{"id":47584,"date":"2024-06-08T14:49:09","date_gmt":"2024-06-08T09:19:09","guid":{"rendered":"https:\/\/yanamtakshashila.com\/?p=47584"},"modified":"2024-11-16T21:22:56","modified_gmt":"2024-11-16T15:52:56","slug":"april-2024-tamil-nadu-polytechnic-board-exam-basic-mathematicsquestion-paper-with-solutions","status":"publish","type":"post","link":"https:\/\/yanamtakshashila.com\/?p=47584","title":{"rendered":"April-2024 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\\ f(x)\\ =\\ 3\\ x\\ +\\ 2\\ and\\ A =\\begin{pmatrix}\n1 &amp; 0 \\\\\n2 &amp; -1\\\\\n\\end{pmatrix},}\\  \\color {green}{find\\ f(A).}\\ \\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\">\\[\\hspace{3cm}\\ Given\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{3cm}\\ A =\\begin{pmatrix}\n1 &amp; 0 \\\\\n2 &amp; -1\\\\\n\\end{pmatrix}\\ and\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{3cm}\\ f(x)\\ =\\ 3\\ x\\ +\\ 2\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[f(A) = \\ 3A\\ +\\ 2\\ I, Where\\ I\\ is\\ a\\ Identity\\ matrix\\ of\\ order\\ 2\\ &#8212;&#8212;&#8212;- (1)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[3A\\ =\\ 3\\begin{pmatrix}\n1 &amp; 0 \\\\\n2 &amp; -1\\\\\n\\end{pmatrix}\\ \\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[3A\\ =\\ \\begin{pmatrix}\n3 &amp; 0 \\\\\n6 &amp; -3\\\\\n\\end{pmatrix}\\ \\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[I\\ =\\ \\begin{pmatrix}\n1 &amp; 0 \\\\\n0 &amp; 1\\\\\n\\end{pmatrix}\\ \\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2I\\ =\\ \\begin{pmatrix}\n2 &amp; 0 \\\\\n0 &amp; 2\\\\\n\\end{pmatrix}\\ \\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[3A\\ +\\ 2I\\ = \\begin{pmatrix}\n3 &amp; 0 \\\\\n6 &amp; -3\\\\\n\\end{pmatrix}\\ +\\ \\begin{pmatrix}\n2 &amp; 0 \\\\\n0 &amp; 2\\\\\n\\end{pmatrix}\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\begin{pmatrix}\n3\\ +\\ 2 &amp; 0\\ +\\ 0 \\\\\n6\\ +\\ 0\\ &amp; -3\\  +\\  2\\\\\n\\end{pmatrix}\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[3A\\ +\\ 2I = \\begin{pmatrix}\n5 &amp; 0 \\\\\n6 &amp; -1\\\\\n\\end{pmatrix}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\color{green}{\\therefore\\ f(A)\\ = \\begin{pmatrix}\n5 &amp; 0 \\\\\n6 &amp; -1\\\\\n\\end{pmatrix}}}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2. \\ \\color{green}{Find\\ the\\ value\\ of\\ x\\ if\\ \\begin{vmatrix}\nx &amp; 1 \\\\\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{2cm}\\ \\begin{vmatrix}\nx &amp; 1 \\\\\n4 &amp; x \\\\\n\\end{vmatrix}\\ = 0 \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[x(x) &#8211; 4 (1) = 0\\ \\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[x^2 &#8211; 4 = 0\\ \\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[x^2  = 4\\ \\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[x  = \\pm\\sqrt{4}\\ \\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[x  = \\pm 2\\ \\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color{green}{\\boxed{x\\ =\\ 2\\ or\\ x\\ =\\ -\\ 2}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[3.\\ \\color{green}{Prove\\ that\\ \\begin{pmatrix}\n1 &amp; 3 \\\\\n2 &amp; 6 \\\\\n\\end{pmatrix}\\ is\\ a\\ singular\\ matrix}\\ \\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\">\\[\\begin{vmatrix}\nA \\\\\n\\end{vmatrix}\\ =\\ 1(6)\\ -\\ 2(3) \\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 6\\ -\\ 6 \\hspace{12cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= 0 \\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[A\\ is\\ a\\ singular\\ matrix\\ \\hspace{10cm}\\]<\/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\\ cofactor\\ of\\ &#8216;2&#8217;\\ in\\  \\begin{pmatrix}\n3 &amp; 0 &amp; 2 \\\\\n5 &amp; 1 &amp; 7 \\\\\n4 &amp; 5 &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\">\\[cofactor\\ of\\ 2\\ =\\ (-1)^{1\\ +\\ 3}\\ \\begin{vmatrix}\n5 &amp; 1 \\\\\n4 &amp; 5 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1)^4 (25\\  -\\ 4)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 1(21)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color{green}{\\boxed{cofactor\\ of\\ 2\\ =\\  21}}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[5.\\ \\color{green}{Convert\\ 30^0\\  to\\  radians}\\ \\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}\\ 30^0\\ \\times \\frac{\\pi\\ radians}{180^0}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\hspace{2cm}\\ \\frac{\\pi}{6}\\ radians\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[6.\\ \\color{green}{Write\\ any\\ two\\ characteristics\\ of\\ the\\ function\\ y\\ =\\  sin\\ 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{Periodicity}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ The\\ sine\\ function\\ is\\ periodic\\ with\\ a\\ period\\ of\\ 2\\ \\pi.\\  This\\ means\\ that\\ the\\ \\hspace{2cm}\\\\ function\\  repeats\\ its\\ values\\ every\\ 2\\ \\pi\\ nuts.\\ Mathematically\\ sin(x\\ +\\  2\\ \\pi)\\ =\\ sin(x)\\ for\\ all\\ values\\ of\\ x.