_i}{N}})^2, \\ where\\ d_i\\ =\\ x_i\\ -\\ A\\ and\\ N\\ =\\ \\Sigma {f_i}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[A\\ =\\ \\bar{x}\\ =\\ \\frac{5\\ +\\ 15\\ +\\ 25\\ +\\ 35}{4}\\  =\\ \\frac{80}{4}\\ =\\ 20\\]<\/div>\n\n\n\n<figure class=\"wp-block-table is-style-regular has-medium-font-size\" style=\"margin-top:0;margin-right:0;margin-bottom:0;margin-left:0;padding-top:0;padding-bottom:0\"><table style=\"border-width:1px\"><tbody><tr><td>x<sub>i<\/sub><\/td><td>f<sub>i <\/sub><\/td><td>d<sub>i<\/sub> = x<sub>i<\/sub> &#8211; A<\/td><td>d<sub>i <\/sub><sup>2<\/sup><\/td><td>f<sub>i<\/sub> d<sub>i<\/sub><\/td><td>f<sub>i<\/sub> d<sub>i<\/sub><sup> 2<\/sup><\/td><\/tr><tr><td>5<\/td><td>2<\/td><td>&#8211; 15<\/td><td>225<\/td><td>&#8211; 30<\/td><td>450<\/td><\/tr><tr><td>15<\/td><td>1<\/td><td>&#8211; 5<\/td><td>25<\/td><td>&#8211; 5<\/td><td>25<\/td><\/tr><tr><td>25<\/td><td>1<\/td><td>5<\/td><td>25<\/td><td>5<\/td><td>25<\/td><\/tr><tr><td>35<\/td><td>3<\/td><td>15<\/td><td>225<\/td><td>45<\/td><td>675<\/td><\/tr><tr><td><\/td><td>N=7<\/td><td><\/td><td><\/td><td><mathml>\\[\\Sigma f_i\\ d_i\\ =\\ 15\\]<\/mathml><\/td><td><mathml>\\[\\Sigma f_i\\ d_i^2\\ =1175\\]<\/mathml><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\sigma\\ =\\ \\sqrt{\\frac{\\Sigma f_i\\ d_i^2}{N}\\ -\\ (\\frac{\\Sigma f_i\\ d_i}{N}})^2\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\sqrt{\\frac{1175}{7}\\ -\\ (\\frac{15}{7}})^2\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\sqrt{\\frac{1175}{7}\\ -\\ \\frac{225}{7\\ \\times\\ 7}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\sqrt{\\frac{8225\\ -\\ 225}{7\\ \\times\\ 7}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\sqrt{\\frac{8000}{49}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\sqrt{163.265}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\sigma\\ =\\ 12.79\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ (c)\\ \\hspace{1cm} \\color{green}{Using\\ the\\ method\\ of\\ least\\ square\\ ,\\  fit\\ a\\ straight\\ line\\ to\\ the\\ following\\ data}\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<figure class=\"wp-block-table\"><table><tbody><tr><td>x<\/td><td>0<\/td><td>1<\/td><td>2<\/td><td>3<\/td><td>4<\/td><\/tr><tr><td>y<\/td><td>10<\/td><td>14<\/td><td>19<\/td><td>26<\/td><td>30<\/td><\/tr><\/tbody><\/table><\/figure>\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\\ line\\ be\\ y\\ =\\ a\\ x\\ +\\ b\\ &#8212;&#8212;&#8211;\\ (1)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[The\\ Normal\\ equations\\ are\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[a\\ \\Sigma x_i\\ +\\ nb\\ =\\ \\Sigma y_i\\ &#8212;&#8212;&#8211;\\ (2)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[a\\ \\Sigma x_i^2\\ +\\ b\\ \\Sigma\\ x_i\\ =\\ \\Sigma x_i\\ y_i\\ &#8212;&#8212;&#8211;\\ (3)\\]<\/div>\n\n\n\n<figure class=\"wp-block-table is-style-regular has-medium-font-size\" style=\"margin-top:0;margin-right:0;margin-bottom:0;margin-left:0;padding-top:0;padding-bottom:0\"><table style=\"border-width:1px\"><tbody><tr><td>x<sub>i<\/sub><\/td><td>y<sub>i <\/sub><\/td><td>x<sub>i<\/sub><sup> 2<\/sup><\/td><td>x<sub>i<\/sub> y<sub>i<\/sub><\/td><\/tr><tr><td>0<\/td><td>10<\/td><td>0<\/td><td>0<\/td><\/tr><tr><td>1<\/td><td>14<\/td><td>1<\/td><td>14<\/td><\/tr><tr><td>2<\/td><td>19<\/td><td>4<\/td><td>38<\/td><\/tr><tr><td>3<\/td><td>26<\/td><td>9<\/td><td>78<\/td><\/tr><tr><td>4<\/td><td>30<\/td><td>16<\/td><td>120<\/td><\/tr><tr><td><mathml>\\[\\Sigma x_i\\ =\\ 10\\]<\/mathml><\/td><td><mathml>\\[\\Sigma y_i\\ =\\ 99\\]<\/mathml><\/td><td><mathml>\\[\\Sigma {x_i}^2\\ =\\ 30\\]<\/mathml><\/td><td><mathml>\\[\\Sigma x_i\\ y_i\\ =\\ 250\\]<\/mathml><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(2)\\ becomes\\ a\\ (10)\\ +\\ 5b\\ =\\ 99\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2\\ a\\ +\\ b\\ =\\ 19.8\\ &#8212;&#8212;&#8211;\\ (4)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(3)\\ becomes\\ a\\ (30)\\ +\\ b\\ (10)\\ =\\ 250\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[3\\ a\\ +\\ b\\ =\\ 25\\ &#8212;&#8212;&#8211;\\ (5)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Solving\\ (4)\\ and\\ (5)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2\\ a\\ +\\ b\\ =\\ 19.8\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[3\\ a\\ +\\ b\\ =\\ 25\\]<\/div>\n\n\n\n<p class=\"has-text-align-center\">&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;-<\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[a\\ =\\ 5.2\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[put\\ a\\ =\\ 5.2\\ in\\ (4)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2(5.2)\\ +\\ b\\ =\\ 19.8\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[10.4\\ +\\ b\\ =\\ 19.8\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[b\\ =\\ 19.8\\ -\\ 10.4\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Eqn\\ (1)\\ becomes\\ y\\ =\\ 5.2\\ x\\ +\\ 9.4\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[25\\ \\hspace{1cm}\\ (a)\\ \\hspace{1cm} \\color{green}{Three\\ coins\\ are\\ tossed\\ simultaneously.