{"id":52567,"date":"2024-11-25T10:46:10","date_gmt":"2024-11-25T05:16:10","guid":{"rendered":"https:\/\/yanamtakshashila.com\/?p=52567"},"modified":"2024-12-10T20:00:16","modified_gmt":"2024-12-10T14:30:16","slug":"october-2024-tamil-nadu-polytechnic-board-exam-basic-mathematics-question-paper-held-on-20-11-2024with-solutions","status":"publish","type":"post","link":"https:\/\/yanamtakshashila.com\/?p=52567","title":{"rendered":"October-2024 TAMIL NADU POLYTECHNIC BOARD EXAM BASIC MATHEMATICS QUESTION PAPER (held on 20-11-2024)WITH SOLUTIONS"},"content":{"rendered":"\n<figure class=\"wp-block-gallery has-nested-images columns-default is-cropped wp-block-gallery-1 is-layout-flex wp-block-gallery-is-layout-flex\">\n<figure class=\"wp-block-image size-large\"><img data-recalc-dims=\"1\" fetchpriority=\"high\" decoding=\"async\" width=\"630\" height=\"840\" data-attachment-id=\"52570\" data-permalink=\"https:\/\/yanamtakshashila.com\/?attachment_id=52570\" 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13&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;1732119640&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;5.1&quot;,&quot;iso&quot;:&quot;64&quot;,&quot;shutter_speed&quot;:&quot;0.02&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;1&quot;}\" data-image-title=\"IMG_0562\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2024\/11\/IMG_0562-rotated.jpg?fit=630%2C840&amp;ssl=1\" data-id=\"52569\" src=\"https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2024\/11\/IMG_0562.jpg?resize=630%2C840&#038;ssl=1\" alt=\"\" class=\"wp-image-52569\" srcset=\"https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2024\/11\/IMG_0562-rotated.jpg?resize=630%2C840&amp;ssl=1 630w, https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2024\/11\/IMG_0562-rotated.jpg?resize=274%2C365&amp;ssl=1 274w, 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https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2024\/11\/IMG_0563-rotated.jpg?resize=274%2C365&amp;ssl=1 274w, https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2024\/11\/IMG_0563-rotated.jpg?resize=768%2C1024&amp;ssl=1 768w, https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2024\/11\/IMG_0563-rotated.jpg?resize=1152%2C1536&amp;ssl=1 1152w, https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2024\/11\/IMG_0563-rotated.jpg?resize=1536%2C2048&amp;ssl=1 1536w, https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2024\/11\/IMG_0563-rotated.jpg?resize=900%2C1200&amp;ssl=1 900w, https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2024\/11\/IMG_0563-rotated.jpg?resize=600%2C800&amp;ssl=1 600w, https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2024\/11\/IMG_0563-rotated.jpg?resize=450%2C600&amp;ssl=1 450w, https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2024\/11\/IMG_0563-rotated.jpg?resize=300%2C400&amp;ssl=1 300w, https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2024\/11\/IMG_0563-rotated.jpg?resize=150%2C200&amp;ssl=1 150w, https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2024\/11\/IMG_0563-rotated.jpg?resize=1200%2C1600&amp;ssl=1 1200w, https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2024\/11\/IMG_0563-rotated.jpg?w=2000&amp;ssl=1 2000w, https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2024\/11\/IMG_0563-rotated.jpg?w=3000&amp;ssl=1 3000w\" sizes=\"(max-width: 630px) 100vw, 630px\" \/><\/figure>\n<\/figure>\n\n\n\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{pmatrix}\n1 &amp; 2 \\\\\n3 &amp; 5\\\\\n\\end{pmatrix}\\  and\\ B =\\begin{pmatrix}\n&#8211; 5 &amp; 7 \\\\\n0 &amp; 4\\\\\n\\end{pmatrix}},\\ \\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{pmatrix}\n1 &amp; 2 \\\\\n3 &amp; 5\\\\\n\\end{pmatrix}\\  \\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2A = \\begin{pmatrix}\n2 &amp; 4 \\\\\n6 &amp; 10\\\\\n\\end{pmatrix}\\ &#8212;&#8212;&#8212;- (1)\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[B = \\begin{pmatrix}\n-5 &amp; 7 \\\\\n0 &amp; 4\\\\\n\\end{pmatrix}\\ &#8212;&#8212;&#8212;- (2)\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2A\\ +\\ B = \\begin{pmatrix}\n2 &amp; 4 \\\\\n6 &amp; 10\\\\\n\\end{pmatrix}\\ +\\ \\begin{pmatrix}\n-5 &amp; 7 \\\\\n0 &amp; 4\\\\\n\\end{pmatrix}\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\begin{pmatrix}\n2\\ -\\ 5 &amp; 4\\ +\\ 7 \\\\\n6\\ +\\ 0 &amp; 10\\  +\\ 4\\\\\n\\end{pmatrix}\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2A\\ +\\ B = \\begin{pmatrix}\n-3 &amp; 11 \\\\\n6 &amp; 14\\\\\n\\end{pmatrix}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2. \\ \\color{green}{Find\\ the\\ cofactor\\ of\\ &#8216;2&#8217;\\ in}\\  \\begin{pmatrix}\n1 &amp; 0 &amp; &#8211; 1 \\\\\n2 &amp; 3 &amp; 4 \\\\\n7 &amp; 8 &amp; &#8211; 2 \\\\\n\\end{pmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ \\hspace{22cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 2\\ =\\ (-1)^{2\\ +\\ 1}\\ \\begin{vmatrix}\n0 &amp; &#8211; 1 \\\\\n8 &amp; &#8211; 2 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1)^3 (0\\  -\\ (- 8)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ &#8211; 1(8)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color{green}{\\boxed{cofactor\\ of\\ 2\\ =\\  &#8211; 8}}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[3.