\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ 2.\\ \\textbf{Range and amplitude}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ \\text{The range of the sine function is the set of all possible values it can}\\ \\hspace{2cm}\\\\ \\hspace{2cm}\\ \\text{take, which is [1, -1]. This means that sinx oscillates between -1 and}\\ \\hspace{2cm}\\\\\\hspace{2cm}\\ \\text{1.  The amplitude, which is the maximum absolute value of the function, is 1.}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[7.\\ \\color{green}{Find\\ the\\ value\\ of\\  sin\\ {40}^0\\  cos\\ {20}^0\\ +\\ cos\\ {40}^0\\ sin\\ {20}^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\">\\[sin\\ {40}^0\\  cos\\ {20}^0\\ +\\ cos\\ {40}^0\\ sin\\ {20}^0\\ =\\ Sin\\ ({40}^0\\ + {20}^0)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ Sin \\ {60}^0\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{\\sqrt{3}}{2}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color{green}{\\boxed{\\therefore\\ sin\\ {40}^0\\  cos\\ {20}^0\\ +\\ cos\\ {20}^0\\ sin\\ {40}^0\\ =\\ \\frac{\\sqrt{3}}{2}}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[8.\\ \\color{green}{Prove\\ that\\  \\frac{sin\\ 2A}{1\\ +\\ 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\\ 2A}{1\\ +\\ cos\\ 2A}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{2\\ sin\\ A\\ cos\\ A}{2\\ cos^2\\ 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}{cos\\ 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\">\\[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}\\ +\\  \\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\">\\[\\hspace{3cm}\\ Given\\ \\hspace{17cm}\\]<\/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}\\ +\\  \\overrightarrow{b}\\ =\\ 4(5\\overrightarrow{i}\\ + 2\\overrightarrow{j}\\ -\\ 3\\overrightarrow{k})\\ +\\  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}\\ +\\   3\\overrightarrow{i}\\ -\\ 2\\overrightarrow{j} +\\ 5\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[4\\overrightarrow{a}\\ +\\  \\overrightarrow{b}\\ =\\ 23\\overrightarrow{i}\\ +\\  6\\overrightarrow{j}\\ -\\ 7\\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\">\\[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\\ 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}\\ (i)\\  \\overrightarrow{k}\\ . \\  \\overrightarrow{i}\\  and\\ (ii)\\ \\overrightarrow{k}\\ \\times\\ \\overrightarrow{i}\\ \\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}\\   (i)\\ \\overrightarrow{k}\\ . \\  \\overrightarrow{i}\\ =\\ 0\\  and\\  (ii)\\ \\overrightarrow{k}\\ \\times\\ \\overrightarrow{i}\\ =\\ \\overrightarrow{j}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[13.\\ \\color{green}{Calculate\\ the\\ arithmetic\\ mean\\ of\\  10,\\ 12,\\ 14,\\ 16\\ and\\ 18}\\ \\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\\ =\\ 5\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Sigma x_i\\ =\\ 10\\ +\\ 12\\ +\\ 14\\ +\\ 16\\ +\\ 18\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Sigma x_i\\ =\\ 70\\]<\/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{70}{5}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bar{x}\\ =\\ 14\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[14.\\ \\color{green}{The\\ arithmetic\\ mean\\ of\\ 6\\ values\\ is\\ 45\\ .\\  If\\ 3\\ is\\ added\\ to\\ each\\ of\\ the\\ \\hspace{5cm}\\\\ numbers,\\ then\\ find\\ the\\ arithmetic\\ mean\\ of\\ the\\ new\\ set\\ of\\ values.}\\ \\hspace{3cm}\\]<\/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{3cm}\\  Given\\ n\\ =\\ 6\\ and\\ \\bar{x}\\ =\\ 45\\ \\hspace{14cm}\\]<\/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\">\\[45\\ =\\ \\frac{\\Sigma X}{6}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\therefore\\ \\Sigma X\\ =\\ 270\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[corrected\\ \\Sigma X\\ =\\ 270\\ +\\ 18\\]\n<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[corrected\\ \\Sigma X\\ =\\ 288\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[correct\\ mean\\ =\\ \\frac{288}{6}\\ =\\ 48\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[15.\\ \\color{green}{\\text{If the variance of a data is 100, then find its standard deviation}}\\ \\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{3cm}\\ Given\\ variance\\ =\\ 100\\ \\hspace{14cm}\\]<\/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\">\\[\\sigma\\ =\\ \\sqrt{variance}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\sigma\\ =\\ \\sqrt{100}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{3cm}\\ =\\ 10\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ \\boxed{standard\\ deviation\\ =\\ 10}\\]<\/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\\ randomly.\\ Find\\ the\\ probability}\\ \\hspace{8cm}\\]\\[\\color{green}{of\\ getting\\ a\\ queen\\ card}\\  \\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 queens(one from each suit: hearts, diamonds, clubs, and spades).