\\  Find\\ the\\ probability\\ of\\ getting}\\ \\hspace{5cm}\\]\\[\\color{green}{(i)\\ exactly\\ one\\ head}\\ \\hspace{13cm}\\]\\[\\color{green}{(ii)\\ exactly\\ two\\ heads}\\ \\hspace{13cm}\\]\\[\\color{green}{(iii)\\ at least\\ two\\ heads}\\ \\hspace{13cm}\\]<\/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\">\\[\\text{When three coins are tossed simultaneously, each coin has two possible outcomes: heads (H) or tails (T).}\\]\\[\\therefore\\  \\text{the total number of possible outcomes when three coins are tossed is:}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2^3\\ =\\ 8\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{Total number of outcomes = 8}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{{HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(i)\\ \\bf{\\color{green}{exactly\\ one\\ head}}\\ \\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{To find the probability of getting exactly one head, we need to count the}\\ \\hspace{5cm}\\]\\[\\text{outcomes where there is exactly one head.}\\ \\hspace{10cm}\\]\n<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{The favourable number of outcomes are:}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{{HTT, THT, TTH}}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[P(Exactly\\ one\\ head)\\ =\\ \\frac{\\text{Number of favourable outcomes}}{\\text{Total Number of outcomes}}\\ =\\ \\frac{3}{8}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(ii)\\ \\bf{\\color{green}{exactly\\ two\\ heads}}\\ \\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{To find the probability of getting exactly two heads, we need to count the}\\ \\hspace{5cm}\\]\\[\\text{outcomes where there are exactly two heads.}\\ \\hspace{10cm}\\]\n<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{The favourable number of outcomes are:}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{{HHT, HTH, THH}}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[P(Exactly\\ two\\ heads)\\ =\\ \\frac{\\text{Number of favourable outcomes}}{\\text{Total Number of outcomes}}\\ =\\ \\frac{3}{8}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(ii)\\ \\bf{\\color{green}{exactly\\ two\\ heads}}\\ \\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{To find the probability of getting exactly one head, we need to count the}\\ \\hspace{5cm}\\]\\[\\text{outcomes where there are exactly two heads.}\\ \\hspace{10cm}\\]\n<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{The favourable number of outcomes are:}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{{HHT, HTH, THH}}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[P(Exactly\\ two\\ heads)\\ =\\ \\frac{\\text{Number of favourable outcomes}}{\\text{Total Number of outcomes}}\\ =\\ \\frac{3}{8}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(iii)\\ \\bf{\\color{green}{at\\ least\\ two\\ heads}}\\ \\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{To find the probability of getting atleast two heads, we need to count the}\\ \\hspace{5cm}\\]\\[\\text{outcomes where there are either two or three heads.}\\ \\hspace{10cm}\\]\n<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{The favourable number of outcomes are:}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{{HHT, HTH, THH, HHH}}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[P(at\\ least\\ two\\ heads)\\ =\\ \\frac{\\text{Number of favourable outcomes}}{\\text{Total Number of outcomes}}\\ =\\ \\frac{4}{8}\\ =\\ \\frac{1}{2}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ (b)\\ \\hspace{1cm} \\color{green}{Two\\ dice\\ are\\ thrown\\ simultaneously\\  Find\\ the\\ probability}\\ \\hspace{10cm}\\\\\\ \\color{green}{of\\ getting\\ a\\ sum\\ 6\\ or\\ same\\ number\\ on\\ both\\ dice}\\ \\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\">\\[When\\ two\\ dice\\ are\\ thrown\\ simultaneously,\\ each\\ die\\ has\\ 6\\ faces,\\ so\\ there\\ are\\ a\\ \\hspace{2cm}\\\\\\ total\\ of\\ 6\\ \\times\\ 6\\ =\\ 36\\ possible\\ outcomes\\ \\hspace{12cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[1.\\ \\bf{\\color{green}{probability\\ of\\ getting\\ a\\ sum\\ of\\ 6}}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{The possible outcomes where the sum of the two dice is 6 are:}\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bullet\\ (1,\\ 5)\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bullet\\ (2,\\ 4)\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bullet\\ (3,\\ 3)\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bullet\\ (4,\\ 2)\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bullet\\ (5,\\ 1)\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{There are 5 such outcomes.