\\ \\color{green}{Prove\\ that\\ the\\ matrix \\begin{pmatrix}\n1 &amp; &#8211; 2 \\\\\n&#8211; 2 &amp; 4\\\\\n\\end{pmatrix}\\ is\\ a\\ singular\\ matrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\  Given\\ A =\\begin{pmatrix}\n1 &amp; &#8211; 2 \\\\\n&#8211; 2 &amp; 4\\\\\n\\end{pmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\begin{vmatrix}\nA \\\\\n\\end{vmatrix}\\ =\\ 1(4) &#8211; (-2)(- 2) \\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 4\\ -\\ 4 \\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\\ adjoint\\ of\\ \\begin{pmatrix}\n1 &amp; 5 \\\\\n3 &amp; 6 \\\\\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:}\\ A\\ =\\begin{pmatrix}\n1 &amp; 5 \\\\\n3 &amp; 6 \\\\\n\\end{pmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 1 = (-1)^{1\\ +\\ 1}\\ (6)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (1) (6)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 1\\ = \\ 6\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 5 = (-1)^{1\\ +\\ 2}\\ (3)\\ \\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\\ 3\\ = (-1)^{2\\ +\\ 1}\\ (5)\\ \\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\\ 3\\ =\\ -\\  5\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 6\\ = (-1)^{2\\ +\\ 2}\\ (1)\\ \\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\\ 6\\ = 1\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\therefore\\ cofactor\\ matrix\\ = \\begin{pmatrix}\n6 &amp; -\\ 3 \\\\\n-\\ 5 &amp; 1 \\\\\n\\end{pmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Adj.\\ A = \\begin{pmatrix}\n6 &amp; -\\ 5 \\\\\n-\\ 3 &amp; 1 \\\\\n\\end{pmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[5.\\ \\color{green}{Convert\\ \\frac{\\pi}{4}\\  in\\ to\\  degrees}\\ \\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}\\ \\frac{\\pi}{4}\\ \\times \\frac{180^0}{\\pi}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\hspace{2cm}\\ 45^0\\]<\/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}{Find\\ the\\ value\\ of\\  \\frac{Tan\\ {25}^0\\  +\\ Tan\\ {20}^0}{1\\ -\\ Tan\\ {25}^0\\ Tan\\ {20}^0}}\\  \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ W.\\ K.\\ T\\ \\frac{Tan A\\ +\\ Tan B}{1\\ -\\ Tan A\\ Tan B}\\  =\\  Tan(A + B)\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{Tan\\ {25}^0\\  +\\ Tan\\ {20}^0}{1\\ -\\ Tan\\ {25}^0\\ Tan\\ {20}^0}\\ =\\ Tan({25}^0\\  +\\  {20}^0)\\ =\\ Tan{45}^0\\ =\\ 1\\ \\hspace{10cm}\\]<\/div>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/3FP6c77U_OY\" title=\"Determining the trigonometrical problem using tan (A +B) formula - 1\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[8.\\ \\color{green}{Find\\ the\\ value\\ of\\  2\\ Sin\\ 15^{0}\\ Cos\\ 15^{0}}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ 2\\ Sin\\ 15^{0}\\ Cos\\ 15^{0}\\ =\\ Sin\\ 2(15^{0})\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\  Sin\\ 30^{0}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{1}{2}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed {2\\ Sin\\ 30^{0}\\ Cos\\ 30^{0}\\ =\\ \\frac{1}{2}}\\]<\/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},}\\ \\color {green} {find\\   the\\ value\\ of\\ 4\\overrightarrow{a}\\ +\\  \\overrightarrow{b}}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ Given\\ \\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}\\ +\\  \\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\\  3\\overrightarrow{i}\\ + 2\\overrightarrow{j}\\ +\\ 4\\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}\\ =\\ 3\\overrightarrow{i}\\ + 2\\overrightarrow{j}\\ +\\ 4\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[r =\\overrightarrow{|a|} = \\sqrt{(3)^2 + (2)^2 + (4)^2 }\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\sqrt{(9\\ +\\ 4\\ +\\ 16) }\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[r =\\sqrt{29}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ Direction\\  cosines\\  are \\frac{3}{\\sqrt(29)},  \\frac{2}{\\sqrt(29)},    \\frac{4}{\\sqrt(29)}  \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\n\\[11.\\ \\color{green}{Find\\ the\\ projection\\ of\\ the\\ vector\\ 2\\overrightarrow{i}\\ + \\overrightarrow{j}\\ -\\ 2\\overrightarrow{k} on\\   \\overrightarrow{i}\\ -\\ 2\\overrightarrow{j}\\ -\\ 2\\overrightarrow{k}} \\ \\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}\\ +\\ \\overrightarrow{j}\\ -\\ 2\\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\">\\[Projection\\  of\\ \\overrightarrow{a} on \\overrightarrow{b} =\\frac{\\overrightarrow{a}.