<\/p>\n\n\n\n<p>The probability&nbsp;PP&nbsp;of drawing a queen 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\\ queens}{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 queenfrom 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\\ a\\ prime\\ 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. No. of possible outcomes = 6}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{The prime numbers on a die are 2, 3, and 5. No. of favourable outcomes = 3}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{A standard die has 6 faces numbered from 1 to 6. The prime numbers on a die are 2, 3, and 5.}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{The probability P of rolling a prime number can be calculated as follows:}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[P(prime\\ number)\\ =\\ \\frac{Number\\ of\\ favourable\\ outcomes}{Number\\ of\\ possible\\ outcomes}\\ =\\ \\frac{3}{6}\\ =\\ \\frac{1}{2}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{So, the probability of getting a prime number on a rolling die is:}\\ \\boxed{\\frac{1}{2}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[19.\\ \\color{green}{If\\ P(A)\\ =\\ 0.42,\\ and\\ P(B)\\ =\\ 0.48},\\ \\hspace{10cm}\\]\\[\\color{green}{find\\ P(\\bar{A})\\ and\\ P(\\bar{B})}\\ \\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\">\\[W.\\ K.\\ T\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[1.\\ P(\\bar{A})\\ =\\ 1\\ -\\ P(A)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2.\\ P(\\bar{B})\\ =\\ 1\\ -\\ P(B)\\]<\/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)\\ =\\ 0.42\\ \\hspace{10cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bullet\\ P(B)\\ =\\ 0.48\\ \\hspace{10cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[P(\\bar{A})\\ =\\ 1\\ -\\ P(A)\\ =\\ 1\\ -\\ 0.42\\ =\\ 0.58\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[P(\\bar{B})\\ =\\ 1\\ -\\ P(B)\\ =\\ 1\\ -\\ 0.48\\ =\\ 0.52\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[20.\\ \\color{green}{Find\\ P(A\/B),\\ If\\ P(B)\\ =\\ 0.5,\\ and\\ P(A\\ \\cap\\ B)\\ =\\ 0.2}\\ \\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\">\\[To\\ find\\ the\\ conditional\\ probability\\ P(A\/B),\\ \\text{We use the definitions of conditional probability.}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[P(A\/B)\\ =\\ \\frac{P(A\\ \\cap\\ B)}{P(B)}\\]<\/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(B)\\ =\\ 0.5\\ \\hspace{10cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bullet\\ P(A\\ \\cap\\ B)\\ =\\ \\ 0.2\\ \\hspace{10cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[P(A\/B)\\ =\\ \\frac{P(A\\ \\cap\\ B)}{P(B)}\\ =\\ \\frac{0.2} {0.5}\\ =\\  \\frac{2}{5}\\ =\\ 0.4\\]<\/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}{If\\ A =\\begin{bmatrix}\n1 &amp; 2 &amp; 2 \\\\\n2 &amp; 1 &amp;  2 \\\\\n2 &amp; 2 &amp; 1 \\\\\n\\end{bmatrix}\\ then\\ show\\ that\\ A^2\\ -\\ 4\\ A\\ -\\ 5I\\ =0}\\ \\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; 2 &amp; 2 \\\\\n2 &amp; 1 &amp;  2 \\\\\n2 &amp; 2 &amp; 1 \\\\\n\\end{bmatrix}\\  \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[A^2 =\\begin{bmatrix}\n1 &amp; 2 &amp; 2 \\\\\n2 &amp; 1 &amp;  2 \\\\\n2 &amp; 2 &amp; 1 \\\\\n\\end{bmatrix}\\  \\begin{bmatrix}\n1 &amp; 2 &amp; 2 \\\\\n2 &amp; 1 &amp;  2 \\\\\n2 &amp; 2 &amp; 1 \\\\\n\\end{bmatrix}\\  \\hspace{12cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[  = \\begin{bmatrix}\n1\\ \u00d7\\ 1\\ +\\ 2 \u00d7\\ 2\\ +\\ 2\\ \u00d7\\ 2 &amp; 1\\ \u00d7\\ 2\\ +\\ 2 \u00d7\\ 1\\ +\\ 2\\ \u00d7\\ 2 &amp; 1\\ \u00d7\\ 2\\ +\\ 2\\ \u00d7\\ 2\\ +\\ 2\\ \u00d7\\ 1\\\\\n2\\ \u00d7\\ 1\\ +\\ 1 \u00d7\\ 2\\ +\\ 2\\ \u00d7\\ 2\\ &amp; 2\\ \u00d7\\ 2\\ +\\ 1 \u00d7\\ 1\\ +\\ 2\\ \u00d7\\ 2\\ &amp; 2\\ \u00d7\\ 2\\ +\\ 1 \u00d7\\ 2\\ +\\ 2\\ \u00d7\\ 1\\\\\n2\\ \u00d7\\ 1\\ +\\ 2 \u00d7\\ 2\\ +\\ 1\\ \u00d7\\ 2\\ &amp; 2\\ \u00d7\\ 2\\ +\\ 2 \u00d7\\ 1\\ +\\ 1\\ \u00d7\\ 2\\ &amp; 2\\ \u00d7\\ 2\\ +\\ 2 \u00d7\\ 2\\ +\\ 1\\ \u00d7\\ 1\\\\\n\\end{bmatrix}\\  \\hspace{9cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[  = \\begin{bmatrix}\n1\\ +\\ 4\\ +\\ 4\\ &amp; 2\\ +\\ 2\\ +\\ 4 &amp; 2\\ +\\ 4  +\\ 2\\\\\n2\\ +\\ 2\\ +\\ 4\\ &amp; 4\\ +\\ 1\\ +\\ 4\\ &amp; 4\\ +\\ 2\\ +\\ 2\\\\\n2\\ +\\ 4\\ +\\ 2\\ &amp; 4\\ +\\  2\\ +\\ 2\\ &amp; 4\\ +\\ 4\\ +\\ 1\\\\\n\\end{bmatrix}\\  \\hspace{9cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[A^2 = \\begin{bmatrix}\n9 &amp; 8 &amp; 8 \\\\\n8 &amp; 9 &amp; 8 \\\\\n8 &amp; 8 &amp; 9 \\\\\n\\end{bmatrix}\\  \\hspace{12cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[4\\ A\\ =\\ 4\\ \\begin{bmatrix}\n1 &amp; 2 &amp; 2 \\\\\n2 &amp; 1 &amp;  2 \\\\\n2 &amp; 2 &amp; 1 \\\\\n\\end{bmatrix}\\  \\hspace{12cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[4\\ A\\ = \\begin{bmatrix}\n4 &amp; 8 &amp; 8 \\\\\n8 &amp; 4 &amp; 8 \\\\\n8 &amp; 8 &amp; 4 \\\\\n\\end{bmatrix}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[5\\ I\\ =\\ 5\\ \\begin{bmatrix}\n1 &amp; 0 &amp; 0 \\\\\n0 &amp; 1 &amp;  0 \\\\\n0 &amp; 0 &amp; 1 \\\\\n\\end{bmatrix}\\  \\hspace{12cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[5\\ I\\ = \\begin{bmatrix}\n5 &amp; 0 &amp; 0 \\\\\n0 &amp; 5 &amp; 0 \\\\\n0 &amp; 0 &amp; 5 \\\\\n\\end{bmatrix}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[A^2\\ -\\ 4\\ A\\ -\\ 5I\\ =\\ \\begin{bmatrix}\n9 &amp; 8 &amp; 8 \\\\\n8 &amp; 9 &amp; 8 \\\\\n8 &amp; 8 &amp; 9 \\\\\n\\end{bmatrix}\\ -\\ \\begin{bmatrix}\n4 &amp; 8 &amp; 8 \\\\\n8 &amp; 4 &amp; 8 \\\\\n8 &amp; 8 &amp; 4 \\\\\n\\end{bmatrix}\\ -\\ \\begin{bmatrix}\n5 &amp; 0 &amp; 0 \\\\\n0 &amp; 5 &amp; 0 \\\\\n0 &amp; 0 &amp; 5 \\\\\n\\end{bmatrix}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[  = \\begin{bmatrix}\n9\\ -\\ 4\\ -\\ 5\\ &amp; 8\\ -\\ 8\\ -\\ 0 &amp; 8\\ -\\ 8  -\\ 0\\\\\n8\\ -\\ 8\\ -\\ 0\\ &amp; 9\\ -\\ 4\\ -\\ 5\\ &amp; 8\\ -\\ 8\\ -\\ 0\\\\\n8\\ -\\ 8\\ -\\ 0\\ &amp; 8\\ -\\  8\\ -\\ 0\\ &amp; 9\\ -\\ 4\\ -\\ 5\\\\\n\\end{bmatrix}\\  \\hspace{9cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= \\begin{bmatrix}\n0 &amp; 0 &amp; 0 \\\\\n0 &amp; 0 &amp; 0 \\\\\n0 &amp; 0 &amp; 0 \\\\\n\\end{bmatrix}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color{green}{\\boxed{\\therefore\\ A^2\\ -\\ 4\\ A\\ -\\ 5I\\ =\\ 0}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ (b)\\ \\hspace{1cm} \\color{green}{Solve\\ the\\ following\\ equations\\ x + y\\ +\\  z\\ =\\ 3,\\ 2x\\ -\\ y\\ +\\ z\\ =\\ 2\\ and}\\  \\hspace{15cm}\\\\ \\color{green}{3\\ x\\ +\\ 2\\ y\\ -\\ 2\\ z\\ =\\ 3\\ using\\ Cramers\\ Rule}\\ \\hspace{17cm}\\]<\/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\">\\[x + y\\ +\\  z\\ =\\ 3\\ &#8212;&#8212;&#8212;&#8212;&#8212; (1) \\hspace{7cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2x\\ -\\ y\\ +\\ z\\ =\\ 2\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[3\\ x\\ +\\ 2\\ y\\ -\\ 2\\ z\\ =\\ 3\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta = \\begin{vmatrix}\n1 &amp; 1 &amp; 1 \\\\\n2 &amp; &#8211; 1 &amp; 1 \\\\\n3 &amp; 2 &amp; -2 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta =1\\begin{vmatrix}\n-1 &amp; 1 \\\\\n2 &amp; &#8211; 2 \\\\\n\\end{vmatrix}\\ -\\ 1\\begin{vmatrix}\n2 &amp; 1 \\\\\n3 &amp; -2 \\\\\n\\end{vmatrix}\\ +\\ 1\\begin{vmatrix}\n2 &amp; -1\\\\\n3 &amp;  2 \\\\\n\\end{vmatrix}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta =\\ 1(2\\ -\\ 2)\\ &#8211; 1 (-4\\ -\\ 3)\\ +\\ 1(4\\ +\\ 3)\\ \n\\hspace{9cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta =1(0)\\ &#8211; 1\\  (-\\ 7)\\ +\\ 1(7)\\ \n\\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta =\\ 0\\ +\\ 7\\  +\\ 7\\ \n\\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\Delta =\\ 14}\\ \n\\hspace{17cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_x = \\begin{vmatrix}\n3 &amp; 1 &amp; 1 \\\\\n2 &amp; -1 &amp; 1 \\\\\n3 &amp; 2 &amp; -2 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_x =\\ 3\\begin{vmatrix}\n-1 &amp; 1 \\\\\n2 &amp; -2 \\\\\n\\end{vmatrix}\\ -\\ 1\\begin{vmatrix}\n2 &amp; 1 \\\\\n3 &amp;  -2 \\\\\n\\end{vmatrix}\\ +\\ 1\\begin{vmatrix}\n2 &amp; -1\\\\\n3 &amp; 2 \\\\\n\\end{vmatrix}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_x\\ =\\ 3(2\\ -\\ 2)\\ -\\ 1 (-4\\ -\\ 3)\\ + \\ 1(4\\ +\\ 3)\\ \n\\hspace{9cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_x =\\ -3(0)\\ &#8211; 1 (-7)\\  +\\ 1 (7)\\ \n\\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_x\\ =\\ 0\\ +\\ 7\\ +\\ 7\\ \n\\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\Delta_x\\ =\\ 14}\\ \n\\hspace{17cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_y = \\begin{vmatrix}\n1 &amp; 3 &amp; 1 \\\\\n2 &amp; 2 &amp; 1 \\\\\n3 &amp; 3 &amp; -2 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_y =1\\begin{vmatrix}\n2 &amp; 1 \\\\\n3 &amp; -2 \\\\\n\\end{vmatrix}\\ -\\ 3\\begin{vmatrix}\n2 &amp; 1 \\\\\n3 &amp; -2 \\\\\n\\end{vmatrix}\\ +\\ 1\\begin{vmatrix}\n2 &amp; 2\\\\\n3 &amp;  3 \\\\\n\\end{vmatrix}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_y =1(-\\ 4\\ -\\ 3)\\ -\\ 3\\ (-\\ 4\\ -\\ 3)\\ +\\ 1(6\\ -\\ 6)\\ \n\\hspace{9cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_y =1(-7)\\ -\\ 3\\ (-\\ 7)\\  +\\ 1(0)\\ \n\\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_y\\ =\\ -\\ 7\\ +\\ 21\\ +\\ 0\\ \n\\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\Delta_y\\ =\\ 14}\\ \n\\hspace{17cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_z = \\begin{vmatrix}\n1 &amp; 1 &amp; 3 \\\\\n2 &amp; -1 &amp; 2 \\\\\n3 &amp; 2 &amp; 3 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_z =1\\begin{vmatrix}\n-1 &amp; 2 \\\\\n2 &amp; 3 \\\\\n\\end{vmatrix}\\ -\\ 1\\begin{vmatrix}\n2 &amp; 2 \\\\\n3 &amp; 3 \\\\\n\\end{vmatrix}\\ +\\ 3\\begin{vmatrix}\n2 &amp; -1\\\\\n3 &amp;  2 \\\\\n\\end{vmatrix}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_z\\ =\\ 1(-\\ 3\\ -\\ 4)\\ -\\ 1(6\\ -\\ 6)\\  +\\ 3(4\\ +\\ 3)\\ \n\\hspace{9cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_z\\ =\\ 1(-\\ 7)\\ -\\ 1 (0)\\ +\\ 3(7)\\ \n\\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_z =\\ -\\ 7\\ -\\ 0\\ +\\ 21\\ \n\\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\Delta_z\\ =\\ 14}\\ \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{14}{14} =\\ 1\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[y=\\ \\frac{\\Delta_y}{\\Delta} =\\ \\frac{14}{14} =\\ 1\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[z=\\ \\frac{\\Delta_z}{\\Delta} =\\ \\frac{14}{14}\\ =\\ 1\\ \\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\\ =\\ 1\\ and\\ z\\ =\\ 1\\ in\\ equation (1)\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[LHS\\ =\\ 1 +\\ 1\\ +\\ 1\\]\\[ =\\ 3\\]\\[ = 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{5}{13},\\  then\\ find\\ 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{5}{13}\\ \\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{5}{13})^2\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{3cm}\\ =\\  1\\ -\\  \\frac{25}{169}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{3cm}\\ =\\ \\frac{169\\ -\\ 25}{169}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{3cm}\\ =\\ \\frac{144}{25}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ \\boxed{sin\\ \\theta\\ =\\ \\pm\\ \\frac{12}{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{12}{5}}{\\frac{5}{13}}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ \\boxed{tan\\ \\theta\\ =\\ \\pm\\ \\frac{12}{13}}\\ \\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{12}{13}}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ \\boxed{cot\\ \\theta\\ =\\ \\pm\\ \\frac{13}{12}}\\ \\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{5}{13}}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ \\boxed{sec\\ \\theta\\ =\\ \\frac{13}{5}}\\ \\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{12}{5}}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ \\boxed{sec\\ \\theta\\ =\\ \\pm\\ \\frac{5}{12}}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ (b)\\ \\hspace{1cm} \\color{green}{If\\ A\\ and\\ B\\ are\\ acute\\ angles\\ such\\ that\\ sin\\ A\\ =\\ \\frac{8}{17} \\  and\\ sin\\ B\\ =\\ \\frac{5}{13},}\\\\ \\hspace{3cm}\\ \\color{green}{then\\ prove\\ that\\ sin(A\\ +\\ B)\\ =\\ \\frac{171}{221}}\\ \\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\">\\[Given\\ Sin\\ A\\ =\\ \\frac{8}{17} \\  and\\ Sin\\ B\\ =\\ \\frac{5}{13}\\  \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[W.\\ K.\\ T\\ Sin ( A + B )\\  =\\  Sin A\\  Cos B\\ +\\  Cos A\\  Sin B\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Cos\\ A\\ =\\ ?\\ ,\\ Cos\\ B\\ = ?\\  \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Cos\\  A\\ =\\ \\sqrt{1\\ -\\ Sin^2\\ A}\\  \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ \\sqrt{1\\ -\\ (\\frac{8}{17})^2}\\  \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ \\sqrt{1\\ -\\ \\frac{64}{289}}\\  \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ \\sqrt{\\frac{289\\ -\\ 64}{289}}\\  \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ \\sqrt{\\frac{225}{289}}\\  \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Cos\\ A\\ =\\ \\frac{15}{17}\\  \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Cos\\  B\\ =\\ \\sqrt{1\\ -\\ Sin^2\\ B}\\  \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ \\sqrt{1\\ -\\ (\\frac{5}{13})^2}\\  \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ \\sqrt{1\\ -\\ \\frac{25}{169}}\\  \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ \\sqrt{\\frac{169\\ -\\ 25}{169}}\\  \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ \\sqrt{\\frac{144}{169}}\\  \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Cos\\ B\\ =\\ \\frac{12}{13}\\  \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Sin ( A + B )\\  =\\  (\\frac{8}{17})\\  (\\frac{12}{13})\\ +\\  (\\frac{15}{17})\\  (\\frac{5}{13})\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{96}{221}\\ +\\ \\frac{75}{221}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{96\\ +\\ 75}{221}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{171}{221}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed {\\therefore\\ Sin ( A + B )\\ =\\ \\frac{171}{221}}\\ \\hspace{10cm}\\]<\/div>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/3iLnBG5gZN0?list=PLQIom4Rz29vyq1oqZYXCXY1iwGHslbqzY\" title=\"Determining the trigonometrical problems using Sin (A + B) formula - 4\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\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}{4\\overrightarrow{i}\\ + 2\\overrightarrow{j}\\ +\\ 3\\overrightarrow{k},  2\\overrightarrow{i}\\ +\\ 3\\overrightarrow{j}\\ +\\ 4\\overrightarrow{k} and\\ 3\\overrightarrow{i}\\ +\\ 4\\overrightarrow{j}\\ +\\ 2\\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}= 4\\overrightarrow{i}\\ + 