}\\ \\hspace{10cm}\\]\n<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2.\\ \\bf{\\color{green}{probability\\ of\\ getting\\ the\\ same\\ number\\ on\\ both\\ dice}}\\ \\hspace{7cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{The possible outcomes where the numbers on both dice are the same are:}\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bullet\\ (1,\\ 1)\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bullet\\ (2,\\ 2)\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bullet\\ (3,\\ 3)\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bullet\\ (4,\\ 4)\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bullet\\ (5,\\ 5)\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bullet\\ (6,\\ 6)\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{There are 6 such outcomes.}\\ \\hspace{10cm}\\]\n<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[3.\\ \\bf{\\color{green}{probability\\ of\\ either\\ getting\\ a\\ \\ sum\\ of\\ 6\\ or\\ the\\ same\\ number\\ on\\ both\\ dice}}\\ \\hspace{2cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[A\\ denotes\\ the\\ event\\ of\\ getting\\ a\\ sum\\ of\\ 6\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[B\\ denotes\\ the\\ event\\ of\\ getting\\ the\\ same\\ number\\ on\\ both\\ dice\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[we\\ needed\\ to\\ find\\ P(A\\ \\cup\\ B) =\\ ?\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[P(A\\ \\cup\\ B)\\ =\\ P(A)\\ +\\ P(B)\\ -\\ P(A\\ \\cap\\ B)\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[P(A)\\ =\\ \\frac{5}{36}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[P(B)\\ =\\ \\frac{6}{36}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[P(P(A\\ \\cap\\ B))\\ =\\ \\frac{1}{36}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[P(A\\ \\cup\\ B)\\ =\\ \\frac{5}{36}\\ +\\ \\frac{6}{36}\\ -\\ \\frac{1}{36}\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{10}{36}\\ =\\ \\frac{5}{18}\\ \\hspace{3cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{So, the probability of getting a sum of 6 or the same number on both dice is:}\\ =\\ \\frac{5}{18}\\ \\hspace{1cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ (c)\\ \\hspace{1cm} \\color{green}{A\\ problem\\ in\\ statistics\\ is\\ given\\ to\\ two\\ students\\ A\\ and\\ B.\\  The}\\ \\hspace{10cm}\\\\\\ \\color{green}{probability\\ of\\ A\\ solves\\ the\\ problem\\ is\\ \\frac{1}{2}\\ and\\ that\\ of\\ B\\ solves\\ the}\\ \\hspace{8cm}\\\\\\ \\color{green}{problem\\ is\\ \\frac{2}{3}.\\ If\\ the\\ students\\ solve\\ the\\ problems\\ independently.}\\ \\hspace{8cm}\\\\\\ \\color{green}{find\\ the\\ probability\\ that\\ the\\ problem\\ is\\ solved.}\\ \\hspace{12cm}\\] <\/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,\\ P(A)\\ =\\ \\frac{1}{2}\\  and\\  P(B)\\ =\\ \\frac{2}{3}\\ \\hspace{10cm}\\\\\\ \\text{Probability that the problem is solved = Probability that A solves the problem or B solves the problem}\\]&nbsp;<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ P(A\\ \\cup\\ B)\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ P(A)\\ +\\  P(B)\\ -\\ (P(A\\ \\cap\\ B)\\ \\hspace{2cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ P(A)\\ +\\ P(B)\\ -\\ P(A)\\ \\cdot \\ P(B)\\ Since A\\ and\\ B\\ are\\ independent\\ \\hspace{2cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{1}{2}\\ +\\ \\frac{2}{3}\\ -\\ \\frac{1}{2}\\ \\cdot \\ \\frac{2}{3}\\ \\hspace{2cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{1}{2}\\ +\\ \\frac{2}{3}\\ -\\ \\frac{1}{3}\\ \\hspace{2cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{3\\ +4\\ -\\ 2}{6}\\ \\hspace{2cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{5}{6}\\ \\hspace{2cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{Thus, the probability that the problem is solved by either student A, student B, or both is:}\\ =\\ \\frac{5}{6}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n","protected":false},"excerpt":{"rendered":"<p>1. Answer any fifteen questions in PART- A. All questions carry equal marks. (15 X 2 =30) 2. Answer all questions, choosing any two sub-divisions each question under Part-B. All questions carry equal marks. 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Answer any fifteen questions in PART- A. All questions carry equal marks. (15 X 2 =30) 2. Answer all questions, choosing any two sub-divisions each question under Part-B. All questions carry equal marks. 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