\\overrightarrow{b}}{\\overrightarrow{|b|}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{(2\\overrightarrow{i}\\ + \\overrightarrow{j}\\ -\\ 2\\overrightarrow{k}).(\\overrightarrow{i}\\ -\\ 2\\overrightarrow{j}\\ -\\ 2\\overrightarrow{k})}{\\sqrt{(1)^2 + (-2)^2 + (-2)^2 }}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\  \\frac{2(1)\\ +\\ 1(-2)\\ +\\ (-2) (-2)}{\\sqrt{(1 + 4 + 4 }}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=  \\frac{2\\ -\\ 2\\ +\\ 4}{\\sqrt{9}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{Projection\\  of\\ \\overrightarrow{a} on \\overrightarrow{b} =\\frac{4}{3}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[12.\\ \\color{green}{Show\\ that\\ the\\ vectors\\ 4\\overrightarrow{i}\\ -\\  2\\overrightarrow{j}\\ -\\ 6\\overrightarrow{k}\\ and\\ 2\\overrightarrow{i}\\ -\\ \\overrightarrow{j}\\ -\\ 3\\overrightarrow{k}\\  are\\ parallel}.\\ \\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\\ show\\ \\overrightarrow{a}\u00d7\\overrightarrow{b} =\\ 0\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{a}\\ =\\ 4\\overrightarrow{i}\\ -\\  2\\overrightarrow{j}\\ -\\ 6\\overrightarrow{k} \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{b}\\ =\\ 2\\overrightarrow{i}\\ -\\ \\overrightarrow{j}\\ -\\ 3\\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}\\\\\n4 &amp; -2 &amp; -6\\\\\n2 &amp; -1 &amp; -3\\\\\n\\end{vmatrix}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\overrightarrow{I}(6\\ -\\ 6)\\  -\\overrightarrow{j}(-12\\ +\\ 12)\\ +\\ \\overrightarrow{k}(-4\\ +\\ 4)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\overrightarrow{i}(0) -\\overrightarrow{j}(0)+\\overrightarrow{k}(0)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\overrightarrow{a}\u00d7\\overrightarrow{b} =\\ 0}\\ \\hspace{7cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\therefore\\ The\\ given\\ vectors\\ \\overrightarrow{a}\\ and\\ \\overrightarrow{b}\\ are\\ parallel\\]<\/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:}\\ 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\\ 10\\ numbers\\ is\\ 20\\ ,\\  find\\ the\\ sum\\ of\\ the}\\ \\hspace{10cm}\\]\\[\\color{green}{values}\\ \\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\\ down\\ the\\ normal\\ equations\\ to\\ fit\\ a\\ 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}{\\text{ An integer is chosen at random from the integers 1 to 10. }}\\ \\hspace{10cm}\\]\\[\\color {green} {Find\\ the\\ probability\\ that\\ it\\ is\\  an\\ even\\ 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{Given an integer is chosen at random from 1 to 10. No. of possible outcomes = 10}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{The even numbers from 1 to 10 are 2, 4, 6, 8, 10.  No. of favourable outcomes = 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(even\\ number)\\ =\\ \\frac{Number\\ of\\ favourable\\ outcomes}{Number\\ of\\ possible\\ outcomes}\\ =\\ \\frac{5}{10}\\ =\\ \\frac{1}{2}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{So, the probability of getting an even number from the integers 1 to 10 is:}\\ \\boxed{\\frac{1}{2}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[18.\\ \\color{green}{While\\ a\\ die\\ is\\ rolled\\ once,}\\ \\hspace{15cm}\\]\\[\\color {green} {Find\\ the\\ probability\\ of\\ getting\\ an\\ even\\ 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 even numbers on a die are 2, 4, and 6. No. of favourable outcomes = 3}\\]<\/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(even\\ number)\\ =\\ \\frac{No.\\ of\\ favourable\\ outcomes}{No.\\ 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 an odd number on a rolling die is:}\\ \\boxed{\\frac{1}{2}}\\]<\/div>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/SSgiSQEg1Cc\" title=\"Probability-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\">\\[19.\\ \\color{green}{If\\ A\\ and\\ B\\ are\\ two\\ events\\ such\\ that\\ 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}{If\\ A\\ and\\ B\\ are\\ two\\ independent\\ events\\ such\\ that\\ P(A)\\ =\\ 0.4\\ and\\ P(A\\ \\cup\\ B)\\ =\\ 0.9.