2\\overrightarrow{j}+ 3\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{OB}= 2\\overrightarrow{i}\\ +3\\overrightarrow{j} + 4\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{OC}= 3\\overrightarrow{i}\\ + 4\\overrightarrow{j} + 2\\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}+ 3\\overrightarrow{j} + 4\\overrightarrow{k}- (4\\overrightarrow{i}+ 2\\overrightarrow{j}+ 3\\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=2\\overrightarrow{i}+ 3\\overrightarrow{j} + 4\\overrightarrow{k}- 4\\overrightarrow{i}- 2\\overrightarrow{j}- 3\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{AB}=  -2\\overrightarrow{i} +\\overrightarrow{j} + \\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[AB =\\overrightarrow{|AB|} = \\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\">\\[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\">\\[=3\\overrightarrow{i}+ 4\\overrightarrow{j} + 2\\overrightarrow{k}- (2\\overrightarrow{i}+ 3\\overrightarrow{j}+ 4\\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=3\\overrightarrow{i}+ 4\\overrightarrow{j} + 2\\overrightarrow{k}- 2\\overrightarrow{i}- 3\\overrightarrow{j}- 4\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{BC}=  \\overrightarrow{i} +\\overrightarrow{j} -2\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[BC =\\overrightarrow{|BC|} = \\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\">\\[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\">\\[=3\\overrightarrow{i}+ 4\\overrightarrow{j} + 2\\overrightarrow{k}- (4\\overrightarrow{i}\\ + 2\\overrightarrow{j}+ 3\\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=3\\overrightarrow{i}+ 4\\overrightarrow{j} + 2\\overrightarrow{k}- 4\\overrightarrow{i}- 2\\overrightarrow{j}- 3\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{AC}=  -\\overrightarrow{i} +2\\overrightarrow{j} -\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[AC =\\overrightarrow{|AC|} = \\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\">\\[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<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/OsM0lZ2G2FA\" title=\"Vector Introduction - Part - 8\" 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}\\ (b)\\ \\hspace{1cm} \\color{green}{Show\\ that\\ the\\ vectors\\ 2\\overrightarrow{i}\\ +\\ 2\\overrightarrow{j}\\ + \\overrightarrow{k},\\  \\overrightarrow{i}\\ -\\ 2\\overrightarrow{j}\\ + 2\\overrightarrow{k}\\ \\hspace{10cm}\\\\ and\\\n  2\\overrightarrow{i}\\ -\\overrightarrow{j}\\ -\\ 2\\overrightarrow{k} are\\ mutually\\ perpendicular.}\\ \\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\">\\[\\overrightarrow{a}\\ =\\ 2\\overrightarrow{i}\\ +\\ 2\\overrightarrow{j}\\ + \\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{b}\\ =\\ \\overrightarrow{i}\\ -\\ 2\\overrightarrow{j}\\ + 2\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{c}\\ =\\ 2\\overrightarrow{i}\\ -\\overrightarrow{j}\\ -\\ 2\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\overrightarrow{a}.\\overrightarrow{b}= (2\\overrightarrow{i}\\ +\\ 2\\overrightarrow{j}\\ + \\overrightarrow{k}) .(\\overrightarrow{i}\\ -\\ 2\\overrightarrow{j}\\ + 2\\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 2(1)\\ +\\ 2(- 2)\\ +\\ 1(2)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 2\\ -\\ 4\\ +\\ 2\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 0\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\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}\\ -\\ 2\\overrightarrow{j}\\ + 2\\overrightarrow{k}) .(2\\overrightarrow{i}\\ -\\overrightarrow{j}\\ -\\ 2\\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 1(2)\\ +\\ -\\ 2(-1)\\ +\\ 2(-2)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 2\\ +\\ 2 -\\ 4\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 0\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\overrightarrow{b} and\\ \\overrightarrow{c}\\  are\\  perpendicular\\  vectors}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\overrightarrow{c}.\\overrightarrow{a}= (2\\overrightarrow{i}\\ -\\overrightarrow{j}\\ -\\ 2\\overrightarrow{k}) .(2\\overrightarrow{i}\\ +\\ 2\\overrightarrow{j}\\ + \\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 2(2)\\ +\\ -\\ 1(2)\\ +\\ -\\ 2(1)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 4\\ -\\ 2 -\\ 2\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 0\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\overrightarrow{c} and\\ \\overrightarrow{a}\\  are\\  perpendicular\\  vectors}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\therefore\\ The\\ three\\ vectors\\  are\\ mutually\\ perpendicular.\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ (c)\\ \\hspace{1cm} \\color{green}{Find\\ the\\ area\\ of\\ the\\ triangle\\ whose\\ adjacent\\ sides\\ are\\ 3\\overrightarrow{i}\\ +\\  \\overrightarrow{j}\\ +\\ 2\\overrightarrow{k}}\\ \\hspace{8cm}\\\\ \\color{green}{and\\  \\overrightarrow{i}\\ -\\  2\\overrightarrow{j}\\ +\\ 4\\overrightarrow{k}}\\ \\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\">\\[\\overrightarrow{a}\\ =\\ 3\\overrightarrow{i}\\ +\\  \\overrightarrow{j}\\ +\\ 2\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{b}\\ =\\  \\overrightarrow{i}\\ -\\  2\\overrightarrow{j}\\ +\\ 4\\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}\\\\\n3 &amp;  1 &amp; 2\\\\\n1 &amp; -2 &amp; 4\\\\\n\\end{vmatrix}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\overrightarrow{i}(1 . 4\\ -\\ 2(-2))\\ -\\ \\overrightarrow{j}(3 . 4\\ -\\ 2(1))\\ +\\ \\overrightarrow{k}(3 . (-2)\\ -\\ 1(1))\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\overrightarrow{i}(4\\ +\\ 4)\\  -\\ \\overrightarrow{j}(12\\ -\\ 2)\\ +\\ \\overrightarrow{k}(-6\\ -\\ 1)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\overrightarrow{i}(8)\\  -\\ \\overrightarrow{j}(10)\\ +\\ \\overrightarrow{k}(-7)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\overrightarrow{a}\u00d7 \\overrightarrow{b}\\ =\\ 8\\overrightarrow{i}\\  -\\ 10\\overrightarrow{j}\\ -\\ 7\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ Area\\ of \\ triangle\\ =\\ \\frac{1}{2} |\\overrightarrow{a}\\ \u00d7\\  \\overrightarrow{b}|\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\  \\frac{1}{2} \\sqrt{((8)^2\\ +\\ (-10)^2 + (-7)^2 }\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\  \\frac{1}{2}\\ \\sqrt{(64\\ +\\ 100\\ +\\ 49)}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{1}{2}\\ \\sqrt{213}\\]<\/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 class=\"has-fixed-layout\"><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><\/tr><tr><td>Frequency<\/td><td>2<\/td><td>6<\/td><td>9<\/td><td>7<\/td><td>4<\/td><td>2<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\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>2<\/td><td class=\"has-text-align-left\" data-align=\"left\">5<\/td><td>10<\/td><\/tr><tr><td>10-20<\/td><td>6<\/td><td class=\"has-text-align-left\" data-align=\"left\">15<\/td><td>90<\/td><\/tr><tr><td>20-30<\/td><td>9<\/td><td class=\"has-text-align-left\" data-align=\"left\">25<\/td><td>225<\/td><\/tr><tr><td>30-40<\/td><td>7<\/td><td class=\"has-text-align-left\" data-align=\"left\">35<\/td><td>245<\/td><\/tr><tr><td>40-50<\/td><td>4<\/td><td class=\"has-text-align-left\" data-align=\"left\">45<\/td><td>220<\/td><\/tr><tr><td>50-60<\/td><td>2<\/td><td class=\"has-text-align-left\" data-align=\"left\">55<\/td><td>110<\/td><\/tr><tr><td><\/td><td>N = 30<\/td><td class=\"has-text-align-left\" data-align=\"left\"><\/td><td><mathml>\\[\\Sigma f_i\\ x_i\\ =\\ 860\\]<\/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{860}{30}\\ =\\ 28.66\\]<\/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}{Fit\\ a\\ straight\\ line\\ to\\ the\\ following\\ data\\ by\\ the\\ method\\ of\\ least\\ squares}\\ \\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>1<\/td><td>1<\/td><td>3<\/td><td>4<\/td><td>6<\/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>1<\/td><td>0<\/td><td>0<\/td><\/tr><tr><td>1<\/td><td>1<\/td><td>1<\/td><td>1<\/td><\/tr><tr><td>2<\/td><td>3<\/td><td>4<\/td><td>6<\/td><\/tr><tr><td>3<\/td><td>4<\/td><td>9<\/td><td>12<\/td><\/tr><tr><td>4<\/td><td>6<\/td><td>16<\/td><td>24<\/td><\/tr><tr><td><mathml>\\[\\Sigma x_i\\ =\\ 10\\]<\/mathml><\/td><td><mathml>\\[\\Sigma y_i\\ =\\ 15\\]<\/mathml><\/td><td><mathml>\\[\\Sigma {x_i}^2\\ =\\ 30\\]<\/mathml><\/td><td><mathml>\\[\\Sigma x_i\\ y_i\\ =\\ 43\\]<\/mathml><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(2)\\ becomes\\ a\\ (10)\\ +\\ 5b\\ =\\ 15\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2\\ a\\ +\\ b\\ =\\ 3\\ &#8212;&#8212;&#8211;\\ (4)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(3)\\ becomes\\ a\\ (30)\\ +\\ b\\ (10)\\ =\\ 43\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[3\\ a\\ +\\ b\\ =\\ 4.3\\ &#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\\ =\\ 3\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[3\\ a\\ +\\ b\\ =\\ 4.3\\]<\/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\\ =\\ 1.3\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[put\\ a\\ =\\ 1.3\\ in\\ (4)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2(1.3)\\ +\\ b\\ =\\ 3\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2.6\\ +\\ b\\ =\\ 3\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[b\\ =\\ 3\\ -\\ 2.6\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[b\\ =\\ 0.4\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Eqn\\ (1)\\ becomes\\ y\\ =\\ 1.3\\ x\\ +\\ 0.4\\]<\/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}\\]\\[(i)\\ at\\ least\\ one\\ head\\ \\hspace{13cm}\\]\\[(ii)\\ at\\ most\\ one\\ head\\ \\hspace{13cm}\\]\\[(iii)\\ exactly\\ one\\ head.