}\\ \\hspace{10cm}\\]\\[\\color {green} {Find\\ P(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\">\\[Given\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bullet\\ P(A)\\ =\\ 0.4\\ \\hspace{10cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bullet\\ P(A\\ \\cup\\ B)\\ =\\ 0.9\\ \\hspace{10cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bullet\\ P(A\\ \\cap\\ B)\\ =\\ 0\\ (A\\ and\\ B\\ are\\ independent\\ events)\\ \\hspace{5cm}\\] <\/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\">\\[0.9\\ =\\ 0.4\\ +\\ P(B)\\ -\\ 0\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[P(B)\\ =\\ 0.9\\ -\\ 0.4\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{P(B)\\ =\\ 0.5}\\]<\/div>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/_exTvmM8uaM\" title=\"Probability - 11\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\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{pmatrix}\n4 &amp; 6 &amp; 2 \\\\\n0 &amp; 1 &amp; 5 \\\\\n0 &amp; 3 &amp; 2 \\\\\n\\end{pmatrix}\\ ,\\ B =\\begin{pmatrix}\n0 &amp; 1 &amp; &#8211; 1 \\\\\n3 &amp; &#8211; 1 &amp; 4 \\\\\n&#8211; 1 &amp; 2 &amp; 1 \\\\\n\\end{pmatrix}}\\ \\hspace{7cm}\\]\\[\\color {green}{Prove\\ that\\ (A\\ +\\ B)^T\\ =\\ A^T\\ +\\ B^T}\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ Given\\  A =\\ \\begin{pmatrix}\n4 &amp; 6 &amp; 2 \\\\\n0 &amp; 1 &amp; 5 \\\\\n0 &amp; 3 &amp; 2 \\\\\n\\end{pmatrix}\\ ,\\ B =\\begin{pmatrix}\n0 &amp; 1 &amp; &#8211; 1 \\\\\n3 &amp; &#8211; 1 &amp; 4 \\\\\n&#8211; 1 &amp; 2 &amp; 1 \\\\\n\\end{pmatrix}\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[A + B=\\begin{pmatrix}\n4 + 0 &amp; 6 + 1 &amp; 2 &#8211; 1 \\\\\n0 + 3 &amp; 1 &#8211; 1 &amp; 5 + 4 \\\\\n0 &#8211; 1 &amp; 3 + 2 &amp; 2 + 1 \\\\\n\\end{pmatrix}\\ \\hspace{6cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[A + B=\\ \\begin{pmatrix}\n4 &amp; 7 &amp; 1 \\\\\n3 &amp; 0 &amp; 9\\\\\n-1 &amp; 5  &amp; 3\\\\\n\\end{pmatrix}\\ \\hspace{6cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(A + B)^T\\ =\\ \\begin{pmatrix}\n4 &amp; 3 &amp; &#8211; 1 \\\\\n7 &amp; 0 &amp; 5 \\\\\n1 &amp; 9  &amp; 3\\\\\n\\end{pmatrix}\\ \\hspace{2cm}\\ &#8212;&#8212;&#8211; (1)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[A^T =\\begin{pmatrix}\n4 &amp; 0 &amp; 0 \\\\\n6 &amp; 1 &amp; 3 \\\\\n2 &amp; 5 &amp; 2 \\\\\n\\end{pmatrix}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[B^T =\\begin{pmatrix}\n0 &amp; 3 &amp; &#8211; 1 \\\\\n1 &amp; &#8211; 1 &amp; 2 \\\\\n&#8211; 1 &amp; 4 &amp; 1 \\\\\n\\end{pmatrix}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[A^T\\ +\\ B^T\\ =\\begin{pmatrix}\n4 + 0 &amp; 0\\ +\\ 3 &amp;  0\\ -\\ 1\\\\\n6\\ +\\ 1 &amp; 1 &#8211; 1 &amp; 3 + 2 \\\\\n2 &#8211; 1 &amp; 5 + 4 &amp; 2 + 1 \\\\\n\\end{pmatrix}\\ \\hspace{6cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[A^T\\ +\\ B^T\\ =\\begin{pmatrix}\n4 &amp; 3 &amp;  &#8211; 1\\\\\n7 &amp; 0 &amp; 5 \\\\\n1 &amp; 9 &amp; 3 \\\\\n\\end{pmatrix}\\ \\hspace{2cm}\\ &#8212;&#8212;&#8212;- (2)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[From\\ (1)\\ and\\ (2), It\\ is\\ concluded\\ that  (A\\ +\\ B)^T\\ =\\ A^T\\ +\\ B^T\\]<\/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}{4\\ x\\ +\\ y\\ +\\  z\\ =\\ 6,\\ 2\\ x\\ -\\ y\\ -\\  2\\ z\\ =\\ -\\ 6\\ and\\  x\\ +\\ y\\ +\\ z\\ =\\  3}\\ \\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\">\\[4\\ x\\ +\\ y\\ +\\  z\\ =\\ 6\\ &#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\">\\[x\\ +\\ y\\ +\\ z\\ =\\  3\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta = \\begin{vmatrix}\n4 &amp; 1 &amp; 1 \\\\\n2 &amp; -1 &amp; -2\\\\\n1 &amp; 1 &amp; 1 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta =4\\begin{vmatrix}\n-1 &amp; -2 \\\\\n1 &amp; 1 \\\\\n\\end{vmatrix}\\ -\\ 1\\begin{vmatrix}\n2 &amp; -2 \\\\\n1 &amp; 1 \\\\\n\\end{vmatrix}\\ +\\ 1\\begin{vmatrix}\n2 &amp; -1\\\\\n1 &amp; 1 \\\\\n\\end{vmatrix}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta =4(-1\\ +\\ 2)\\ &#8211; 1 (2\\ +\\  2)\\ +\\ 1(2\\ +\\ 1)\\ \n\\hspace{9cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta\\ =\\ 4(1)\\ &#8211; 1 (4)\\ +\\ 1(7)\\ \n\\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta =4\\ -\\ 4\\ +\\  3\\ \n\\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\Delta\\ =\\ 3}\\ \n\\hspace{17cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_x = \\begin{vmatrix}\n6 &amp; 1 &amp; 1 \\\\\n-6 &amp; -1 &amp; -2 \\\\\n3 &amp; 1 &amp; 1 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_x =6\\begin{vmatrix}\n-1 &amp; -2 \\\\\n1 &amp; 1 \\\\\n\\end{vmatrix}\\ -\\ 1\\begin{vmatrix}\n-6 &amp; -2 \\\\\n3 &amp; 1 \\\\\n\\end{vmatrix}\\ +\\ 1\\begin{vmatrix}\n-6 &amp; -1\\\\\n3 &amp; 1 \\\\\n\\end{vmatrix}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_x\\ =\\ 6(-1\\ +\\ 2) &#8211; 1 (\\ -6\\ +\\ 6)\\ +\\ 1(-6\\ +\\ 3)\\ \n\\hspace{9cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_x\\ =\\ 6(1)\\ &#8211; 1 (0)\\ +\\ 1(-3)\\ \n\\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_x = 6\\ +\\ 0\\ -3\\ \n\\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\Delta_x\\ =\\ 3}\\ \n\\hspace{17cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_y = \\begin{vmatrix}\n4 &amp; 6 &amp; 1 \\\\\n2 &amp; -6 &amp; -2 \\\\\n1 &amp; 3 &amp; 1 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_y\\ =\\ 4\\begin{vmatrix}\n-6 &amp;  -2 \\\\\n3 &amp; 1\\\\\n\\end{vmatrix}\\ -\\ 6\\begin{vmatrix}\n2 &amp; -2 \\\\\n1 &amp; 1 \\\\\n\\end{vmatrix}\\ +\\ 1\\begin{vmatrix}\n2 &amp; -6\\\\\n1 &amp;  3 \\\\\n\\end{vmatrix}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_y\\ =\\ 4(-\\ 6\\ +\\ 6)\\ -\\ 6 (2\\ +\\ 2)\\ +\\ 1(6\\ +\\ 6)\\ \n\\hspace{9cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_y\\ =\\ 4(0)\\ -\\ 6 (4)\\ +\\ 1(12)\\ \n\\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_y\\ =\\ -\\ 0\\ -\\ 24\\ +\\ 12\\ \n\\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\Delta_y\\ =\\ -12}\\ \n\\hspace{17cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_z = \\begin{vmatrix}\n4 &amp; 1 &amp; 6 \\\\\n2 &amp; -1 &amp; -6 \\\\\n1 &amp; 1 &amp; 3 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_z\\ =\\ 4\\begin{vmatrix}\n-1 &amp; -6 \\\\\n1 &amp; 3 \\\\\n\\end{vmatrix}\\ -\\ 1\\begin{vmatrix}\n2 &amp; -6 \\\\\n1 &amp; 3 \\\\\n\\end{vmatrix}\\ +\\ 6\\begin{vmatrix}\n2 &amp; &#8211; 1\\\\\n1 &amp;  1 \\\\\n\\end{vmatrix}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_z\\ =\\ 4(-\\ 3\\ +\\ 6)\\ -\\ 1 (6\\ +\\ 6)\\ +\\ 6(2\\ +\\ 1)\\ \n\\hspace{9cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_z\\ =\\ 4(3)\\ -\\ 1 (12)\\ +\\ 6(3)\\ \n\\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_z\\ =\\ 12\\  -\\  12\\ +\\  18\\ \n\\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\Delta_z\\ =\\ 18}\\ \n\\hspace{17cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[The\\ Solution\\ is\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[x=\\ \\frac{\\Delta_x}{\\Delta} =\\ \\frac{3}{3} =\\ 1\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[y=\\ \\frac{\\Delta_y}{\\Delta} =\\ \\frac{-12}{3} =\\ -4\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[z=\\ \\frac{\\Delta_z}{\\Delta} =\\ \\frac{18}{3} =\\ 6\\ \\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\\ =\\ -4\\ and\\ z = 6\\ in\\ equation (1)\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[LHS\\ =\\ 4(1) &#8211; 4 + 6\\]\\[ = 4 &#8211; 4 + 6 = 6\\]\\[ = RHS\\] <\/div>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/I8vKziN5qco?list=PLQIom4Rz29vxcUyYeKo2Sw4cmEJ0RbBA2\" title=\"Determinants and Matrices - Part - 7\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ (c)\\ \\hspace{1cm} \\color{green}{Find\\ the\\ inverse\\ of\\ \\begin{pmatrix}\n2 &amp; 1 &amp; 1 \\\\\n1 &amp; 0 &amp; 2 \\\\\n4 &amp; 2 &amp;  2 \\\\\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}\n2 &amp; 1 &amp; 1 \\\\\n1 &amp; 0 &amp; 2 \\\\\n4 &amp; 2 &amp;  2 \\\\\n\\end{pmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\begin{vmatrix}\nA \\\\\n\\end{vmatrix}\\ =\\ 2\\begin{vmatrix}\n0 &amp; 2 \\\\\n2 &amp; 2 \\\\\n\\end{vmatrix}\\ -\\ 1\\begin{vmatrix}\n1 &amp; 2 \\\\\n4 &amp; 2 \\\\\n\\end{vmatrix}\\ +\\ 1\\begin{vmatrix}\n1 &amp; 0\\\\\n4 &amp;  2 \\\\\n\\end{vmatrix}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ 2(0\\ -\\ 4)\\ -\\ 1 (2\\ -\\ 8)\\ +\\ 1(2\\ -\\ 0)\\ \n\\hspace{9cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ 2(-4)\\ -\\ 1 (-6)\\ +\\ 1(2)\\ \n\\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ -8\\ +\\ 6\\ + 2\\ \n\\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\begin{vmatrix}\nA \\\\\n\\end{vmatrix}\\ =\\ 0\\ \n\\hspace{17cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\therefore\\ Inverse\\ of\\ A\\ does\\ not\\ exist\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[22\\ \\hspace{1cm}\\ (a)\\ \\hspace{1cm} \\color{green}{If\\ sin\\ \\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\\ sin\\ \\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}\\ cos^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}{13}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ \\boxed{cos\\ \\theta\\ =\\ \\pm\\ \\frac{12}{13}}\\ \\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{5}{13}}{\\frac{12}{13}}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ \\boxed{tan\\ \\theta\\ =\\ \\pm\\ \\frac{5}{12}}\\ \\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{5}{12}}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ \\boxed{cot\\ \\theta\\ =\\ \\pm\\ \\frac{12}{5}}\\ \\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}\\ =\\  \\pm\\ \\frac{1}{\\frac{12}{13}}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ \\boxed{sec\\ \\theta\\ =\\ \\pm\\ \\frac{13}{12}}\\ \\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}{\\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}\\ (b)\\ \\hspace{1cm} \\color{green}{If\\ A\\ and\\ B\\ are\\ acute\\ angles\\ such\\ that\\ Sin\\ A\\ =\\ \\frac{3}{5} \\  and\\ Cos\\ B\\ =\\ \\frac{12}{13}},\\hspace{15cm}\\\\  \\color {green} {then\\ prove\\ that\\  Cos(A\\ +\\ B)\\ =\\ \\frac{33}{65}}\\  \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ Given\\ Sin\\ A\\ =\\ \\frac{3}{5} \\  and\\ Cos\\ B\\ =\\ \\frac{12}{13}\\  \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[W.