\\ \\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}{at\\ least\\ one\\ head}}\\ \\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{To find the probability of getting atleast one head, we need to count the}\\ \\hspace{5cm}\\]\\[\\text{outcomes where there are either one, 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{{HTT, HHT, HTH, THH, HHH}}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[P(at\\ least\\ one\\ head)\\ =\\ \\frac{\\text{Number of favourable outcomes}}{\\text{Total Number of outcomes}}\\ =\\ \\frac{5}{8}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(ii)\\ \\bf{\\color{green}{at\\ most\\ one\\ head}}\\ \\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{To find the probability of getting at most one head, we need to count the}\\ \\hspace{5cm}\\]\\[\\text{outcomes where there is zero or 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{{TTT, HTT}}\\ \\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{2}{8}\\ =\\ \\frac{1}{4}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(iii)\\ \\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\">\\[\\hspace{2cm}\\ (b)\\ \\hspace{1cm} \\color{green}{A\\ card\\ is\\ drawn\\ at\\ random\\ from\\ a\\ pack\\ of\\ 52\\ cards.\\ Find\\ the}\\ \\hspace{8cm}\\]\\[\\color{green}{probability\\ that\\ the\\ card\\ is\\ either\\ a\\  black\\ card\\ or\\ a\\ card\\ with\\ number\\ 6.}\\ \\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\">\\[\\text{To find the probability of drawing either a black card or a card with the number 6 from a standard deck of 52 cards,}\\\\ \\text{we need to consider both events and their possible overlap.}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[1.\\ \\text{Total number of cards in a deck: 52}\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{3cm}\\ 2.\\ \\text{Number of black cards: There are 26 black cards in the deck (13 spades +}\\\\ \\text{13 clubs}).\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{3cm}\\ 3.\\ \\text{Number of cards with the number 6: There are 4 cards with the number 6}\\\\ \\text{(one in each suit: hearts, diamonds, clubs, and spades}).\\ \\hspace{2cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[A\\ denotes\\ the\\ event\\ of\\ drawing\\ a\\ black\\ card\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[B\\ denotes\\ the\\ event\\ of\\ drawing\\ a\\ card\\ with\\ on\\ number\\ 6\\ \\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{26}{52}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[P(B)\\ =\\ \\frac{4}{52}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[P(P(A\\ \\cap\\ B))\\ =\\ \\frac{2}{52}\\ (2\\ such\\ cards\\ with \\ black\\ card\\ and\\ number\\ 6\\ from\\ spades\\ and\\ clubs)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[P(A\\ \\cup\\ B)\\ =\\ \\frac{26}{52}\\ +\\ \\frac{4}{52}\\ -\\ \\frac{2}{52}\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{28}{52}\\ =\\ \\frac{7}{13}\\ \\hspace{3cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{So, the probability of drawing either a black card or a card with the}\\\\ \\text{number 6 from a standard deck of 52 cards is:}  \\frac{7}{13}\\ \\hspace{5cm}\\]<\/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}{4}\\ and\\ that\\ of\\ B\\ solves\\ the}\\ \\hspace{8cm}\\\\\\ \\color{green}{problem\\ is\\ \\frac{2}{5}.\\ 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\">\n\\[\\color {blue} {Soln:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Given,\\ P(A)\\ =\\ \\frac{1}{4}\\  and\\  P(B)\\ =\\ \\frac{2}{5}\\ \\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}{4}\\ +\\ \\frac{2}{5\n}\\ -\\ \\frac{1}{4}\\ \\cdot \\ \\frac{2}{5}\\ \\hspace{2cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{1}{4}\\ +\\ \\frac{2}{5}\\ -\\ \\frac{1}{10}\\ \\hspace{2cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{5\\ +\\ 8\\ -\\ 2}{20}\\ \\hspace{2cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{11}{20}\\ \\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{11}{20}\\]<\/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. (5 X 14 = 70) ( 7 + 7) The Normal equations are In a standard deck of 52 playing cards, [&hellip;]<\/p>\n","protected":false},"author":187055548,"featured_media":52305,"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","default_image_id":0,"font":"","enabled":false},"version":2},"_wpas_customize_per_network":false,"jetpack_post_was_ever_published":false},"categories":[1],"tags":[],"class_list":["post-47584","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-uncategorized"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>April-2024 TAMIL NADU POLYTECHNIC BOARD EXAM BASIC MATHEMATICS 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=47584\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"April-2024 TAMIL NADU POLYTECHNIC BOARD EXAM BASIC MATHEMATICS QUESTION PAPER WITH SOLUTIONS - YANAMTAKSHASHILA\" \/>\n<meta property=\"og:description\" content=\"1. 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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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