\\ K.\\ T\\ Cos( A + B )\\  =\\  Cos A\\  Cos B\\ -\\  Sin A\\  Sin B\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Cos\\ A\\ =\\ ?\\ ,\\ Sin\\ 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{3}{5})^2}\\  \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ \\sqrt{1\\ -\\ \\frac{9}{25}}\\  \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ \\sqrt{\\frac{25\\ -\\ 9}{25}}\\  \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ \\sqrt{\\frac{16}{25}}\\  \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Cos\\ A\\ =\\ \\frac{4}{5}\\  \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Sin\\  B\\ =\\ \\sqrt{1\\ -\\ Cos^2\\ B}\\  \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ \\sqrt{1\\ -\\ (\\frac{12}{13})^2}\\  \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ \\sqrt{1\\ -\\ \\frac{144}{169}}\\  \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ \\sqrt{\\frac{169\\ -\\ 144}{169}}\\  \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ \\sqrt{\\frac{25}{169}}\\  \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Sin\\ B\\ =\\ \\frac{5}{13}\\  \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Cos( A + B )\\  =\\  (\\frac{4}{5})\\  (\\frac{12}{13})\\ -\\  (\\frac{3}{5})\\  (\\frac{5}{13})\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{48}{65}\\ -\\ \\frac{15}{65}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{48\\ -\\ 15}{65}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{33}{65}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed {Cos ( A + B )\\ =\\ \\frac{33}{65}}\\ \\hspace{10cm}\\]<\/div>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/lXkJ9SmH3Kg\" title=\"Determining the trigonometrical problem using Cos (A + B) formula - 1\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ (c)\\ \\hspace{1cm} \\color{green}{Show\\ that\\  \\frac{1\\ +\\ cos\\ 2A\\ +\\ sin\\ 2A}{1\\ -\\ cos\\ 2A\\ +\\ sin\\ 2A}\\ =\\ cot\\ A}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ LHS\\ =\\ \\frac{1\\ +\\ cos\\ 2A\\ +\\ sin\\ 2A}{1\\ -\\ cos\\ 2A\\ +\\ sin\\ 2A}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{2\\ cos^2\\ A\\ +\\ 2\\ sin\\ A\\ cos\\ A}{2\\ sin^2\\ A\\ +\\ 2\\ sin\\ A\\ cos\\ A}\\ \\hspace{1cm}\\ W.\\ K.\\ T.\\ sin\\ 2A\\ =\\  2\\ sin\\ A\\ cos\\ A\\ ,\\ 1\\ +\\ cos\\ 2A\\ =\\ 2\\ cos^2\\ A\\ and\\ 1\\ -\\ cos\\ 2A\\ =\\ 2\\ sin^2\\ A\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{2\\ cos\\ A(cos\\ A\\ +\\ sin\\ A)}{2\\ sin\\ A(sin\\ A\\ +\\ cos\\ A)}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{con\\ A}{sin\\ A}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ cot\\ 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}{3\\overrightarrow{i}\\ -\\  \\overrightarrow{j}\\ -\\  2\\overrightarrow{k}\\ ,\\  5\\overrightarrow{i}\\ +\\  \\overrightarrow{j}\\ -\\  3\\overrightarrow{k} and\\  6\\overrightarrow{i}\\  -\\  \\overrightarrow{j}\\  -\\  \\overrightarrow{k}\\  form\\  an\\ isosceles\\ triangle}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln\\ :}\\ Given\\ \\hspace{17cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{OA}\\ =\\ 3\\overrightarrow{i}\\ -\\  \\overrightarrow{j}\\ -\\  2\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{OB}\\ =\\ 5\\overrightarrow{i}\\ +\\  \\overrightarrow{j}\\ -\\  3\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{OC}\\ =\\ 6\\overrightarrow{i}\\  -\\  \\overrightarrow{j}\\  -\\  \\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\">\\[=\\ 5\\overrightarrow{i}\\ +\\  \\overrightarrow{j}\\ -\\  3\\overrightarrow{k}\\ &#8211; (3\\overrightarrow{i}\\ -\\  \\overrightarrow{j}\\ -\\  2\\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 5\\overrightarrow{i}\\ +\\  \\overrightarrow{j}\\ -\\  3\\overrightarrow{k}\\ -\\ 3\\overrightarrow{i}\\ +\\  \\overrightarrow{j}\\ -\\  2\\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{AB}\\ =\\  2\\overrightarrow{i}\\ +\\  2\\overrightarrow{j}\\ -\\  \\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[AB =\\overrightarrow{|AB|} = \\sqrt{(2)^2\\ +\\  (2)^2\\ +\\ (-1)^2 }\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\sqrt{4\\ +\\ 4\\ +\\ 1}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\  \\sqrt{9}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[AB =\\ 3 \\]<\/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\">\\[=\\ 6\\overrightarrow{i}\\  -\\  \\overrightarrow{j}\\  -\\  \\overrightarrow{k}\\ &#8211; (5\\overrightarrow{i}\\ +\\  \\overrightarrow{j}\\ -\\  3\\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 6\\overrightarrow{i}\\  -\\  \\overrightarrow{j}\\  -\\  \\overrightarrow{k}\\ -\\ 5\\overrightarrow{i}\\ -\\  \\overrightarrow{j}\\ +\\  3\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{BC}\\ =\\  \\overrightarrow{i}\\ -\\  2\\overrightarrow{j}\\ +\\  2\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[BC =\\overrightarrow{|BC|} = \\sqrt{(1)^2\\ +\\  (-2)^2\\ +\\ (2)^2 }\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\sqrt{1\\ +\\ 4\\ +\\ 4}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\  \\sqrt{9}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[BC =\\ 3 \\]<\/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\">\\[=\\ 6\\overrightarrow{i}\\  -\\  \\overrightarrow{j}\\  -\\  \\overrightarrow{k}\\ &#8211; (3\\overrightarrow{i}\\ -\\  \\overrightarrow{j}\\ -\\  2\\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 6\\overrightarrow{i}\\  -\\  \\overrightarrow{j}\\  -\\  \\overrightarrow{k}\\ -\\ 3\\overrightarrow{i}\\ +\\  \\overrightarrow{j}\\ +\\  2\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{AC}\\ =\\  4\\overrightarrow{i}\\ +\\ \\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[AC =\\overrightarrow{|AC|} = \\sqrt{(4)^2\\ +\\  (0)^2\\ +\\ (1)^2 }\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\sqrt{16\\ +\\  0\\ +\\  1}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[AC = \\sqrt{17}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{AB = BC\\ \\neq\\ AC}\\]<\/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 class=\"has-text-align-center\" data-align=\"center\">0-10<\/td><td class=\"has-text-align-center\" data-align=\"center\">10-20<\/td><td class=\"has-text-align-center\" data-align=\"center\">20-30<\/td><td class=\"has-text-align-center\" data-align=\"center\">30-40<\/td><\/tr><tr><td>Frequency<\/td><td class=\"has-text-align-center\" data-align=\"center\">2<\/td><td class=\"has-text-align-center\" data-align=\"center\">5<\/td><td class=\"has-text-align-center\" data-align=\"center\">1<\/td><td class=\"has-text-align-center\" data-align=\"center\">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<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>5<\/td><td class=\"has-text-align-left\" data-align=\"left\">15<\/td><td>75<\/td><\/tr><tr><td>20-30<\/td><td>1<\/td><td class=\"has-text-align-left\" data-align=\"left\">25<\/td><td>25<\/td><\/tr><tr><td>30-40<\/td><td>3<\/td><td class=\"has-text-align-left\" data-align=\"left\">35<\/td><td>105<\/td><\/tr><tr><td><\/td><td>N = 11<\/td><td class=\"has-text-align-left\" data-align=\"left\"><\/td><td><mathml>\\[\\Sigma f_i\\ x_i\\ =\\ 215\\]<\/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{215}{11}\\ =\\ 19.55\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ (b)\\ \\hspace{1cm} \\color{green}{\\text{Find  the standard deviation  of the following data:}}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<figure class=\"wp-block-table\"><table><tbody><tr><td>Class interval<\/td><td>4 &#8211; 8<\/td><td>8 &#8211; 12<\/td><td>12 &#8211; 16<\/td><td>16 &#8211; 20<\/td><\/tr><tr><td>frequency<\/td><td>3<\/td><td>6<\/td><td>4<\/td><td>7<\/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\">\\[\\sigma\\ =\\ \\sqrt{\\frac{\\Sigma f_i\\ x_i^2}{N}\\ -\\ (\\frac{\\Sigma f_i\\ x_i}{N})^2}, \\  where\\ N\\ =\\ \\Sigma {f_i}\\]<\/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>f<sub>i<\/sub><\/td><td>x<sub>i <\/sub><\/td><td>f<sub>i<\/sub>x<sub>i<\/sub><\/td><td>x<sub>i<\/sub><sup> 2<\/sup><\/td><td>f<sub>i<\/sub>xi<sup>2<\/sup><\/td><\/tr><tr><td>4 -8<\/td><td>3<\/td><td>6<\/td><td>18<\/td><td>36<\/td><td>108<\/td><\/tr><tr><td>8 -12<\/td><td>6<\/td><td>10<\/td><td>60<\/td><td>100<\/td><td>600<\/td><\/tr><tr><td>12 &#8211; 16<\/td><td>4<\/td><td>14<\/td><td>56<\/td><td>196<\/td><td>784<\/td><\/tr><tr><td>16 &#8211; 20<\/td><td>7<\/td><td>18<\/td><td>126<\/td><td>324<\/td><td>2268<\/td><\/tr><tr><td><\/td><td>N = 20<\/td><td><\/td><td><mathml>\\[\\Sigma f_i\\ x_i\\ =\\ 260\\]<\/mathml><\/td><td><\/td><td><mathml>\\[\\Sigma\\ f_i\\ x_i^2\\ =\\ 3760\\]<\/mathml><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\sigma\\ =\\ \\sqrt{\\frac{\\Sigma f_i\\ x_i^2}{N}\\ -\\ (\\frac{\\Sigma f_i\\ x_i}{N})^2}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\sqrt{\\frac{3760}{20}\\ -\\ (\\frac{-260}{20})^2}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\sqrt{188\\ -\\ (13)^2}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\sqrt{188\\ -\\ 169}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\sqrt{19}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\sigma\\ =\\ 4.36\\ (approximately)\\]<\/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:}\\ \\hspace{15cm}\\]<\/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>31<\/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>31<\/td><td>16<\/td><td>124<\/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\\ =\\ 254\\]<\/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)\\ =\\ 254\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[3\\ a\\ +\\ b\\ =\\ 25.4\\ &#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\\ =\\ 25.4\\]<\/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\\ =\\ 5.6\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[put\\ a\\ =\\ 5.6\\ in\\ (4)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2(5.6)\\ +\\ b\\ =\\ 19.8\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[11.2\\ +\\ b\\ =\\ 19.8\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[b\\ =\\ 19.8\\ -\\ 11.2\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[b\\ =\\ 8.6\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Eqn\\ (1)\\ becomes\\ y\\ =\\ 5.6\\ x\\ +\\ 8.6\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[25\\ \\hspace{1cm}\\ (a)\\ \\hspace{1cm} \\color{green}{Two\\ dice\\ are\\ rolled\\ together.\\ Find\\ the\\ probability}\\ \\hspace{10cm}\\\\\\ \\color{green}{for\\ getting\\ the\\ sum\\ of\\ the\\ numbers\\ on\\ on\\ the\\ faces\\ are\\ 10\\ and\\ 7}\\ \\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\\ 10}}\\ \\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\\ (4,\\ 6)\\ \\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,\\ 4)\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{There are 3 such outcomes.}\\ \\hspace{10cm}\\]\n<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[A\\ denotes\\ the\\ event\\ of\\ getting\\ a\\ sum\\ of\\ 10\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[P(A)\\ =\\ \\frac{3}{36}\\  \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2.\\ \\bf{\\color{green}{probability\\ of\\ getting\\ a\\ sum\\ of\\ 7}\\ \\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,\\ 6)\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bullet\\ (2,\\ 5)\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bullet\\ (3,\\ 4)\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bullet\\ (4,\\ 3)\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bullet\\ (5,\\ 2)\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bullet\\ (6,\\ 1)\\ \\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\">\\[B\\ denotes\\ the\\ event\\ of\\ getting\\ a\\ sum\\ of\\ 10\\ \\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\">\\[3.\\ \\bf{\\color{green}{probability\\ of\\ sum\\ of\\ the\\ numbers\\ on\\ on\\ the\\ faces\\ are\\ 10\\ and\\ 7}}\\ \\hspace{2cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/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{3}{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))\\ =\\ 0\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[P(A\\ \\cup\\ B)\\ =\\ \\frac{3}{36}\\ +\\ \\frac{6}{36}\\ -\\ 0\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{9}{36}\\ =\\ \\frac{1}{4}\\ \\hspace{3cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{So, the probability of getting a sum of 10 and 7 on both dice is:}\\ =\\ \\frac{1}{4}\\ \\hspace{1cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ (b)\\ \\hspace{1cm} \\color{green}{A\\ card\\ is\\ selected\\ 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\\ they\\ solve\\ the\\ problem\\ 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}{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\n\n\n<script async src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js?client=ca-pub-9453835310745500\"\n     crossorigin=\"anonymous\"><\/script>\n<!-- Leader board 1 -->\n<ins class=\"adsbygoogle\"\n     style=\"display:inline-block;width:728px;height:90px\"\n     data-ad-client=\"ca-pub-9453835310745500\"\n     data-ad-slot=\"8769628924\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\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 = 1(1) + 2(1) + 1(-3) = 0 [&hellip;]<\/p>\n","protected":false},"author":187055548,"featured_media":52583,"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":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","enabled":false},"version":2},"_wpas_customize_per_network":false,"jetpack_post_was_ever_published":false},"categories":[711788674],"tags":[],"class_list":["post-52567","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-basic-mathematics"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>October-2024 TAMIL NADU POLYTECHNIC BOARD EXAM BASIC MATHEMATICS QUESTION PAPER (held on 20-11-2024)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=52567\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"October-2024 TAMIL NADU POLYTECHNIC BOARD EXAM BASIC MATHEMATICS QUESTION PAPER (held on 20-11-2024)WITH SOLUTIONS - YANAMTAKSHASHILA\" \/>\n<meta property=\"og:description\" content=\"1. 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